Quantum Metrology: Foundation of Units and...

Ernst O. Göbeland Uwe Siegner

Quantum Metrology: Foundation

of Units and Measurements

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Ernst O. Göbel and Uwe Siegner

Quantum Metrology: Foundation of Units

and Measurements

Authors

Professor Dr. Ernst O. Göbel

Physikalisch-Technische Bundesanstalt

Bundesallee 100

38116 Braunschweig

Germany

Dr. Uwe Siegner

Physikalisch-Technische Bundesanstalt

Bundesallee 100

38116 Braunschweig

Germany

Cover

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Bundesanstalt, Braunschweig, Germany.

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V

Contents

Foreword IXPreface XIList of Abbreviations XIIIColor Plates XVII

1 Introduction 1References 3

2 Some Basics 52.1 Measurement 52.1.1 Limitations of Measurement Uncertainty 52.1.1.1 The Fundamental Quantum Limit 62.1.1.2 Noise 72.2 The SI (Système International d’Unités) 102.2.1 The Second: Unit of Time 112.2.2 The Meter: Unit of Length 142.2.3 The Kilogram: Unit of Mass 152.2.4 The Ampere: Unit of Electrical Current 172.2.5 The Kelvin: Unit of Thermodynamic Temperature 182.2.6 The Mole: Unit of Amount of Substance 192.2.7 The Candela: Unit of Luminous Intensity 20

References 21

3 Laser Cooling, Atomic Clocks, and the Second 233.1 Techniques for Laser Cooling 253.1.1 Doppler Cooling, Optical Molasses, and Magneto-Optical Traps 263.1.2 Cooling Below the Doppler Limit 283.1.3 Optical Lattices 293.1.4 Ion Traps 303.2 The Cs Fountain Clock 323.3 Optical Clocks 363.3.1 Femtosecond Frequency Combs 383.3.2 Neutral Atom Clocks 43

VI Contents

3.3.3 Atomic Ion Clocks 463.3.4 Possible Variation of the Fine-Structure Constant, 𝛼 49

References 51

4 Superconductivity, Josephson Effect, and Flux Quanta 614.1 Josephson Effect and Quantum Voltage Standards 614.1.1 Brief Introduction to Superconductivity 614.1.2 Basics of the Josephson Effect 634.1.2.1 AC and DC Josephson Effect 644.1.2.2 Mixed DC and AC Voltages: Shapiro Steps 654.1.3 Basic Physics of Real Josephson Junctions 664.1.4 Josephson Voltage Standards 684.1.4.1 General Overview: Materials and Technology of Josephson

Arrays 694.1.4.2 DC Josephson Voltage Standards: The Conventional Volt 704.1.4.3 Programmable Binary AC Josephson Voltage Standards 734.1.4.4 Pulse-Driven AC Josephson Voltage Standards 764.1.4.5 Applications of AC Josephson Voltage Standards 794.2 Flux Quanta and SQUIDs 814.2.1 Superconductors in External Magnetic Fields 824.2.1.1 Meissner–Ochsenfeld Effect 824.2.1.2 Flux Quantization in Superconducting Rings 844.2.1.3 Josephson Junctions in External Magnetic Fields and Quantum

Interference 854.2.2 Basics of SQUIDs 874.2.3 Applications of SQUIDs in Measurement 904.2.3.1 Real DC SQUIDs 914.2.3.2 SQUID Magnetometers and Magnetic Property Measurement

Systems 924.2.3.3 Cryogenic Current Comparators: Current and Resistance Ratios 944.2.3.4 Biomagnetic Measurements 964.2.4 Traceable Magnetic Flux Density Measurements 97

References 99

5 Quantum Hall Effect 1035.1 Basic Physics of Three- and Two-Dimensional Semiconductors 1035.1.1 Three-Dimensional Semiconductors 1045.1.2 Two-Dimensional Semiconductors 1065.2 Two-Dimensional Electron Systems in Real Semiconductors 1085.2.1 Basic Properties of Semiconductor Heterostructures 1085.2.2 Epitaxial Growth of Semiconductor Heterostructures 1105.2.3 Semiconductor QuantumWells 1115.2.4 Modulation Doping 1125.3 The Hall Effect 1145.3.1 The Classical Hall Effect 114

Contents VII

5.3.1.1 The Classical Hall Effect in Three Dimensions 1145.3.1.2 The Classical Hall Effect in Two Dimensions 1155.3.2 Physics of the Quantum Hall Effect 1165.4 Quantum Hall Resistance Standards 1195.4.1 DC Quantum Hall Resistance Standards 1205.4.1.1 Comparison Between Classical and Quantum-Based Resistance

Metrology 1205.4.1.2 The Conventional Ohm 1215.4.1.3 Technology of DC Quantum Hall Resistance Standards and

Resistance Scaling 1225.4.1.4 Relation Between the von Klitzing Constant and the Fine-Structure

Constant 1245.4.2 AC Quantum Hall Resistance Standards 125

References 127

6 Single-Charge Transfer Devices and the New Ampere 1316.1 Basic Physics of Single-Electron Transport 1326.1.1 Single-Electron Tunneling 1326.1.2 Coulomb Blockade in SET Transistors 1336.1.3 Coulomb Blockade Oscillations and Single-Electron Detection 1356.1.4 Clocked Single-Electron Transfer 1376.2 Quantized Current Sources 1396.2.1 Metallic Single-Electron Pumps 1396.2.2 Semiconducting Quantized Current Sources 1416.2.2.1 GaAs Based SET Devices 1426.2.2.2 Silicon-Based SET Devices 1456.2.3 Superconducting Quantized Current Sources 1456.2.4 A Quantum Standard of Current Based on Single-Electron

Transfer 1486.3 Consistency Tests: QuantumMetrology Triangle 149

References 151

7 The Planck Constant, the New Kilogram, and the Mole 1557.1 The Avogadro Experiment 1587.2 TheWatt Balance Experiment 1657.3 The Mole: Unit of Amount of Substance 169

References 170

8 Boltzmann Constant and the New Kelvin 1758.1 Primary Thermometers 1768.1.1 Dielectric Constant Gas Thermometry 1778.1.2 Acoustic Gas Thermometry 1788.1.3 RadiationThermometry 1808.1.4 Doppler BroadeningThermometry 1818.1.5 Johnson Noise Thermometry 183

VIII Contents

8.1.6 Coulomb BlockadeThermometry 1858.2 Realization and Dissemination of the New Kelvin 186

References 186

9 Single-Photon Metrology and Quantum Radiometry 1919.1 Single-Photon Sources 1939.1.1 (NV) Color Centers in Diamond 1949.1.2 Semiconductor Quantum Dots 1969.2 Single-Photon Detectors 1989.2.1 Nonphoton-Number Resolving Detectors 1999.2.2 Photon-Number-Resolving Detectors 1999.3 Metrological Challenge 201

References 202

10 Outlook 207References 208

Index 209

IX

Foreword

Measurements with ever-increasing precision and reliability are the fundamentalbasis of science and indispensable for progress in science, economy, and society.To ensure worldwide comparability of measurements, the Meter Convention wassigned by 17 states in 1875 and its organs, the General Conference on Weightsand Measures (Conférence Générale des Poids et Mesures, CGPM), the Interna-tional Committee for Weights and Measures (Comité International des Poids etMesures, CIPM), and the International Bureau of Weights and Measures (BureauInternational des Poids et Mesures, BIPM), were established.

As of today, the Meter Convention counts 96 member states or associated states,includes 95 National Metrology Institutes and 150 Designated Institutes alongwith four international organizations, and, thus, represents not less than 97.6%of the total world economic power. Therefore, the International System of Units(Système International d’Unités), the SI, put in place by the 11th CGPM in 1960as the basis for any measurement of any relevant quantity, provides a worldwideharmonized measurement and quality infrastructure enabling and fostering inter-national trade around the globe.

The SI and its defined units, however, are not static but evolve as measurementsare becoming more and more demanding and as science and technology proceeds.The remarkable progress in science, in particular in the area of laser physics, quan-tum optics, solid-state physics, and nanotechnology, has now paved the way for anupcoming fundamental revision of the SI. In the future, all SI units will be based onseven “defining constants,” among them are fundamental constants of nature likePlanck’s constant, the speed of light, or the charge of the electron, and, thus, willbe independent of space and time with a relative accuracy below 10−16 accordingto state-of-the-art experiments.

The present book explains and illustrates the physics and technology behindthese new SI definitions and their realization as well as their impact on measure-ments based on quantum physics phenomena. The book thus is essential, timely,and indeed urgently needed to communicate the envisaged revision of the SI andits consequences to the broad scientific community and other interested reader-ship, including lecturers and teachers.

X Foreword

The authors are well qualified for this undertaking. Both have a long experiencein metrology: Ernst Göbel has been the president of the PTB, the National Metrol-ogy Institute of Germany, for more than 16 years. He had been a member of theCIPM for more than 15 years and its president from 2004 to 2010. Uwe Siegnerjoined the PTB in 1999, working on metrological applications of femtosecond lasertechnology and on electrical quantum metrology. He is the head of the electricitydivision of the PTB since 2009. Both authors are experienced university lectur-ers, and in fact, the book is based on lectures given at the Technical University ofBraunschweig.

I have studied the book with great interest and pleasure, and I wish the same toa broad readership.

Braunschweig, Prof. Dr. Joachim Ullrich,November 2014 President of PTB,

Vice President of CIPM,President of the Consultative Committee for Units (CCU)

XI

Preface

The concept of some indivisible discrete single particles that are the basic buildingblocks of all matter goes back to philosophers many centuries BC. In particular,the Greek philosopher Demokrit and his students specified the idea of atoms (fromthe Greek àtomos) as the base elements of all matter.

These concepts found support in natural science beginning in the eighteenthcentury. This was particularly driven by chemistry (e.g., A. Lavoisier, J. Dalton,and D. Mendeleev), kinetic gas theory (e.g., J. Loschmidt and A. Avogadro), andstatistical physics (e.g., J. Stefan, L. Boltzmann, and A. Einstein).

The discovery of the electron by J.J. Thomson (1897) and the results of the scat-tering experiments by J. Rutherford and his coworkers (1909) opened a new erain physics, based on their conclusions that atoms are not indivisible but insteadcomposite species. In the atomic model developed by N. Bohr in 1913, the atomconsists of electrons carrying a negative elementary charge (−e) and a tiny nucleuswhich carries almost all the mass of an atom composed of positively charged (+e)protons and electrically neutral neutrons. In Bohr’s model, the electrons in anatom can only occupy discrete energy levels, consistent with the experimentalfindings of atom spectroscopy.

In the standard model of modern particle physics, electrons are in fact elemen-tary particles belonging to the group of leptons. Protons and neutrons are com-posite particles composed of fractionally charged elementary particles, namedquarks, which are bound together by the strong force.

In the last 50 years or so, scientists have learned to handle single quantumobjects, for example, atoms, ions, electrons, and Cooper pairs, not least due to thetremendous progress in laser physics and nanotechnology. This progress has alsolaid the base for what we call “quantum metrology.” The paradigm of quantummetrology is to base measurements on the counting of discrete quanta (e.g.,charge or magnetic flux quanta). In contrast, in classical metrology, the valuesof continuous variables are determined. Proceeding from classical to quantummetrology, the measurement of real numbers is replaced by counting of integers.

The progress in quantum metrology has also stimulated the discussion abouta revision of the present International System of Units (Système Internationald’Unités), the SI. In particular, quantum metrology allows for a new definitionof the base units of the SI in terms of constants of nature. This envisaged new

XII Preface

definition of the SI as well as future possible revisions have actually set the frameof the present book.

The discrete nature of a physical system is sometimes obvious, for example,when considering microwave or optical transitions between discrete energy statesin atoms or ions. The discrete quantum character of solid-state systems is lessobvious because their single-particle energy spectra are quasicontinuous energybands. Discrete quantum entities can then result from collective effects calledmacroscopic quantum effects.

The paradigm of quantum metrology becomes particularly obvious whenthe proposed new definition of the electrical units (ampere, volt, and ohm)is considered. We therefore give a more comprehensive description of theunderlying solid-state physics and the relevant macroscopic quantum effects. Forexample, we partly summarize the textbook knowledge and deduce results startingfrom general principles in Chapter 4 where we introduce superconductivity, theJosephson effect, and quantum interference phenomena in superconductors.

This book addresses advanced students, research workers, scientists, practi-tioners, and professionals in the field of modern metrology as well as a generalreadership interested in the foundations of the forthcoming new SI definition.However, we consider this book as an overview which shall not cover all subjectsin the same detail as it covers the electrical units. For further reading, we shallrefer to the respective literature.

This book would not have been possible without the support of many colleaguesand friends. We would like to especially mention Stephen Cundiff (JILA, nowUniversity of Michigan) and Wolfgang Elsäßer (University of Darmstadt) as wellas our PTB colleagues Franz Ahlers, Peter Becker, Ralf Behr, Joachim Fischer,Frank Hohls, Oliver Kieler, Johannes Kohlmann, Stefan Kück, Ekkehard Peik,Klaus Pierz, Hansjörg Scherer, Piet Schmidt, Sibylle Sievers, Lutz Trahms, andRobert Wynands. We are also grateful for the technical support provided byAlberto Parra del Riego and Jens Simon. We further acknowledge the support ofthe present and former Wiley-VCH staff members, in particular Valerie Moliere,Anja Tschörtner, Heike Nöthe, and Andreas Sendtko.

Braunschweig, Ernst Göbel and Uwe SiegnerApril 2015

XIII

List of Abbreviations

2DEG two-dimensional electron gasAGT acoustic gas thermometer/thermometryAIST National Institute of Advanced Industrial Science and Technology

(National Metrology Institute of Japan)APD avalanche photo diodeBIPM International Bureau for Weights and Measures (Bureau

International des Poids et Mesures)CBT Coulomb blockade thermometer/thermometryCCC cryogenic current comparatorCCL Consultative Committee for LengthCCM Consultative Committee for MassCCT Consultative Committee for TemperatureCCU Consultative Committee for UnitsCERN European Organization for Nuclear ScienceCGPM General Conference on Weights and Measures (Conférence Générale

des Poids et Mesures)CIPM International Committee for Weights and Measures (Comité

International des Poids et Mesures)CODATA International Council for Science: Committee on Data for Science

and TechnologyCVGT constant volume gas thermometer/thermometryDBT Doppler broadening thermometer/thermometryDCGT dielectric constant gas thermometer/thermometryECG electrocardiographyEEG electroencephalographyEEP Einstein’s equivalence principleFQHE fractional quantum Hall effectGUM Guide to the Expression of Uncertainty in MeasurementsHEMT high electron mobility transistorIDMS isotope dilution mass spectroscopyINRIM National Institute of Metrology of Italy (Istituto Nazionale di Ricerca

Metrologia)ISO International Organization for Standards

XIV List of Abbreviations

ITS International temperature scaleJNT Johnson noise thermometer/thermometryKRISS Korea Research Institute of Standards and ScienceLED light-emitting diodeLNE French Metrology Institute (Laboratoire national de métrologie et

d’essaisMBE molecular beam epitaxyMCG magnetocardiographyMEG magnetoencephalographyMETAS Federal Institute of Metrology, SwitzerlandMOCVD metalorganic chemical vapor depositionMODFET modulation-doped field-effect transistorMOS metal-oxide-semiconductorMOSFET metal-oxide-semiconductor field-effect transistorMOT magneto-optical trapMOVPE metalorganic vapor phase epitaxyMSL Measurement Standards Laboratory of New ZealandNIM National Institute of Metrology (National Metrology Institute of

China)NININ normal metal/insulator/normal metal/insulator/normal metalNIST National Institute of Standards and Technology (National Metrology

Institute of the United States)NMR nuclear magnetic resonanceNPL National Physical Laboratory (National Metrology Institute of the

United Kingdom)NRC National Research Council, CanadaPMT photomultiplier tubePTB Physikalisch–Technische Bundesanstalt (National Metrology

Institute of Germany)QED quantum electrodynamicsQHE quantum Hall effectQMT quantum metrology triangleQVNS quantized voltage noise sourceRCSJ resistively–capacitively shunted junctionRHEED reflection high-energy electron diffractionRIGT refractive index gas thermometer/thermometryrms root-mean-squareRT radiation thermometrySEM scanning electron microscope/microscopySET single-electron transportSI International System of Units (Système International d’Unités)SINIS superconductor/insulator/normal metal/insulator/superconductorSIS superconductor/insulator/superconductorSNS superconductor/normal metal/superconductorSOI silicon-on-insulator

List of Abbreviations XV

SPAD single-photon avalanche diodeSQUID superconducting quantum interference deviceTES transition-edge sensorTEM transmission electron microscope/microscopyTPW triple point of waterUTC coordinated universal timeXRCD X-ray crystal densityYBCO yttrium barium copper oxide

Color Plates

Figure 3.7 Yb+ ions (seen by their fluorescence) in a linear Paul trap. The distancebetween the ions is about 10–20 μm. (Courtesy of T. Mehlstäubler, PTB.)

(a) (b) (c) (d)

Detectionlaser

Microwavecavity

Figure 3.10 Principle of operation of theatomic fountain clock illustrating the essen-tial four steps in a measurement cycle: (a)preparation of the cloud of cold atoms, (b)launch of the cloud toward the microwavecavity and subsequent passage through thestate selection cavity and Ramsey cavity, (c)

free flight and turnaround of the cloud andsubsequent second passage through theRamsey cavity, and (d) detection of the num-ber of atoms in the |Fg = 3⟩ and |Fg = 4⟩states, respectively. (Courtesy of R. Wynands,PTB.)

XVIII Color Plates

980 nm

WDM

80/20Coupler

Er:fiber

CL

4 2λ

4λ

4PD

LFFI

λ

λ/4λ/2

L

λ

PBS CL

Exit beam(a)

(b)

Inputoscillator

‘‘Stretcher’’

‘‘Pick-off’’-mirror

Variable Si-prismcompressor

Exit beam

(negative GVD)

Er:fiber(Positive GVD)

Pump diode

980 nm

WDM WDM

Pump diode lasers

Figure 3.15 Mode-locked Er fiber laseroscillator (a) and amplifier (b) for opticalfrequency comb generation (WDM = wave-length division multiplexer, PBS = polar-izing beam splitter, PD = photo diode,

FI = Faraday isolator, LF = spectral filter,CL = collimating lens, GVD = group velocitydispersion). (Courtesy of F. Adler and A. Leit-enstorfer, University Konstanz.)

Color Plates XIX

760 nm

370 nm436 nm

935 nm

467 nm

F = 4

F = 2

F = 1

F = 21[3/2]3/2

3[3/2]1/2

F = 1

F = 0

F = 1

F = 1

F = 0

F = 0

F = 1

F = 3

2F7/2

2D3/2

2P1/2

2S1/2

Figure 3.18 Partial energy scheme of171Yb+indicating the cooling transitionat 370 nm (dashed blue arrow), the elec-tric quadrupole transition at 436 nm, aswell as the electric octupole transition at467 nm (green arrows). The other transi-tions to higher D states (red arrow) are used

for repumping. The dotted arrows indicatespontaneous transitions. The notation atthe upper left refers to a specific couplingscheme (JK or J1L2 coupling) particularlyapplied, for example, for rare earth atoms.(Courtesy of N. Huntemann and E. Peik, PTB.)

0−10 −5 0

Laser detuning (Hz)

5 10

2.4 Hz

0.2

0.4

0.6

Excitation p

robabili

ty

0.8

1

Figure 3.19 Excitation spectrum of the 2S1∕2(F = 0) →2 F7∕2(F = 3) transition in 171Yb+ .(Courtesy of N. Huntemann, PTB.)

XX Color Plates

7

8

9

6

5

4

3

2

1

Figure 4.6 Schematic layout of aNb/Al–Al2O3/Nb Josephson array. Shownare four junctions embedded in a microstripline whose ground plate is seen at the top ofthe structure. (1) Silicon substrate; (2) sput-tered Al2O3 layer, typical thickness 30 nm;(3) niobium tunnel electrode, 170 nm; (4)Al2O3 barrier, 1.5 nm, fabricated by thermal

oxidation of an Al layer; (5) niobium tunnelelectrode, 85 nm; (6) wiring layer, 400 nm; (7)niobium ground plane, 250 nm; (8) Nb oxideedge protection, 80 nm. The Nb groundplane (7) rests on a 2 μm thick Si oxidedielectric layer (9). (Courtesy of J. Kohlmann,PTB.)

Figure 4.7 Photograph of a 10 V Josephson array mounted onto a chip carrier. The size ofthe array is 24 mm by 10 mm. (Courtesy of PTB.)

Color Plates XXI

15

10

5

0

−5

−10

−150 10 20

Time (ms)

Voltage (

V)

30

Figure 4.9 Stepwise approximated 50 Hz sine wave generated by a programmable binaryAC Josephson voltage standard. (Courtesy of R. Behr, PTB.)

Figure 4.22 Eighty-three-channel SQUID system aligned above a patient for MCGmeasurements. (Courtesy of PTB.)

XXII Color Plates

++

+

++

Buffer

Substrate

(a)

(b)

Doping layer

Si+

Ec

EF

EV

2DEG

Cap layer

n-Al0.3

Ga0.7

As [Si] = 1 × 1018

cm−3

Al0.3

Ga0.7

As

GaAs

Semi-insul.GaAs

Doping layer

Spacer layer

Spacerlayer

GaAs

Figure 5.5 Modulation-dopedAl0.3Ga0.7As/GaAs heterostructure. (a) Layersequence showing the GaAs substrateand buffer layer, the undoped Al0.3Ga0.7Asspacer (thickness on the order of 10 nm),the Al0.3Ga0.7As[Si] doping layer (typicalthickness 50 nm, doped with Si donorsat a typical concentration of 1018 cm−3),and the typically 10 nm thick GaAs cap

layer. (b) Schematic band profile. In theconduction band, a triangular potential wellis formed at the interface of the GaAs andthe Al0.3Ga0.7As spacer layer. EF is the Fermilevel. Shown in red in the band profile is thelowest quantized energy state of the triangu-lar potential well. It holds a 2DEG as shown(in red) in the layer sequence. (Courtesy of K.Pierz, PTB.)

Color Plates XXIII

Figure 5.11 Photograph of a GaAs/AlGaAs quantum Hall resistance standard showing twoHall bars mounted in a chip carrier. (Courtesy of PTB.)

Figure 6.10 Schematic layout of aGaAs/AlGaAs SET pump. Typical parame-ters: width of the one-dimensional channel700 nm, gate width 100 nm, gate separation

250 nm. An AC and a DC voltage are appliedto the left gate, while only a DC voltage isapplied to the right gate. (Courtesy of A.Müller, PTB.)

XXIV Color Plates

(a) (b)

(c) (d)

Figure 6.11 Schematic representation ofthe pumping cycle of a GaAs/AlGaAs SETpump. Shown is the temporally varyingpotential of the quantum dot (red), theenergy levels of the dot (yellow), and the

Fermi level (green). The transferred electronis shown as blue dot. The different phases a,b, c, d of the pumping cycle are explained inthe text. (Courtesy of A. Müller, PTB.)

Figure 7.2 Photo of a single crystal Si sphere used in the Avogadro experiment. Thediameter and mass, respectively, are about 10 cm and 1 kg. (Courtesy of PTB.)

Color Plates XXV

Figure 7.4 Diameter variations of a single crystal Si sphere. (Courtesy of A. Nicolaus, PTB.)

Mo Kα

Opticalinterferometer

Multianodedetector

Displacement

PhotodiodeNd:YAG

Fixed crystal Analyzer

Figure 7.5 Schematic layout of the combined optical and X-ray interferometer for measur-ing the lattice constant of crystalline Si. (Courtesy of E. Massa, G. Mana, INRIM.)

XXVI Color Plates

Figure 7.7 Photo of the central part ofthe INRIM X-ray interferometer showingthe Si crystal plates right in the center. Theexperiment is performed in a temperature-stabilized vacuum chamber. A Mo Kα X-ray

source and an iodide-stabilized single-modeHe–Ne laser are used for the X-ray and opti-cal interferometer, respectively. (Courtesy ofE. Massa, G. Mana, INRIM.)

Knife edge Balance wheel

Multi-filamentband

Countermass

Velocity modemotor

Moving coil

Interferometer(1 of 3)

NorthTrimcoil

West

z

1 m

Uppersuperconductingsolenoid

Spider

Mass

Interfero-meter

Trimcoil

Stationarycoils

Lowersuper-conductingsolenoid

Figure 7.9 Schematic drawing of the NISTwatt balance. The magnetic field is createdby two superconducting solenoids wired inseries opposition creating a magnetic flux

density at the moving coil radius of about0.1 T. The trim coil is used to achieve a 1/rdependence of the field [52]. (Courtesy ofNIST.)

Color Plates XXVII

Figure 8.1 Photo of the assembled National Physics Laboratory (NPL) 1 l copper AGTresonator. (Courtesy of NPL).

R

TcT

ΔT = εEυ/Ce

υ

Figure 9.7 Operation principle of a TESdepicting the resistance, R, versus tempera-ture T close to the superconductor transitiontemperature TC. ΔT is the increase of the

temperature due to the absorption of a pho-ton (ε is the detection efficiency, Eν = hν isthe energy of the photon, and Ce the elec-tronic heat capacity).

1

1

Introduction

Metrology is the science of measurement including all theoretical andexperimental aspects, in particular the experimental and theoretical inves-tigation of the uncertainty of measurement results. According to Nobel Prizewinner J. Hall, “metrology truly is the mother of science” [1].

Metrology, actually, is almost as old as mankind. As men began to exchangegoods, they had to agree on commonly accepted standards as a base for their trade.And indeed, many of the ancient cultures like China, India, Egypt, Greece, and theRoman Empire had a highly developed measurement infrastructure. Examples arethe Nippur cubit from the third millennium BCE found in the ruins of a templein Mesopotamian and now exhibited in the archeology museum in Istanbul andthe famous Egyptian royal cubit as the base length unit for the construction ofthe pyramids. Yet, the culture of metrology got lost during the Middle Ages whenmany different standards were in use. In Germany, for instance, at the end of theeighteenth century, 50 different standards for mass and more than 30 standards forlength were used in different parts of the country. This of course had been a bar-rier to trade and favored abuse and fraud. It was then during the French revolutionthat the French Académie des Sciences took the initiative to define standards inde-pendent of the measures taken from the limbs of royal representatives. Instead,their intent was to base the standards on stable quantities of nature available foreveryone at all times. Consequently, in 1799, the standard for length was definedas one 10 million part of the quadrant of the earth, and a platinum bar was fabri-cated to represent this standard (Mètre des Archives). Subsequently, the kilogram,the standard of mass, was defined as the mass of one cubic decimeter of pure waterat the temperature of its highest density at 3.98 ∘C. This can be seen as the birth ofthe metric system which, however, at that time was not generally accepted throughEurope or even in France. It was only with the signature of the Meter Conventionin 1875 by 17 signatory countries that the metric system based on the meter andthe kilogram found wider acceptance [2]. At the time of this writing, the MeterConvention has been signed by 55 states with another 41 states being associatedwith the General Conference on Weights and Measures (Conférence Générale desPoids et Mesures, CGPM). At the General Conferences, following the first one in1889, the system of units was continuously extended. Finally, at the 11th CGPMin 1960, the present SI (Système International d’Unités) (see Section 2.2) with the

Quantum Metrology: Foundation of Units and Measurements, First Edition.Ernst O. Göbel and Uwe Siegner.© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Introduction

kilogram, second, meter, ampere, kelvin, and candela as base units was defined.The mole, unit of amount of substance, was added at the 14th CGPM in 1971.Within the SI, the definition of some of the units has been adopted according toprogress in science and technology; for example, the meter was defined in 1960on the basis of the wavelength of a specific emission line of the noble gas kryp-ton. But then, in 1983, it was replaced by the distance light travels in a given timeand by assigning a fixed value to the speed of light in vacuum. Likewise, the sec-ond, originally defined as the ephemeris second, was changed by the 13th CGPMand defined via an electronic transition in the Cs isotope 133. Thus, today, themeter and the second are defined by constants of nature. At present, efforts tonewly define the system of all units shall be based on constants of nature [3–7].In fact, in this context, single quanta physics has a decisive role as will be outlinedin this book.

We shall begin with introducing some basic principles of metrology inChapter 2. We start in Section 2.1 by repeating some basic facts related tomeasurement and will discuss in particular the limitations for measurementuncertainty. The SI, in its present form, is summarized in Section 2.2 togetherwith the proposed new definitions.

Chapter 3 is entitled laser cooling, atom clocks, and the second. Here, wedescribe the realization of the present definition of the second based on thehyperfine transition in the ground state of 133Cs employing laser-cooled atoms.We further describe recent developments of the so-called optical clocks, whichhave the potential for higher accuracy and stability than the present microwaveclocks and will definitely lead to a revised definition of the second in theforeseeable future.

Chapter 4 is devoted to superconductivity and its utilization in metrology.Because of its prominent role for electrical metrology, we give an introduction tosuperconductivity, the Josephson effect, magnetic flux quantization, and quantuminterference. By means of the Josephson effect, the volt (the unit for the electricalpotential difference) is traced back to the Planck constant and the electron chargeas realized in today’s most precise voltage standards. Magnetic flux quantizationand quantum interference allow the realization of quantum magnetometers(superconducting quantum interference devices) with unprecedented resolutionand precision.

The underlying solid-state physics and the metrological application of the quan-tum Hall effect are discussed in Chapter 5. In Chapter 6, we describe the physicsof single-electron transport devices and their potential for realizing the unit ofelectrical current, the ampere. The ampere is then traced back to the charge of theelectron and frequency. Finally, the so-called metrological triangle experiment willbe described.

Chapter 7 is then devoted to the envisaged new definition of the kilogram basedon the Planck constant. In particular, we will present the watt balance and thesilicon single crystal experiment for a precise determination of the Planck constantand the realization of the newly defined kilogram.

References 3

The envisaged new definition of the kelvin and various experiments to deter-mine precisely the value of the Boltzmann constant are discussed in Chapter 8.

In Chapter 9, finally, we take an even further look into the future of the SI whenwe discuss the prospect of single-photon emitters for a possible new definition ofradiometric and photometric quantities, for example, for (spectral) irradiance andluminous intensity.

References

1. Hall, J. (2011) Learning from the timeand length redefinition, and the metredemotion. Philos. Trans. R. Soc. A, 369,4090–4108.

2. for a review on the development of mod-ern metrology see e.g.: Quinn, T. andKovalevsky, J. (2005) The developmentof modern metrology and its role today.Philos. R. Soc. Trans. A, 363, 2307–2327.

3. Discussion Meeting issue “The new SIbased on fundamental constants”, orga-nized by Quinn, T. (2011) Philos. Trans.R. Soc. A, 369, 3903–4142.

4. Mills, I.M., Mohr, P.J., Quinn, T.J., Taylor,B.N., and Williams, E.R. (2006) Redef-inition of the kilogram, ampere, kelvinand mole: a proposed approach to

implementing CIPM recommendation1 (CI-2005). Metrologia, 43, 227–246.

5. Flowers, J. and Petley, B. (2004) inAstrophysics, Clocks and FundamentalConstants (eds S.G. Karshenboim and E.Peik), Springer, Berlin, Heidelberg, pp.75–93.

6. Okun, L.B. (2004) in Astrophysics, Clocksand Fundamental Constants (eds S.G.Karshenboim and E. Peik), Springer,Berlin, Heidelberg, pp. 57–74.

7. Leblond, J.-M. (1979) in Problems inthe Foundations of Physics; Proceedingsof the International School of Physics“Enrico Fermi” Course LXXXII (ed. G.Lévy Toraldo di Francia), North Holland,Amsterdam, p. 237.

5

2Some Basics

2.1Measurement

Measurement is a physical process to determine the value (magnitude) of aquantity. The quantity value can be expressed as

Q = {q} • [Q], (2.1)

where {q} is the numerical value and [Q] the unit (see following chapter). Repeatedmeasurements of the same quantity, however, generally will result in slightlydifferent results. In addition, systematic effects affecting the measurement resultmight be present and have to be considered. Thus, any measurement resultmust be completed by an uncertainty statement. This measurement uncertaintyquantifies the dispersion of the quantity values being attributed to a measurand,based on the information used. Measurement uncertainty comprises, in general,many components. Some of these may be evaluated by type A evaluation ofmeasurement uncertainty from the statistical distribution of the quantity valuesfrom series of measurements and can be characterized by standard deviations.The other components, which may be evaluated by type B evaluation of measure-ment uncertainty, can also be characterized by standard deviations, evaluatedfrom probability density functions based on experience or other information.For the evaluation of uncertainties of measurements, an international agreedguide has been published jointly by ISO and the Bureau International des Poidset Mesures (BIPM), the Guide to the Expression of Uncertainty in Measurement(GUM) [1, 2]. Precision measurements generally are those with the smallestmeasurement uncertainty.

2.1.1Limitations of Measurement Uncertainty

One might tend to believe that measurement uncertainty can be continuouslydecreased as more and more effort is put in the respective experiment. However,this is not the case since there are fundamental as well as practical limitations for


6 2 Some Basics

measurement precision. The fundamental limit is a consequence of the Heisen-berg uncertainty principle of quantum mechanics, and the major practical limit isdue to noise.

2.1.1.1 The Fundamental Quantum LimitNote that throughout this book, we will use the letter f to denote technical fre-quencies, while the Greek letter ν is used to denote optical frequencies.

The Heisenberg uncertainty principle is a fundamental consequence of quan-tum mechanics stating that there is a minimum value for the physical quantityaction, H:

ΔHmin ≈ h, (2.2)

where h is the Planck constant. Action has the dimensions of energy multipliedby time and its unit is joule seconds. From the Heisenberg uncertainty princi-ple, it follows that conjugated variables, like position and momentum or time andenergy, cannot be measured with ultimate precision at a time. For example, if Δxand Δp are the standard deviation for position, x, and momentum, p, respectively,the inequality relation

ΔxΔp ≥12ℏ (2.3)

holds (ℏ = h∕2𝜋). Applied to measurement, the argument is as follows: in thecourse of a measurement, an exchange of information takes place between themeasurement system and the system under consideration. Related to this is anenergy exchange. For a given measurement time, 𝜏 , or bandwidth of the measure-ment system, Δf = 1∕𝜏 , the energy which can be extracted from the system islimited according to Eq. (2.2) [3]:

Emin • 𝜏 =EminΔf

≈ h. (2.4)

Let us now consider, for example, the relation between inductance, L, and, respec-tively, magnetic flux, Φ, and current, I (see Figure 2.1). The energy is given byE = (1∕2)LI2 = (1∕2)(Φ2∕L), and consequently,

Imin ≈√

2h𝜏 • L

; Φmin ≈√

2h • L𝜏

. (2.5)

These relations are depicted also in Figure 2.1. The gray area corresponds to theregime which is accessible by measurement. Please note that this is a heuristicapproach which does not consider a specific experiment. Nevertheless, it mayprovide useful conclusions on how to optimize an experiment. For instance, ifan ideal coil (without losses) shall be applied to measure a small current, theinductivity should be large (e.g., L = 1 H, 𝜏 = 1 s, and then Imin = 3.5 × 10−17 A).If instead the coil is applied to measure magnetic flux, L should be small(e.g., L = 10−10 H, 𝜏 = 1 s, and then Φmin = 4 × 10−22 Vs = 2 × 10−7 × Φ0 whereΦ0 = h∕2e is the flux quantum = 2.067 × 10−15 Vs).

2.1 Measurement 7

Components Quantum limit

Electrical current, I Magnetic flux, Φ

Imin ≈

Imin

I0

I0 = e / τ ; L0 = 2τh / e2 Φ0 = h / 2e ; L′0 = (τ / 8)h / e2

L′0 L0

Φ0

Φmin

L0 L0

2h 1

LτΦmin ≈

2hL

τ

Inductance

L

L I

Φ

Figure 2.1 Components and quantities considered (left) and the minimum current, Imin,and the minimum magnetic flux, Φmin, versus inductance, L, for an ideal coil. (From [3],with kind permission from Wiley-VCH.)

Likewise, for a capacitor with capacitance, C, the energy is given by

E = 12

Q2∕C = 12

U2 • C, (2.6)

and thus,

Qmin ≈√

2h • C𝜏

; Umin ≈√

2h𝜏 • C

. (2.7)

Finally, for a resistor with resistance, R, the energy is given by

E = I2 • R • 𝜏 = U2

R• 𝜏 , (2.8)

and thus, for the minimum current and voltage, respectively, we obtain

Imin ≈ 1𝜏

•

√hR; Umin ≈ 1

𝜏

•√

h • R. (2.9)

2.1.1.2 NoiseIn this chapter, we briefly summarize some aspects of noise theory. For a moredetailed treatment of this important and fundamental topic, the reader is referredto, for example, [4].

Noise limits the measurement precision in most practical cases. The noisepower spectral density, P(T , f )/Δf , can be approximated by (Planck formula)

P(T , f )Δf

= h • f +h • f

ehf ∕kBT − 1, (2.10)

where f is the frequency, kB the Boltzmann constant, and T the temperature. Twolimiting cases can be considered as follows.

8 2 Some Basics

Thermal Noise (Johnson Noise) (kBT ≫ hf )

Pth(T)Δf

= kB • T . (2.11)

According to this “Nyquist relation,” the thermal noise power spectral density isindependent of frequency (white noise) and increases linearly with temperature.Thermal noise was first studied by Johnson [5]. It reflects the thermal agitation of,for example, carriers (electrons) in a resistor.

Quantum Noise (hf ≫ kBT)

Pqu(f )Δf

= h • f . (2.12)

The quantum noise power spectral density in this limit is determined by the zeropoint energy, hf , and is independent of temperature and increases linearly withfrequency.

Thermal noise dominates at high temperatures and low frequencies (seeFigure 2.2). The transition frequency, f c(T), where both contributions are equaldepends on temperature and is given by

fc(T) =kBT

h• ln 2. (2.13)

This transition frequency amounts to 4.3 THz at T = 300 K and 60.6 GHz at thetemperature of liquid He at T = 4.2 K.

10−15

10−18

10−21

10−24

10−27

10−30

1 103 106

Pth (T) / Δf

Thermal noise

T = 300 K

T = 4.2 K

T = 0.4 K

Quantumnoise

PQu (f) / Δf

P(T

, f)

/ Δ

f

f

109 1012 1015 1018Hz

Figure 2.2 Noise power spectral density, P(T , f ), versus frequency for different tempera-tures. (From [3], with kind permission from Wiley-VCH.)

2.1 Measurement 9

The thermal noise in an electrical resistor at temperature T generates underopen circuit or shortcut, respectively, a voltage or current with effective values:

Ueff =

√⟨u2(t)⟩Δf

=√

4kBT • R, (2.14)

Ieff =

√⟨i2(t)⟩Δf

=√

4kBT∕R. (2.15)

To keep the noise level low, the detector equipment should be cooled to low tem-peratures to reduce thermal noise. Going from room temperature (300 K) to liquidHe temperature (4.2 K) actually reduces the thermal noise power by a factor ofabout 70. In addition, both thermal and quantum noise can be reduced by reduc-ing the bandwidth, that is, integrating over longer times, 𝜏 . This, however, requiresstable conditions during the measurement time, 𝜏 . Unfortunately, however, othernoise contributions may take over like shot noise and at low frequencies the socalled 1/f noise.

Shot Noise Shot noise originates from the discrete nature of the species carry-ing energy (e.g., electrons, photons). It was first discovered by Schottky [6] whenstudying the fluctuations of current in vacuum tubes. Shot noise is observed whenthe number of particles is small such that the statistical nature describing theoccurrence of independent random events is described by the Poisson distribu-tion. The Poisson distribution transforms into a normal (Gaussian) distributionas the number of particles increases. At low frequencies, shot noise is white, thatis, the noise spectral density is independent of frequency and in contrast to thethermal noise also independent of temperature. The shot noise spectral density ofan electrical current, Sel, at sufficiently low frequencies is given by

Sel = 2eI, (2.16)

where I is the average current. Likewise, for a monochromatic photon flux, wehave for the shot noise spectral density of photon flux, Sopt,

Sopt = 2hνP, (2.17)

where h𝜈 is the photon energy and P the average power.

Low Frequency Noise (1/f Noise) 1/f noise (sometimes also called pink noise orflicker noise) occurs widely in nature but nevertheless might have quite differ-ent origin. More precisely, the relation between noise power spectral density andfrequency often is given by

P(f )Δf

∝ 1∕f β (0.5 ≤ β ≤ 2) (2.18)

with 𝛽 mostly close to 1. In contrast to thermal or quantum noise, the noise powerof 1/f noise decreases with increasing frequency (by 3 dB per octave of frequency).Figure 2.3 shows, for example, the noise power spectral density as measured for

10 2 Some Basics

10−21

10−22

10−23

0.1 1

1/f noise

f

P(f)

Δf

10

Thermal noise

100Hz

J

Figure 2.3 Noise power spectral density as measured for a SQUID magnetometer versusfrequency. (From [7], with kind permission from Wiley-VCH.)

a superconducting quantum interference device (SQUID) magnetometer versusfrequency [7].

2.2The SI (Système International d’Unités)

The present SI (Système International d’Unités) consists of seven base units and 22derived units with specific names. The system is called coherent which means thatthe derived units are given as a product of powers of the base units with only “1”as the numerical factor (e.g., 1Ω = 1 m2 kg s−3 A−2). As a consequence, numericalequations do have the same format as quantity equations.

We now shall briefly describe the seven base units, second, meter, kilogram,kelvin, ampere, mol, and candela. For further reading, we refer to the SI brochureof the BIPM [8] and [9].

Before doing so, we will recall a few basics about the nomenclature of electronicstates in atoms, since this will repeatedly encounter us through the book.

Let us take the ground state of the Cs atom “6 2S1/2” as an example. The firstnumber, 6, indicates the main quantum number. The capital letter, S, gives theangular momentum in terms of ℏ = h∕2𝜋, where S, P, D, F, and so on stand for,respectively, 0, 1, 2, 3, 4, and so on. The small number on the upper left stands forthe multiplicity which is given by (2S+ 1), where S is the resulting electron spinof the atom in units ℏ. The lower right number finally corresponds to the totalangular momentum of the atom, 𝐉 = 𝐋 + 𝐒 (for Russell–Saunders coupling). Forthe notation of a specific quantum state, we shall use the ⟨bra|ket⟩ notation. Notethat for optical dipole transitions, we have the selection rules ΔJ = 0,±1 with theexception that |0⟩ → |0⟩ transition are also forbidden.

2.2 The SI (Système International d’Unités) 11

2.2.1The Second: Unit of Time

The second was originally defined as the 86 400th part of the duration ofa mean solar day. However, at the 11th Conférence Générale des Poids etMesures (CGPM) in 1960, after it had been shown that the rotation of theearth is not really stable, the second was referred to the duration of the trop-ical year in 1900 (ephemeris second). Since 1968, however, the second is nolonger based on astronomic time scale but refers to the frequency of electro-magnetic radiation of a magnetic dipole transition in the hyperfine splitted|F = 3,mF = 0⟩ ↔ |F = 4,mF = 0⟩ ground state 6 2S1/2 of the isotope 133Cs:

The second is the duration of 9 192 631 770 periods of the radiation cor-responding to the transition between the two hyperfine levels of the groundstate of the caesium 133 atom.

It follows that the hyperfine splitting in the ground state of the caesium 133atom is exactly 9 192 631 770 Hz, ν (hfs Cs) = 9 192 631 770 Hz.

At its 1997 meeting the CIPM affirmed that:

This definition refers to a caesium atom at rest at a temperature of 0 K.

This note was intended to make it clear that the definition of the SI secondis based on a caesium atom unperturbed by black body radiation, that is,in an environment whose thermodynamic temperature is 0 K. The frequen-cies of all primary frequency standards should therefore be corrected for theshift due to ambient radiation, as stated at the meeting of the ConsultativeCommittee for Time and Frequency in 1999.

By this definition, it had been ensured that at the time of the definition the new“atomic clock second” did agree with the ephemeris second. Further, according togeneral relativity, different gravitational potential has to be considered. In orderto keep the astronomic time scale and the atomic time scale (coordinated univer-sal time (UTC)) identical, leap seconds are added (or subtracted) to the atomictime scale occasionally whenever their difference becomes larger than 0.9 s. Up totoday, 25 leap seconds have been added to UTC since 1972. The responsibility foradding or subtracting leap seconds lies with the International Earth Rotation andReference Systems Service (IERS), and actually, it is discussed presently to changethe procedure by taking much longer time intervals for coordinating these twotime scales.

The definition of the second is put into praxis, that is, realized as metrologistsuse to say, by atomic clocks (see also Chapter 3). The basic concept of atomicclocks is to lock the frequency of a local oscillator to the frequency of an elec-tronic resonance of the respective atom, which in the classical Cs atomic clockslies in the microwave regime. The setup of a Cs atomic clock with a thermallygenerated atom beam is schematically shown in Figure 2.4. In these clocks, the Cs

12 2 Some Basics

Polarizer Magnetic shield

Microwave resonator

Frequencysynthesizer

Quartzoscillator

1

2

3

4

56

7

8

9

10

1112

Frequencycontrol

Analyzer lD

C -fieldE1 E1

E2

fp

fn Ur

lD

fr fp

E2

lonisationdetector𝑙 𝑙

L

Vacuum tank

Cs oven

Figure 2.4 Schematic representation of a “thermal” Cs atomic clock. In the lower left asection of a Ramsey resonance curve is shown. ID is the current of the ionization detector.Courtesy of A. Bauch, PTB.

atoms are generated by evaporation of Cs in an oven. The atoms in this “thermalbeam” are then state selected with respect to their quantum state by an inhomo-geneous magnetic field, the polarizer (Stern–Gerlach technique). Alternatively,optical pumping can be applied for state selection. Subsequently, the atoms entera microwave Ramsey resonator where the resonant transition between the twohyperfine states is induced.

The method of separated oscillatory fields applied here has been first proposedby Ramsey [10, 11] in the frame of atomic beam magnetic resonance spectroscopy.As shown in Figure 2.4, the microwave interaction regime is not a homogeneousmicrowave cavity but instead is splitted into two separated interaction regimes(each of width l) separated by an interaction free regime of length L. The majorresult of this arrangement is to increase the effective interaction time between theatom and the microwave which thus according to the Heisenberg uncertainty rela-tion results in a respective decrease of the linewidth of the resonant transition. TheRamsey technique has several advantages even as compared to a single interactionzone of the same total length 2l + L. For example, the linewidth is narrower (by afactor of 0.6), the requirements on the homogeneity of the magnetic field are con-siderably relaxed, and the first-order Doppler effect is absent provided the phasedifference of the microwave field in the two sections is constant [12, 13].

There are different approaches to describe phenomenologically the action ofthe Ramsey resonator. One is in terms of coherent interaction of the atoms withthe microwave field by two subsequent π∕2 pulses in the two regions. When the


frequency of the microwave field exactly matches the Cs hyperfine frequency split-ting of the |F = 3,mF = 0⟩ and |F = 4,mF = 0⟩ states, atoms are placed by the π∕2pulse in a superposition with equal probability for both states. This state then canevolve freely with a frequency corresponding to the energy difference of these twostates and enter into the second interaction zone. Since the phase evaluation hasbeen dictated by the microwave field in the first zone, the interaction with the sec-ond π∕2 pulse is fully coherent (provided no phase relaxation occurs during the freetravel), that is, after the interaction in the second zone, the probability of findingthe atom in either state (F = 3 or F = 4) depends on the phase of the rf field withrespect to the atomic oscillator. Thus, as the frequency of the rf field is changed, thenumber of atoms in either state oscillates, giving rise to the Ramsey interference.Alternatively, the action of the Ramsey arrangement can be described in analogyto an optical double-slit experiment [12]. A calculation of the transition proba-bility, P(𝜏), for monochromatic atoms and T ≫ 𝜏 , where T is the time traveledfreely between the two interaction sections, T = L∕v (v is the velocity), and τ isthe interaction time with the microwave field in each section, 𝜏 = l∕v, yields [13]:

P(𝜏) = 12

sin2b𝜏(1 + cos(𝜔𝜇W − 𝜔HF)T + 𝜑), (2.19)

where 𝜔𝜇W is the angular frequency of the microwave field and 𝜔HF is the angular

frequency of the hyperfine splitting, b is the Rabbi frequency, b = 𝜇B𝜇W∕ℏ (𝜇 is the

magnetic dipole moment and B𝜇W the amplitude of the microwave magnetic

field), and 𝜑 is the phase difference between the two microwave fields in the twointeraction sections. This as a function of the detuning, 𝛿 = 𝜔

𝜇W − 𝜔HF, describesan interference structure as shown in Figure 2.5, the central part of which is alsoshown in Figure 2.4.

1.00

0.95

0.90

0.85−4 −2 0

Frequency detuning (kHz)

De

tecto

r sig

na

l

2 4

Figure 2.5 Measured Ramsey fringe pattern for the PTB’s CS1 thermal-beam clock. Thecurve appears upside down compared to the result of Eq. (2.19) due to the special opera-tion configuration of PTB CS1. Courtesy of A. Bauch, R. Wynands, PTB.

14 2 Some Basics

Equation (2.19) is valid for a given velocity of the atoms, and although the cen-tral peak of the Ramsey fringes does not exhibit first-order Doppler broadening,the fringe pattern is smeared out at larger detuning as a result of the velocity dis-tribution, causing the so-called Ramsey pedestal.

Finally, as shown in Figure 2.4, the atoms leaving the Ramsey resonator passa second state-selecting magnet (analyzer) and hit a detector with its signalintensity proportional to the number of atoms that have undergone a resonancetransition. In addition, a small constant magnetic field (C field) is applied to splitthe otherwise energetically degenerated mF states in order to excite only the|F = 3,mF = 0⟩ ↔ |F = 4,mF = 0⟩ transitions. Yet, the magnetic field-induced(quadratic) shift of the mF = 0 states has to be accounted for. The detector signalthen is used through a feedback loop to stabilize the oscillator to the clocktransition frequency. The relative uncertainty for these thermal-beam clocks withmagnetic state selection is of the order of 10−14 or slightly below as, for example,for the PTB’s CS1 with a relative uncertainty of 8 × 10−15 [14]. Slightly smalleruncertainties have been achieved using optical pumping for state selection andlaser-induced fluorescence for detection [15, 16].

The presently most precise Cs clocks (fountain clocks; see Section 3.2), how-ever, apply laser cooling to reduce the temperature and hence the speed of theatoms considerably, thus allowing for much longer interaction times to probe thetransition. Due to the corresponding reduction of the linewidth of the resonancetransition, its center frequency can be detected with higher precision. Still, con-siderable smaller uncertainties of relative resonance frequency determination canbe achieved with optical clocks where electronic transitions in the visible or near-UV spectral regime are probed (see Section 3.3). In the presently discussed newdefinition of the SI, however, the second will be unchanged yet with a modifiedwording to be consistent with the other base units [17]:

The second, s, is the unit of time; its magnitude is set by fixing the numericalvalue of the unperturbed ground state hyperfine splitting frequency of thecaesium 133 atom to be equal to exactly 9 192 631 770 when it is expressedin the unit s−1, which is equal to hertz.

2.2.2The Meter: Unit of Length

Since 1983, the definition of the meter is as follows:

The metre is the length of the path traveled by light in vacuum during a timeinterval of 1/299 792 458 of a second.

It follows that the speed of light in vacuum is exactly 299 792 458 m s−1,c0 = 299 792 458 m s−1.

Though customs in astronomy to measure distances in the path length light trav-els in a given time (e.g., light-year), it is not very convenient for daily life purposes.


The Consultative Committee for Length (CCL) of the International Committeefor Weights and Measures (CIPM) therefore recommended three different waysto realize the meter:

(i) According to its definition by measuring the distance light travels within acertain time interval.

(ii) Via radiation sources (in particular lasers) with known wavelength (or fre-quency). A list of respective radiation sources (Mise en Pratique) is pub-lished by the CCL and frequently updated [18, 19].

(iii) Via the vacuum wavelength, 𝜆, of a plane electromagnetic wave withfrequency f . The wavelength then is obtained according to λ = c0∕f .

According to procedures (ii) and (iii), interferometry then can be applied to cal-ibrate a length gauge block [20]. Gauge blocks made out of metals or ceramicsexhibit two opposing precisely flat parallel surface. For calibration of their length,usually, the gauge block is wrung on an auxiliary platen forming one of the endmirrors of a modified Michelson interferometer (Twyman–Green interferometer,Kösters comparator). Since then interference can be obtained from both end sur-faces of the gauge block, its length can be measured in terms of the wavelength ofthe used radiation by counting the interference orders. Quite frequently, iodide-stabilized He–Ne lasers are applied for this purpose. For the highest precision,the interferometer is placed in vacuum to avoid uncertainties due to the refractiveindex of air. In addition, the temperature must be precisely known and stable. Inany case, the frequency (and hence the wavelength) of the respective laser has tobe known in terms of the frequency of the Cs hyperfine transition frequency whichdefines the second. Today, these many orders of frequency are bridged by opticalfrequency combs. This technique for which T. Hänsch and J. Hall were awardedthe 2005 Nobel Prize in Physics can be considered as a gear which transfers themicrowave frequency of the Cs atomic clock into the visible and adjacent spectralregimes. The name “optical frequency comb” refers to the emission spectrum ofmode locked lasers generating ultrafast (fs) laser pulses. Femtosecond frequencycombs will be discussed in more detail in Section 3.3.1.

Presently, the meter can be realized according to recommendations (ii) and (iii)of the CCL with a relative uncertainty of the order of 10−11, and gauge calibrationscan reach fractional uncertainties as low as 10−8 [20, 21].

The definition of the meter in the new SI will be unchanged, however; as for thesecond, the wording will be modified [17]:

The metre, m, is the unit of length; its magnitude is set by fixing the numer-ical value of the speed of light in vacuum to be equal to exactly 299 792 458when it is expressed in the unit meters per second.

2.2.3The Kilogram: Unit of Mass

Since the first CGPM in 1889, the kilogram is defined by the international plat-inum/iridium prototype (Figure 2.6) stored in the premises of the BIPM in Sévres

16 2 Some Basics

Figure 2.6 The Pt/Ir kilogram prototype as stored at the Bureau International des Poids etMesures (BIPM). Courtesy of BIPM.

in the suburban of Paris. At the third CGPM in 1901, this definition was confirmedby stating:

The kilogram is the unit of mass; it is equal to the mass of the internationalprototype of the kilogram.

It follows that the mass of the international kilogram prototype is always 1 kgexactly, m(K)= 1 kg. However, due to the inevitable accumulation of contaminantson surfaces, the international prototype is subject to reversible surface contamina-tion that approaches 1 μg per year in mass. For this reason, the CIPM declared that,pending further research, the reference mass of the international prototype is thatimmediately after cleaning and washing by a specific method. This reference massis then used to calibrate national standards of platinum/iridium alloy or stainlesssteel.

However, repeated comparisons of the prototype with its copies indicate thatthere might also be some irreversible change of its mass, which indeed is the majordriving force for the presently discussed new definition of mass and the other SI


units [22]. Within this new SI, the mass will be related to the Planck constant, h,and the new definition then will read [17]:

The kilogram, kg, is the unit of mass; its magnitude is set by fixingthe numerical value of the Planck constant to be equal to exactly6.626 06X × 10−31 when it is expressed in the unit s−1 m2 kg, which is equalto joule seconds.

The symbol X represents one or more additional digits to be added to the numer-ical value of h using values based on the most recent International Council forScience: Committee on Data for Science and Technology (CODATA) adjustmentat the time of the new definition.

2.2.4The Ampere: Unit of Electrical Current

The ampere was defined at the ninth CGPM in 1948 as follows:

The ampere is that constant current which, if maintained in two straightparallel conductors of infinite length, of negligible circular cross-section, andplaced 1 metre apart in vacuum, would produce between these conductorsa force equal to 2 × 10−7 N m−1 of length.

This definition actually fixes the value of 𝜇0, the permeability of vacuum or mag-netic constant, according to Faraday’s law to exactly 4π × 10−7 Hm−1.

A realization of the ampere precisely according to its definition obviously isnot possible. Closest to its definition, the ampere is realized by the so-calledcurrent balances, where the force between two coils passed by a given currentis balanced by gravitational force. This allows realization of the ampere with anuncertainty of order 10−6. Alternatively, the ampere is reproduced accordingto Ohm’s law through the units volt and ohm. The volt and the ohm can bereproduced with high reproducibility of the order of 10−9 or even better byquantum standards (see Chapters 4 and 5). Actually, this is presently the standardmethod to reproduce the ampere, yet, it has to be kept in mind that this is asidefrom the present SI. In a forthcoming new definition of the SI, the ampere willbe defined through fixing the exact value of the elementary charge. Realizationof the ampere with the required uncertainty then seems feasible thru single-electron transport devices (see Chapter 6). The new definition of the ampere thenwill read [17]:

The ampere, A, is the unit of electric current; its magnitude is set by fixingthe numerical value of the elementary charge to be equal to exactly 1.60217X ×10−19 when it is expressed in the unit second ampere, which is equalto C.

18 2 Some Basics

2.2.5The Kelvin: Unit of Thermodynamic Temperature

The definition of the unit of thermodynamic temperature has been decided at the10th CGPM in 1954 by choosing the triple point of water as the basic fix pointand assigning the temperature of 273.16 K to it. The name kelvin, however, wasonly accepted at the 13th CGPM in 1967/1968. The definition of the kelvin is asfollows:

The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 ofthe thermodynamic temperature of the triple point of water.

It follows that the thermodynamic temperature of the triple point of wateris exactly 273.16 K, Ttpw = 273.16 K.

However, since the triple-point temperature depends on the isotopic composi-tion of the water, the CIPM at its 2005 meeting affirmed that:

This definition refers to water having the isotopic composition definedexactly by the following amount of substance ratios: 0.000 155 76 mol of 2Hper mole of 1H, 0.000 379 9 mol of 17O per mole of 16O, and 0.002 005 2 molof 18O per mole of 16O. (Vienna Standard Mean Ocean Water).

Additionally, the triple-point temperature is affected by impurities dissolved.Further, in the definition of the kelvin, it is stated:

Because of the manner in which temperature scales used to be defined, itremains common practice to express a thermodynamic temperature, symbolT, in terms of its difference from the reference temperature T0 = 273.15 K, theice point. This difference is called the Celsius temperature, symbol t, whichis defined by the quantity equation:

t = T –T0.

The unit of celsius temperature is the degree celsius, symbol ∘C, which isby definition equal in magnitude to the kelvin. A difference or interval oftemperature may be expressed in kelvins or in degrees celsius (13th CGPM,1967/68)

The triple-point-of-water temperature is realized in especially constructedtriple-point cells with a reproducibility of 2 × 10−7. To set up a temperature scaledefining temperatures other than the triple point of water, primary thermometersbased on a well-understood physical system whose temperature may be derivedfrom traceable measurements of other quantities (like e.g., volume, pressure,speed of sound, etc.) have to be used (see Chapter 8). However, since primarythermometers are difficult to use, a practical international temperature scale(ITS) is defined which is supposed to be as close as possible to the thermodynamictemperature scale. The ITS is defined and represented by a number of fix points


and respective measurement procedures to interpolate between these fix points.Presently, the ITS-90 (as decided by the CGPM in 1990) is valid with fix pointslike the triple points of hydrogen, neon, oxygen, argon, mercury, and water (ofcourse) and melting points of, for example, gallium and other metals like indiumand copper (at 1357 K). The ITS-90 presently extends from 0.65 K up to thehighest temperature accessible by radiation thermometry applying Planck’s law.For temperatures in the range of 1 K to 0.902 mK, also a practical temperaturescale on the basis of the melting pressure curve of 3He has been defined, theprovisional low-temperature scale, PLTS 2000.

The forthcoming new definition of the SI will also include the kelvin, since thetriple point of water depends not only on isotopic composition but also on purityof the water. To become independent of the specific properties of water, the tem-perature then will be defined by fixing the value of the Boltzmann constant, kB,using the proportionality between energy, E, and T , E = kBT [17]:

The kelvin, K, is the unit of thermodynamic temperature; its magnitude isset by fixing the numerical value of the Boltzmann constant to be equal toexactly 1.380 6X× 10−23 when it is expressed in the unit s−2 m2 kg K−1, whichis equal to joule per kelvin.

2.2.6The Mole: Unit of Amount of Substance

The quantity used by chemists to quantify the amount of elements or chemicalcompounds taking part in a chemical reaction is called amount of substance. Thisquantity is proportional to the number of elementary units of a sample with theproportionality constant being the same universal constant for all samples. Theunit of amount of substance is the mole:

1) The mole is the amount of substance of a system which contains as manyelementary entities as there are atoms in 0.012 kg of carbon 12; its sym-bol is “mol.”

2) When the mole is used, the elementary entities must be specified andmay be atoms, molecules, ions, electrons, other particles, or specifiedgroups of such particles.

It follows that the molar mass of carbon 12 is exactly 12 g mol−1,M(12C) = 12 g mol−1.

In this definition, it was understood that unbound atoms of carbon 12, at restand in their ground state, are referred to. The constant that relates the number ofentities, n(X), to the amount of substance, N(X), is called the Avogadro constant,NA, n(X) = NA • N(X). The unit of the Avogadro constant is thus reciprocal mole.

The realization of the mole is done by primary measurement techniques (likegravimetry, coulombmetry, or isotope dilution mass spectroscopy) with specifiedmeasurands and uncertainties traced back to the SI (see, e.g., in the Appendix ofthe SI brochure [8]).

20 2 Some Basics

Apart from the fact that a realization of the mole according to its definition isdifficult, it at least sounds strange to relate a quantity which basically refers to anumber of entities to the unit of mass. In the forthcoming new SI definition, themole thus will be directly related to a fixed value of the Avogadro constant [17]:

The mole, mol, is the unit of amount of substance of a specified elementaryentity, which may be an atom, molecule, ion, electron, any other particle or aspecified group of such particles; its magnitude is set by fixing the numericalvalue of the Avogadro constant to be equal to exactly 6.022 14X × 1023 whenit is expressed in the unit mol−1.

2.2.7The Candela: Unit of Luminous Intensity

The candela is a photometric unit which defines the value of luminous intensityat the maximum of the spectral response of human eyes for daylight seeing, V (𝜆),at a wavelength of about 555 nm corresponding to a frequency of 540 × 1012 Hz.The justification as one of the base units in the present SI is, among others, dueto the immense economic importance of the quantitative characterization of illu-minating light sources. Its present definition has been set by the 16th CGPM in1979:

The candela is the luminous intensity, in a given direction, of a source thatemits monochromatic radiation of frequency 540 × 1012 Hz and that has aradiant intensity in that direction of 1/683 watt per steradian.It follows that the spectral luminous efficacy for monochromatic radiationof frequency of 540 × 1012 Hz is exactly 683 lm W−1, K = 683 lm W−1 =683 cd sr W−1.

The realization of the candela can be achieved on the basis of lasers with dif-ferent emission wavelength whose optical emission power is measured absolutelyby, for example, a cryoradiometer and subsequently transferred to the so-calledtrap detectors functioning as transfer standards. These trap detectors then cal-ibrate photometers with spectral response corresponding to V (𝜆). Finally, thesephotometers then are applied to characterize under well-specified conditions theluminous intensity of special standard lamps which serve to maintain and dissem-inate the unit.

The definition of the candela will also be unchanged in the new SI; however, thewording again will be modified to be consistent with the other units [17]:

The candela, cd, is the unit of luminous intensity in a given direction; itsmagnitude is set by fixing the numerical value of the luminous efficacy ofmonochromatic radiation of frequency 540 × 1012 Hz to be equal to exactly683 when it is expressed in the unit s3 m−2 kg−1 cd sr, or cd sr W−1, which isequal to lumens per watt.

References 21

When comparing the present and envisaged new definitions of the base units,we note that the new definitions are less specific. They only fix the numerical valueof seven constants (“defining constants”) which in fact set up the entire SI includ-ing base and derived units. Their distinction therefore becomes partly obsolete.The requirements these constants have to fulfill are that (i) they are really con-stant, respectively—their possible variation is insignificant for the present require-ments of measurements; (ii) their numerical values have been determined withthe required uncertainty (precision); and (iii) the link between the unit and therespective defining constant must be feasible experimentally in order to realizethe unit. However, it should be noted that the definitions in the new SI leave roomfor different realizations of the respective unit.

To conclude this section, we state that the present SI has proven to be the mostsuccessful in providing the base for a harmonized, comparable, and traceable mea-surement system worldwide. Yet, as the quality of measurements improves andscience progresses, the system of units has to follow or actually go ahead, and thisis the major reason for the present discussion about a new SI as will be describedin the remaining of this book.

References

1. Siebert, B.R.L. and Sommer, K.D.(2010) in Uncertainty in Handbook ofMetrology, vol. 2 (eds M. Gläser and M.Kochsiek), Wiley-VCH Verlag GmbH,Weinheim, pp. 415–462.

2. Weise, K. and Wöger, W. (1999) Meßun-sicherheit und Messdatenauswertung,Wiley-VCH Verlag GmbH, Weinheim (inGerman).

3. Kose, V. and Melchert, F. (1991) Quan-tenmaße in der elektrischen Meßtechnik,Wiley-VCH Verlag GmbH, Weinheim (inGerman).

4. (a) van der Ziel, A. (1954) Noise,Prentice-Hall; (b) Vasilescu, G. (2005)Electronic Noise and Interfering Signals:Principals and Applications, Springer,Berlin, Heidelberg, New York.

5. Johnson, J.B. (1928) Thermal agitation ofelectricity in conductors. Phys. Rev., 32,97–109.

6. Schottky, W. (1918) Über spontaneStromschwankungen in verschiede-nen Elektrizitätsleitern. Ann. Phys., 57,541–567 (in German).

7. Gutmann, P. and Kose, V. (1987) Opti-mum DC current resolution of aferromagnetic core flux transformer

coupled SQUID instrument. IEEE Trans.Instrum. Meas., IM-36, 267–270.

8. BIPM http://www.bipm.org/en/publications/si-brochure/ (accessed15 November 2014).

9. Discussion Meeting Issue “The funda-mental constants of physics, precisionmeasurements and the base units ofthe SI”, organized by Quinn, T. andBurnett, K. (2005) Philos. Trans. R. Soc.London, Ser. A, 363, 2097–2327.

10. Ramsey, N.F. (1950) A molecular beamresonance method with separated oscil-lating fields. Phys. Rev., 78, 695–699.

11. Ramsey, N.F. (1990) Experimentswith separated oscillatory fields andhydrogen masers. Rev. Mod. Phys., 62,541–552.

12. Wynands, R. (2009) in Time in Quan-tum Mechanics, Lecture Notes onPhysics, Vol. 789, vol. 2 (eds G. Muga,A. Ruschhaupt, and A. del Campo),Springer, Berlin, Heidelberg, pp.363–418.

13. Vanier, J. and Audo, C. (2005) Theclassical caesium beam frequency stan-dard: fifty years later. Metrologia, 42,S31–S42.

http://www.bipm.org/en/

22 2 Some Basics

14. Bauch, A. (2005) The PTB primaryclocks CS1 and CS2. Metrologia, 42,S43–S54.

15. Makdissi, A. and de Clercq, E. (2001)Evaluation of the accuracy of the opti-cally pumped caesium beam primaryfrequency standard of BNM-LPTF.Metrologia, 38, 409–425.

16. Hasegawa, A., Fukuda, K., Kajita, M., Ito,H., Kumagai, M., Hosokawa, M., Kotake,N., and Morikawa, T. (2004) Accuracyevaluation of optically pumped primaryfrequency standard CRL-O1. Metrologia,41, 257–262.

17. BIPM The Proposed New Definitionsare Preliminary as Listed in the DraftChapter 2 of the 9th SI-brochure,http://www.bipm.org/en/measurement-units/new-si/ (accessed 15 November2014)

18. Quinn, T.J. (2003) Practical realizationof the definition of the metre, includingrecommended radiations of other optical

frequency standards. Metrologia, 40,103–132.

19. BIPM http://www.bipm.org/en/publications/mises-en-pratique/standard-frequencies.html (accessed 15 November2014).

20. see e.g. Schödel, R. (2009) in Handbookof Optical Metrology; Principals andApplications (ed T. Yoshizawa), CRCPress, pp. 365–390.

21. Schödel, R., Walkov, A., Zenker, M.,Bartl, G., Meeß, R., Hagedorn, D.,Gaiser, C., Thummes, G., and Heitzel, S.(2012) A new ultra precision interferom-eter for absolute length measurementsdown to cryogenic temperatures. Meas.Sci. Technol., 23, 094004 (19 pp).

22. Mills, I.M., Mohr, P.J., Quinn, T.J.,Taylor, B.N., and Williams, E.R. (2005)Redefinition of the kilogram: a decisionwhose time has come. Metrologia, 42,71–80.

http://www.bipm.org/en/measurement-units/new-si/




23

3Laser Cooling, Atomic Clocks, and the Second

Even though the definition of the second is supposed to be unchanged inthe forthcoming revision of the International System of Units (SI), we shallnow consider atomic clocks again, since the implementation of laser coolingtechniques not only has considerably improved the standard Cs clock but alsoenabled the development of a new generation of atom clocks (so-called opticalclocks) with unprecedented properties. Let us recall again briefly what a clock is.Simply speaking, it is the combination of a frequency standard, a counter, and anindicator or display. In a classical pendulum clock, for example, the frequencystandard is the pendulum, the counter is the clockwork consisting of a gear trainof cogwheels (usually combined with the escapement which transfers energy tothe pendulum to keep its frequency constant), and the indicator usually is a clockface with rotating hands (neglecting the power source which is a weight on a cordor chain or a mainspring). In a quartz watch, the frequency standard is a quartzcrystal oscillating at a specific frequency, the counter is an electronic circuit, andthe display could be both, analog and digital. In the classical Cs atomic clockdescribed in Section 2.2.1, the frequency standard is the ground-state microwavehyperfine transition of 133Cs, and the counter again is an electronic circuitwhose input is the microwave signal received from the stabilized microwaveoscillator.

The quality of a frequency standard is quantified by its accuracy and frequencystability or vice versa its uncertainty and frequency instability. The accuracy ofa frequency standard characterizes how well the output of a clock agrees withthe SI definition of the second. Different systematic effects may be the cause fordifferences of the instantaneous frequency output with respect to the nominaltransition frequency of the unperturbed individual atom like finite temperature,external magnetic, or electric fields. A careful estimate of the uncertainty of a clockconsidering all possible contributions thus is of utmost importance in particularfor primary clocks that claim highest accuracy and stability. The evaluation of thefrequency stability, which reflects statistical (noise) fluctuations of the output fre-quency of a standard, could in principle follow the standard statistical procedureby computing, for example, the standard deviation of a series of clock readingsrelative to a perfect or much better clock. However, this would result in mislead-ing conclusions in some cases. Consider, for instance, a very stable clock with a


24 3 Laser Cooling, Atomic Clocks, and the Second

constant frequency offset. In this case, the standard deviation incorrectly wouldassign a high instability to the standard, and even worse, the standard deviationwould grow with time. Therefore, clocks and their frequency standards are usuallycharacterized by the so-called Allan deviation and its square, the Allan variance,respectively [1] as discussed briefly in the following.

Consider the output voltage of a frequency standard

U(t) = U0 sin(2𝜋ν(t) • t) = U0 sin(2𝜋ν0 • t + 𝜑(t)), (3.1)

where U0 is the amplitude (which we have assumed to be stable), ν(t) the instan-taneous frequency, ν0 the nominal frequency, and 𝜑(t) the instantaneous phase.The relative frequency deviation, also called fractional frequency, is then given by

y(t) ≡ν(t) − ν0

ν0= 1

2𝜋ν0

d𝜑dt , (3.2)

the relative frequency drift by

y(t) ≡ ddt y(t), (3.3)

and the normalized phase fluctuation by

x(t) ≡ 𝜑(t)2𝜋ν0

. (3.4)

Assuming that the time scale is divided into contiguous sections with width 𝜏 , themean relative frequency deviation in section n, yn(𝜏), is given by

yn(𝜏) =1𝜏 ∫

tn+𝜏

tn

y(t)dt. (3.5)

The fluctuation of the instantaneous frequency of a clock, that is, its stability orinstability, is characterized by the two-sample variance, also called Allan variance:

𝜎2y (𝜏) =

12

⟨(yn+1 − yn

)2⟩

. (3.6)

For a finite series of measurements, this can be approximated by

𝜎2y (𝜏) =

12(k − 1)

k−1∑

n=1

(yn+1 − yn

)2, (3.7)

where k, the number of samples taken, has to be sufficiently large to achieve highsignificance.

The Allan standard deviation, 𝜎y(𝜏), is defined as the square root of the Allanvariance. A double logarithmic plot of 𝜎y(𝜏) versus 𝜏 allows to identify possiblecauses of instability. If, for example, shot noise (white frequency noise) is the dom-inating contribution, 𝜎y(𝜏) decreases like 𝜏−1∕2 , for 1/f frequency noise 𝜎y(𝜏) turnsconstant at higher 𝜏 and may even increase again, for example, if frequency driftis present.

For white frequency noise, the Allan standard deviation scales as

𝜎y(𝜏) ∝1Q

1(S∕N)

𝜏−1∕2, (3.8)

3.1 Techniques for Laser Cooling 25

where Q, the line quality factor, Q = ν∕Δν, is given by the frequency of the transi-tion, ν, with respect to its measured linewidth, Δν, and S/N is the signal to noiseratio.

A more elaborated discussion of the properties of frequency standards can befound in [2].

As already mentioned when describing the realization of the present defini-tion of the second by means of the thermal Cs clock, the second-order (relativis-tic) Doppler effect and the limited interaction time are limiting factors for bothclock accuracy and frequency stability even when the Ramsey scheme is applied(always assuming that the intrinsic, recombination lifetime-limited linewidth ismuch narrower). Both scale with the velocity of the atoms, v, (for the second-order Doppler effect, which is a consequence of the relativistic time dilatation, wehave Δν∕ν = (1∕2)(v∕c)2 which at room temperature (v ∼ 100 m s−1) is of order10−13). Thus, the ultimate choice would be to use atoms with lower velocity. Thefirst proposal in this sense by Zacharias [see e.g. in 3] was to use a vertical geometrywith one microwave interaction regime where the atoms are launched upward andstill interact with the microwave twice, first when flying upward and second whenfalling down due to the action of gravity. The transit time would again be deter-mined by the velocity of the atoms, and thus, a considerable increase in interactiontime could be expected for the slowest atoms within the thermal distribution. Yet,the early approaches failed due to the weakness of the signal. However, with theprogress of laser cooling techniques, the concept was successfully realized in theso-called fountain clocks (see Section 3.2).

Finally, since a clock is more than a frequency standard, qualifying of a clockneeds at least a second clock to compare with. In fact, clock comparisons are atthe heart of clock metrology as briefly mentioned at the end of Section 3.3.4.

3.1Techniques for Laser Cooling

Laser irradiation as a method for cooling atoms and ion gases was first proposedin 1975 by, respectively, Hänsch and Schawlow [4] and Wineland and Dehmelt [5].Cooling of atoms or ions relies on the presence of an (strong) allowed optical tran-sition, as it is the case in Cs, for example, for the 62S1/2 to 62P3/2 transition (seeFigure 3.9). However, cooling and trapping techniques differ for an ensemble ofatoms or single ions. We shall briefly describe the techniques relevant for atomicclocks, that is, Doppler and sub-Doppler cooling and trapping of an atom cloudin Sections 3.1.1 and 3.1.2 and cooling and trapping of single ions in Section 3.1.4.For a more detailed reading on techniques and application of laser cooling, thereader is referred to, for example, [6–8]. The specific case of trapping of atoms inoptical lattices which takes use of the advantages of both previously mentionedtechniques is treated separately in Section 3.1.3. Subsequently, we will describethe improvement of the classical Cs atomic clock by applying laser cooling tech-niques in Section 3.2 before we turn over to optical clocks in Section 3.3 including


the description and application of femtosecond combs for frequency metrologyin Section 3.3.1. Finally, we discuss in Section 3.3.4 the application of optical fre-quency standards to study possible variations of the fine-structure constant whichis considered as one of the fundamental constants of nature.

3.1.1Doppler Cooling, Optical Molasses, and Magneto-Optical Traps

If an atom gas with a given temperature and hence velocity distribution – inthermal equilibrium, this is a Maxwell–Boltzmann distribution – is irradiatedby a laser with wavelength 𝜆, slightly tuned to the red with respect to the reso-nance transition, only atoms with the right velocity opposing the laser beam areable to absorb a photon, as the frequency is shifted by the proper amount dueto the Doppler effect. With the absorption process, a recoil momentum ℏk = h∕𝜆pointing along the propagation direction of the incoming laser is transferred to theatom. As the atom recombines back into its ground state by emission of a photon,it again will receive a recoil momentum. However, since the emission will be ina random direction, in repeating absorption and emission processes, the “emis-sion” recoil momentum averages out to zero but not the “absorption” momentum.Thus, as a consequence, the velocity of the atoms traveling toward the incominglaser beam will be lowered, and due to thermalization by scattering among eachother, the entire atom gas will become cooler actually maintaining approximatelya Maxwell–Boltzmann distribution. However, as the atom gas cools, the Dopplershift will be reduced, and eventually, the incoming laser will not be in resonanceanymore. There are basically two ways to overcome this problem: (i) tune (sweep)the laser frequency [9] or (ii) change the resonance frequency, for example, byapplying a DC magnetic field due to the Zeeman effect [10]. The Zeeman methodrequires, however, that the shifts of the ground state and excited state are differentas it is the case for the cooling transition in Cs. In this case, the field can graduallybe changed along the path of the atoms, thus always keeping a subset of atoms inresonance (Zeeman slower). A limit for this cooling procedure is set by the natu-ral (homogeneous) linewidth,Δν = 1∕2𝜋T (T is the excited-state phase relaxationtime) of the resonance transition, resulting in a minimum temperature achievableby Doppler cooling in a two-level system of [11]:

TD = hΔν2kB

. (3.9)

Consider next two laser beams opposing each other and still detuned slightly to thered. Then for each atom, there is a laser beam, which travels in the opposite direc-tion as the atom (for the one-dimensional case to which we restrict the discussionfor the moment). The atomic resonance frequency then is shifted toward the laserfrequency of the opposing beam such that absorption can take place. Thus, thereis a resulting force decelerating the atoms. With three intersecting pairs of orthog-onal laser beams as illustrated in Figure 3.1, a so-called optical molasses (OM) canbe realized.


Figure 3.1 Laser beam arrangement for an optical molasses.

The name “optical molasses” reflects the fact that the motion of atoms is similarto a particle (or body) in a viscous medium. Note, however, that an OM does notprovide a trap for atoms, since there is no restoring force for atoms leaving thecenter of the intersecting laser beams. Trapping and cooling of atoms are realizedby magneto-optical traps (MOTs).

In an MOT, the combined action of a spatially inhomogeneous magnetic fieldand laser light performs both cooling and trapping. It requires for the resonancetransition being involved that the angular momentum of the ground state, Jg,and the excited state, Je, differ by one unit, Je = Jg + 1, that is, Jg = 0 and Je = 1in the simplest case as, for example, for 40Ca and 88Sr. Taking this simplest case,the action of a DC magnetic field, B, will leave the ground state, Jg = 0, unaf-fected, while the excited state, Je = 1, splits into 2Je + 1 = 3 substates with mJ =0,−1, and + 1, respectively. The energy of the mJ = 0 state is almost independentof the magnetic field strength, while the energies of the mJ = ±1 vary linearly withmagnetic field:

ΔE = ±gJ𝜇BB, (3.10)

where gJ is the Landé factor and 𝜇B the Bohr magneton. Optical transitionsbetween the ground state to the mJ = +1 and mJ = −1 excited state can beinduced by 𝜎

+ and 𝜎− circular polarized light, respectively.

We next consider a magnetic field which varies linearly along the z directionlike Bz(z) = bz from a center at z = 0 and two counterpropagating laser beamsof opposite circular polarization, 𝜎+ and 𝜎

−, respectively, slightly tuned to the redwith respect to the Jg → Je, mJ = 0 transition as shown in Figure 3.2. Atoms trav-eling from z = 0 to the right opposing the 𝜎− laser beam will have their energy level


E

Ee

Eg

ħω0

ħωL

σ+ σ−

mJ = +1

mJ = −1

mJ = 0

J = 0

0 z

J = 1

Figure 3.2 Energy levels and laser beam arrangement in a (1D) MOT. (From [2], with kindpermission from Wiley-VCH.)

mJ = −1 shifted toward the laser frequency, resulting in an increase of absorp-tion. Opposite, for the 𝜎

+ laser beam, the absorption even further decreases. Asa consequence, we have a resulting cooling. But in contrast to the OM, we nowhave a redriving force toward the center at z = 0 due to the spatial gradient ofthe magnetic field. On the other side, the same arguments hold with the role ofthe 𝜎+ and 𝜎

− beam reversed. A three-dimensional (3D) trap consequently can berealized like in the OM (see Figure 3.1) but now with three pairs of opposing laserbeams with opposite circular polarization [12]. The magnetic field with propertiesas required is a quadrupole field that usually is generated by a pair of Helmholtzcoils with opposite direction of the electrical current (“anti-Helmholtz coils”).

3.1.2Cooling Below the Doppler Limit

The first observation of cooling well below the Doppler limit has been made in BillPhillips’s group at the National Institute of Standards and Technology (NIST) inGaithersburg where in a sodium gas they measured temperatures as low as T =43μK while the Doppler limit was at TD = 240μK [13]. As subsequently shownby Dalibad and Cohen-Tannoudji [14], this is due to the multilevel character ofthe alkali atoms combined with a spatially varying light field. Even though in themean time different schemes for sub-Doppler cooling (see, e.g., in [8]) have beendemonstrated here, we shall briefly describe only this mechanism, called “Sisy-phus cooling.” It is based on the standing wave pattern generated by opposing laserbeams (in z direction) with the same wavelength but orthogonal linear polariza-tion (lin ⊥ lin). In this standing wave pattern, the polarization state varies spatiallywith a period corresponding to the wavelength of the laser beams. The polar-ization changes across half a wavelength are shown in Figure 3.3. Starting, forexample, at z = 0 where we have a resulting linear polarization under 45∘ withrespect to the incoming beams, the polarization changes to 𝜎

− circular polariza-tion at z = 𝜆∕8, then becomes linear again at z = 𝜆∕4, and subsequently changes to𝜎+ circular polarization at z = 3∕8𝜆 and so forth. An atom with a J = 1∕2 ground


0

k1

E0 xk2

E0 y

σ−

σ− σ+ σ+σ−

σ+

λ/8

λ/8(a) (b)

mg = −1/2

mg = +1/2

me = ±1/2

λ/4 3/8λ

3/8λ 5/8λ

λ/2 z

z

x

y

Figure 3.3 Illustration of Sisyphus cooling. (a) The polarization along the z directionin a “lin ⊥ lin” standing wave configuration. (b) The corresponding light shift of the mg =+1∕2 and mg = −1∕2 ground state. (From [2], with kind permission from Wiley-VCH.)

state will experience spatially varying shift of its mg = −1∕2 and mg = +1∕2 statesdue to the Stark effect as shown in Figure 3.3b. Consider an atom at the z = 𝜆∕8position, where the polarization is 𝜎−, traveling along the z direction. While mov-ing forward, it has to climb up the hill at the expense of its kinetic energy. At thetop of the hill at z = 3∕8𝜆, the polarization has changed to 𝜎

+ resulting in strongtransitions into the excited state me = +1∕2 (Δm = +1) from where the atom canrecombine in either the mg = −1∕2, 0, or + 1∕2 state, resulting in a net transferof atoms from the mg = −1∕2 into the mg = +1∕2 state where it ends up in thepotential minimum. Moving forward, it has to climb up the potential hill again,and as it reaches the top at z = 5∕8𝜆, it will be pumped by the 𝜎

− light back intothe mg = −1∕2 state and the story starts all over again. In principle, cooling couldproceed until the recoil limit for atoms with total mass M

T = (ℏk)2

2kBM (3.11)

is reached (k is the wave vector of the laser light). This fundamental limit is set bythe spontaneous emission of a single photon, that is, the momentum transferredby the last photon emitted before the final temperature is reached.

3.1.3Optical Lattices

Trapping and manipulation of neutral particles by means of laser radiation werefirst demonstrated by Ashkin [15]. Particularly, neutral atoms can be trapped ina standing wave light field generated by interference between two (or more) laserbeams due to the intensity-dependent “light shift” of the energy levels (Stark shift)and the resulting dipole force [16]. The potential landscape of a two-dimensional(2D) optical lattice is shown in Figure 3.4. As can be seen, a periodic lattice ofpotential minima is formed. The period of the minima is Δ = 𝜆∕2, while theirdepth, which depends on the laser intensity, typically is of the order of 10 μK.


U

y

Xλ/2

Figure 3.4 Potential landscape of a two-dimensional optical lattice.

3.1.4Ion Traps

Electrically charged particles can be trapped spatially by the combined action ofelectric and magnetic fields. The so-called Penning trap uses a combination ofstatic electric and magnetic fields [17], while the Paul trap uses an AC electricfield (rf trap) [18]. In either case, the particles are trapped in vacuum, possiblywith the addition of some buffer gas.

A Penning trap uses a static spatially homogeneous magnetic field generatedby a cylindrical magnet and a static spatially inhomogeneous electric field gener-ated by a quadrupole–ring electrode configuration. The magnetic field confinesthe particles in the plane perpendicular to the magnetic field direction, and theelectric field hinders them to escape along the direction of the magnetic field. Pen-ning traps are very successfully applied to measure properties of ions and particles(like mass, g-factor, etc.). For spectroscopy applications in conjunction with lasercooling, Paul traps are at an advantage.

The Paul trap or quadrupole ion trap as it is sometimes called can be realizedin a linear and 3D configuration. The electrode configuration of the 3D Paul trapis the same as for the Penning trap and is shown in Figure 3.5. It consists of twohyperbolic electrodes (a) and a hyperbolic ring electrode (b). The two hyperbolicelectrodes are centered in the ring electrode facing each other. The rf electric fieldis applied between the ring electrode and the hyperbolic electrodes, generatingan oscillating electric quadrupole field. A charged particle faces oscillating forcesunder such conditions; in the first half cycle of the field, the ions are focused inthe axial direction and defocused in the perpendicular direction, while in the sec-ond half cycle, they are defocused in the axial and focused in the perpendiculardirection. Since both effects alternate with a high frequency (typically of the orderof megahertz), the ions are trapped in the space between these three electrodes.Mathematically, the motion of the charged particles in such a field is describedby Mathieu’s differential equations. A rigorous treatment can be found in [2]. Anintuitive understanding has been presented by Wolfgang Paul himself: the motion


+

+ –a

b b

a a

a

FE

FE

bb

+ +

−

−+

+ +

+

E E

FE

FE

E E

−

− −

Figure 3.5 Two-dimensional sectional view of the electrode configuration and electric fielddistribution (for two half cycles of the rf field) of a three-dimensional Paul trap. The arrowslabelled E and FE denote the resulting instantaneous electric field and force, respectively.

of the charged particle can be seen analogously to the motion of a mechanicalparticle, let’s say a ball, in a 3D saddle point landscape; putting the ball on top ofthe saddle point would result in an unstable situation, resulting in the ball rollingdown the hill. Yet, if the saddle point landscape is rotated with sufficiently highfrequency around its symmetry axis piercing through the top of the saddle point,the ball will be stabilized near the top because there is not enough time for it toroll down before the potential has changed due to its rotation.

In a linear Paul trap, the electrodes are formed by metallic rods arranged in arectangular configuration (see Figure 3.6). Axial confinement then can be achievedby either including additional ring electrodes as shown in Figure 3.6a or by usingsegmented rods with three isolated parts (Figure 3.6b) and a DC potential appliedto the outer parts.

It must be noted that generally a Doppler broadening of optical transitions oftrapped ions in a rf trap occurs due to the oscillatory micromotion which for thefirst-order Doppler effect can be avoided, however, when the ion is confined toa regime smaller than the wavelength of the interacting laser field (Lamb–Dickeregime) [19].

U0

(a) (b)

Figure 3.6 Configuration of linear Paul traps with additional ring electrodes (a) or seq-mented rods (b) for axial confinement. (From [2], with kind permission from Wiley-VCH.)


Figure 3.7 Yb+ ions (seen by their fluorescence) in a linear Paul trap. The distancebetween the ions is about 10–20 μm. (Courtesy of T. Mehlstäubler, PTB.) (Please find a colorversion of this figure on the color plate section.)

If more than one ion is trapped in a linear trap, Coulomb repulsion betweenthem has to be considered. If the kinetic energy of the ions is lower than the repul-sion energy, this leads to the formation of crystalline structures, a linear chain inthe simplest case (see Figure 3.7). These and more complex (2D, 3D) quasicrys-talline structures can be used for quantum information processing [20] and tomimic solid-state phenomena otherwise difficult to study (see, e.g., [21]), similarto what has been reported for atoms in optical lattices (see, e.g., [22, 23]). As inreal crystals, collective motion of the ions can be excited, resulting in a harmonic-oscillator-like discrete vibrational excitation spectrum. This actually can be usedfor cooling of ions in the trap (sideband cooling) [5, 24] and provides a means forentanglement of ions [20, 25].

3.2The Cs Fountain Clock

The Cs fountain clock [26] relies on the same principle as the “thermal-beam” Csclock described in Section 2.2.1, that is, the |F = 3,mF = 0⟩ → |F = 4,mF = 0⟩hyperfine microwave transition of the 133Cs ground state is the reference transi-tion according to the definition of the second, and a Ramsey scheme is applied forthe interrogation of the Cs atoms with the rf field. Only that now laser-cooledCs atoms with much lower velocity (about 1 cm s−1) than for a thermal beam(of order 100 m s−1) are used, and thus, the interaction time increases accord-ingly, resulting in a considerably reduced linewidth. The setup of a Cs fountainclock is schematically shown in Figure 3.8. It consists of three major parts: thepreparation zone, the detection zone, and the microwave interaction zone. TheCs atoms are released into the cooling chamber of the preparation zone froma Cs reservoir held at a constant temperature close to room temperature and apartial pressure of some 10−6 Pa. Laser cooling is achieved in an MOT in combi-nation with an OM (sometimes also only an OM is used). The cloud of Cs atomswith about 107 –108 atoms is cooled below the Doppler limit to temperatures oforder 1 μK. The strong dipole-allowed |62S1∕2, Fg = 4⟩ to the |62P3∕2, Fe = 5⟩ tran-sition (labeled 2 in Figure 3.9) is used for laser cooling. However, since duringthe cooling cycle some atoms also relax – though ideally forbidden – into the|Fg = 3⟩ state, in addition to the cooling laser beams, a repumping laser (labeled1 in Figure 3.9) is required to pump these atoms back into the |Fg = 4⟩ state viathe |Fe = 4⟩ state. At the end of the cooling phase, a moving OM is generated byslightly detuning the vertical laser beams. If the upward pointing laser beam is

3.2 The Cs Fountain Clock 33

Magneticshields

To pumpand window

C-field coil

Vacuum tank

Ramsey cavity

State-selectioncavity

Detection zone

Preparationof cold atoms

Caesium reservoir

Figure 3.8 Schematic setup (simplified) of an atomic fountain clock. (Courtesy of R.Wynands, PTB.)

detuned to the blue by an amount 𝛿𝜈 and vice versa the downward pointing laserto the red by the same amount, the resulting interference pattern moves upward(moving molasses), and the atom cloud is accelerated to velocities of typically sev-eral meters per second. When subsequently all laser beams are turned off, theatoms fly on into the microwave resonator after passing the detection zone andthe state selection cavity. As the atoms enter the state selection cavity, they are inthe |Fg = 4⟩ state with the nine |mF⟩ states (mF = −4,−3, … ,+4) equally popu-lated. In the state selection cavity, atoms are transferred from the |Fg = 4,mF = 0⟩state into the |Fg = 3,mF = 0⟩ state by applying a microwave pulse tuned at theclock transition frequency, and subsequently, all other atoms which remained inthe |Fg = 4⟩ state are pushed away by a laser beam tuned to the |Fg = 4⟩ → |Fe =5⟩ transition. Thus, only |Fg = 3,mF = 0⟩ atoms enter the Ramsey cavity where


6 2P3/2

Fe = 5

mF

Zeeman-splittingin kHz/μT

Fe = 4

251.4 MHz

+5

−5

0 5.6

3.73

−00.6

−9.34

3.5

−3.51

+4

−4

0

−3

+3

0

−2

+2

0

+4

−4

0

−3

+3

0

201.5 MHz

151.3 MHz

Fe = 3

Fe = 2

Fg = 4

9192 MHz

Fg = 3

D21 2

λ = 852.1 nm

6 2S1/2

Figure 3.9 Energy level scheme of the lower states in 133Cs.

the clock transition is excited as described in Section 2.2.1. The interaction time isnow given by the time of flight of the atoms up to their apogee, where due to grav-itational force they come to rest and subsequently fall back down to cavity. Thistime of flight (T = 2

√2h∕g) for an h ∼ 1 m apogee above the cavity is of order

1 s, which consequently determines the repetition rate for the entire cycle. Dur-ing the flight, the atomic cloud will expand according to its thermal velocity, andthus, only a fraction of the atoms will hit the aperture of the Ramsey resonator andundergo the second Ramsey transition. However, as the atoms are cooled belowthe Doppler limit, this spread for a T = 1μK cloud is small such that about 50%of the atoms enter again the interaction regime. If the atoms had been cooled tothe Doppler limit only (T = 125μK), this fraction would have been reduced toabout 1%. Finally, the atoms after leaving the microwave section pass through thedetection zone where the number of atoms in the |Fg = 4⟩ and |Fg = 3⟩ statesis detected separately by means of laser-induced fluorescence [27]. The detec-tion zone has to be viewed as three spatially subsequent zones. In the first part,the atoms pass a standing wave laser field exciting the |Fg = 4⟩ → |Fe = 5⟩ tran-sition. Relaxation into the |Fg = 4⟩ state results in the emission of fluorescence.

3.2 The Cs Fountain Clock 35

Microwavecavity

(a) (b) (c) (d)

Detectionlaser

Figure 3.10 Principle of operation ofthe atomic fountain clock illustrating theessential four steps in a measurementcycle: (a) preparation of the cloud of coldatoms, (b) launch of the cloud toward themicrowave cavity and subsequent pas-sage through the state selection cavity and

Ramsey cavity, (c) free flight and turnaroundof the cloud and subsequent second passagethrough the Ramsey cavity, and (d) detectionof the number of atoms in the |Fg = 3⟩ and|Fg = 4⟩ states, respectively. (Courtesy of R.Wynands, PTB.) (Please find a color version ofthis figure on the color plate section.)

Repeating this process many times (cycling) results in a fluorescence signal whichcan be detected by a photodetector with its overall signal strength being pro-portional to the number of atoms originally in the |Fg = 4⟩ state. Under this firstdetection stage, the atoms in the |Fg = 4⟩ state are then pushed away by a strongunidirectional laser beam again tuned to the |Fg = 4⟩ → |Fe = 5⟩ transition so thatonly |Fg = 3⟩ atoms reach the third zone. There, they are pumped in a standingwave field into the |Fg = 4⟩ state, and then again, the cycling transition is employedwith the fluorescence signal now being proportional to the number of |Fg = 3⟩atoms entering into the zone. The entire cycle from cooling to detection is sum-marized in Figure 3.10.

As the major result, the Cs fountain clock has improved stability and accuracy ofthe realization of the second by roughly 1 order of magnitude as compared to thethermal-beam Cs clock. Resulting values for the Allan standard deviation as low as𝜎y(𝜏) = 1.6 × 10−14( 𝜏∕s)−

1∕2 and fractional uncertainties of 2–7 × 10−16 have beenreported [28–30].

Also, fountain clocks operating in continuous mode rather than the previouslydescribed pulse mode have been developed [31]. Further, 87Rb fountain frequencystandard has been realized at the SYRTE laboratory in Paris [32] and at Penn StateUniversity [33].

At the end of this section dealing with microwave clocks and frequencystandards, the hydrogen maser has to be mentioned because of its excellentshort-term stability (Allan standard deviation <10−14 at 1 s of averaging time) andits use in timing laboratories as the so-called flywheel to increase the short timestability of the time scales. The hydrogen maser is based on transitions betweenthe two |F = 1,mF = 0⟩ and |F = 0,mF = 0⟩ ground states of the hydrogen atomat 1.42 GHz. From a beam of atoms, the |F = 1,mF = 0⟩ and |F = 1,mF = 1⟩


states are selected by a Stern–Gerlach magnet and transferred into a storage bulbinside a microwave cavity resonant with the 1.42 GHz transition frequency. Allatoms in the other ground states |F = 1,mF = −1⟩ and |F = 0,mF = 0⟩ do notreach the storage bulb. Thus, a population inversion exists resulting in stimulatedemission of the |F = 1,mF = 0⟩ → |F = 0,mF = 0⟩ transition, and self-sustainingmaser oscillation may build up. A small antenna then can pick up this oscillation(active hydrogen maser). In contrast, in the passive hydrogen maser, a microwavesignal at the resonance frequency is amplified by the population invertedhydrogen atom gas.

For further description and discussion of the properties of active and passivehydrogen masers, the reader is referred to [2].

3.3Optical Clocks

Since the short-term stability of a frequency standard scales with the quality fac-tor, Q, of the respective transition (see Eq. (3.8)), it would be favorable to movethe “clock transition” to higher frequencies. In particular, moving to the visiblerange of the electromagnetic spectrum at several hundred THz would, in prin-ciple, result in an improvement of as much as 105 as compared to the 9.2 GHzfrequency of the microwave clock transition in Cs provided the linewidth remainsthe same. This in fact has driven the development of the so-called optical clocks(we shall use the wording “clocks” even though most of the developments reportedso far rather should be considered as “optical frequency standards” because theyhave not yet fulfill the requirements for a clock, for example, long-term contin-uous operation to establish a time scale, etc.). The remarkable and still ongoingprogress in recent years, however, can again only be touched here. For furtherreading, see, for example, [34–36]. It has been particularly the development oflaser cooling and trapping techniques for both neutral atoms and ions which pro-moted the development of optical clocks. One of the essential requirements forsuitable atoms or ions is the presence of a strong dipole-allowed optical transition(e.g., S → P) for laser cooling together with a transition with narrow homogeneous(sometime also called “natural”) linewidth since the narrower the linewidth, themore precise a determination of the line center frequency is possible. The halfwidth of the homogeneous, that is, Lorentzian-shaped, optical transition, Δν, isdetermined by the phase relaxation time, T , of the coherent polarization createdby the driving laser field

Δν = 12𝜋T (3.12)

with T given according to

1T = 1

2T1+ 1

T2, (3.13)

3.3 Optical Clocks 37

where T1 is the recombination (population) lifetime and T2 accounts for all otherphase relaxation processes like collisions. For pure recombination damping, wehave T = 2T1.

Consequently, narrow linewidth requires long excited-state lifetime as it isthe case, for example, for dipole-forbidden optical transitions (e.g., a S → D(quadrupole transition), S → F (octupole transition)), or appropriate intercombi-nation transitions. Intercombination transitions involve a change of the spin state(ΔS ≠ 0; e.g., singlet → triplet). They are forbidden since the electric field cannotinduce a spin flip. This is strictly true for light atoms but for heavier atoms, thetransitions become weakly allowed.

For the choice of the atom or ion species, several sometimes contradicting issueshave to be considered, like energy of the respective transitions and accessibilityby available stable laser systems, robustness against external perturbations likemagnetic and electric fields, and so on.

The principle of operation (Figure 3.11) of an optical clock is very much thesame as for a microwave clock. A sufficiently stable local oscillator realized by anarrowband laser source [37] is needed to perform the spectroscopy of the atomsor ions which have been prepared, for example, by laser cooling and trapping.During the spectroscopy phase, the state preparation system generally is switchedoff to avoid perturbations. Fluorescence is used for detection in most cases pro-viding the internal state information and thus the feedback signal for the localoscillator. Femtosecond frequency combs (see Section 3.3.1) finally are used asfrequency dividers providing the microwave output signal. For the spectroscopy,naturally high-resolution techniques like free-space saturated absorption [38–40]and free-space Bordé–Ramsey atom interferometry are employed.

For saturated absorption spectroscopy, two counterpropagating laser beamswith the same frequency are directed into the atomic cloud. Due to the Dopplereffect, atoms with different velocities with respect to the beam directions areprobed if the laser is tuned slightly off resonance. If, for example, the laser is

Statepreparation

Localoscillator

Counter

Output signal

Servosystem

Spectroscopyatoms or ion

Detector

Figure 3.11 Principle of operation of an optical clock.


detuned toward lower frequencies with respect to the resonance frequency,absorption can occur only if atoms with suitable velocity move opposite to thedirection of a laser beam; thus, while one of the laser beams will be absorbed byatoms with a velocity component +v in the direction of the laser beams, the otherbeam will be absorbed by atoms with −v. Only when the laser frequency is exactlyat resonance, the same subgroup of atoms with zero velocity component in thebeam directions will be addressed. Since both beams then have to share the atomsfor absorption, the total absorption will be reduced and a dip in the absorptionprofile occurs. Under ideal conditions, the spectral width of this so-called Lambdip can approach the natural linewidth of the transition.

The Bordé–Ramsey atom interferometry [41, 42] can be seen as the extensionof the Ramsey separated field technique to the optical regime. The basics of thistechnique can be understood as follows: consider a two-level atom. Absorptionof a resonant photon not only transfers the atom from the ground state |g⟩ tothe excited state |e⟩ but also transfers the recoil momentum ℏ𝐤 with |𝐤| = 2𝜋∕𝜆 tothe atom. As a consequence, the trajectory of the excited-state atoms is slightlychanged with respect to those remaining in the ground state. In the matter wavepicture, where the atoms with mass M and velocity v are represented by their deBroglie wavelength 𝜆dB = h∕Mv, this process can be viewed as a beam splitter. For aproperly (pulse duration and amplitude) chosen excitation pulse (𝜋∕2 pulse), theamplitudes of the two partial matter waves are equal. By setting up for respectiveinteraction zones with temporal delay between these zones, the atom beam canbe split and recombined by the lasers as long as the interaction is still coherentand a time-domain Mach–Zehnder interferometer can be set up as illustratedin Figure 3.12. This interferometer has two output ports where the atoms leave,respectively, in the ground and excited state. The probability to find the atomin either port depends on the phase difference of the partial waves, and thus,the output signal detected, for example, by fluorescence will exhibit interferencefringes as a function of detuning with fringe width, Δν, inversely proportional tothe flight time, T f (Δν = 1∕(4Tf )). Also, the phase shift can be affected externally,for example, by gravitation or by rotation of the interferometer (Sagnac effect),which makes atom interferometers a very sensitive measurement instrument [43].

Finally, to complete an optical clock, a technique for counting the high-frequency optical cycles had to be developed. Today, this is accomplished byfemtosecond frequency combs, which will briefly be described in the followingsection.

3.3.1Femtosecond Frequency Combs

For an absolute measurement of an optical frequency, let’s say at 500 THz, this hasto be traced back to the frequency of definition of the second at 9.2 GHz. Thus,approximately 5 orders of magnitude have to be bridged [44].

For this purpose, coherent frequency dividing or multiplying techniques hadbeen used originally. In particular, technically very demanding frequency chains


| e,me⟩| g,2⟩

| e,1⟩

| g,2⟩

| g,0⟩

| e,−1⟩

II

I

| g,0⟩

| g,0⟩

| e,−1⟩| e,1⟩| e,1⟩

Laser beams

D d D

Laser beams

II

I

| g,mg⟩

Figure 3.12 Scheme of a Bordé–Ramseyatom interferometer. The laser beams areshown as oscillatory lines with the arrowindicating the direction. The labeling |i,m⟩

refers to the state of the atom where i = gand e refer to the ground and excited states,respectively, and m = 0, 1, and −1 stand forthe number of photon momenta transferred

to the atom. The transition from |e, 1⟩ to|g, 0⟩ at the second interaction zone reflectsstimulated emission. By the interaction asshown, two equivalent interferometers areset up. The two output ports of each inter-ferometer labeled I and II correspond toatoms leaving the interferometer in theexcited and ground states, respectively.

had been developed at several national metrology labs. In the case of PTB’s(National Metrology Institute of Germany) frequency chain, a delicate setup fill-ing three laser laboratories consisting of seven intermediate oscillators and sevennonlinear mixing steps had been established providing a direct link between aCa optical frequency standard at 455.9 THz, that is, a dye laser stabilized to the3P1 → 1S0 intercombination transition in 40Ca, and the Cs frequency [45].

A major breakthrough in absolute optical frequency measurements has beenthe development of optical frequency combs based on mode-locked femtosecondlasers [46–48] for which T. Hänsch and J. Hall received the Nobel Prize in Physicsin 2005.

Mode locking refers to the phase-coherent superposition of the longitudinalmodes of a laser resonator supported by the respective gain medium. For atextbook description, see, for example, [49]. For active mode locking, intracavityelectro-optic or acousto-optic modulators are applied to modulate the loss ofthe laser cavity periodically with a frequency corresponding to the round-triptime of light in the laser resonator (T = 2L∕vg, vg is the group velocity) or higherharmonics. Alternatively, the gain can be periodically modulated (synchronouspumping). For passive mode locking, a nonlinear device is placed inside the laserresonator which then by itself causes the periodic modulation, for example, a sat-urable absorber. A saturable absorber exhibits a nonlinear transmission becomingfully transparent at high irradiance. It thereby forces the longitudinal modes ofthe laser resonator to add up constructively to achieve highest irradiance. The


total electric field then can be written as

E(t) =∑

qAqei(𝜔0+qΔ𝜔)t + cc, (3.14)

where q is the mode number and the mode spacing is given by Δ𝜔 = 2𝜋∕T = 2𝜋frepwith the repetition rate f rep. The output in the time domain thus correspondsto a pulse train with pulses separated by the round-trip time, T , and a width of∼ (NΔ𝜔)−1, where N is the number of longitudinal modes. (NΔ𝜔) correspondsto the effective gain bandwidth. In this ideal case, the individual pulses wouldjust be time-shifted copies, that is, E(t) = E(t − T). In reality, however, in par-ticular for lasers with large gain bandwidth generating extremely short pulses inthe femtosecond or even subfemtosecond regime, intracavity dispersion has to beconsidered resulting in different group and phase velocities (in lowest order). As aconsequence, the carrier wave exhibits a constant phase shift,ΔΦgpo, for each sub-sequent pulse, as seen in the upper part of Figure 3.13. In the frequency domain,this results in an offset with respect to zero frequency 𝜔ceo = ΔΦgpo∕T when extrap-olating the frequency comb spanned by the longitudinal modes to zero frequency.The frequency of an individual laser mode, m, is then given by

𝜔m = 𝜔ceo + mΔ𝜔. (3.15)

Any frequency falling in between two adjacent laser modes then can be measuredby detecting in the radio-frequency regime the beat note between the unknownfrequency and the adjacent laser mode, provided the mode spacing, Δ𝜔, the mode

ΔΦgpo

Δω = 2π frepω(m)

ω(m) = ωceo + m Δω ωceo = ΔΦgpo/T

ωceoω = 0Frequency

Time

(a)

(b)

T = 1/frep

frep

Figure 3.13 Time trace of a mode-locked laser pulse train (a) and corresponding frequencyspectrum (b, schematically).


number, m, and the carrier envelope offset frequency, 𝜔ceo, are known. Since themode spacing, which corresponds to the repetition rate, typically is of order of100 MHz, it can easily be measured with a photodiode calibrated by a conve-nient frequency standard in the microwave regime. The mode number, m, canbe obtained by a crude estimate of the unknown frequency with a wavemeterproviding a resolution of the order of the mode spacing. The carrier envelopeoffset frequency can be measured by beating the second harmonic of an indi-vidual mode with mode number m, 2𝜔m, with the mode with mode number 2m,𝜔2m (self-referencing). The second harmonic is given by 2𝜔m = 2𝜔ceo + 2mΔ𝜔,while the frequency of mode 2m is 𝜔2m = 𝜔ceo + 2mΔ𝜔. The beat note 2𝜔m −𝜔2m = 𝜔ceo thus yields directly the carrier envelope offset frequency. This proce-dure requires, however, that the frequency comb spans at least one octave. As𝜔ceo and Δ𝜔 are in the radio-frequency regime, they can be locked to a stablemicrowave oscillator traced back to the Cs clock. Frequency combs then can beapplied for absolute optical frequency measurement by measuring the beat notebetween the unknown optical frequency and its adjacent comb line [50–52], fordirect comparison of optical transition lines [53], for precise measurement of fre-quency ratios [54], and as a clockwork in an optical clock to transfer the opticalfrequency into the microwave regime. It meanwhile has been established thatoptical frequency combs generated with femtosecond lasers can generate opticalfrequencies with a fractional accuracy better than 10−18.

The mode-locked lasers most frequently used for frequency comb generationare titanium–sapphire lasers (Ti:Al2O3) and fiber lasers. The active medium in thetitanium–sapphire laser is a sapphire (Al2O3) crystal heavily doped (about 0.1%in weight) with Ti3+ ions replacing Al ions. The titanium–sapphire laser exhibitsa large gain bandwidth from about 670 nm to 1.1 μm due to the crystal field split-ting of the electronic states involved. The setup of a mode-locked Ti:Al2O3 laser isshown in Figure 3.14. Excitation of the Ti:Al2O3 crystal is usually by a frequency-doubled Nd:YAG laser with a few watt pump power. Mode locking is achieved bythe so-called Kerr lens mode locking [55] due to the optical Kerr effect, that is, anonlinear dependence of the refractive index, n, of the sapphire crystal on optical

Chirpedmirror

Ti:Sa

Prism

PrismBlue

Red

Beam

End mirror

Pump

Outputcoupler

Figure 3.14 Setup of a mode-locked Ti–sapphire laser with a linear resonator.


irradiance, I, n ≅ n0 + n2I. This Kerr effect causes a spatial variation of the phasein transverse direction, resulting in self-focusing similar to an optical lens. Thiseffect of course is stronger for pulsed light with higher intensity as compared toCW light. A sufficiently small aperture behind the sapphire crystal, which also canbe provided by the narrow focus of the pump laser itself, thus acts in the same wayas a saturable absorber, resulting in self-starting mode locking. This self-startingcan be supported by a semiconductor saturable absorber. The pair of intracav-ity prisms together with the chirped mirror [56] compensate for group velocitydispersion and self-phase modulation imposed on the spectrum of the pulses.

Pulse trains with pulse width of the order of some 10 fs and below at a rep-etition rate of about 100 MHz up to 10 GHz [57] can be readily obtained withmode-locked Ti:Al2O3 lasers in a wavelength regime of about 700–900 nm. Themode comb spectral width of Ti–sapphire lasers with pulse width of some 10 fs,however, does not cover a full octave. The comb width can be broadened exter-nally applying the so-called holey fibers [58]. These silica fibers consist of a 2Dperiodic array of bores close to their core providing a very small waveguide with ahigh refractive index contrast allowing compensation of material dispersion by tai-lored waveguide dispersion. With these fibers, coherent supercontinuum spectracovering the entire visible spectrum and the near-infrared and frequency combswith more than one octave spectral width can be generated [59].

Mode-locked Er-doped fiber laser systems are very attractive to realize com-pact optical frequency comb generators [60]. Figure 3.15 shows the setup of amode-locked Er-doped fiber laser oscillator (a) with an amplifier stage (b). Theunidirectional Er fiber oscillator is pumped by diode lasers emitting at 980 nm.Mode locking is achieved by nonlinear polarization rotation. Nonlinear polariza-tion rotation is again due to the optical Kerr effect causing self-phase and cross-phase modulation. As a result, the polarization state in an optical fiber dependson the irradiance. Combined with a linear polarizer, this causes an intensity-dependent loss again similar to a saturable absorber. By properly adjusting thetwo polarization controlling stages, self-starting mode locking can be initiated.The center wavelength is at about 1.55 nm and a few milliwatts average outputpower can be generated. The oscillator output is then coupled into an amplifierstage pumped by two diode laser chips. The pulses are first stretched by a fiberwith negative group velocity dispersion and then amplified whereby due to thepositive group velocity dispersion of the Er-doped fiber, the prechirped pulsesare shortened during amplification. In addition, a Si prism pulse compressor isused to control the group velocity dispersion of the output. Pulse widths are inthe order of 50–100 fs at an average power of about 200 mW. The output pulsetrain then can be used for supercontinuum generation and second harmonicgeneration for self-referencing of the comb [60].

Finally, it should be mentioned that also alternative nonlinear techniqueshave been applied to generate frequency combs like the use of microresonators[61], which, however, due to their short length consequently exhibit very largemode spacing.


980 nm

WDM

80/20Coupler

Er:fiber

CL

4 2λ

4λ

4PD

LFFI

λ

λ/4λ/2

L

λ

PBS CL

Exit beam(a)

(b)

Inputoscillator

‘‘Stretcher’’

‘‘Pick-off’’-mirror

Variable Si-prismcompressor

Exit beam

(negative GVD)

Er:fiber(Positive GVD)

Pump diode

980 nm

WDM WDM

Pump diode lasers

Figure 3.15 Mode-locked Er fiberlaser oscillator (a) and amplifier (b) foroptical frequency comb generation(WDM=wavelength division multiplexer,PBS=polarizing beam splitter, PD=photodiode, FI= Faraday isolator, LF= spectral filter,

CL= collimating lens, GVD=group velocitydispersion). (Courtesy of F. Adler and A. Leit-enstorfer, University Konstanz.) (Please find acolor version of this figure on the color platesection.)

3.3.2Neutral Atom Clocks

Atom clocks based on clouds of cold neutral atoms can be operated with high sig-nal to noise ratio (as compared to single ions) due to the large number of atoms(up to 108). However, a shift of the respective clock transition may occur due tointeraction of the atoms, in particular collision-related frequency shifts, in addi-tion to all other processes which could cause frequency shifts, like magnetic andelectric fields, blackbody radiation, and so on.

Promising neutral atom standards to date are the 1S → 2S two-photon tran-sition in hydrogen [62]; intercombination transitions in alkaline earth atomslike strontium (88Sr), calcium (40Ca), and ytterbium (Yb); and dipole-forbidden


1P1

423 nm

657 nm

3P1

1S0

Figure 3.16 Simplified diagram of the rel-evant energy levels of 40Ca.

transitions in 87Sr. We shall next describe briefly the Sr and Ca frequencystandards, for example.

Optical transitions in the neutral 40Ca have been investigated for clock applica-tions mainly at the NIST [63, 64] and PTB [65, 66].

A simplified diagram of the relevant energy levels indicating the cooling transi-tion at 423 nm and intercombination transition at 657 nm is shown in Figure 3.16.The 657 nm clock transition has a natural linewidth of about 400 Hz. The Ca atomsare cooled in an MOT to temperatures of a few millikelvin well below the Dopplerlimit of the 1S0 → 1P1 transition by involving also the forbidden 1S0 → 3P1 tran-sition [67, 68]. For the spectroscopy phase, the trapping laser and the magneticfield of the MOT are turned off, and the free-falling and expanding atoms areexcited by the 657 nm laser radiation in a Bordé–Ramsey interferometer config-uration. To obtain the absorption dip profile, the number of atoms in the excited3P1 state has to be measured as a function of frequency. Since the weak 3P1 → 1S0fluorescence is difficult to detect, the so-called electron shelving techniques arefrequently applied [63, 65, 69]. In this technique, the strong 1P1 → 1S0 fluores-cence is used to monitor the 3P1 state population. Since both transitions sharethe same ground state, atoms excited to the long-lived 3P1 state will reduce the1P1 → 1S0 fluorescence; atoms in the 3P1 state are shelved for some time.

The precise frequency of the 657 nm probe laser stabilized to the central fringeof the interferometer has been measured with a femtosecond comb [64, 66, 70]with a fractional uncertainty well below 10−13. As a result, the Ca 657 nm inter-combination transition has been recommended by the CIPM (International Com-mittee for Weights and Measures) as one of the radiations for the realization of themeter [71].

It should be mentioned finally that also a transportable Ca frequency standardhas been developed at the PTB [72] and used for the comparison of standards atthe PTB and NIST.


461 nm

689 nm

(7.6 kHz)

1P1

3P2

3P1

3P0

1S0

Figure 3.17 Simplified term scheme of neutral 88Sr, indicating the cooling transition at461 nm, the 3P1 intercombination transition at 689 nm, and the doubly forbidden 3P0 tran-sition. The natural linewidth of the intercombination transition is given in brackets.

The frequency of the 1S0 → 3P1 intercombination transition in 88Sr (Figure 3.17)has been measured by saturated absorption spectroscopy in a thermal beam byFerrari et al. [73] and in a free-falling ultracold atom beam by Ido et al. [40]. Thelatter authors achieved a relative uncertainty of the order of 10−15 by particularlyaccounting for the collision-induced frequency shift.

Very promising results have been further achieved with the doubly forbidden1S0 → 3P0 transition in 87Sr atoms trapped and stored in optical lattices [74–80].This transition becomes weakly allowed in 87Sr because of hyperfine interactionwith the large nuclear spin (𝐈 = 9∕2). Yet, the lifetime-limited natural linewidth isstill expected to be extremely narrow (∼1 mHz). Trapping and storing the atomsin an optical lattice are very attractive because a large number of atoms can beinvolved and kept in the Lamb–Dicke regime but still avoiding collisional fre-quency shifts by proper design of the lattice spacing. However, in general, therather strong light field creating the optical lattice would also cause frequencyshifts due to the AC Stark effect. It has been proven, however, that it is possibleto find a wavelength for the trapping light where the AC Stark shift of the groundstate and the excited state is the same (magic wavelength) due to the nonresonantcoupling of these states to higher energy levels [81–83]. At present, the resultsfor the 1S0 → 3P0 transition in 87Sr atoms agree on a level of 10−16. Given its highstability [84–86] and low uncertainty [87, 88], the Sr lattice clock becomes a seri-ous candidate for the new definition of the second. The 1S0 → 3P0 transition in87Sr is also already recommended as one of the secondary representations of the


second [71]. Besides Sr, also spin-polarized Yb lattice clocks with excellent stabilityhave been realized [89].

3.3.3Atomic Ion Clocks

The major advantage of single-ion frequency standards [90] as compared to atomclouds is the absence of interaction effects and their amazingly long storage timewhich easily can be several months. This means practically unlimited interro-gation times for the probe laser. However, the price to pay is the lower signalintensity and thus lower signal to noise ratio. Trapped single-ion frequency stan-dards have been realized with the 1S0 → 3P0 transition in 115In+ [91] and 27Al+, the2S1∕2 → 2D5∕2 electric quadrupole transition in 199Hg+ [92–94], the 2S1∕2 → 2D5∕2electric quadrupole transition in 88Sr+ at 674 nm [95, 96] and 40Ca+ [97–99], aswell as 171Yb+. We will describe briefly in the following the results obtained with171Yb+ and 27Al+.

Ion frequency standards usually start with a neutral atom beam created by evap-oration and subsequent ionization by either electron impact or optical radiationfollowed by the cooling and trapping procedure.

The ytterbium ion (171Yb+) is a particularly interesting candidate for an opticalclock since besides the electric quadrupole transition (2S1∕2 → 2D3∕2) at 436 nm, ithas a second highly forbidden octupole transition (2S1∕2 → 2F7∕2) at 467 nm withan extremely long excited-state lifetime of several years. A partial energy schemeis shown in Figure 3.18.

Measurements of the 2S1∕2(F = 0) → 2D3∕2(F = 2) transition have beenreported by Tamm et al. [100] and of the 2S1∕2(F = 0) → 2D5∕2(F = 0) transitionat 411 nm by Roberts et al. [101] using quantum jump fluorescence detection[102]. Quantum jump fluorescence detection utilizes that the fluorescence at thecooling transition at 370 nm is quenched (dark) whenever the ion is excited to the2D3/2 or 2D5/2 state by the probe laser. The absolute frequency of the 436 nm tran-sition has been determined with a femtosecond comb with a relative uncertaintyof 1.1 × 10−15 [103] and by using cross-linked optical and microwave oscillatorswith an uncertainty of 1.1 × 10−16 [104]. A comparison of the frequencies of ionsin two independent traps has obtained agreement on a level of 4 × 10−16 [105].

The electric octupole transition in 171Yb+ is not only of special interest becauseof its long excited-state lifetime and thus narrow homogeneous linewidth in thenanohertz regime but also because the quadrupole transition and the octupoletransition exhibit quite different relativistic corrections [106], thus providing anideal probe to investigate possible time variations of the fine-structure constant,𝛼 (see Section 3.3.4). The 2S1∕2(F = 0) → 2F7∕2(F = 3) transition had been firstinvestigated at the National Physical Laboratory (NPL) [107]. A difficulty encoun-tered with the octupole transition is the high intensity of the probe laser needed todrive this very weak transition, resulting in a considerable AC Stark shift. Carefulextrapolation schemes are thus required to determine the unperturbed transition


760 nm

370 nm436 nm

935 nm

467 nm

F = 4

F = 2

F = 1

F = 21[3/2]3/2

3[3/2]1/2

F = 1

F = 0

F = 1

F = 1

F = 0

F = 0

F = 1

F = 3

2F7/2

2D3/2

2P1/2

2S1/2

Figure 3.18 Partial energy scheme of171Yb+indicating the cooling transition at370 nm (dashed vertical arrow), the elec-tric quadrupole transition at 436 nm, aswell as the electric octupole transition at467 nm (solid gray arrows). The other tran-sitions to higher D states (solid gray arrows,labelled 760 nm and 935 nm) are used for

repumping. The dotted arrows indicatespontaneous transitions. The notation atthe upper left refers to a specific couplingscheme (JK or J1L2 coupling) particularlyapplied, for example, for rare earth atoms.(Courtesy of N. Huntemann and E. Peik, PTB.)(Please find a color version of this figure onthe color plate section.)

frequency. Yet, recently, a special excitation scheme (generalized Ramsey excita-tion) has been proposed [108] and demonstrated [109] that suppresses the light-induced shift and thus opens the way to even more precise measurements on thelevel of a few parts in 1018.

Recent measurements of the octupole transition frequency show agreementon the 10−15 level [110, 111]. Figure 3.19 shows an excitation spectrum of theoctupole transition using again the quantum jump detection scheme. The tran-sition is excited by a laser system with excellent stability better than 2 × 10−15 at1 s averaging time [112].

The aluminum ion (27Al+) is also a promising candidate for an optical clockbecause its 1S0 → 3P0 intercombination transition exhibits a narrow linewidth of8 mHz [102, 113] and has low sensitivity to electromagnetic perturbations andblackbody radiation. Yet, the 27Al+ ion does not possess an accessible strong opti-cal transition for laser cooling and detection. However, using quantum logic spec-troscopy [25], an Al+ ion frequency standard has been constructed at the NIST forthe first time [114]. In quantum logic spectroscopy, an auxiliary atom (logic atom)is used to cool the vibrational motion and probe the internal state of the atomto be investigated (spectroscopy atom). For this purpose, both ions are trappedtogether in a linear Paul trap, and the ion pair forms a two-ion linear Coulombcrystal along the axis of the trap due to their repulsive Coulomb interaction. The


0−10 −5 0

Laser detuning (Hz)

5 10

2.4 Hz

0.2

0.4

0.6

Excitation p

robabili

ty0.8

1

Figure 3.19 Excitation spectrum of the 2S1∕2(F = 0) → 2F7∕2

(F = 3) transition in 171Yb+ .

(Courtesy of N. Huntemann, PTB.) (Please find a color version of this figure on the colorplate section.)

quantum mechanical state transfer is brought about by their joint motion givingrise to vibronic sidebands. The spectroscopy ion is cooled through the laser-cooledlogic ion via their Coulomb interaction (sympathetic cooling). The spectroscopyof the ion is then performed with a suitable laser, and the internal state is trans-ferred to the logic ion by coherent interaction with a sequence of laser pulses onboth ions. The outcome of the spectroscopy is then detected on the logic ion usingthe quantum jump technique [102].

In the first spectroscopy experiment, the 1S0 → 3P0 clock transition in 27Al+was probed by a 9Be+ ion [114]. When driving the 1S0 → 3P0 transition by aclock laser, this will modulate also the 1S0 → 3P1 transition due to electronshelving, and this state then is transferred to the Be+ ion. This experimentenabled the first precise measurement of the 1S0 → 3P0 transition frequencywith a fractional uncertainty of 5 × 10−15 and the determination of the 3P0state lifetime of 20.6 ± 1.4 s. Further, the frequency ratio of the Al+ and Hg+single-ion optical clock has been measured at the NIST with an uncertaintybetter than 10−16 [94]. A second version of the Al+ ion clock constructed recentlyat the NIST uses Mg+ as logic ions which better match the mass of the Al+ ion,thus enabling more efficient sympathetic cooling [115]. A comparison of thisAl+–Mg+ clock with the Al+–Be+ clock showed fractional agreement of bothmeasured frequencies on the 10−17 level and a relative stability of a few times10−15

𝜏−1/2 [116]. This demonstrates their potential not only for clock applications

but maybe even more for fundamental physics studies like the investigationof possible changes of fundamental constants, particularly the fine-structureconstant (see Section 3.3.4), relativity, and geodesy applications [117–119].


3.3.4Possible Variation of the Fine-Structure Constant, 𝜶

The dimensionless fine-structure constant, 𝛼,

𝛼 = e2

4𝜋ε0ℏc ≈ 1137

(3.16)

is considered a fundamental constant of nature. According to quantum electrody-namics (QED), 𝛼 is a measure of the strength of the electromagnetic interaction.Its value, however, cannot be calculated within QED but has to be determined byexperiment. Presently, the most precise value comes from the determination ofthe Landé g-factor of the electron [120, 121], and based on these results com-bined with QED calculations, 𝛼 has been determined with a relative standarduncertainty of 7 × 10−10 [122], while the latest CODATA(International Councilfor Science: Committee on Data for Science and Technology) result lists an uncer-tainty of 3.2 × 10−10 [123]. The fine-structure constant can also be determined viathe quantum Hall effect (see Section 5.4.1.4).

Recently, considerable attention has been paid to search for possible temporalvariations of fundamental constants, in particular the fine-structure constant andthe proton-to-electron mass ratio [124]. According to the present understandingof the laws of physics including quantum theory and relativity, the nongravita-tional constants do not vary in time. This is a consequence of Einstein’s equivalenceprinciple (EEP). In particular, the local time and position invariance states that inany local free-falling reference frame, the result of a nongravitational measure-ment is independent of space and time. On the other hand, theories beyond thestandard model of physics which aim at unification of the theory of all forces andbringing together quantum theory and gravitation allow for space-time variationof fundamental constants. This would mean that the frequency of an optical tran-sition might vary in time. Whether this contradicts the EEP is not obvious butdepends on the detailed physics behind. Further, recent studies of the absorptionlines of interstellar clouds in the light of distant quasars have been interpretedas evidence for a variation of the fine-structure constant 𝛼 on cosmological timescales of some 10 billion years [125]. According to their interpretation, an increaseof the fine-structure constant Δ𝛼∕𝛼 of the order of 10−6 should have occurred in thefirst half of the evolution of the universe. Assuming a linear variation with timethat would continue up to today, this would extrapolate to a relative increase of𝛼 of some 10−16 per year. In contrary, other studies seem to rule out a changeof 𝛼 [126].

It is only with the development of the most recent optical frequency standardsthat this order of magnitude is accessible by laboratory experiment in reasonabletime intervals [94, 104, 127–130]. To analyze frequency measurements inrespect to possible variations of 𝛼, the electronic transition frequency can beexpressed as [131]

ν = const • Ry • F(𝛼), (3.17)


where the Rydberg frequency Ry = mee4∕8ε0h3 is the common scaling factor for theenergy levels in atoms and the dimensionless factor F(𝛼) accounts for relativis-tic corrections of the energy levels. The constant prefactor depends only on thequantum numbers of the atomic states involved and is independent of time. Therelative temporal variation of 𝜈 then is given by

d ln νdt =

d ln Ry

dt + A d ln 𝛼

dt with A = d ln Fd ln 𝛼

. (3.18)

A variation of the Rydberg frequency given by the first term would be commonfor all transition frequencies. In contrast, the second term is specific to the atomictransition considered. The so-called sensitivity factor A accounts for the sign andstrength of the effect of a variable α on the transition frequency and has beencalculated for several transitions of interest [132, 133].

A summary of recent results is depicted in Figure 3.20 where the estimatedvariation of the frequency measured in SI units, that is, referred back to the Csstandards, is plotted versus the sensitivity factor A. On the base of these data, aconstraint for the time variation of

d ln 𝛼

dt = (−0.7 ± 1.1) × 10−16 per year (3.19)

has been obtained. An even more stringent constraint based on the comparisonof the Hg+ and Al+ optical clocks which does not involve the Cs standard of

d ln 𝛼

dt = (−1.6 ± 2.3) × 10−17 per year (3.20)

has been reported in [94]. So, the present results of laboratory studies of possiblechanges of 𝛼 using the latest state of the art optical clocks do not give evidence

−4−10

−8

−6d I

n ν

/dt (

10

−16 p

er

ye

ar)

−4

−2

0

2

4

6

8

10

−2 0

A

2

Srworld

Yb+

PTB

Hg+

NIST

Figure 3.20 Time variation of measuredtransition frequencies in different atomsand ions versus their respective sensitivityparameter. (Courtesy of E. Peik, PTB; for the

original data, see (the respective transition isgiven in brackets) 199Hg+ (2S1/2 –2D5/2), Ref.[134]; 87Sr (1S0 –3P0), Ref. [135]; and 171Yb+

(2S1/2 –2D3/2), Ref. [104].)

References 51

for a change on the uncertainty level of order of 10−17 per year. This, however,cannot exclude possible changes on cosmic time scales. Further improvement ofoptical clocks including possibly clocks based on nuclear transitions [136] andclock assemblies in space [137] definitely will provide even more accurate tests offundamental physics and thereby contribute to solving the remaining puzzles inthe understanding of nature.

As at the time of writing, the second is still defined by the Cs hyperfinetransition one might ask about the “when and how” of a new definition [138].Concerning the “when,” one must state that the development of ultrapreciseand stable optical clocks is primarily driven by basic science. Present techni-cal and industrial requirements are generally satisfied by the best Cs clocksnotwithstanding that the availability of improved technologies often results innew applications. So a new definition of the second should only be consideredwhen the physical grounds for the definition and its realization are proven andgenerally accepted. This brings us to “how” the new second should be defined.If we consider optical clocks, it must be decided on which atom or ion is chosenas the primary realization of the second with the femtosecond comb techniqueproviding traceability of other optical clocks. Alternatively, it has been suggested[138] to use an assembly of clocks defining a weighed mean frequency valueand a set of frequency ratios. In either case, an extensive intercomparison ofthe frequencies of the optical clocks has to take place to develop a robust andgenerally accepted base for a new definition. On-site clock comparisons todaycan be performed at the 10−19 uncertainty level using femtosecond combs pro-vided the (eventually remaining) height difference and the related gravitationalredshift are carefully considered. The comparison of remote clocks with therequired uncertainty level, however, turns out to be a much bigger problem,since the established two-way satellite microwave frequency transfer techniquesare limited to some 10−15 uncertainty at averaging times of a day. Establishingdirect optical (laser) links via satellites [139, 140] would bear the potential foroptical clock comparisons; however, these techniques are much dependent onenvironmental conditions and not yet fully established. Alternatively, frequencytransfer by optical fibers has been demonstrated over almost 2000 km with anuncertainty of 4 × 10−19 at only 100 s averaging time [141]. So, at least within acontinent, optical clock comparisons with the required uncertainty seem feasibleusing existing fiber links. Finally, further improvement of transportable opticalclocks may also provide a means for remote clock comparison [142].

References

1. Allan, D.W. (1966) Statistics of fre-quency standards. Proc. IEEE, 54,221–230.

2. Riehle, F. (2004) Frequency Standards,Wiley-VCH Verlag GmbH, Weinhein.

3. Kasevich, M.A., Riis, E., Chu, S., andDeVoe, R.G. (1989) Atomic fountains

and clocks. Opt. News, 15 (12),31–32.

4. Hänsch, T. and Schawlow, A. (1975)Cooling of gases by laser radiation. Opt.Commun., 13, 68–69.

5. Wineland, D. and Dehmelt, H. (1975)Proposed 1014

𝛿ν < ν laser fluorescence


spectroscopy on Ti+ mono-ion oscilla-tor. Bull. Am. Phys. Soc., 20, 637.

6. Campbell, G.K. and Phillips, W.D.(2011) Ultracold atoms and precise timestandards. Philos. Trans. R. Soc. London,Ser. A, 369, 4078–4089.

7. Metcalf, H.J. and van der Straten, P.(1999) Laser Cooling and Trapping,Springer, New York, Berlin, Heidelberg.

8. Phillips, W.D. (1989) Laser cooling andtrapping of neutral atoms. Rev. Mod.Phys., 70, 721–741.

9. Letokhov, V.S., Minogin, V.G., andPavlik, B.D. (1976) Cooling and trappingof atoms and molecules by a resonantlaser field. Opt. Commun., 19, 72–75.

10. Phillips, W. and Metcalf, H. (1982)Laser deceleration of an atomic beam.Phys. Rev. Lett., 48, 596–599.

11. Lett, P.D., Phillips, W.D., Rolston,S.L., Tanner, C.E., Watts, R.N.,and Westbrook, C.I. (1989) Opti-cal molasses. J. Opt. Soc. Am. B, 6,2084–2107.

12. Raab, E.L., Prentiss, M., Cable, A., Chu,S., and Pritchard, D.E. (1987) Trappingof neutral atoms with radiation pres-sure. Phys. Rev. Lett., 59, 2631–2634.

13. Lett, P.D., Watts, R.N., Westbrook, C.I.,Phillips, W.D., Gould, R.L., and Metcalf,H.J. (1988) Observation of atoms lasercooled below the Doppler limit. Phys.Rev. Lett., 61, 169–172.

14. Dalibad, J. and Cohen-Tannoudji, C.(1989) Laser cooling below the Dopplerlimit by polarization gradients: simpletheoretical models. J. Opt. Soc. Am. B,6, 2023–2045.

15. Ashkin, A. (1970) Acceleration andtrapping of particles by radiation pres-sure. Phys. Rev. Lett., 24 (4), 156–159.

16. Letokhov, V.S. (1968) Narrowing of theDoppler width in a standing light wave.JETP Lett., 7, 272–275.

17. Demelt, H.G. (1967) in Advances inAtomic and Molecular Physics, vol. 3(eds D.R. Blates and I. Esterman), Aca-demic Press, New York, London, pp.53–72.

18. Paul, W. and Steinwedel, H. (1953)Ein neues massenspektrometer ohnemagnetfeld. Z. Naturforsch. A, 8 (7),448–450 (in German).

19. Dicke, R.H. (1953) The effect of col-lisions upon the Doppler width ofspectral lines. Phys. Rev., 89, 472–473.

20. Cirac, J.I. and Zoller, P. (1955) Quantumcomputations with cold trapped ions.Phys. Rev. Lett., 74, 4091–4094.

21. Pyka, K., Keller, J., Partner, H.L.,Nigmatullin, R., Burgermeister, T.,Meier, D.M., Kuhlmann, K., Retzker,A., Plenio, M.B., Zurek, W.H., delCampo, A., and Mehlstäubler, T.E.(2013) Topological defect formation andspontaneous symmetry breaking in ionCoulomb crystals. Nat. Commun., 4,Article no. 2291.

22. Aidelsburger, M., Atala, M., Lohse, M.,Barreiro, J.T., Paredes, B., and Bloch,I. (2013) Realization of the HofstadterHamiltonian with ultracold atoms inoptical lattices. Phys. Rev. Lett., 111,185301-1–185301-5.

23. Hirokazu, M., Siviloglou, G.A., Kennedy,C.J., Burton, W.C., and Ketterle, W.(2013) Realizing the Harper Hamilto-nian with laser-assisted tunneling inoptical Lattices. Phys. Rev. Lett., 111,185302-1–185302-5.

24. Dietrich, F., Berquist, J.C., Bollinger, J.J.,Itano, W.M., and Wineland, D.J. (1989)Laser cooling to the zero-point energyof motion. Phys. Rev. Lett., 62, 403–406.

25. Schmidt, P.O., Rosenband, T., Langer,C., Itano, W.M., Bergquist, C., andWineland, D.J. (2005) Spectroscopyusing quantum logic. Science, 309,749–752.

26. For review see e.g.: Wynands, R. andWeyers, S. (2005) Atomic fountainclock. Metrologia, 42, S64–S79.

27. Weyers, S., Bauch, A., Hübner, U.,Schröder, R., and Tamm, C. (2000) Firstperformance results of PTB’s atomiccaesium fountain and a study of contri-butions to its frequency instability. IEEETrans. Ultrasound Ferroelectr. Freq.Control, 47, 432–437.

28. Bize, S., Laurent, P., Abgrall,M., Marion, H., Maksimovic, I.,Cacciapuoti, L., Grünert, J., Vian, C.,PereiraDosSantos, F., Rosenbusch, P.,Lemonde, P., Santarelli, G., Wolf, P.,Clairon, A., Luiten, A., Tobar, M., andSalomon, C. (2004) Advances in atomicfountains. C. R. Phys., 5, 829–843.

References 53

29. Guéna, J., Abgrall, M., Rovera, D.,Laurent, P., Chupin, B., Lours, M.,Santarelli, G., Rosenbusch, P., Tobar,M.E., Li, R., Gibble, K., Clairon, A.,and Bize, S. (2012) Progress in atomicfountains at LNE-SYRTE. IEEE Trans.Ultrasound Ferroelectr. Freq. Control,59, 391–410.

30. Heavner, T.P., Jefferts, S.R., Donley,E.A., Shirley, J.H., and Parker, T.E.(2004) Recent improvements inNIST-Fl and resulting accuracies ofδf/f<7x10−16. IEEE Trans. Instrum.Meas., 54, 498–499.

31. Dudle, G., Mileti, G., Joyet, A., Fretel,E., Berthoud, P., and Thomann, P. (2000)An alternative cold cesium frequencystandard: the continuous fountain. IEEETrans. Instrum. Meas., 47, 438–442.

32. Bize, S., Sortais, Y., Santos, M.S.,Mandache, C., Clairon, A., andSalomon, C. (1999) High-accuracymeasurement of the 87Rb ground-state hyperfine splitting in an atomicfountain. Europhys. Lett., 45, 558–564.

33. Fertig, C. and Gibble, K. (1999) Lasercooled 87Rb clock. IEEE Trans. Instrum.Meas., 48, 520–523.

34. Hollberg, L., Oates, C.W., Curtis, E.A.,Ivanov, E.N., Diddams, S.A., Udem, T.,Robinson, H.G., Berquist, J.C., Rafac,R.J., Itano, W.M., Drullinger, R.E., andWineland, D.J. (2001) Optical frequencystandards and measurements. IEEE J.Quantum Electron., 37, 1502–1513.

35. Gill, P., Barwood, G.P., Klein, H.A.,Huang, G., Webster, S.A., Blythe, P.J.,Hosaka, K., Lea, S.N., and Margolis,H.S. (2003) Trapped ion frequencystandards. Meas. Sci. Technol., 14,1174–1186.

36. Gill, P. (2005) Optical frequency stan-dards. Metrologia, 42, S125–137.

37. For a recent development see e.g.:Kessler, T., Hagemann, C., Grebing, C.,Legero, T., Sterr, U., Riehle, F., Martin,M.J., Chen, L., and Ye, J. (2012) A sub-40-mHz-linewidth laser based on asilicon single-crystal optical cavity. Nat.Photonics, 6, 687–692.

38. Letokhov, V.S. (1976) in High ResolutionLaser Spectroscopy, Topics in AppliedPhysics, vol. 13 (ed. K. Shimoda),

Springer, Berlin, Heidelberg, NewYork, pp. 95–171.

39. Oates, C.W., Wilpers, G., and Hollberg,L. (2005) Observation of large atomic-recoil-induced asymmetries in coldatom spectroscopy. Phys. Rev. A, 71,023404-1–023404-6.

40. Ido, T., Loftus, T.H., Boyd, M.M.,Ludlow, A.D., Holman, K.W., andYe, J. (2005) Precision spectroscopyand density-dependent frequency shiftsin ultracold Sr. Phys. Rev. Lett., 94,153001-1–153001-4.

41. Borde, C. (1989) Atomic interferometrywith internal state labeling. Phys. Lett.A, 140, 10–12.

42. Helmcke, J., Zevgolis, D., and Yen, B.Ü.(1982) Observation of high contrast,ultra narrow optical Ramsey fringesin saturated absorption utilizing fourinteraction zones of travelling waves.Appl. Phys. B, 28, 83–84.

43. Riehle, F., Kisters, T., Witte, A.,Helmcke, J., and Bordé, C. (1991) Opti-cal Ramsey spectroscopy in a rotatingframe: Sagnac effect in a matter-waveinterferometer. Phys. Rev. Lett., 67,177–180.

44. Holberg, L., Diddams, S., Bartels, A.,Forier, T., and Kim, K. (2005) Themeasurement of optical frequencies.Metrologia, 42, S105–124.

45. Schnatz, H., Lipphardt, B., Helmcke, J.,Riehle, F., and Zinner, G. (1996) Firstphase-coherent frequency measurementof visible radiation. Phys. Rev. Lett., 76,18–21.

46. Hall, J.L. (2006) Defining and measuringoptical frequencies. Rev. Mod. Phys., 78,1279–1295.

47. Hänsch, T.W. (2006) Passion for preci-sion. Rev. Mod. Phys., 78, 1297–1309.

48. Ye, J. and Cundiff, S.T. (2005) Fem-tosecond Optical Frequency CombTechnology: Principle, Operation, andApplications, Springer Science + Busi-ness Media, Inc., New York, ISBN:0-387-23790-9.

49. Diels, J.-C. and Rudolph, W. (1996)Ultrashort Laser Pulse Phenomena: Fun-damentals, Techniques, and Applicationson a Femtosecond Timescale, AcademicPress, San Diego, CA.


50. Udem, T., Reicher, J., Holzwarth, R.,and Hänsch, T.W. (1999) Absoluteoptical frequency measurement of thecesium D-1 line with a mode-lockedlaser. Phys. Rev. Lett., 82, 3568–3571.

51. Reichert, J., Nierig, M., Holzwarth,R., Weitz, M., Udem, T., and Hänsch,T.W. (2000) Phase coherent vacuum-ultraviolet to radio frequency compari-son with a mode-locked laser. Phys. Rev.Lett., 84, 3232–3235.

52. Diddams, S.A., Jone, D.J., Ye, J., Cundiff,S.T., and Hall, J.L. (2000) Directlink between microwave and opticalfrequencies with a 300 THz femtosec-ond laser comb. Phys. Rev. Lett., 84,5102–5105.

53. Udem, T., Reichert, J., Holzwarth, R.,and Hänsch, T.W. (1999) Accurate mea-surement of large optical frequencydifferences with a mode-locked laser.Opt. Lett., 24, 881–883.

54. Stenger, J., Schnatz, H., Tamm, C., andTelle, H.R. (2002) Ultra-precise mea-surement of optical frequency ratios.Phys. Rev. Lett., 88, 073601-1–073601-4.

55. See e.g.: Siegner, U. and Keller, U.(2001) in Handbook of Optics (eds M.Bass, J.M. Enoch, E.W. Van Stryland,and W.L. Wolfe), McGraw-Hill, NewYork, pp. 18.1–18.30.

56. Szipöcs, R., Spielmann, C., Krausz, F.,and Ferencz, K. (1994) Chirped multi-layer coatings for broadband dispersioncontrol in femtosecond lasers. Opt.Lett., 19, 201–203.

57. Bartels, A., Heinecke, D., and Diddams,S.A. (2008) Passively mode-locked10 GHz femtosecond Ti: sapphire laser.Opt. Lett., 33, 1905–1907.

58. Russell, P. (2003) Photonic crystal fibers.Science, 299, 358–362.

59. Husakou, A., Kalosha, V.P., andHermann, J. (2003) in Optical Soli-tons. Theoretical and ExperimentalChallenges, Lecture Notes in Physics(eds K. Porsezian and V.C. Kuirakose),Springer, Heidelberg, New York, pp.299–326.

60. Tauser, F., Leitenstorfer, A., and Zinth,W. (2003) Amplified femtosecond pulsesfrom an Er:fiber system: nonlinearpulse shortening and self-referencing

detection of the carrier-envelope phaseevolution. Opt. Express, 11, 594–600.

61. Del Haye, P., Schliesser, A., Arcizet,O., Wilken, T., Holzwarth, R., andKippenberg, T.J. (2007) Optical fre-quency comb generation from amonolithic microresonator. Nature,450, 1214–1217.

62. Niering, M., Holzwarth, R., Reichert,J., Pokasov, P., Udem, T., Weitz, M.,Hänsch, T.W., Lemonde, P., Santarelli,G., Abgrall, M., Laurent, P., Salomon,C., and Clairon, A. (2000) Measure-ment of the hydrogen 1S- 2S transitionfrequency by phase coherent compari-son with a microwave cesium fountainclock. Phys. Rev. Lett., 84, 5496–5499.

63. Oates, C.W., Bondu, F., Fox, R.W.,and Hollberg, L. (1999) A diode-laseroptical frequency standard based onlaser-cooled Ca atoms: sub-kilohertzspectroscopy by optical shelving detec-tion. Eur. Phys. J. D, 7, 449–460.

64. Udem, T., Diddams, S.A., Vogel, K.R.,Oates, C.W., Curtis, E.A., Lee, W.D.,Itano, W.M., Drullinger, R.E., Berquist,J.C., and Hollberg, L. (2001) Absolutefrequency measurements of the Hg+and Ca optical clock transitions with afemtosecond laser. Phys. Rev. Lett., 86,4996–4999.

65. Riehle, F., Schnatz, H., Lipphardt, B.,Zinner, G., Trebst, T., and Helmcke, J.(1999) The optical calcium frequencystandard. IEEE Trans. Instrum. Meas.,48, 613–617.

66. Schnatz, H., Lipphardt, B., Degenhardt,C., Peik, E., Schneider, T., Sterr, U., andTamm, C. (2005) Optical frequencymeasurements using fs-comb genera-tors. IEEE Trans. Instrum. Meas., 54,750–753.

67. Curtis, E.A., Oates, C.W., and Hollberg,L. (2001) Quenched narrow-line lasercooling of 40Ca to near the photonrecoil limit. Phys. Rev. A, 64, 031403-1–031403-4.

68. Binnewiss, T., Wilpers, G., Sterr, U.,Riehle, F., Helmcke, J., Mehlstäubler,T.E., Rasel, E.M., and Ertmer, W. (2001)Doppler cooling and trapping on for-bidden transitions. Phys. Rev. Lett., 87,123002-1–123002-4.

References 55

69. Wilpers, G., Binnewies, T., Degenhardt,C., Sterrr, U., Helmcke, J., and Riehle,F. (2002) Optical clock with ultracoldneutral atoms. Phys. Rev. Lett., 89,230801-1–230801-5.

70. Stenger, J., Binnewies, T., Wilpers,G., Riehle, F., Telle, H.R., Ranka, J.K.,Windeler, R.S., and Stenz, A.J. (2001)Phase-coherent frequency measurementof the Ca intercombination line at 657nm with a Kerr-lens mode-locked laser.Phys. Rev. A, 63, 021802-1–021802-4.

71. BIPM http://www.bipm.org/en/publications/mises-en-pratique/standard-frequencies.html (accessed15 November 2014).

72. Kersten, P., Mensin, F., Sterr, U., andRiehle, F. (1999) A transportable opticalcalcium frequency standard. Appl. Phys.B, 68, 27–38.

73. Ferrari, G., Cancio, P., Drullinger,R., Giusfredi, G., Poli, N., Prevedelli,M., Toninelli, C., and Tino, G.M.(2003) Precision frequency mea-surement of visible intercombinationlines of strontium. Phys. Rev. Lett., 91,243002–243005.

74. Boyd, M.M., Ludlow, A.D., Blatt, S.,Foreman, S.M., Ido, T., Zelevinsky, T.,and Ye, J. (2007) 87Sr Lattice clock withinaccuracy below 10-15. Phys. Rev. Lett.,98, 083002-1–083002-4.

75. Baillard, X., Fouché, M., Le Targat,R., Westergaard, P.G., Lecallier, A.,Chapelet, F., Abgrall, M., Rovera, G.D.,Laurent, P., Rosenbusch, P., Bize, S.,Santarelli, G., Clairon, A., Lemonde, P.,Grosche, G., Lipphardt, B., and Schnatz,H. (2008) An optical lattice clock withspin-polarized 87Sr atoms. Eur. Phys. J.,48, 11–17.

76. Campbell, G.K., Ludlow, A.D., Blatt,S., Thomsen, J.W., Martin, M.J., deMiranda, M.H.G., Zelevinsky, T., Boyd,M.M., Ye, J., Diddams, S.A., Heavner,T.P., Parker, T.E., and Jefferts, S.R.(2008) The absolute frequency of the87Sr optical clock transition. Metrologia,45, 539–548.

77. Takamoto, M., Hong, F.-L., Higashi, R.,and Katori, H. (2005) An optical latticeclock. Nature, 435, 321–324.

78. Katori, H. (2011) Optical lattice clocksand quantum metrology. Nat. Photonics,5, 203–210.

79. Middelmann, T., Falke, S., Lisdat, C.,and Sterr, U. (2012) High accuracy cor-rection of blackbody radiation shift inan optical lattice. Phys. Rev. Lett., 109,263004-1–263004-5.

80. Lemonde, P. (2009) Optical latticeclocks. Eur. Phys. J. Spec. Top., 172,81–96.

81. Ido, T. and Katori, H. (2003) Recoil-freespectroscopy of neutral Sr atoms in theLamb-Dicke regime. Phys. Rev. Lett., 91,053001-1–053001-4.

82. Takamoto, M. and Katori, H. (2003)Spectroscopy of the 1S0 -3P0 clock tran-sition in 87Sr in an optical lattice. Phys.Rev. Lett., 91, 223001-1–223001-4.

83. Ye, J., Kimble, H.J., and Katori, H.(2008) Quantum state engineeringand precision metrology using state-insensitive light traps. Science, 320,1734–1738.

84. Hagemann, C., Grebing, C., Kessler, T.,Falke, S., Lisdat, C., Schnatz, H., Riehle,F., and Sterr, U. (2013) Providing 1E-16short-term stability of a 1.5 μm laserto optical clocks. IEEE Trans. Instrum.Meas., 62, 1556–1562.

85. Jiang, Y.Y., Ludlow, A.D., Lemke, N.D.,Fox, R.W., Sherman, J.A., Ma, L.-S.,and Oates, C.W. (2011) Making opticalatomic clocks more stable with 10−16-level laser stabilization. Nat. Photonics,5, 158–161.

86. Takamoto, M., Takano, T., and Katori,H. (2011) Frequency comparison ofoptical lattice clocks beyond the Dickelimit. Nat. Photonics, 5, 288–292.

87. Bloom, B.J., Nicholson, T.L., Williams,J.R., Campell, S.L., Bishof, M., Zhang,X., Zhang, W., Bromley, S.L., and Ye,J. (2014) An optical lattice clock withaccuracy and stability at the 10 −18

level. Nature, 506, 71–75.88. Ludlow, A.D., Zelevinsky, T., Campbell,

G.K., Blatt, S., Boyd, M.M., de Miranda,M.H.G., Martin, M.J., Thomsen, J.W.,Foreman, S.M., Ye, J., Fortier, T.M.,Stalnaker, J.E., Diddams, S.A., Le Coq,Y., Barber, Z.W., Poli, N., Lemke, N.D.,Beck, K.M., and Oates, C.W. (2008) SrLattice clock at 1 × 10−16 fractional



uncertainty by remote optical evalu-ation with a Ca clock. Science, 319,1805–1808.

89. Hinkley, N., Sherman, J.A., Phillips,N.B., Schloppo, M., Lembke, N.D.,Beloy, K., Pizzocaro, M., Oates, C.W.,and Ludlow, A.D. (2013) An atomicclock with 10−18 instability. Science, 341(6151), 1215–1218.

90. Margolis, H.S. (2009) Trapped ion opti-cal clocks. Eur. Phys. J. Spec. Top., 172,97–107.

91. Becker, T., van Zanthier, J., Nevsky,A.Y., Schwedes, C., Skvortsov, M.N.,Walther, H., and Peik, E. (2001) High-resolution spectroscopy of a singleIn+ ion: progress towards an opticalfrequency standard. Phys. Rev. A, 63,051802–051805.

92. Berkeland, D.J., Miller, J.D., Bergquist,J.C., Itano, W.M., and Wineland, D.J.(1998) Laser-cooled mercury-ion fre-quency standard. Phys. Rev. Lett., 80,2089–2092.

93. Diddams, S.A., Udem, T., Bergquist,J.C., Curtis, E.A., Drullinger, R.E.,Hollberg, L., Itano, W.M., Lee, W.D.,Oates, C.W., Vogel, K.R., and Wineland,D.J. (2001) An optical clock based on asingle trapped 199Hg+ ion. Science, 293,825–828.

94. Rosenband, T., Hume, D.B., Schmidt,P.O., Chou, C.W., Brusch, A., Lorini, L.,Oskay, W.H., Drullinger, R.E., Fortier,T.M., Stalnaker, J.E., Diddams, S.A.,Swann, W.C., Newbury, N.R., Itano,W.M., Wineland, D.J., and Bergquist,J.C. (2008) Frequency ratio of Al+ andHg+ single-ion optical clocks; metrologyat the 17th decimal place. Science, 319(5871), 1808–1812.

95. Margolis, H.S., Barwood, G.P., Huang,G., Klein, H.A., Lea, S.N., Szymaniec,K., and Gill, P. (2004) Hertz-level mea-surement of the optical clock frequencyin a single 88Sr+ ion. Science, 306,1355–1358.

96. Madej, A.A., Dubé, P., Zhou, Z.,Bernard, J.E., and Gertsvolf, M. (2012)88Sr+ 445-THz single-ion referenceat the 10−17 level via control andcancellation of systematic uncertain-ties and its measurement against the SI

second. Phys. Rev. Lett., 109, 203002-1-203002-4.

97. Chwalla, M., Benhelm, J., Kim, K.,Kirchmair, G., Monz, T., Riebe,M., Schindler, P., Villar, A., Hänsel,W., Roos, C., Blatt, R., Abgrall, M.,Santarelli, G., Rovera, G., and Laurent,Ph. (2009) Absolute frequency measure-ment of the 40Ca+ 4s S1/22-3d D5/22clock transition. Phys. Rev. Lett., 102,023002-1-023002-4.

98. Matsubara, K., Hachisu, H., Li, Y.,Nagano, S., Locke, C., Nogami, A.,Kajita, M., Hayasaka, K., Ido, T., andHosokawa, M. (2012) Direct comparisonof a Ca+ single-ion clock against a Srlattice clock to verify the absolute fre-quency measurement. Opt. Express, 20,22034–22041.

99. Huang, Y., Liu, P., Bian, W., Guan, H.,and Gao, K. (2014) Evaluation of thesystematic shifts and absolute frequencymeasurement of a single Ca+ ion fre-quency standard. Appl. Phys. B, 114,189–201.

100. Tamm, C., Engelke, D., and Buehner,V. (2000) Spectroscopy of the electric-quadrupole transition 2S1/2 (F=0)-2D3/2(F=2) in trapped 171Yb+. Phys. Rev.A, 61, 053405-1–053405-9.

101. Roberts, M., Taylor, P., Gateva-Kostova,S.V., Clarke, R.B.M., Rowley, W.R.C.,and Gill, P. (1999) Measurement ofthe 2S1/2-2D5/2 clock transition in asingle 171Yb+ ion. Phys. Rev. A, 60,2867–2872.

102. Dehmelt, H. (1975) Proposed 1014 δν/νlaser fluorescence spectroscopy on Tl+mono-ion oscillator II (spontaneousquantum jumps). Bull. Am. Phys. Soc.,20, 60.

103. Tamm, C., Weyers, S., Lipphardt, B.,and Peik, E. (2009) Stray-field-inducedquadrupole shift and absolute frequencyof the 688-THz 171Yb+ single-ion opti-cal frequency standard. Phys. Rev. A, 80,043403-1–043403-7.

104. Tamm, C., Huntemann, N., Lipphardt,B., Gerginov, V., Nemitz, N., Kazda,M., Weyers, S., and Peik, E. (2014) ACs-based optical frequency measure-ment using cross-linked optical andmicrowave oscillators. Phys. Rev. A, 89,023820-1–023820-8.

References 57

105. Schneider, T., Peik, E., and Tamm, C.(2005) Sub-hertz optical frequencycomparisons between two trapped171Yb+ ions. Phys. Rev. Lett., 94,230801-1–230801-4.

106. Dzuba, V.A. and Flambaum, V.V. (2009)Atomic calculations and search for vari-ation of the fine-structure constant inquasar absorption spectra. Can. J. Phys.,87 (1), 25–35.

107. Roberts, M., Taylor, P., Barwood, G.P.,Gill, P., Klein, H.A., and Rowley, W.R.C.(1997) Observation of an electricoctupole transition in a single ion.Phys. Rev. Lett., 78, 1876–1879.

108. Yudin, V.I., Taichenachev, A.V., Oates,C.W., Barber, Z.W., Lemke, N.D.,Ludlow, A.D., Sterr, U., Lisdat, C., andRiehle, F. (2010) Hyper-Ramsey spec-troscopy of optical clock transitions.Phys. Rev. A, 82, 011801-1–011801-4.

109. Huntemann, N., Lipphardt, B.,Okhapkin, M., Tamm, C., and Peik,E. (2012) Generalized Ramsey excita-tion scheme with suppressed light shift.Phys. Rev. Lett., 109, 213002-1–213002-5.

110. Huntemann, N., Okhapkin, M.,Lipphardt, B., Weyers, S., Tamm, C.,and Peik, E. (2012) High-accuracyoptical clock based on the octupoletransition in 171Yb+. Phys. Rev. Lett.,108, 090801-1–090801-5.

111. King, S.A., Godun, R.M., Webster,S.A., Margolis, H.S., Johnson, L.A.M.,Szymaniec, K., Baird, P.E.G., and Gill, P.(2012) Absolute frequency measurementof the 2S 1/2 – 2F7/2 electric octupoletransition in a single ion of 171Yb +

with 10−15 fractional uncertainty. New J.Phys., 14, 013045.

112. Sherstov, I., Okhapkin, M., Lipphardt,B., Tamm, C., and Peik, E. (2010)Diode-laser system for high-resolutionspectroscopy of the 2S1/2→

2F7/2octupole transition in 171Yb+. Phys.Rev. A, 81, 021805-1–021805-5.

113. Yu, N., Dehmelt, H., and Nagourney, W.(1992) The 1S0 - 3P0 transition in thealuminum isotope ion 26Al+: a poten-tially superior passive laser frequencystandard and spectrum analyzer. Proc.Natl. Acad. Sci. U.S.A., 89, 7289.

114. Rosenband, T., Schmidt, P.O., Hume,D.B., Itano, W.M., Fortier, T.M.,Stalnaker, J.E., Kim, K., Diddams, S.A.,Koelemeij, J.C.J., Bergquist, J.C., andWineland, D.J. (2007) Observation ofthe 1S0→

3P0 clock transition in 27Al+.Phys. Rev. Lett., 98, 220801-1–220801-4.

115. Wübbena, J.B., Amairi, S., Mandel, O.,and Schmidt, P.O. (2012) Sympatheticcooling of mixed-species two-ion crys-tals for precision spectroscopy. Phys.Rev. A, 85, 043412-1-043412-13.

116. Chou, C.W., Hume, D.B., Koelemeij,J.C.J., Wineland, D.J., and Rosenband,T. (2013) Frequency comparison of twohigh-accuracy Al+ optical clocks. Phys.Rev. Lett., 104, 070802-1–070802-4.

117. See e.g. Chou, C.W., Hume, D.B.,Rosenband, T., and Wineland, D.J.(2010) Optical clocks and relativity.Science, 329, 1630–1633.

118. Blatt, S., Ludlow, A.D., Campbell, G.K.,Thomsen, J.W., Zelevinsky, T., Boyd,M.M., Ye, J., Baillard, X., Fouche, M.,Le Target, R., Brusch, A., Lemonde, P.,Takamoto, M., Hong, F.-L., Katori, H.,and Flambaum, V.V. (2008) New limitson coupling of fundamental constants togravity using 87Sr optical lattice clocks.Phys. Rev. Lett., 100, 140801-1–140801-4.

119. see e.g.Bjerhammar, A. (1985) On arelativistic geodesy. Bull. Géodé., 59 (3),207–220.

120. Hanneke, D., Fogwell Hoogerheide,S., and Gabrielse, G. (2011) Cavitycontrol of a single-electron quantumcyclotron: measuring the electronmagnetic moment. Phys. Rev. A, 83,052122-1–052122-26.

121. Odom, B., Hanneke, D., D’Urso, B., andGabrielse, G. (2006) New measurementof the electron magnetic moment usinga one-electron quantum cyclotron. Phys.Rev. Lett., 97, 030801-1–030801-4.

122. Gabrielse, G., Hanneke, D., Kinoshita,T., Nio, M., and Odom, B. (2007) Erra-tum: new determination of the finestructure constant from the electron gvalue and QED. Phys. Rev. Lett. (2006)97, 030802, Phys. Rev. Lett. (2006) 99,039902-1–039902-2.

123. Mohr, P.J., Taylor, B.N., and Newell,D.B. (2012) CODATA recommendedvalues of the fundamental physical


constants: 2010. Rev. Mod. Phys., 84,1527–1605.

124. see e.g. Karshenboim, S.G. and Peik, E.(eds) (2004) Astrophysics, Clocks andFundamental Constants, Lecture Noteson Physics, vol. 648, Springer, Berlin,Heidelberg.

125. Webb, J.K., Murphy, M.T., Flambaum,V.V., Dzuba, V.A., Barrow, J.D.,Churchill, C.W., Prochaska, J.X., andWolfe, A.M. (2001) Further evidencefor cosmological evolution of the finestructure constant. Phys. Rev. Lett., 87,091301-1–091301-4.

126. see e.g. Srianand, R., Chand, H.,Petitjean, P., and Aracil, B. (2004) Limitson the time variation of the electromag-netic fine-structure constant in the lowenergy limit from absorption lines inthe spectra of distant quasars. Phys. Rev.Lett., 92, 121302-1–121302-4.

127. Peik, E., Lipphardt, B., Schnatz, H.,Schneider, T., and Tamm, C. (2004)Limit on the Present Temporal Vari-ation of the Fine Structure Constant.Phys. Rev. Lett., 93, 170801-1–170801-4.

128. Bize, S., Diddams, S.A., Tanaka, U.,Tanner, C.E., Oskay, W.H., Drullinger,R.E., Parker, T.E., Heavner, T.P., Jefferts,S.R., Hollberg, L., Itano, W.M., andBergquist, J.C. (2003) Testing the stabil-ity of fundamental constants with the199Hg single-ion optical clock. Phys. Rev.Lett., 90, 150802-1–150802-4.

129. Marion, H., Pereira Dos Santos, F.,Abgrall, M., Zhang, S., Sortais, Y.,Bize, S., Maksimovic, I., Calonico, D.,Grünert, J., Mandache, C., Lemonde,P., Santarelli, G., Laurent, P., Clairon,A., and Salomon, C. (2003) Search forvariations of fundamental constantsusing atomic fountain clocks. Phys. Rev.Lett., 90, 150801-1–150801-4.

130. Fischer, M., Kolachevsky, N.,Zimmermann, M., Holzwarth, R.,Udem, T., Hänsch, T.W., Abgrall, M.,Grünert, J., Maksimovic, I., Bize, S.,Marion, H., Pereira Dos Santos, F.,Lemonde, P., Santarelli, G., Laurent,P., Clairon, A., Salomon, C., Haas,M., Jentschura, U.D., and Keitel, C.H.(2004) New limits on the drift of fun-damental constants from laboratory

measurements. Phys. Rev. Lett., 92,230802-1–230802-4.

131. Karshenboim, S.G. and Peik, E. (2008)Astrophysics, atomic clocks and fun-damental constants. Eur. Phys. J. Spec.Top., 163, 1–7.

132. Dzuba, V.A., Flambaum, V.V., andWebb, J.K. (1999) Calculations of therelativistic effects in many-electronatoms and space-time variation of fun-damental constants. Phys. Rev. A, 59,230–237.

133. Dzuba, V.A., Flambaum, V.V., andMarchenko, M.V. (2003) Calculations ofthe relativistic effects in many-electronatoms and space-time variation of fun-damental constants. Phys. Rev. A, 68,022506-1–022506-5.

134. Fortier, T.M., Ashby, N., Bergquist, J.C.,Delaney, M.J., Diddams, S.A., Heavner,T.P., Hollberg, L., Itano, W.M., Jefferts,S.R., Kim, K., Levi, F., Lorini, L., Oskay,W.H., Parker, T.E., Shirley, J., andStalnaker, J.E. (2007) Precision atomicspectroscopy for improved limits onvariation of the fine structure constantand local position invariance. Phys. Rev.Lett., 98, 070801-1–070801-4.

135. Le Targat, R., Lorini, L., Le Coq, Y.,Zawada, M., Guéna, J., Abgrall, M.,Gurov, M., Rosenbusch, P., Rovera, D.G.,Nagorny, B., Gartman, R., Westergaard,P.G., Tobar, M.E., Lours, M., Santarelli,G., Clairon, A., Bize, S., Laurant, P.,Lemonde, P., and Lodewyck, J. (2013)Experimental realization of an opticalsecond with strontium lattice clocks.Nat. Commun., 4, 2109-1–2109-8.

136. Peik, E. and Tamm, C. (2003) Nuclearlaser spectroscopy of the 3.5 eV transi-tion in Th-229. Europhys. Lett., 61 (2),181–186.

137. See e.g.: Cacciapuoti, L., Dimarcq, N.,Santarelli, G., Laurent, P., Lemonde, P.,Clairon, A., Berthoud, P., Jornod, A.,Reina, F., Feltham, S., and Salomon, C.(2007) Atomic clock ensemble in space:scientific objectives and mission status.Nucl. Phys. B, 166, 303–306.

138. See e.g.: Gill, P. (2011) When shouldwe change the definition of the sec-ond? Philos. Trans. R. Soc. A, 369,4109–4130.

References 59

139. Djerroud, K., Acef, O., Clairon, A.,Lemonde, P., Man, C.N., Samain, E., andWolf, P. (2010) Coherent optical linkthrough the turbulent atmosphere. Opt.Lett., 35, 1479–1481.

140. Giorgetta, F.R., Swann, W.C., Sinclair,L.C., Baumann, E., Coddington, I.,and Newbury, N.R. (2013) Opticaltwo-way time and frequency trans-fer over free space. Nat. Photonics, 7,434–438.

141. Droste, S., Ozimek, F., Udem, T.,Predehl, K., Hänsch, T.W., Schnatz,H., Grosche, G., and Holzwarth, R.(2013) Optical-frequency transfer overa single-span 1840 km fiber link. Phys.Rev. Lett., 111, 110801-1–110801-5.

142. Schiller, S., Görlitz, A., Nevsky,A., Alighanbari, S., Vasilyev, S.,Abou-Jaoudeh, C., Mura, G., Franzen,T., Sterr U., Falke, S., Lisdat, C.,Rasel, E., Kulosa, A., Bize, S.,Lodewyck, J., Tino, G.M., Poli, N.,Schioppo, M., Bongs, K., Singh, Y.,Gill, P., Barwood, G., Ovchinnikov, Y.,Stuhler, J., Kaenders, W., Braxmaier, C.,Holzwarth, R., Donati, A., Lecomte, S.,Calonico, D., and Levi, F. (2012) Thespace optical clocks project: develop-ment of high-performance transportableand breadboard optical clocks andadvanced subsystems. Proceedings ofthe 2012 European Frequency and TimeForum (EFTF 2012), arXiv:1206.3765.

61

4Superconductivity, Josephson Effect, and Flux Quanta

Superconductivity is a macroscopic quantum effect, which is observed incertain solid-state systems at low temperatures. The superconducting state can bedescribed by a single wave function, which extends over macroscopic distances inreal space. The composite quasiparticles that occupy the macroscopic quantumstate are bosonic Cooper pairs consisting of two electrons, which are weaklybound to each other. If a superconducting ring is placed in a magnetic field, themagnetic flux penetrating the ring is found to be quantized in integer multiplesof the flux quantum. Quantum metrology takes advantage of Cooper pairs andflux quanta. The tunneling of Cooper pairs between two superconductors isreferred to as the Josephson effect [1]. It links the macroscopic physical quantityvoltage and the unit volt to the counting of flux quanta per time interval. Thus,flux quanta contribute to the foundation of units. This aspect will be addressedin Section 4.1. In Section 4.2, we will discuss that flux quanta also contribute tothe foundation of measurements. The ratio between the magnetic flux and theflux quantum determines the outcome of interference effects in superconductingquantum interference devices (SQUIDs). This interference allows highly sensitivemeasurements of magnetic flux and, in turn, of magnetic field and magneticmoment to be performed linked to the flux quantum. It is the aim of this chapterto introduce the basic physics and metrological applications of superconductors,the Josephson effect, and SQUIDs.

4.1Josephson Effect and Quantum Voltage Standards

4.1.1Brief Introduction to Superconductivity

Superconductivity was discovered by the Dutch physicist Heike Kamerlingh Onnesin 1911 after he had succeeded in liquefying 4He in 1908. For his achievements inlow-temperature physics, Kamerlingh Onnes received the Nobel Prize in Physicsin 1913. Superconductivity is characterized by the disappearance of the electricalresistance below a critical temperature Tc and the expulsion of magnetic fields


62 4 Superconductivity, Josephson Effect, and Flux Quanta

from the interior of a superconducting material (Meissner–Ochsenfeld effect;see Section 4.2.1). Kamerlingh Onnes discovered superconductivity by studyingthe temperature dependence of the resistance of mercury (Hg), which has a crit-ical temperature Tc = 4.2 K. Subsequently, superconductivity was found in othermetals, for example, tin (Sn) (Tc = 3.7 K), lead (Pb, Tc = 7.2 K), and niobium (Nb,Tc = 9.5 K).

After several classical or semiclassical approaches to describe superconductivity[2, 3], superconductivity was quantum mechanically described by Bardeen et al.in 1957 [4, 5]. In 1972, Bardeen, Cooper, and Schrieffer received the Nobel Prizein Physics for their “BCS theory.”

The basic ingredients of the BCS theory are the forming of Cooper pairs fromelectrons close to the Fermi surface and their condensation into a macroscopicquantum state described by a single wave function. Cooper pairs consist of twoelectrons with opposite spin S and wave vector 𝐤 resulting in S = 0, 𝐤 = 0, anda total charge eS = −2e if e denotes the elementary charge. In the classical low-temperature superconductors, the attractive force to bind two electrons togetheris mediated by the electron–phonon interaction, which overcomes the repulsionof the negatively charged electrons. We note, however, that the BCS theory is inde-pendent of the nature of the attractive force between the electrons.

The Cooper pairs are separated in energy from the single-particle electron statesby an energy gap 2Δ(T). With increasing temperature,Δ(T)decreases fromΔ(T =0) = 1.76 kBTc to zero at Tc according to

Δ(T) = Δ(T = 0)

√√√√cos

(𝜋

2

(TTc

)2)

. (4.1)

The continuous decrease of the order parameter is characteristic of a second-order phase transition. At any temperature, superconductivity is unstable againstexternal magnetic fields and disappears at some critical magnetic field strength,different for type I and II superconductors. Moreover, superconductivity breaksdown in electric fields that cause a potential drop over the superconductor com-parable to its energy gap 2Δ(T).

For the forthcoming discussion of the Josephson effect, the key element is thewave function Ψ, which describes the macroscopic quantum state of a supercon-ductor according to the BCS theory. This wave function can be written as

Ψ =√

nSei𝜃 (4.2)

with nS = ΨΨ∗ being the density of Cooper pairs and 𝜃 the phase of the macro-scopic wave function. The asterisk indicates the complex conjugate.

The BCS theory describes metallic low-temperature superconductors, on whichtoday’s most advanced applications in metrology are based. For completeness, welike to mention that in 1986 K. Alexander Müller and J. Georg Bednorz at the IBMlaboratories in Rüschlikon discovered superconductivity in a perovskite ceramicmaterial (a Ba–La–Cu oxide) at a temperature of 35 K [6]. In 1987, they receivedthe Nobel Prize in Physics for this discovery. Their work set the starting point for

4.1 Josephson Effect and Quantum Voltage Standards 63

intense research in the so-called high-temperature superconductors with the mostprominent cuprate material being yttrium barium copper oxide (YBCO). YBCOwas the first material in which superconductivity was observed at a critical tem-perature Tc = 93 K above the temperature of liquid nitrogen [7]. So far, the highestcritical temperature of Tc = 133 K has been achieved with a Hg–Ba–Ca–Cu–O-based cuprate [8]. The theoretical description of the high-Tc superconductors isstill a matter of discussion. Yet, it seems clear that the CuO planes and the preciseoxygen content play a decisive role.

More recently, iron-based materials (so-called pnictides, like Sm(O1−xFx)FeAs)with critical temperatures up to Tc = 55 K have been discovered [9]. Even thoughtheir critical temperatures are still considerably lower than the ones of thecuprates, the mechanical properties of some oxygen-free pnictides, like SrFe2As2,are superior to the brittle cuprates. This property may allow easier fabrication of,for example, cables.

4.1.2Basics of the Josephson Effect

The Josephson effect was theoretically predicted by Brian D. Josephson in 1962 [1].It refers to the tunneling of Cooper pairs without resistance between two super-conductors, which are separated by a thin tunnel barrier. This arrangement, calledJosephson junction, is schematically shown in Figure 4.1.

A key element of a Josephson junction is the tunnel barrier, which can be aninsulator, a normal metal, or a semiconductor. Its thickness is typically a fewnanometers, chosen to be large enough to prevent direct exchange of Cooperpairs. On the other hand, the barrier is thin enough to allow the macroscopicwave function Ψ1 of superconductor 1 to couple into superconductor 2, andvice versa. Such a barrier is said to provide a weak link. The coupling is a purelyquantum mechanical phenomenon. It reflects the fact that a quantum mechanicalwave function does not end abruptly at the edge of a sample or structure but leaksinto the neighboring region, in which it decays exponentially.

The supercurrent of Cooper pairs across the tunnel barrier is determined by thetime-dependent Schrödinger equation. More specifically, two separate equationshave to be written for the superconductors 1 and 2, which read in short notation

iℏ∂Ψ1,2(t)

∂t= E1,2Ψ1,2(t) + KΨ2,1. (4.3)

E1Ψ1(t) + KΨ2(t) E2Ψ2(t) + KΨ1(t)

Figure 4.1 Schematic drawing of a Josephson junction. Two superconductors are separatedby a thin tunnel barrier (gray) with a typical thickness of a few nanometers. Ψi is the wavefunction and Ei the energy of superconductor i. With K , we denote the coupling constant,which depends on barrier thickness and height.


The coupling constant K describes the quantum mechanical coupling between thesuperconductors and, hence, couples the two equations. If an external voltage U isapplied across the junction, the energies E1 and E2 denote the potential of super-conductors 1 and 2, respectively, arising from the voltage. Thus, |E2 − E1| = 2eU .If the two superconductors are identical, the voltage drop is symmetric, E2 = eUand E1 = −eU .

Solving the coupled Schrödinger Eq. (4.3) with an ansatz for the wave functionsaccording to Eq. (4.2) shows that the Cooper pair densities n1 and n2 of supercon-ductors 1 and 2, respectively, are time dependent. The time dependence results ina Cooper pair current

IS(t) ∝∂∂t

n1(t) = − ∂∂t

n2(t) (4.4)

given by

IS(t) = ISmax sin(𝜃1(t) − 𝜃2(t)). (4.5)

In this equation, ISmax is the critical current, which is proportional to the cou-pling constant K , and 𝜃i(t) is the phase of the wave function of the superconductorlabeled i. The time evolution of the Cooper pair current is determined by the timeevolution of the phase terms. The time dependence of the phase difference

𝜑(t) = 𝜃1(t) − 𝜃2(t) (4.6)

is given by∂𝜑(t)∂t

= 2eℏ

U . (4.7)

Equations (4.5) and (4.7) are called the Josephson equations, and the prefactor2e∕h (not 2e∕ℏ) is known as the Josephson constant KJ. Its inverse, h∕2e, is the fluxquantum Φ0. Combining Eq. (4.5) till Eq. (4.7), we obtain for the tunnel current ofCooper pairs (or Josephson current)

IS(t) = ISmax sin(

2eℏ ∫

t

0U (𝜏) d𝜏 + 𝜑0

)

. (4.8)

The constant phase 𝜑0 is the integration constant determined by the initial con-ditions of the experiment. In the following, we will analyze the predictions ofEq. (4.8) for different types of external voltages across the Josephson junction.

4.1.2.1 AC and DC Josephson EffectIf a constant DC voltage U ≠ 0 is applied across a Josephson junction, the Cooperpair current amounts to

IS(t) = ISmax sin(2eℏ

U •t + 𝜑0

). (4.9)

Thus, IS(t) is a high-frequency AC current with angular frequency 𝜔J = 2eU∕ℏand frequency fJ = 2eU∕h. Equation (4.9) describes the so-called AC Josephsoneffect, that is, the conversion of voltage to frequency. The temporal average of theAC Josephson current is zero since it contains no DC contribution.


U

0

IS−ISmax

+ISmax

Figure 4.2 Voltage–current characteristic of anideal Josephson junction illustrating the DC andAC Josephson effect. The current axis shows thetime-averaged current.

If no voltage is applied, U = 0, a DC Cooper pair current is generated whosemagnitude and direction depend on the constant phase term 𝜑0. For 𝜑0 ≠ 0, acurrent flows without voltage drop. This is referred to as the DC Josephson effect.For the ideal Josephson junction treated so far, the DC and AC Josephson effect areillustrated in Figure 4.2, in which the voltage is plotted versus the time-averagedcurrent.

4.1.2.2 Mixed DC and AC Voltages: Shapiro StepsThe essence of Josephson voltage standards becomes apparent if a mixed voltagecontaining DC and AC contributions is inserted in Eq. (4.8). We write this mixedvoltage as

U(t) = U + uM cos(𝜔Mt), (4.10)

where 𝜔M is the angular frequency of the AC part. For the Josephson current onethen finds

IS(t) = ISmax

∞∑

n=−∞(−1)nJn

(2euMℏ𝜔M

)

sin((𝜔J − n𝜔M)t + 𝜑0), (4.11)

where Jn is the Bessel function of order n. Equation (4.11) shows that the Joseph-son junction carries a DC Cooper pair current whenever 𝜔J − n𝜔M = 0 holds,that is, if

Un = n h2e

fM =nfMKJ

, (4.12)

where n is an integer number. The discrete voltages Un are called Shapiro steps,named after Shapiro who experimentally observed them for the first time in 1963[10]. The integer n is referred to as the step number. Figure 4.3 illustrates thevoltage–current characteristic.

From a physical point of view, the discrete voltages Un are the result of thefrequency modulation of the AC Josephson current with frequency 𝜔J by theapplied AC voltage with frequency 𝜔M. This frequency modulation generatessidebands, among which DC terms are found for 𝜔J − n𝜔M = 0. In the contextof quantum metrology, Eq. (4.12) can be interpreted such that the DC voltage isgiven by the number of flux quanta transported through the Josephson junctionper time interval. Here, the Josephson junction acts as an ideal frequency–voltage


U

0

n = 4

n = 3

n = 2

n = 1

IS

Figure 4.3 Voltage–current characteristic ofan ideal Josephson junction for the applicationof a mixed voltage illustrating the generationof quantized voltages. The current axis showsthe time-averaged current.

converter, which can be viewed as the inverse of the AC Josephson effect treatedin Section 4.1.2.1.

4.1.3Basic Physics of Real Josephson Junctions

When proceeding from ideal Josephson junctions, which have been dealt within Section 4.1.2, to real ones, other current contributions have to be taken intoaccount. In addition to the Cooper pair current IS, a displacement current IC hasto be considered due to the finite capacitance C of the junction. Moreover, a single-electron tunnel current IN flows across the junction at finite temperatures. Thetreatment of real Josephson junctions should also consider that a current bias isapplied in experiments on Josephson junctions. These considerations are takeninto account by the resistively–capacitively shunted junction model (RCSJ model)proposed by Stewart and McCumber [11, 12]. The RCSJ model describes the realJosephson junction by the electrical circuit of Figure 4.4. In this parallel circuit,the bias current Ibias is split into the Cooper pair current IS of an ideal Josephsonjunction, the displacement current IC through the capacitance C, and the single-electron current IN, which is expressed as current through an ohmic resistanceR. Thus, for finite temperatures below Tc, the RCSJ model yields the followingequation for the dynamic behavior of a real Josephson junction

Ibias(t) = ISmax sin(𝜑(t)) + UR

+ C dUdt

. (4.13)

Ibias

IN IC IS

Figure 4.4 RCSJ model to describe a realJosephson junction accounting for displace-ment (IC) and single-electron (IN) currents.


The RCSJ model is useful to distinguish between two different types of realJosephson junctions, that is, junctions that show either hysteretic or nonhys-teretic dynamic behavior. This becomes apparent if in a linear approximationto Eq. (4.13), the term sin(𝜑) is replaced by 𝜑, the Josephson inductanceLJ = ℏ∕(2eISmax) is introduced, and the Josephson equation (4.7) is used toreplace U by the time derivative of 𝜑. One then obtains the equation of anRLC oscillator. Using the general expression for the eigenfrequency of an RLCoscillator, the eigenfrequency or plasma frequency of a real Josephson junctioncan be written as

fP = 12𝜋

√LJC

=√

eISmax𝜋hC

. (4.14)

The quality factor of an RLC oscillator, defined as the ratio between its eigenfre-quency and the full width at half maximum of its resonance, is given by

Q = 2𝜋fPRC. (4.15)

For the description of a real Josephson junction in the frame of the RCSJ model,the so-called McCumber parameter, 𝛽C, is introduced as the square of the qualityfactor

𝛽C = Q2 = 2eISmaxR2C

ℏ

. (4.16)

The McCumber parameter is used to distinguish between hysteretic and nonhys-teretic junctions. If 𝛽C > 1 holds, the junction is underdamped and shows the hys-teretic behavior illustrated in the upper part of Figure 4.5. If only a DC bias currentis applied, the supercurrent increases till the critical current is reached for increas-ing bias. For higher bias currents, the junction switches to the normal conductingstate, and the voltage–current characteristics approaches the normal-state resis-tance. If the bias current is decreased again, hysteretic behavior is observed.

If microwave excitation is added to the DC bias current, different Shapirosteps can be observed, which overlap around zero-bias current. As explained inmore detail in Section 4.1.4.2, this behavior is utilized for DC Josephson voltagestandards. The current range, over which a constant-voltage step extends, andthe observed step number depend on the applied microwave power. MetastableShapiro steps are observed for certain ranges of microwave power and modula-tion frequency 𝜔M. Beyond these ranges, the Josephson junction shows chaoticbehavior [13]. Underdamped junctions satisfying the relation 𝛽C > 1 are realizedif the tunnel barrier is an insulator with a large resistance R and for a finitecapacitance C of the junction, as shown by Eq. (4.16). This type of Josephsonjunction is often referred to as superconductor/insulator/superconductor (SIS)junction.

Overdamped Josephson junctions satisfy the relation 𝛽C ≤ 1. They can be real-ized by lowering the junction resistance using a normal metal (N) or a combina-tion of a normal metal and insulating layers as tunnel barrier. These junctions arereferred to as superconductor/normal metal/superconductor (SNS) and super-conductor /insulator/normal metal/insulator/superconductor (SINIS) junctions,


Vo

lta

ge

Vo

lta

ge

Current

ISmax

n = +5

n = +1

n = 0

n = −1

n = −5

ISmax

Current

Figure 4.5 Schematic voltage–currentcharacteristic of real Josephson junctions.Upper part: hysteretic (underdamped) junc-tions. Lower part: nonhysteretic (over-damped) junctions. Left-hand side: behaviorif only a DC bias current is applied. Right-hand side: behavior for excitation with

microwaves in addition to the DC bias cur-rent. Note that the characteristic of hysteretic(underdamped) junctions is observed whenmicrowave power and DC bias are tuned andthe different voltages generated during thetuning are superimposed. (Courtesy of PTB.)

respectively. As illustrated in the lower part of Figure 4.5, overdamped junctionsshow nonhysteretic behavior. In particular, for excitation with microwaves, anunambiguous relation between the DC bias current and the voltage step numberis obtained. This feature provides the basis for the development of AC Josephsonvoltage standards treated in Sections 4.1.4.3 and 4.1.4.4.

An in-depth analysis of the physics of real Josephson junctions can be found,for example, in Refs [14–17].

4.1.4Josephson Voltage Standards

The Josephson effect links voltage to frequency, the fundamental constants h ande, and an integer number only, as seen from Eq. (4.12). Since the frequency canbe realized with extremely high precision with atomic clocks (nowadays with arelative uncertainty smaller than 10−15; see Chapters 2 and 3), Eq. (4.12) bearspotential for the very precise realization of voltages. Moreover, already in the1960s, it was experimentally demonstrated that the Josephson effect itself washighly reproducible at the level of 1 part in 108 [18]. Subsequent measurementsdemonstrated even better reproducibility up to the level of parts in 1016 [19] andparts in 1019 [20]. These findings prompted substantial efforts to construct volt-age standards based on the Josephson effect. The main obstacle encountered was


the small magnitude of the voltage generated by a single Josephson junction evenfor frequencies in the gigahertz range. For example, at a frequency of 70 GHz,the voltage of the lowest Shapiro step is only 145 μV. Therefore, a considerableamount of work was spent on the development of arrays of Josephson junctions.A Josephson array is a series circuit of many junctions, in which the voltages ofthe junctions add up to reach the practical voltage level of 1 V and nowadays10 V. Today, Josephson arrays generating DC voltages of 10 V are commerciallyavailable. Besides lasers, they are one of the few quantum technologies that havealready entered the market. Josephson arrays for AC measurements at the 10 Vlevel have also progressed substantially. They are in use at several national metrol-ogy institutes and are being commercialized at present. In Section 4.1.4, we willreview the technology of Josephson voltage standards and the main ideas thatadvanced their development, as well as their present state of the art and impacton metrology. More details of the development of Josephson voltage standardsare presented in several review papers. The interested reader is referred to, forexample, Refs [21–23].

4.1.4.1 General Overview: Materials and Technology of Josephson ArraysAs mentioned earlier, a single Josephson junction generates voltages in the sub-millivolt range. Therefore, the voltages of thousands or tens of thousands junctionsmust be added up in a DC series circuit to obtain practical voltage levels. To thisend, integrated circuits are fabricated using thin-film technology, including sput-ter deposition of superconducting layers and dielectrics, patterning by photo orelectron-beam lithography, and etching processes.

In the 1980s, integrated Josephson arrays were based on lead/lead alloy tech-nology [24]. Yet, this technology did not provide the required long-term stabilitysince lead alloys can be damaged by humidity and thermal cycling between roomtemperature and low temperatures. Niobium proved to be a better choice of thesuperconducting material of Josephson junctions in an array. This metal com-bines chemical stability with a large critical temperature of 9.5 K. Niobium caneasily be covered with aluminum serving as a normal metal, which even has astable natural oxide that can be used to form an insulating layer. Thus, Nb/Al/Aloxide technology [25] provides all ingredients required to fabricate SIS and SINISJosephson arrays. For the fabrication of SNS Josephson arrays, niobium can becombined with, for example, PdAu [26] or NbSi [27]. Thus, niobium is chosenas the superconducting material for Josephson voltage standards nowadays. Nio-bium standards can be operated at the temperature of liquid helium, that is, 4.2 K.Operation at higher temperature around 10 K becomes feasible with NbN arrays[28]. The higher temperature allows the NbN arrays to be operated in cryocool-ers. The high-temperature superconductor cuprate materials have not yet led to amajor breakthrough in voltage standards since their inhomogeneity precludes thedevelopment of highly integrated circuits with a large number of uniform Joseph-son junctions.

The choice of material determines the operating margins of a Josephson array.The maximum frequency (named characteristic frequency), f c, at which the array


can be operated, is limited by the so-called characteristic voltage V c according to

fc = Vc2eh

. (4.17)

The characteristic voltage V c is given by the critical current and the normal-stateresistance:

Vc = ISmaxR. (4.18)

The critical current depends on the critical current density jSmax of the supercon-ducting material. Only in theory the critical current ISmax = jSmaxA can be adjustedat will by the adjustment of the area A of the Josephson junction for a given mate-rial with given critical current density jSmax. In practice, if the area A is increased,the size of the Josephson array increases, which compromises the uniformity of thejunctions across the array and complicates the microwave design. Thus, to obtainthe desired output voltage of a Josephson array, the driving frequency, numberof junctions, step number, material parameters, and dimensions have to be cho-sen carefully. In the following sections, we will discuss in more detail how theseconstraints are dealt with for the different types of Josephson voltage standards.

As concluding remark regarding general technological aspects, we have toaddress microwave issues. The design of a Josephson array must be chosensuch that all junctions are excited by almost the same microwave power. Tothis end, the Josephson junctions are embedded in high-frequency transmissionlines, such as low-impedance microstrip lines or 50 Ω coplanar waveguidesand coplanar striplines. Measures must be taken to avoid reflections and theformation of standing waves. The number of Josephson junctions, which a singlemicrowave transmission line can accommodate, is limited by the attenuationof the microwave power along the line. The transmission line must not be toolong since otherwise power losses become too large. This constraint limits thenumber of Josephson junctions per line. To excite a large number of junctionsequally with microwaves, several microwave branches can be operated in parallel.The microwave is split and the resulting partial waves are fed into the differentbranches. Microwave components for power splitting are available for sinusoidalmicrowaves with a narrow frequency spectrum.

4.1.4.2 DC Josephson Voltage Standards: The Conventional VoltJosephson voltage standards for the generation and dissemination of DC voltagesof 1 and 10 V have been developed since the 1980s and were the first Josephsonstandards that substantially impacted metrology. Nowadays, they are routinelyused by national metrology institutes around the world to reproduce and maintainthe DC voltage scale. They are also commercially available and commercial cali-bration laboratories have started to use them. Their development was advancedby two ideas. The first one was the suggestion to take advantage of the overlap-ping Shapiro steps of highly hysteretic SIS Josephson junctions around zero biascurrent [29]. This concept eliminates the need to bias different junctions individ-ually, which facilitates the integration of a large number of Josephson junctions in


a series array. The second important idea was to embed the junctions in a high-frequency transmission line to ensure uniform microwave excitation.

In 1984, the first Josephson array providing 1 V output was demonstrated. Itwas based on lead/lead alloy technology and a microstrip line was employed todistribute the microwave power [24]. Nowadays, Nb/Al/Al oxide technology isused to fabricate SIS Josephson arrays for the generation of DC voltages of 10 V.The typical current step width is some tens of microamperes. In most designs,microstrip lines are chosen as high-frequency transmission lines. A schematiclayout of such an array is shown in Figure 4.6. Thanks to the large normal-stateresistance of the SIS junctions, the characteristic voltage is so large that drivingfrequencies f of 70 GHz can be applied and the arrays can be operated on higher-order Shapiro steps. For typical parameters, such as f = 70 GHz and step numbern= 5, 14 000 junctions are sufficient to obtain 10 V output.

Figure 4.7 shows a photograph of a 10 V Josephson array fabricated by thePhysikalisch–Technische Bundesanstalt (PTB). The Josephson chip is mountedonto a chip carrier. The fin-line-taper antenna, seen on the left-hand side, iscoupled to a waveguide (not shown), through which the microwave is transmittedto the Josephson array. For operation, the complete array is immersed into liquidhelium.

In order to calibrate a secondary voltage standard, such as a Zener diode, witha Josephson standard, the output voltage of the Josephson standard is comparedto the output of the secondary standard at room temperature. For this purpose, acompensation technique is used where the difference of the two voltages is mea-sured with a sensitive nanovoltmeter that serves as a null detector. To fine-tune theJosephson voltage, the driving frequency of the Josephson array can be adjusted.

7

8

9

6

5

4

3

2

1

Figure 4.6 Schematic layout of aNb/Al–Al2O3/Nb Josephson array. Shownare four junctions embedded in a microstripline whose ground plate is seen at the top ofthe structure. (1) Silicon substrate; (2) sput-tered Al2O3 layer, typical thickness 30 nm;(3) niobium tunnel electrode, 170 nm; (4)Al2O3 barrier, 1.5 nm, fabricated by thermal

oxidation of an Al layer; (5) niobium tunnelelectrode, 85 nm; (6) wiring layer, 400 nm; (7)niobium ground plane, 250 nm; (8) Nb oxideedge protection, 80 nm. The Nb groundplane (7) rests on a 2 μm thick Si oxidedielectric layer (9). (Courtesy of J. Kohlmann,PTB.) (Please find a color version of thisfigure on the color plate section.)


Figure 4.7 Photograph of a 10 V Josephson array mounted onto a chip carrier. The size ofthe array is 24 mm by 10 mm. (Courtesy of PTB.) (Please find a color version of this figure onthe color plate section.)

For an absolute measurement, the frequency is referenced to an atomic clockstandard. Thermal voltages arise from the temperature difference between theJosephson array and the room-temperature part of the measurement setup. How-ever, they can be compensated by reversing the polarity of the Josephson voltageand the voltage of the secondary standard.

Impact of DC Josephson Standards: The Conventional Voltage Scale. Already inthe 1980s, it was realized that the reproducibility of voltages generated with 1 VJosephson arrays was very high, that is, better than one part in 108, independentof the material and geometry of the array. This result has to be compared to theuncertainty of the realization of the International System of Unit (SI) volt. TheSI volt is realized using a so-called voltage balance [30] and a calculable capaci-tor [31], which yields an SI value of the capacitance traceable to the meter. Thevoltage balance compares the electrostatic force to the gravitational force, therebyrealizing the SI volt with a relative uncertainty of a few parts in 107 [32]. Thus,the reproducibility of voltages generated with the Josephson effect is clearly betterthan the uncertainty of the SI volt. The prime goal of metrology is to ensure world-wide uniformity of units and comparability of measurements. Obviously, this goalcan be achieved by taking advantage of the excellent reproducibility of the Joseph-son effect and the small uncertainty of frequency measurements if a fixed value ofthe Josephson constant K J is agreed upon. Therefore, the General Conference ofthe Meter Convention instructed the International Committee for Weights andMeasures (CIPM) in 1987 to recommend a value of the Josephson constant K Jwhich should be used when analyzing Josephson measurements [33]. In 1988, theCIPM recommended a value that was determined using the best experimentaldata available at that time and should be used from 1 January 1990 [34]. This con-ventional value or agreed-upon value of K J is denoted by KJ−90:

KJ−90 = 483 597.9 GHz V−1. (4.19)

In order to make K J−90 compatible with the SI value of K J, a so-called conventionalrelative uncertainty of four parts in 107 was assigned to K J−90. Today, the relativedeviation between K J and K J−90 is much smaller, namely, 6.3 parts in 108 accordingto the adjustment of the fundamental constants in 2010 [35].


In essence, using the relation

U90 =nf

KJ−90, (4.20)

a new, highly reproducible voltage scale U90 is established. In Eq. (4.20), K J−90 canbe treated as a constant with zero uncertainty since no comparison to SI quanti-ties is made. Equation (4.20) is said to provide a representation of the unit volt,while a realization of the volt is obtained if the definition of the SI is utilized. Thisterminology emphasizes that Eq. (4.20) does not provide the SI volt. Nonetheless,K J−90 is used nowadays to represent, maintain, and disseminate the unit volt, moreprecisely “volt90,” for its superior reproducibility. Today, the uncertainty of voltagemeasurements based on K J−90 is a few parts in 1010 or better [36]. The use of theJosephson effect to harmonize voltage measurements is a major breakthrough ofquantum metrology.

We conclude this section with an outlook on how the new SI will change volt-age metrology. If fixed numerical values with zero uncertainty will be assigned tothe elementary charge and the Planck constant, the Josephson constant K J willalso have a fixed value with zero uncertainty. Josephson measurements will thenproduce the SI volt of the new systems of units if they are analyzed using the newvalue of K J. Thus, it will become possible to reference voltage measurements of thehighest precision to the SI volt. The agreed-upon constant K J−90 will be abrogated,which will put an end to the existence of two different units of the electrical volt-age. Thus, electrical metrology will become conceptually simpler. Yet, one needs tokeep in mind that the new SI volt will most likely differ slightly from the presentlyused “volt90.” The deviation will be determined by the latest adjustment of the fun-damental constants before the redefinition. If the adjustment of 2010 was used,the relative deviation would amount to 6.3 parts in 108 as mentioned previously.Such a deviation is negligible in many measurements. Yet, when comparing high-precision voltage measurements, one must carefully keep track of the voltage scalethey are referenced to.

4.1.4.3 Programmable Binary AC Josephson Voltage StandardsMany important measurements in electrical metrology involve AC voltages. Aprime example is the measurement of electrical power and energy at the line fre-quency of 50 or 60 Hz of the power grid. Conventionally, the AC volt (and also theAC ampere) is realized and disseminated using thermal converters. In a thermalconverter, the heat generated by an AC electrical quantity is compared to the heatproduced by its DC counterpart, which can be determined with high precision.This is a calorimetric approach, which yields the root-mean-square (rms) valueof the AC quantity. Thermal converters can measure the rms value of AC volt-ages in the range from millivolts to kilovolts and over the frequency range from10 Hz to MHz with relative uncertainties as good as one part in 106. Unlike ther-mal converters, AC Josephson voltage standards have the potential to determinethe complete waveform of an AC voltage with high precision. Moreover, they holdpromise to establish quantum-based measurements of other electrical quantities,


such as impedance and electrical power, and quantum-based characterization ofmeasuring instruments, such as analog–digital converters. Consequently, sincethe 1990s, efforts have been made to harness the Josephson effect and its highreproducibility for AC voltage metrology.

Despite their tremendous impact on DC voltage metrology, hysteretic SISJosephson junctions are not suited for AC Josephson standards. The outputvoltage of an AC standard varies in time, which can be achieved in a straight-forward way by rapid and reliable switching between different Shapiro steps.Such switching cannot be realized with hysteretic Josephson junctions due to theambiguity of their voltage–current characteristic, that is, due to the overlap ofthe voltage steps. Therefore, nonhysteretic Josephson junctions are used for ACstandards. Varying the DC bias current, the n= 0, 1, or −1 voltage step can beaddressed in these junctions (see Figure 4.5).

Nonhysteretic SNS or SINIS Josephson arrays are employed for the so-calledprogrammable binary AC Josephson voltage standards treated in this section. Ifm(t) is the number of junctions activated at the time t, that is, with step numbern ≠ 0, the time-dependent output voltage of the array is

U(t) = nm(t)K−1J f . (4.21)

Usually, the Shapiro step number n is±1. As for DC Josephson arrays, f is assumedto be the constant frequency of a sinusoidal microwave. The schematic layoutof a binary AC Josephson array is depicted in Figure 4.8. The array is dividedinto N + 1 segments with 20, 21, 22, … ,2N Josephson junctions. The segmentscan be addressed individually. If each junction generates a voltage U1, any volt-age between −(2N+1 − 1)U1 and +(2N+1 − 1)U1 can be generated with a resolu-tion given by U1. For a typical frequency of 15 GHz, U1 is 31 μV. The binary ACJosephson array can be considered as a multibit digital-to-analog converter. Asan example of an AC voltage generated with a binary Josephson array, Figure 4.9shows a stepwise approximated 50 Hz sine wave with 16 steps per period.

The fabrication of binary AC Josephson arrays entails technological challengesnot encountered with DC arrays. Binary arrays have to be operated on the step

x xx xxxxf

U1 2U1 4U1 2NU1

Output voltage U(t)

Figure 4.8 Schematic layout of a binarydivided AC Josephson array. Each of thebold X represents a single Josephson junc-tion delivering a voltage U1. The numbersof junctions per segment form a binary

sequence. Each segment has its own powersupply, which provides the bias current toselect the Shapiro step number n= 1, −1 (or0 to deactivate the segment).


15

10

5

0

−5

−10

−150 10 20

Time (ms)

Voltage (

V)

30

Figure 4.9 Stepwise approximated 50 Hz sine wave generated by a programmable binaryAC Josephson voltage standard. (Courtesy of R. Behr, PTB.) (Please find a color version ofthis figure on the color plate section.)

number n = ±1, rather than on a step number n > 1 as DC arrays. Therefore, thenumber of Josephson junctions has to be increased to achieve the same voltagelevel as with a DC array that is driven at the same frequency. Other constraintsarise regarding the choice of the driving frequency. If an SNS array is employed, theproduct of critical current ISmax and normal-state resistance R is usually smallerthan for SIS arrays. As a consequence, the characteristic voltage and the driv-ing frequency are reduced, as seen from Eqs (4.17) and (4.18). The decrease ofthe driving frequency has to be compensated by a further increase of the num-ber of junctions. The first practical programmable binary 1 V array of the SNStype contained 32 768 junctions with PdAu barriers and was operated at 16 GHz[26]. Nowadays, programmable binary 10 V arrays for operation at 16–20 GHzare available at the NIST (United States) and AIST (Japan) [28, 37]. Such arraysconsist of 300 000 junctions. With increasing number of junctions, uniformity ofjunctions across an array is more difficult to achieve. Moreover, the microwavedesign becomes more complicated since a larger number of microwave brancheshave to be operated in parallel.

In order to reduce the required number of junctions, the PTB developed SINISarrays based on Nb/Al/Al oxide technology. This technology allows the ISmaxRproduct to be tuned for operation at 70 GHz. In 2007, the PTB presented a pro-grammable binary 10 V SINIS array with “only” 70 000 junctions [38]. Yet, the fab-rication yield of these SINIS arrays was rather low due to their thin, only 1–2 nmthick, and damage-prone Al oxide insulating layers. Therefore, the PTB and NISTjointly developed NbSi as an alternative barrier material [27]. NbSi barriers havea thickness on the order of 10 nm and are less damage prone. Moreover, NbxSi1−xallows large ISmaxR products to be realized when the Nb content is tuned closeto the metal–insulator transition at x= 11%. Nowadays, the PTB employs NbSi


as barrier material for programmable 10 V arrays. The arrays have a current stepwidth of 1 mA or larger and can reliably be operated at 70 GHz.

This brief rundown shows that the fabrication of binary AC Josephson arrayswith 10 V output is a mature technology nowadays. Even arrays with 20 V outputhave been demonstrated recently [39, 40].

A key element of binary AC Josephson standards is the programmable currentsource used to individually address the binary segments of the array. In princi-ple, stepwise approximations of any waveform can be generated. In practice, theswitching time of the current source has to be considered, that is, the time requiredto change the bit pattern that is inputted to the Josephson array. This time togetherwith the number of voltage levels, which is chosen to approximate one cycle of theoutput waveform, limits the frequency of the output voltage. For example, for atypical switching time of 2 μs, corresponding to a rate of 500 kHz, the maximumfrequency of the output voltage is limited to the 10 kHz range considering thateach cycle of the output waveform is composed of several tens of voltage levels.

When settled on a voltage level, the output of a binary Josephson array has thesame high reproducibility as achieved with a DC Josephson standard. Yet, dur-ing the switching between two voltage levels, the output voltage of the Josephsonarray is not determined by Eq. (4.21). It is given by the so-called transients whoseamplitude and shape are not exactly known. The transient regime can be limited toa time window of less than 100 ns using modern electronics with rise times in the10 ns range. Nonetheless, the transients compromise the uncertainty of rms mea-surements with binary AC Josephson arrays. The effect of the transients increaseswith the number of switching events per time, that is, with the number of voltagelevels per cycle and the frequency of the output voltage. As a consequence, thefrequency is limited to the kilohertz range. In Section 4.1.4.5, we will discuss inmore detail how the transients affect different types of measurements. In Section4.1.4.4, a conceptually different approach toward AC Josephson voltage standardsis presented, which completely avoids the problem of undefined transients.

4.1.4.4 Pulse-Driven AC Josephson Voltage StandardsA conceptually straightforward alternative to the variation of the number m ofactivated Josephson junctions for the generation of an AC voltage is the variationof the driving frequency f . Yet, simulations based on the RCSJ model show that thisapproach faces severe limitations if a sinusoidal microwave drive is considered. Forsinusoidal excitation, stable operation of a nonhysteretic Josephson array, that is,a sufficiently large current step width, is only obtained for frequencies close to thecharacteristic frequency (see Eqs (4.17) and (4.18)) [41, 42]. However, frequencytuning can be realized if a train of sufficiently short current pulses is used to drivethe Josephson array [42, 43]. The output voltage of an array with m Josephsonjunction is then given by

U(t) = nmK−1J fR(t), (4.22)

where fR(t) is the repetition frequency of the pulse train, that is, the inverse ofthe temporal spacing between successive pulses, and n = ±1 is the step number.


Stable operation is obtained if the width of the individual pulses is shorter thanthe inverse of the characteristic frequency [42]. The repetition frequency fR(t) canthen be tuned between zero and the characteristic frequency.

The basic physics of pulse-driven operation is seen considering the effect of asingle current pulse. Each current pulse induces a phase change of 2𝜋n acrossthe Josephson junction [44]. This phase change corresponds to the generationof a voltage pulse with an area of n flux quanta Φ0 = h∕(2e) = K−1

J according toEq. (4.7). If this process is repeated at the frequency fR(t), a voltage according toEq. (4.22) is obtained. The voltage is determined by the number of flux quantatransferred through the Josephson junctions per time.

In order to drive the Josephson arrays, pulse pattern generators are used thatallow the generation of various current pulse sequences and, thereby, the variationof the repetition frequency fR(t). State-of-the-art pulse pattern generators outputbipolar current pulse sequences with a maximum repetition rate of 15 GHz. Theuse of bipolar current pulses allows truly alternating voltages to be generated. Ifno bipolar pulse pattern generator is available, the output of a unipolar pulse pat-tern generator can be combined with a sinusoidal microwave to produce a bipolarcurrent pulse train. Yet, this scheme involves the sensitive synchronization of thetwo signals, which compromises the stability of the pulse drive.

Nonhysteretic Josephson junctions of the SNS type with typical characteristicfrequencies on the order of 10 GHz are used for pulse-driven Josephson voltagestandards. Thus, the maximum repetition frequency of the train of current pulsesis of the same order as the characteristic frequency of the junctions.

The microwave design of pulse-driven Josephson voltage standards involvesissues not encountered with DC or binary AC Josephson standards. These issuesresult from the broadband spectrum of the current pulses. The pulse spectrumextends from DC to beyond the maximum repetition frequency, that is, to roughly30 GHz. Today’s microwave splitters do not support such a large bandwidth. Asa consequence, it is not possible to operate several array branches in paralleldriven by the output of a single current source. Therefore, the attenuation of thehigh-frequency components of the pulse spectrum along the array limits thenumber of Josephson junctions per array to 5000–10 000 even if the area of eachjunction is reduced as compared to binary array designs. This constraint restrictsthe maximum output voltage of pulse-driven arrays. A total output voltage of275 mV rms was achieved by combining the outputs of two pulse-driven arrayswith 6400 junctions each [45]. The arrays were driven by two synchronizedcurrent pulse trains [45].

Present work aims at increasing the output voltage to 1 V. To this end, it isnecessary to combine the outputs of more than 2, most likely up to 8, arrays,which are driven by an equal number of synchronized current pulse generators.At the time of writing, both the PTB and the NIST had reported the generationof total output voltages of 1 V rms at conferences. Optoelectronic schemes maybe applied to facilitate the generation of a large number of synchronized currentpulse trains [44]. In the optoelectronic scheme, electrical pulses are convertedto optical ones with electrically pumped lasers. The optical pulses are duplicated


(a)

Arbitrarywaveform

Currentpulses

ΣΔ-modulation

Low-pass filtering

+1

−1

0

Quantizedvoltage pulses

Quantizedwaveform

(b)

(c)

(d)

Figure 4.10 Scheme of the generation ofa quantized waveform with arbitrary shapeusing a pulse-driven Josephson array, shownfor the example of a sine wave. The arbitrarywaveform (a) is encoded in a pulse trainwith a ΣΔ converter, whose code controlsthe output of a pulse pattern generator. Thepulse pattern generator outputs the current

pulse sequence shown in (b). The currentpulses drive a Josephson array, which gen-erates voltage pulses having the area of theflux quantum Φ0 (c). Low-pass filtering pro-duces the quantized waveform of (d) accord-ing the Eq. (4.22). (Courtesy of O. Kieler,PTB.)

with optical beam splitters and then converted back to current pulses withsuitable photodetectors.

As an important advantage, pulse-driven Josephson standards can generatearbitrary waveforms. In particular, they can generate pure sinusoidal voltageswithout undefined transient contributions. The corresponding spectrum consistsof a single narrow line at the fundamental frequency and does not contain higherharmonics. The fundamental frequency can be varied from DC to megahertz.

In order to generate an arbitrary waveform, a ΣΔ converter is usually usedto encode the waveform in a train of short current pulses with variable pulseseparation. Figure 4.10 shows the operation principle for the example of a sinewave. Note that the voltage pulses generated by the pulse-driven Josephson array(Figure 4.10c) have the area of the flux quantum Φ0. The quantized waveform ofFigure 4.10d is obtained by low-pass filtering of the voltage pulses of Figure 4.10c.The filtering removes quantization noise.

The quality of the waveforms achievable with this scheme is demonstrated bythe experimental data of Figure 4.11. This figure shows a sine wave with frequencyof 1.875 kHz, generated by a pulse-driven Josephson array with 8000 junctions.The power spectrum clearly demonstrates that higher harmonics are suppressedby at least −123 dBc (the small peak at 6 kHz originates from an experimentalartifact [46]).

The accuracy of the voltage quantization achievable with a pulse-drivenJosephson array has been tested comparing its output to the output of a binary


200

0.00

0 5 10 15 20

−123 dBc

Frequency (kHz)

25 30 35 40 45 50

0.25 0.50 0.75 1.00

Time (ms)(a)

(b)

Voltage (

mV

)P

ow

er

(dB

m)

1.25 1.50 1.75 2.00

100

0

−100

−200

−40

−60

−80

−100

−120

−140

−160

−180

Figure 4.11 Sine wave with frequencyof 1.875 kHz, synthesized with a pulse-driven Josephson array of the SNS type(Nb/NbSi/Nb with 8000 junctions): (a) tem-poral waveform and (b) power spectrum. In

the spectrum, no higher harmonics are seenabove the noise floor at −171 dBm, whichdemonstrates that higher harmonics are sup-pressed by at least −123 dBc. (Courtesy of O.Kieler, PTB.)

AC Josephson voltage standard. At a frequency of 500 Hz and an rms valueof 104 mV, the fundamental frequency components of the two systems werefound to agree within an uncertainty of three parts in 107 [47]. Sine wavesgenerated with two pulse-driven Josephson arrays were found to agree withinan uncertainty of three parts in 108 [48]. Thus, pulse-driven Josephson voltagestandards provide arbitrary waveforms with high precision. They will certainlyfind many applications in metrology as their voltage amplitude increases.

4.1.4.5 Applications of AC Josephson Voltage StandardsApplications of AC Josephson voltage standards in metrology have mainlyinvolved programmable binary Josephson arrays so far. This is because their highoutput voltage of 10 V facilitates high-precision measurements. Such measure-ments become more difficult if pulse-driven arrays with smaller output voltagesare employed. The development of metrological applications of binary Josephsonarrays is a very active field at present. In this section, we give an overview of themost important measuring techniques that are currently being developed.

Before treating true AC measurements, we like to note that programmablebinary Josephson standards are also used to calibrate secondary DC voltagestandards. Binary Josephson standards allow faster polarity reversal as compared


to DC Josephson standards. The faster polarity switching increases the amountof data that can be taken per time interval and, thus, reduces the measurementuncertainty. Moreover, programmable standards allow measurement proceduresinvolving several voltages to be automated. Of course, DC calibrations are notaffected by the ill-defined transients.

A true AC measurement is realized by the so-called AC quantum voltmeter,which is intended to measure the waveform of an unknown periodic AC voltage[49, 50]. To this end, a binary Josephson standard is used to synthesize an ACreference voltage, which is synchronized and phase locked to the unknown volt-age. The difference between the reference and the unknown voltage is measuredwith a sampling voltmeter, which serves as null detector. This concept is similarto the DC voltage calibration described in Section 4.1.4.2, in which also the differ-ence between reference and measurand is nulled. AC quantum voltmeters can beoperated in the audio frequency range. Thanks to the sampling technique, the out-put of the binary Josephson array is only used as reference if the array has settledon a quantized voltage step. Data taken during the switching are discarded. Thistemporal gating suppresses the adverse effect of the transients. The achievableuncertainty depends on the frequency and complexity of the unknown waveform.For a simple approximation of a sinusoidal voltage (consisting of only four levels),an uncertainty of parts in 109 was demonstrated at frequencies below 400 Hz [21].

The concept of the AC quantum voltmeter was also used to develop aquantum-based standard of electrical power [51]. Another approach toward aquantum-based power standard involves the use of analog–digital converters thatare characterized using a binary Josephson standard [52, 53]. Electrical powerstandards reach uncertainties on the order of 10−6 limited by the uncertainty oftheir voltage and current transformers.

Various approaches are reported in the literature to use binary Josephson stan-dards for measurements of rms values, for example, for the calibration of thermalconverters. A comprehensive summary can be found, for example, in Ref. [21].As a given, rms measurements are affected by the transients, which, so far, haslimited the achievable uncertainty to parts in 107 even under the most favorablemeasuring conditions [54].

Binary Josephson standards have also been employed for impedance measure-ments. Usually, impedance ratios are determined with impedance bridges. Theidea of the bridge measurement is to adjust the ratio of two voltages such that thevoltages drive the same current through the two impedances whose ratio is to bedetermined. A null detector monitors the balancing of the bridge. If the bridgeis balanced, the impedance ratio is given by the voltage ratio. In conventionalbridges, the voltage ratio is manually adjusted using inductive voltage dividers. InJosephson impedance bridges, the voltages are generated by two binary Josephsonarrays [55]. Josephson impedances bridges offer the advantage that the balancingcan be automated over a frequency range from some tens of hertz to several kilo-hertz, which substantially facilitates the calibration of impedances. The effect ofthe transients can be suppressed in bridge measurements by generating a squarewave with the binary Josephson arrays and using phase-locked detection at the

4.2 Flux Quanta and SQUIDs 81

fundamental frequency of this wave [56]. Thereby, the higher harmonics of thefast transients do not affect the measurement. Josephson impedance bridges reachapproximately the same uncertainties as obtained with manually operated con-ventional bridges. For example, for the measurement of two 10 kΩ resistors, anuncertainty of a few parts in 108 was demonstrated [55]. The ratio of two 100 pFcapacitors was determined with uncertainties in the range 10−8 –10−7 dependingon frequency [21]. Current work aims at extending Josephson impedance bridgesto the measurement of ratios that substantially differ from unity and to measure-ments of unlike impedances, for example, resistance and capacitance. The Joseph-son impedance bridge and the AC quantum voltmeter seem to be very promisingapplications of binary Josephson arrays.

A promising application of pulse-driven Josephson voltage standards is John-son noise thermometry. The Josephson standard is used to generate a calculablepseudonoise voltage waveform whose power is compared to the thermal noisepower of a resistor [57] (see Section 8.1.5). The method can be implemented withvoltage amplitudes below 1 μV [57].

Applications of pulse-driven Josephson voltage standards in electrical metrol-ogy have been scarce so far due to their limited amplitudes. Pulse-drivenJosephson voltage standards lend themselves well to measurements of rmsvalues, for example, to the calibration of thermal converters [58]. These measure-ments largely benefit from the absence of undefined voltage transients. Anotherpossible application is the testing of electronic components at higher frequencies(10 kHz and larger), which are not accessible with binary Josephson standards[59]. These fields are likely to expand as pulse-driven Josephson arrays generatinglarger voltage amplitudes become broadly available.

4.2Flux Quanta and SQUIDs

In Section 4.1, the flux quantum Φ0 = h∕(2e) ≈ 2•10−15 V s was introduced andshown to provide the basis of the representation of the unit volt. In this section,we will discuss that flux quanta also enable extremely sensitive measurements ofmagnetic quantities using SQUIDs as already mentioned. In SQUIDs, the physicsof Josephson junctions combines with the physics of flux quantization in a super-conducting ring. Flux quantization refers to the fact that the smallest amount ofmagnetic flux that can be maintained in a superconducting ring is given by theflux quantum. Moreover, the magnetic flux threading a superconducting ring isalways an integer multiple of the flux quantum, very much like an isolated amountof charge is an integer multiple of the elementary charge e. Flux quantization isindeed the rationale for considering Φ0 = h∕(2e) a quantum entity rather than asimple combination of two fundamental constants.

The first SQUID was demonstrated in 1964 [60], only 2 years after Brian D.Josephson had published his seminal paper about supercurrents in superconduct-ing tunnel structures [1]. SQUID technology has substantially matured since then.


Nowadays, SQUIDS are commercially available and used in various applicationsfrom biomagnetism to nondestructive material testing and geophysics. For a com-prehensive, in-depth treatment of SQUID physics, technology, and applicationsthe interested reader is referred to specialized monographs and review articles(e.g., [15, 61–63]). In Section 4.2, we will focus on the basics of SQUIDS in thecontext of quantum metrology and on selected applications in measurement.

4.2.1Superconductors in External Magnetic Fields

A SQUID consists of a superconducting ring, which is interrupted by one or twoJosephson junctions and threaded by magnetic flux. Therefore, we introduce thereader to the physics of superconducting structures in external magnetic fieldsbefore treating SQUIDs in Section 4.2.2. To do so, we start with bulk superconduc-tors in Section 4.2.1.1 and then proceed to superconducting rings to introduce theconcept of flux quantization in Section 4.2.1.2. Finally, we discuss single Josephsonjunctions in external magnetic fields in Section 4.2.1.3.

4.2.1.1 Meissner–Ochsenfeld Effect

The Meissner–Ochsenfeld effect refers to the observation that a magnetic fielddoes not penetrate deeply into a superconductor. The effect was first observed byWalther Meissner and Robert Ochsenfeld at the Physikalisch–Technische Reich-sanstalt, the predecessor of the PTB, in 1933 [64].

In order to describe a superconductor in a magnetic field and, in particular, theMeissner–Ochsenfeld effect, we start from very general grounds, namely, fromthe quantum mechanical electrical current density, 𝐣S(𝐫). The electrical currentdensity is obtained by multiplying the probability current density by the charge ofthe Cooper pair eS = −2e. The electrical current density can then be written as

𝐣S(𝐫) =eSℏ

2mSi[Ψ∗ (𝐫) gradΨ(𝐫) − Ψ(𝐫)gradΨ∗(𝐫)

]−

e2S

mS𝐀(𝐫)Ψ∗(𝐫)Ψ(𝐫) (4.23)

with

Ψ(𝐫) =√

nS(𝐫)ei𝜃(𝐫) (4.24)

being the macroscopic quantum mechanical wave function of the BCS theoryaccording to Eq. (4.2). The mass of the Cooper pair is denoted by mS. As usual,𝐀(𝐫) is the vector potential of the magnetic flux density 𝐁(𝐫), that is,

𝐁(𝐫) = rot 𝐀(𝐫) (4.25)

holds. While the Josephson effect is determined by the temporal variation of thephase difference 𝜑, spatial variations will turn out to be important for the descrip-tion of superconductors in magnetic fields. Therefore, the spatial dependence ofall quantities is explicitly noted in the above equations. Inserting Eq. (4.24) in


Eq. (4.23) yields

𝐣S(𝐫) =nSe2

SmS

[ℏ

eSgrad𝜃 (𝐫) − 𝐀(𝐫)

]

. (4.26)

Taking the curl of either side and keeping in mind that the curl of any gradientfield vanishes, we obtain

rot 𝐣S(𝐫) = − 1𝜇0𝜆

2 𝐁(𝐫) (4.27)

with

𝜆2 =

mS

𝜇0nSe2S

. (4.28)

As usual, 𝜇0 is the magnetic field constant also known as the permeability of vac-uum. Equation (4.27) shows that the Cooper pair current density and the magneticfield are related.

The Meissner–Ochsenfeld effect is derived if Eq. (4.27) is combined withMaxwell’s equations, namely,

𝐣S(𝐫) =1𝜇0

rot𝐁(𝐫) (4.29)

(neglecting the displacement current), and

div𝐁(𝐫) = 0. (4.30)

Inserting Eq. (4.29) in Eq. (4.27) and using the identity for the Laplace operatorΔ = grad(div) − rot(rot) and 4.30, we obtain the following equation:

Δ𝐁(𝐫) − 1𝜆2 𝐁(𝐫) = 0. (4.31)

To extract the physics, that is, the Meissner–Ochsenfeld effect, from Eq. (4.31),we assume that the magnetic field is oriented along the z-axis of a rectangularcoordinate system and depends only on the x-coordinate. It is further assumedthat the superconductor extends from x = 0 to+∞ (and that vacuum extends from−∞ to x = 0). Then the magnetic field in the superconductor is given by

Bz(x) = Bz(x = 0) exp(− x𝜆

). (4.32)

Thus, the magnetic field decays exponentially and is negligibly small in the interiorof a bulk superconductor, where it can be considered to be zero. This damping ofthe magnetic field is known as the Meissner–Ochsenfeld effect. The magnetic fieldis finite only in a narrow edge region whose width is approximately 𝜆. The length 𝜆

is named London penetration depth after Fritz and Heinz London who describeda superconductor in a magnetic field as early as 1935 [2]. The London penetra-tion depth is typically in the range from 10 to 100 nm for type I superconductors.It increases with increasing temperature and diverges as the critical temperatureTc is reached (the specific difference between type I and II superconductors willnot be considered here). An equivalent formulation of the Meissner–Ochsenfeldeffect is to state that a superconductor expels the magnetic field from its interior


and behaves like a perfect diamagnet. The perfect diamagnetism is as characteris-tic of the superconducting state as is the disappearance of the electrical resistance.

The Meissner–Ochsenfeld effect is due to a screening current flowing at thesurface of the superconductor as can be seen from the Maxwell equation (4.29).Taking the curl of Bz(x) yields a screening current in the y direction, that is, per-pendicular to the magnetic field and parallel to the interface between the super-conductor and the vacuum:

jSy(x) =1

𝜇0𝜆Bz(x = 0) exp

(− x𝜆

). (4.33)

Thus, the magnetic field gives rise to a screening current in the edge region, which,in turn, results in a field-free interior of the superconductor.

4.2.1.2 Flux Quantization in Superconducting RingsWe consider a superconducting ring in the x–y plane and a magnetic fieldalong the z-axis, which can be expressed by a vector potential 𝐀(𝐫) according toEq. (4.25). To study the magnetic flux ΦF through the area F , which is enclosed bythe ring, we use Eq. (4.26). Rewriting it in terms of the flux quantum Φ0 = h∕(2e)and the London penetration depth 𝜆 gives

𝜇0𝜆2𝐣S(𝐫) = −

Φ02𝜋

grad𝜃(𝐫) − 𝐀(𝐫). (4.34)

Equation (4.34) can be integrated along a closed path C in the superconductingring. When calculating the integral over 𝐀(𝐫), we can take advantage of Stokes’theorem and write

∮C𝐀(𝐫) • d𝐬 =

∫F(C)rot𝐀(𝐫) • d𝐟 =

∫F(C)𝐁(𝐫) • d𝐟 = ΦF. (4.35)

Thus, the term with the vector potential yields the magnetic flux ΦF through thering area F . When working out the term that contains the phase 𝜃(𝐫), one has tokeep in mind that the macroscopic wave function Ψ(𝐫) has to be defined withoutany ambiguity. This requires that the relation

∮Cgrad𝜃(𝐫) • d𝐬 = −2𝜋n (4.36)

holds with n being an integer number. Then the phases 𝜃(𝐫) before and aftertraversing the closed path C differ only by 2𝜋n, which is meaningless due to the2𝜋 periodicity of the phase. Collecting the results of Eqs (4.35) and (4.36), weobtain

∮C𝜇0𝜆

2𝐣S(𝐫) • d𝐬 + ΦF = nΦ0. (4.37)

Let us assume that the ring is a bulk superconductor, that is, that its width andthickness are much larger than the London penetration depth 𝜆. The integrationpath C can then be chosen to be several 𝜆 away from the surface of the ring. In thiscase, the screening current density 𝐣S(𝐫) is negligible along the integration path,and Eq. (4.37) simplifies to

ΦF = nΦ0. (4.38)


Equation (4.38) states that the magnetic flux through the area enclosed by a super-conducting ring is quantized in units of the flux quantum (if the ring has a suffi-ciently large width and thickness). We also like to emphasize that this result wasderived for an uninterrupted superconducting ring without a Josephson junction.If a Josephson junction is embedded in the ring, Eq. (4.38) does not apply anymore.This case will be studied in Section 4.2.2.

For an uninterrupted superconducting ring, the question arises how ΦF can bequantized even though the external magnetic field 𝐁(𝐫) and the external flux Φextcan vary continuously. The flux quantization is the result of the screening currentthat circulates close to the surface of the ring and gives rise to its field-free interior.The screening current generates a flux ΦS whose magnitude is such that an integermultiple of flux quanta is obtained if ΦS is added to the external flux. Thus, therelation

ΦF = Φext + ΦS = nΦ0 (4.39)

holds. The circulating Cooper pair current and the magnetic flux that it generateswill be reconsidered when treating SQUIDs in Section 4.2.2.

4.2.1.3 Josephson Junctions in External Magnetic Fields and Quantum InterferenceIn this section, we consider a single Josephson junction in an external magneticfield to set the stage for the description of SQUIDs. To this end, it will bediscussed how an external magnetic field modifies the phase difference 𝜑 acrossa Josephson junction and how this field dependence gives rise to quantuminterference.

The quantity of interest is the supercurrent IS across the Josephson junctionaccording to Eq. (4.5). However, in contrast to Section 4.1, we have to considerspatially dependent phases 𝜃1 and 𝜃2 of the wave functions of the two supercon-ductors which form the Josephson junction. The spatially dependent phases 𝜃1(𝐫)and 𝜃2(𝐫) can be obtained from Eq. (4.34) if the vector potential 𝐀(𝐫) and thecurrent density 𝐣S(𝐫) are known. Let us consider a Josephson junction where thetunnel barrier is located around x = 0 as shown in Figure 4.12. Superconductor 1extends from x = −∞ to the tunnel barrier, and superconductor 2 from the tunnelbarrier to x = +∞. The superconductors shall be made of the same material andextend from −a∕2 to +a∕2 in y direction. The magnetic flux density points alongthe z-axis and shall be constant in the tunnel barrier.

x

y

2a

2

•−a

x = 0

Bz

θ1 (r) θ2 (r)

Figure 4.12 Josephson junction in a mag-netic field that points along the z direction.The tunnel barrier around x = 0 is shown ingray. The phase of the wave function to theleft and right of the barrier is 𝜃1(𝐫) and 𝜃2(𝐫),respectively.


In the superconductors, the magnetic flux density decays exponentially due tothe Meissner–Ochsenfeld effect as described by Eq. (4.32). Separately for eachsuperconductor, the vector potential and the current density can be calculatedfrom the magnetic flux density using Eqs (4.25) and (4.29), respectively. Bothquantities have only a y component and the vector Eq. (4.34) simplifies to

d𝜃(y)dy

= − 2𝜋Φ0

(𝜇0𝜆

2jsy (x) + Ay(x))

. (4.40)

As x approaches±∞, the vector potential becomes constant (corresponding to thevanishing magnetic flux density), and the current density is found to decay to zero(see Eq. (4.33)). Since Eq. (4.40) is valid at any point x, it can be readily integratedat x = ±∞ where the current density is zero. Doing so for superconductors 1 and2, one obtains for the phase difference across the Josephson junction

𝜑(y) = 𝜃1(y) − 𝜃2(y) = 𝜑0 +2𝜋Φ0

[Ay (+∞) − Ay(−∞)

]• y (4.41)

with 𝜑0 being the phase difference at y = 0. The term with the vector potential canbe rewritten, and we obtain

𝜑(y) = 𝜑0 +2𝜋Φ0 ∮

𝐀(𝐫)d𝐬 = 𝜑0 +2𝜋Φ0

Φ(y). (4.42)

The integral is taken along a closed loop in the x−y plane (normal to the mag-netic field). The loop has a width y and a length in x direction that mathematicallyextends from −∞ to +∞ yet can be restricted to several times the London pen-etration depth 𝜆 from a physical point of view. The magnetic flux Φ(y) throughthis area depends on the y coordinate. As Φ(y) changes by a flux quantum Φ0, thephase difference 𝜑 changes by 2𝜋. Consequently, the supercurrent density

jS(y) = jSmax sin(𝜑(y)) (4.43)

is a periodic function with period Φ0 and changes its direction depending on theposition y within the Josephson junction.

The supercurrent IS across the Josephson junction is obtained by integration ofthe supercurrent density jS(y) over the area of the tunnel barrier using Eqs (4.41)and (4.43). The integrations yields

IS = ISmax sin 𝜑0

sin(𝜋

ΦAΦ0

)

𝜋ΦAΦ0

(4.44)

with the magnetic flux ΦA = Φ(y = a). Thus, ΦA is the magnetic flux through theJosephson junction. Applying a bias current, the 𝜑0 term can be adjusted, but|sin 𝜑0| ≤ 1 always holds. Therefore, the maximum current or critical current ina magnetic field is given by

ISmax(ΦA) = ISmax

|||||||

sin(𝜋

ΦAΦ0

)

𝜋ΦAΦ0

|||||||

. (4.45)


1

ISmax (ΦA)

ISmax

0−5 −4 −3 −2 −1

ΦA/Φ0

0 1 2 3 4 5

Figure 4.13 Critical current of a Josephson junction in a magnetic field (normalized to thezero-field critical current) versus magnetic flux in units of the flux quantum.

The critical current under magnetic field is shown in Figure 4.13. The modulationinduced by the magnetic flux resembles the optical diffraction pattern observedbehind a slit which is illuminated by coherent light. This observation corroboratesthat interference between differently phased current contributions is at the heartof the modulation of the critical current.

Summarizing the results of Section 4.2.1.3, we can state that:

• Magnetic flux changes the phase difference across a Josephson junction• The natural unit of the magnetic flux is the flux quantum Φ0 since a flux change

of Φ0 gives rise to a phase change of 2𝜋• Quantum interference takes place when current contributions with different

phases are superimposed

4.2.2Basics of SQUIDs

In principle, measurements of magnetic fields could be realized using the mag-netic flux dependence of the critical current of a single Josephson junction asshown in Figure 4.13. Yet, in this approach, the area over which the field is inte-grated is small, which limits the field resolution for a given flux resolution. ASQUID consists of a superconducting loop and thus has an increased area for fieldintegration. The loop is interrupted by one or two Josephson junction. In order tointroduce the basics of SQUID physics in the context of quantum metrology, wewill restrict the discussion to the so-called DC SQUIDs. In DC SQUIDs, the loopis interrupted by two junctions, as shown in Figure 4.14.

Let us consider a symmetric DC SQUID with two identical ideal Josephson junc-tions. We assume that the area of each junction is much smaller than the areaF of the superconducting loop, so that the magnetic flux through each junctionis negligible. The DC SQUID is penetrated by an external magnetic field B nor-mal to the plane of the SQUID loop giving rise to an external magnetic flux Φext


U2

B

J

Ibias

IS1

1

IS2

u2

u1

Figure 4.14 Schematic drawing of a DCSQUID with two Josephson junctions 1 and 2.The SQUID loop is penetrated by a magneticfield normal to the superconducting loop. Asdiscussed in Section 4.2.3.1, voltages are mea-sured between contacts u1 and u2.

through the loop. The SQUID is biased with a DC current Ibias, which splits intotwo currents IS1 and IS2 through the Josephson junctions 1 and 2, respectively. Forthese currents, the Josephson equation (4.5) holds:

IS1,2 = ISmax sin(𝜑1,2). (4.46)

The phase differences across the junctions 1 and 2 are denoted by 𝜑1 and 𝜑2,respectively. Following the line of thought of Section 4.2.1.2, we also have toaccount for a circulating current, termed J in Figure 4.14. The circulating currentcontributes to the currents IS1 and IS2 according to the relations

IS1 =Ibias

2+ J , IS2 =

Ibias2

− J . (4.47)

The SQUID behavior is determined by the relation between the phase differences𝜑1 and 𝜑2 and the magnetic flux through the SQUID loop. This relation can bederived from the integration of Eq. (4.34) using the condition that the wave func-tion is defined without any ambiguity as expressed by Eq. (4.36). Analogous to thetreatment of the superconducting ring in Section 4.2.1.2, it can then be shownthat

2𝜋Φ0

[

∮C𝜇0𝜆

2𝐣S (𝐫) • d𝐬 + ΦF

]

= 2𝜋n + (𝜑1 − 𝜑2). (4.48)

In this equation, the presence of the Josephson junctions manifests itself by theterm (𝜑1 − 𝜑2). Apart from this term, the equation is identical to Eq. (4.37) of anuninterrupted superconducting ring. If the SQUID loop can be considered as abulk superconductor, the integral term can be neglected based on the argumentdeveloped in Section 4.2.1.2. The total flux is then given by ΦF. The flux ΦF is thesum of the external flux Φext and the flux generated by the circulating current Jflowing at the surface of the SQUID loop:

ΦF = Φext + LJ , (4.49)

where L is the inductance of the SQUID loop. As for a single Josephson junctionin a magnetic field, Eq. (4.48) shows that the magnetic flux changes the phase termand should be quantified in terms of the natural unit Φ0.


Next, we will discuss how quantum interference in the SQUID loop provides thebasis for extremely sensitive measurements of magnetic quantities. According toKirchhoff’s law, the bias current must be equal to the sum of IS1 and IS2. Togetherwith Eq. (4.48), in which the integral term is neglected, Kirchhoff’s law yields

Ibias = ISmax[sin(𝜑1) + sin(𝜑2)] = 2ISmax cos(𝜑1 − 𝜑2

2

)sin

(𝜑2 +

𝜑1 − 𝜑22

)

= 2ISmax cos(𝜋ΦFΦ0

)

sin(

𝜑2 +𝜋ΦFΦ0

)

(4.50)

In general, the analysis of Eq. (4.50) is complicated since the flux ΦF depends onthe external flux and the circulating current, which also affects the phase differ-ence across the Josephson junctions. Yet, with regard to highly sensitive SQUIDmeasurements, we can treat the simple case of very small SQUID inductance L.To make a more quantitative argument, the screening parameter

𝛽L =ISmaxLΦ0∕2

(4.51)

is defined as the maximum flux that a supercurrent in the SQUID loop can gen-erate normalized by half a flux quantum. The case of small inductance L is thengiven by the condition 𝛽L ≪ 1. For 𝛽L ≪ 1, we obtain ΦF = Φext, and the currentIbias of Eq. (4.50) is modulated by the external fluxΦext only. This behavior providesthe basis of measurements of the external flux or the external magnetic field. Wealso note that the flux through the SQUID loop is not quantized in this case. Themaximum current is obtained if the sine term of Eq. (4.50) is adjusted to the value±1 by an appropriate choice of 𝜑2. Then the maximum current, that is, the criticalcurrent, is found to be

ISmax(Φext) = 2ISmax

|||||cos

(𝜋ΦextΦ0

)|||||

. (4.52)

As shown in Figure 4.15, the critical current is a periodic function of the externalmagnetic flux with the periodicity given by the flux quantum Φ0. Maxima occurwheneverΦext = nΦ0, while the critical current is zero forΦext = (n + 1∕2)Φ0. Themodulation depth ΔISmax is given by 2ISmax. The pattern of Figure 4.15 resemblesthe optical interference pattern observed behind a double slit, which is illuminatedby coherent light. In the SQUID case, quantum interference occurs between theleft-hand and the right-hand path in the SQUID loop. The outcome of the quan-tum interference, that is, whether it is constructive or destructive, depends on thephases 𝜑1 and 𝜑2 across the Josephson junctions 1 and 2, respectively. Phase dif-ferences result from the magnetic flux as shown by Eq. (4.48). To take the analogyto optics one step further, we note that when plotting versus magnetic field, thepattern of Figure 4.15 has an envelope given by the pattern of a single junction.Likewise, in optics, the double-slit pattern has an envelope given by the single-slitdiffraction profile. When discussing DC SQUIDs, we have neglected this single-junction effect since the area of a single junction is much smaller than the areaof the SQUID loop. As a consequence, the magnetic field period of the single-junction pattern is much larger than the one of the SQUID pattern.


ISmax (Φext)

ISmax

Φext/Φ0

2

0−2 −1 0 1 2

Figure 4.15 Critical current of a DC SQUID (normalized to the zero-field critical current ofa single Josephson junction) versus external magnetic flux in units of the flux quantum fornegligible SQUID inductance.

For completeness, we also briefly discuss the case of large SQUID inductanceL, corresponding to 𝛽L ≫ 1. In this case, even a small circulating current J addsa nonnegligible flux to the external flux. Let us assume that the external flux isincreased from zero. This flux change induces a circulating screening current Jsuch that the associated flux LJ compensates for the increase of the external fluxand the total flux remains zero. As the external flux exceeds Φ0∕2, it is energeti-cally more favorable to change the direction of the screening current J such that itsmagnetic flux adds up to the external one to adjust the total flux to one flux quan-tumΦ0. With further increase of the external flux, this behavior repeats. More fluxquanta are added so that the total magnetic flux is always given by an integer mul-tiple of Φ0. Obviously, this case is not favorable for measurements of the externalmagnetic flux. In fact, for large 𝛽L, the modulation depth ΔISmax can be approxi-mated byΔISmax = Φ0∕L = 2ISmax∕𝛽L, which is much smaller than the modulationdepth ΔISmax = 2ISmax obtained for 𝛽L ≪ 1. The limit 𝛽L ≫ 1 is also characterizedby the inequality

LJ ≤ Φ0∕2 ≪ LISmax. (4.53)

Thus, J ≪ ISmax holds so that the circulating screening current has only a negligibleeffect on the phases 𝜑1 and 𝜑2, which are almost equal in this case. As a conse-quence, the term (𝜑1 − 𝜑2) is small in Eq. (4.48) and aspects of the physics of asuperconducting ring without Josephson junctions are recovered. In particular,the total flux is found to be given by an integer number of flux quanta.

4.2.3Applications of SQUIDs in Measurement

The most sensitive magnetic measuring instruments available today are DCSQUIDs made from low-temperature superconductors, such as niobium. Com-mercial instruments have a noise floor of (1 − 10)𝜇Φ0∕

√Hz and (1 − 10)fT∕

√Hz

for measurements of the magnetic flux and the flux density, respectively. Thesenumbers correspond to an energy resolution of (10−31 − 10−32) J∕Hz. The energy


resolution is not far off the fundamental Heisenberg limit and corresponds to theenergy required to lift an electron by 1 mm to 1 cm in the gravitational field ofthe earth. In this section, we briefly discuss real DC SQUIDS and the scheme fortheir readout, which will picture the SQUID as a highly sensitive flux-to-voltageconverter. It will then be discussed how DC SQUIDS are implemented in mag-netometers and how their high resolution is harnessed for the precise scaling ofcurrents and resistances and for biomagnetic measurements.

4.2.3.1 Real DC SQUIDs

In a real DC SQUID, the Josephson junctions of Figure 4.14 are real junctions asintroduced in Section 4.1.3. The SQUID can then be described by the RCSJ model.Nonhysteretic overdamped junctions with McCumber parameter 𝛽C ≤ 1 areemployed in real DC SQUIDs. The critical current of real SQUIDs is maximumfor external magnetic fluxes Φext = nΦ0 and minimum for Φext = (n + 1∕2)Φ0.Thus, their flux dependence is similar to the one derived for ideal SQUIDs inSection 4.2.2. The SQUIDs are biased with a DC current Ibias, and the time-averaged voltage drop over the SQUID ⟨U⟩ is measured between the contacts u1and u2 as shown in Figure 4.14.

In principle, flux measurements with DC SQUIDs can be realized by increasingthe bias current from zero till a finite voltage ⟨U⟩ is observed. The voltage dropindicates that the bias current equals and starts to exceed the critical current at theapplied flux. If this measurement is repeated for different fluxes, the flux depen-dence of the critical current is obtained, which is a sensitive gauge of the flux. Yet,this is a cumbersome procedure. Therefore, in practice, the SQUID is biased witha current slightly above the maximum critical current (i.e., the critical current forΦext = nΦ0). Then, the voltage drop ⟨U⟩ is measured as the external magnetic fluxis varied. Figure 4.16 schematically shows the voltage drop ⟨U⟩ versus the bias cur-rent Ibias for the two limiting flux conditions Φext = nΦ0 and Φext = (n + 1∕2)Φ0.The operating bias current is Ibias, op. As illustrated in Figure 4.16, the voltage drop⟨U⟩ changes as Φext is varied for a constant operating current Ibias, op. The volt-age drop ⟨U⟩ is a periodic function of Φext with periodicity Φ0 as is the criticalcurrent. However, maxima of ⟨U⟩ correspond to minima of the critical current,and vice versa. Thus, voltage maxima occur for Φext = (n + 1∕2)Φ0, while voltageminima are observed for Φext = nΦ0.

U

Φext = (n + ½)Φ0

Φext = nΦ0

Ibias, op

Ibias

Figure 4.16 Schematic graph of time-averaged voltage versus bias current of areal DC SQUID for the two limiting magneticflux conditions Φext = nΦ0 and Φext = (n +1∕2) Φ0. The operating bias current Ibias, op isindicated by the vertical line.


U UoutRfb

Lfb Figure 4.17 Simplified circuit diagram of a DCSQUID operated in a flux-locked loop. The outputvoltage Uout is proportional to the change of theexternal flux through the SQUID loop.

A SQUID operated in this mode can be considered as a flux-to-voltage con-verter with a resolution of a fraction of the magnetic flux quantum Φ0. Yet, thevoltage ⟨U⟩ does not provide an unequivocal measure of the flux due to the intrin-sic periodicity of the SQUID signal. This problem can be solved by operating theSQUID in a so-called flux-locked loop. A flux-locked loop feeds additional flux inthe SQUID loop in order to keep the flux in the SQUID at a constant value whilethe external flux varies.

This negative feedback scheme is schematically shown in Figure 4.17. The volt-age ⟨U⟩ is amplified and integrated. The resulting signal generates an opposingflux in the SQUID loop with the help of an inductance Lfb and produces the out-put voltage Uout over the resistor Rfb. This scheme linearizes the SQUID responsesince Uout is proportional to the change of the external flux even if this change ismuch larger than a flux quantum. In order to increase the sensitivity, flux modu-lation schemes and lock-in detection are applied (not shown in Figure 4.17). Withthese improvements, DC SQUIDs reach the outstanding magnetic flux resolutionmentioned at the beginning of Section 4.2.3.

4.2.3.2 SQUID Magnetometers and Magnetic Property Measurement Systems

Magnetometers measure the magnetic flux density or the magnetic field. Whena SQUID is employed for these measurements, its effective area has to be takeninto account. For a given flux resolution, the field resolution can be improved ifthe area is increased. Yet, an increase of the area results in an increased SQUIDinductance L, which reduces the modulation depth as discussed in Section 4.2.2.To maintain a sufficiently large modulation depth at a high field resolution, theconcept of flux transformation is employed. A flux transformer consists of a closedsuperconducting loop with primary inductance Lp and secondary inductance Lsas shown in Figure 4.18.

M

LI

ΔΦext

Lp

Ls

Figure 4.18 Superconducting flux trans-former with primary inductance Lp and sec-ondary inductance Ls. The magnetic fluxthrough the transformer is coupled into theSQUID via the mutual inductance M.


As the external flux changes by ΔΦext, a current I is induced in the flux trans-former. Flux quantization requires that the flux generated by this current compen-sates ΔΦext so that the flux through the transformer loop is nΦ0 before and afterthe change of the external flux. Thus, ΔΦext determines the current according to

ΔΦext + (Lp + Ls)I = 0. (4.54)

In turn, the current gives rise to a flux change ΔΦSQUID in the SQUID via themutual inductance M:

ΔΦSQUID = −MI = MLp + Ls

ΔΦext. (4.55)

The flux change ΔΦext increases linearly with the area of the flux transformer fora given magnetic field. As seen from Eq. (4.55), this results in an increase of theflux change ΔΦSQUID in the SQUID (we note without proof that the inductanceterm decreases sublinearly with the area). Thus, the magnetic field sensitivity ofa SQUID magnetometer can be increased by choosing a larger flux transformerarea. The SQUID inductance is not increased thereby, and detrimental effects onthe SQUID modulation depth are avoided. Using flux transformation magnetome-ters with a noise floor in the fT∕

√Hz range can be realized as mentioned earlier.

SQUID magnetometers allow measurements of the magnetic flux density withoutstanding resolution. Yet, they are not quantum standards since the effectiveSQUID area is not quantized. In Section 4.2.4, a quantum-based realization of theunit of the magnetic flux density, tesla, will be discussed. This realization utilizesnuclear magnetic resonance (NMR) techniques, which allow a primary standardof the tesla to be realized. SQUID magnetometers can be calibrated against suchprimary standards to obtain traceability to the SI.

We conclude this section on the use of SQUIDs for magnetic measurementsby a brief discussion of SQUID gradiometers and SQUID-based instruments thatmeasure the magnetic moment of materials. A first-order SQUID gradiometer is aspecial flux transformer, in which the single superconducting sensor loop or coil,shown in Figure 4.18, is replaced by two sensor coils. The arrangement of the coilsis shown in Figure 4.19. The coils have opposite winding directions so that the sig-nals from the two coils cancel each other if the magnetic field has the same valuein both coils. Therefore, first-order gradiometers are only sensitive to the gradi-ent of the magnetic field along the z direction. This is of particular importance forhighly sensitive measurements, for which the effects of background fields have to

M

LI

z

Ls

Figure 4.19 First-order SQUID gradiometer.


be suppressed. The concept works well as long as the background fields are con-stant over the separation of the coils. This is often the case, for example, for themagnetic field of earth. If four coils are used in a so-called second-order gradiome-ter, the signal from field gradients is also suppressed. The instrument is then onlysensitive to the second derivative of the magnetic field in z direction.

Second-order SQUID gradiometers are employed for the measurement of smallmagnetic moments. The magnetic sample under study is moved through the coilarrangement at a constant velocity. Due to the excellent background field sup-pression in the second-order gradiometer, the SQUID signal solely results fromthe magnetic field of the sample. The SQUID signal is recorded versus sampleposition. In order to determine the magnetic moment, the measured curve iscompared to a calibration curve obtained with a reference sample with knownmagnetic moment. Magnetic moments as low a 10−11 Am2 can be detected withSQUID-based magnetic property measurements systems.

4.2.3.3 Cryogenic Current Comparators: Current and Resistance RatiosThe outstanding magnetic flux sensitivity of SQUIDs is also utilized in the so-called cryogenic current comparators (CCCs) [65]. These comparators allow cur-rent and resistance ratios to be determined with relative uncertainties of 10−9 andbetter. The realization of well-known resistance ratios is of utmost importance inelectrical metrology since it allows the electrical resistance scale to be established.The anchor point of this scale is the representation of the ohm provided by thequantum Hall effect. As discussed in Chapter 5, the quantum Hall effect can beused to link electrical resistance to the elementary charge and the Planck constantwith very low uncertainty. However, it provides only a limited set of nondecaderesistance values. For practical applications in electrical engineering, decade resis-tance values are required from the milliohm to the teraohm range. Highly precisedecade resistance values are derived from the quantum Hall resistance using CCCsat national metrology institutes.

Furthermore, precise current ratios, as determined with CCCs, are importantfor the amplification of quantized currents generated with single-electron trans-port devices. Quantized currents are directly linked to the elementary charge, asdiscussed in Chapter 6, yet currents levels are only in the picoampere to nanoam-pere range. Therefore, amplification is required.

In a CCC, current ratio measurements are based on Ampere’s law combinedwith the Meissner–Ochsenfeld effect. The principle can be seen considering asuperconducting tube with a wall thickness that is much larger than the Londonpenetration depth 𝜆. Inside the tube, a wire along the tube axis carries a current I,as shown in Figure 4.20. The current generates a magnetic flux density 𝐁. In orderto prevent the magnetic flux density from penetrating through the superconduc-tor, a screening current Iinner is induced at the inner surface of the tube. ApplyingAmpere’s law to a closed integration contour inside the superconducting tube,where the magnetic flux density is zero, we obtain

∮𝐁 • d𝐬 = 𝜇0(I + Iinner) = 0. (4.56)


I

Iouter

Figure 4.20 Cross-section of a superconduct-ing tube (gray). A current I is passed througha wire inside the tube. The current Iouter at theouter tube surface equals the current I.

The screening current flows back at the outer surface of the superconducting tube,Iouter = −Iinner = I.

Next, we consider a superconducting tube with two wires carrying currents I1and I2. Obviously, the outer-surface current Iouter = I1 + I2 will be zero if (and onlyif ) I1 = −I2, that is, if currents of equal magnitude flow in opposite direction. Thiscondition can be tested with a SQUID device set up to detect the magnetic fieldgenerated by Iouter outside the superconducting tube. Thus, the SQUID serves as avery sensitive null detector. It is important to note that the outer-surface currentdoes not depend on the specific position of the wires inside the tube if the tube islong compared to its diameter. This is the basis of the CCC concept and ensuresits high precision.

The tube arrangement realizes the current ratio I2∕I1 = 1. In a CCC, the tube isreplaced by a superconducting torus, whose ends overlap but are electrically iso-lated from each other. Inside the torus, two coils with opposite winding directionscarry currents I1 and I2. If the winding numbers are n1 and n2, any rational currentratio

I2I1

=n1n2

(4.57)

can be realized. The SQUID null detector is placed in the center of the torus. Itmonitors the magnetic flux and generates a feedback signal, which adjusts one ofthe currents till Eq. (4.57) is fulfilled, that is, till the ampere turns of both coils areequal.

As mentioned earlier, CCCs are widely used for precise resistance compar-isons. A schematic circuit diagram of a CCC-based resistance bridge is shown inFigure 4.21. An unknown resistor RX is compared to a resistance standard RN.The two coils with winding numbers nX and nN have opposite winding directions.The balance of ampere turns is monitored by a SQUID device, and the differencebetween the voltage drops over the resistors is measured by a voltmeter. Whenthe bridge is completely balanced, the equations IXnX = INnN and IXRX = INRNare fulfilled. Thus, the resistance ratio is given by

RXRN

=nXnN

. (4.58)

UIN

RN

nN

RX

nX

IX

Figure 4.21 Schematic circuit diagram of aresistance bridge based on a cryogenic cur-rent comparator.


In practice, an auxiliary circuit (not shown in Figure 4.21) is needed to completelybalance the bridge. A detailed discussion of auxiliary circuits can be found, forexample, in Refs [66, 67]. The CCC-based resistance bridge is operated by peri-odically reversing the current polarity at low frequencies (typically below 1 Hz)to compensate for unwanted thermal electromotive forces. Uncertainties of 10−9

and better are achieved.

4.2.3.4 Biomagnetic Measurements

SQUIDs do not only contribute to the scaling of electrical units but have alsofound “real-world” applications. Examples include geophysical surveying for oiland gas as well as nondestructive material testing, where SQUIDs can be used, forexample, to detect subsurface flaws in aircraft parts. The most challenging real-world application in terms of the required magnetic field resolution is the mea-surement of biomagnetic signals. We briefly discuss biomagnetic measurementsin this section to illustrate how the unprecedented sensitivity of a quantum-basedSQUID device pushes the limits of measurements.

The most intensively investigated biomagnetic signals are those generated bythe human heart (magnetocardiography, MCG) [68] and human brain (magne-toencephalography, MEG) [69]. Their investigation is of particular interest sinceMCG and MEG are noninvasive diagnostic tools.

MCG is the magnetic counterpart of electrocardiography (ECG), in which elec-trical signals are measured that originate from the heartbeat. Their temporal shapeprovides information on the functioning of the heart. In MCG, the correspondingmagnetic field is measured. ECG signals are obtained with electrodes attached tothe thorax and, thus, originate from current contributions at the surface of thebody. In contrast, MCG signals are measured in a contactless mode and resultfrom the total current distribution generated by the heart. Therefore, MCG con-tains additional information not accessible with ECG.

In MEG, the magnetic field distribution is measured that is generated by theelectrical activity of the brain. From the field distribution, information is obtainedon the location of the source (often modeled as current dipole) of the magneticfield and, in turn, on the brain function. MEG combines high temporal resolutionon the order of a millisecond with localization accuracy in the centimeter rangeand is noninvasive, as mentioned previously. This combination makes it an attrac-tive diagnostic tool. For comparison, its electric counterpart, that is, noninvasiveelectroencephalography, provides much less accuracy.

The challenge that MCG and MEG presents to measurement lies in the weak-ness of the magnetic signals. Peak amplitudes of MCG signals are several tens ofpicotesla, and MEG signals are even smaller with signal levels below 1 pT. Themeasurements have to cover a bandwidth of several 100 Hz to obtain the desiredinformation on the temporal shape of the signals. Therefore, magnetic field detec-tors with a noise level in the fT∕

√Hz range must be used to obtain a sufficiently

large signal–noise ratio. As a consequence, SQUIDs are the magnetic field sensorsof choice for MCG and MEG.


Besides the sensitivity of the detectors, screening of external static as well asalternating magnetic stray fields is required. These fields can be on the order ofmicrotesla and would otherwise completely mask the biomagnetic signals. There-fore, the measurements are performed in magnetically shielded rooms. To date,the magnetically shielded room with the highest shielding factor has been built atthe Berlin institute of the PTB. It comprises seven magnetic layers of mu-metalwith varying thickness and one highly conductive eddy current layer consistingof 10 mm aluminum. In the inner measuring chamber, a noise level well below1 fT∕

√Hz is achieved with a white characteristic up to 1 MHz (apart from a 1/f

contribution at low frequencies).The detection system for biomagnetic measurements consists of SQUID arrays

rather than a single SQUID detector. Up to several hundreds of SQUID sensorsare employed to measure the magnetic field distribution caused by the bioelectri-cal currents related to the activity of the heart or brain. The current distributionis then reconstructed from the measured magnetic field distribution. This is aso-called inverse problem, which is much more challenging than the so-calledforward calculations, in which the field is determined from a known source distri-bution. Figure 4.22 shows a photograph of a multichannel SQUID system operatedin a magnetically shielded chamber in the Benjamin Franklin hospital of the Char-ité university clinic in Berlin.

4.2.4Traceable Magnetic Flux Density Measurements

The magnetic flux density can be resolved with outstanding resolution usingSQUID magnetometers as described in Section 4.2.3. However, these instrumentsdo not provide traceability to the international systems of units. A conceptuallystraightforward way to obtain traceability is to use a calculable magnetic fieldcoil and to pass a current through it, whose SI value is known. In practice, thisconcept faces severe limitations since it is difficult to establish the geometry of acoil with the required high precision. Therefore, NMR techniques are employedat many national metrology institutes to realize, maintain, and disseminate the SIunit of the magnetic flux density, tesla. These efforts provide the basis of traceablemagnetic measurements.

NMR measurements link the magnetic flux density to the magnetic moment ofa nucleon, that is, to a fundamental constant. Thus, NMR experiments can be con-sidered as a prime example of quantum metrology. To catch the essence of NMRmeasurements, consider a proton with its spin components sz = ±1∕2ℏ along thequantization axis z. If a DC magnetic field Bz is applied along z, the two spin statesare shifted upward and downward in energy by

E± = ±

(

g e2mp

sz

)

Bz. (4.59)

The expression in parenthesis is the z component of the magnetic moment, g theg-factor, and mp the proton mass. According to Eq. (4.59), the energy splitting


Figure 4.22 Eighty-three-channel SQUID system aligned above a patient for MCG mea-surements. (Courtesy of PTB.) (Please find a color version of this figure on the color platesection.)

between the spin states can be expressed as

ΔE = ℏ𝜔 = ℏγ′pBz, (4.60)

where 𝜔 is the (angular) spin-flip frequency. It is also named precession frequencysince in a classical picture the proton spin precesses around the magnetic field withthe angular frequency 𝜔. The constant γ′p is the gyromagnetic ratio of the protongiven by twice the magnetic moment divided by ℏ. More strictly speaking, theprime put as superscript is supposed to indicate that a proton in a spherical sampleof pure water is considered (at 25 ∘C). Thus, γ′p is the shielded proton gyromag-netic ratio. Its approximate value is 2.675 • 108 s−1 T−1 with a relative uncertainty of2.5 • 10−8 according to the adjustment of the fundamental constants of 2010 [35].Equation (4.60) allows the tesla to be realized based on a frequency measurement,which can be performed with high precision.

Two different approaches are used to determine the frequency. For magneticflux densities of a few millitesla and below, the water sample is polarized by amagnetic field pulse, that is, the upper spin state is populated. After the polariza-tion pulse has been turned off, the free-precession decay is observed in the time

References 99

domain. The oscillating free-precession decay signal directly reveals the preces-sion frequency. The signal decays due to the intrinsic spin–spin relaxation timeand the inhomogeneity of the magnetic flux density across the water sample. Thus,the latter has to be small for the technique to be applicable. To illustrate the appli-cation range of the free-precession technique for the realization of the tesla atnational metrology institutes, we take data from the PTB as an example. At thePTB, the technique is used to realize the unit of the magnetic flux density in therange from 10 μT to 2 mT. The lower boundary is determined by the requirementto precisely compensate the magnetic field of the earth. The relative uncertaintyvaries from 10−4 (at 10 μT) to 10−6 (at 2 mT) [70].

NMR absorption techniques can be used for magnetic flux densities in the mil-litesla range and higher [70]. The absorption of a radiofrequency (RF) magneticfield is monitored with the help of a resonator circuit to determine the precessionfrequency 𝜔 and, hence, the unknown DC magnetic flux density Bz [71]. Since theabsorbed power scales with B2

z [71], the technique cannot be extended to the lowfield range. At the PTB, the absorption technique is used to realize the magneticflux density from 1–2 mT to 2 T with a relative uncertainty on the order of 10−5.

References

1. Josephson, B.D. (1962) Possible neweffects in superconductive tunneling.Phys. Lett., 1, 251–253.

2. London, F. and London, H. (1935) Theelectromagnetic equations of the supra-conductor. Proc. R. Soc. London, Ser. A,149, 71–88.

3. Ginzburg, V.L. and Landau, L.D. (1950)On the theory of superconductivity. Zh.Eksp. Teor. Fiz., 20, 1064–1082. Englishtranslation in Landau, L.D. (1965) Col-lected Papers, Pergamon Press, Oxford,p. 546.

4. Bardeen, J., Cooper, L.N., and Schrieffer,J.R. (1957) Microscopic theory of super-conductivity. Phys. Rev., 106, 162–164.

5. Bardeen, J., Cooper, L.N., and Schrieffer,J.R. (1957) Theory of Superconductivity.Phys. Rev., 108, 1175–1204.

6. Bednorz, J.G. and Müller, K.A. (1986)Possible high Tc superconductivity inthe Ba-La-Cu-O system. Z. Phys. B:Condens. Matter, 64, 189–193.

7. Wu, M.K., Ashburn, J.R., Torng, C.J.,Hor, P.H., Meng, R.L., Gao, L., Huang,Z.J., Wang, Y.Q., and Chu, C.W. (1987)Superconductivity at 93 K in a newmixed-phase Y-Ba-Cu-O compound

system at ambient pressure. Phys. Rev.Lett., 58, 908–910.

8. Schilling, A., Cantoni, M., Guo, J.D.,and Ott, H.R. (1993) Superconductivityabove 130 K in the Hg–Ba–Ca–Cu–Osystem. Nature, 363, 56–58.

9. Ren, Z.-A., Che, G.-C., Dong, X.-L.,Yang, J., Lu, W., Yi, W., Shen, X.-L., Li,Z.-C., Sun, L.-L., Zhou, F., and Zhao, Z.-X. (2008) Superconductivity and phasediagram in iron-based arsenic-oxidesReFeAsO1-δ (Re = rare-earth metal)without fluorine doping. Eur. Phys. Lett.,83, 17002 (4 pp).

10. Shapiro, S. (1963) Josephson currentsin superconducting tunneling: the effectof microwaves and other observations.Phys. Rev. Lett., 11, 80–82.

11. Stewart, W.C. (1968) Current–voltagecharacteristics of Josephson junctions.Appl. Phys. Lett., 12, 277–280.

12. McCumber, D.E. (1968) Effect of ACimpedance on DC voltage–current char-acteristics of superconductor weak-linkjunctions. J. Appl. Phys., 39, 3113–3118.

13. Kautz, R.L. and Monaco, R. (1985) Sur-vey of chaos in the rf-biased Josephsonjunction. J. Appl. Phys., 57, 875–889.


14. Barone, A. and Paterno, G. (eds) (1982)Physics and Applications of the Joseph-son Effect, John Wiley & Sons, Inc.,New York.

15. Likharev, K.K. (1986) Dynamics ofJosephson Junctions and Circuits, Gordonand Breach Science, New York.

16. Kautz, R.L. (1996) Noise, chaos, and theJosephson voltage standard. Rep. Prog.Phys., 59, 935–992.

17. Kadin, A.M. (1999) Introduction toSuperconducting Circuits, John Wiley &Sons, Inc., New York.

18. Clarke, J. (1968) Experimental compari-son of the Josephson voltage-frequencyrelation in different superconductors.Phys. Rev. Lett., 21, 1566–1569.

19. Tsai, J.-S., Jain, A.K., and Lukens, J.E.(1983) High-precision test of the Univer-sality of the josephson voltage-frequencyrelation. Phys. Rev. Lett., 51, 316–319.

20. Jain, A.K., Lukens, J.E., and Tsai, J.-S.(1987) Test for relativistic gravitationaleffects on charged particles. Phys. Rev.Lett., 58, 1165–1168.

21. Behr, R., Kieler, O., Kohlmann, J.,Müller, F., and Palafox, L. (2012) Devel-opment and metrological applicationsof Josephson arrays at PTB. Meas. Sci.Technol., 23, 124002 (19 pp).

22. Harris, R.E. and Niemeyer, J. (2011) in100 Years of Superconductivity (eds H.Rogalla and P.H. Kes), Taylor & Francis,Boca Raton, FL, pp. 515–557.

23. Jeanneret, B. and Benz, S.P. (2009)Applications of the Josephson effect inelectrical metrology. Eur. Phys. J. Spec.Top., 172, 181–206.

24. Niemeyer, J., Hinken, J.H., and Kautz,R.L. (1984) Microwave-induced constantvoltage steps at one volt from a seriesarray of Josephson junctions. Appl. Phys.Lett., 45, 478–480.

25. Gurvitch, M., Washington, M.A., andHuggins, H.A. (1983) High qualityrefractory Josephson tunnel junctionsutilizing thin aluminum layers. Appl.Phys. Lett., 42, 472–474.

26. Benz, S.P., Hamilton, C.A., Burroughs,C.J., Harvey, T.E., and Christian, L.A.(1997) Stable 1 volt programmablevoltage standard. Appl. Phys. Lett., 71,1866–1868.

27. Mueller, F., Behr, R., Weimann, T.,Palafox, L., Olaya, D., Dresselhaus, P.D.,and Benz, S.P. (2009) 1 V and 10 V SNSprogrammable voltage standards for 70GHz. IEEE Trans. Appl. Supercond., 19,981–986.

28. Yamamori, H., Ishizaki, M., Shoji,A., Dresselhaus, P.D., and Benz, S.P.(2006) 10 V programmable Joseph-son voltage standard circuits usingNbN/TiNx/NbN/TiNx/NbN double-junction stacks. Appl. Phys. Lett., 88,042503 (3 pp).

29. Levinsen, M.T., Chiao, R.Y., Feldman,M.J., and Tucker, B.A. (1977) An inverseAC Josephson effect voltage standard.Appl. Phys. Lett., 31, 776–778.

30. Funck, T. and Sienknecht, V. (1991)Determination of the volt with theimproved PTB voltage balance. IEEETrans. Instrum. Meas., IM-40, 158–161.

31. Thompson, A.M. and Lampard, D.G.(1956) A new theorem in electrostat-ics and its application to calculablestandards of capacitance. Nature, 177,888.

32. Flowers, J. (2004) The route to atomicand quantum standards. Science, 306,1324–1330.

33. Giacomo, P. (1988) News from theBIPM. Metrologia, 25, 115–119,(see also Resolution 6 of the 18thmeeting of the CGPM (1987), BIPMhttp://www.bipm.org/en/CGPM/db/18/6/(accessed 22 August 2014)).

34. Quinn, T.J. (1989) News from the BIPM.Metrologia, 26, 69–74.

35. Mohr, P.J., Taylor, B.N., and Newell, D.B.(2012) CODATA recommended valuesof the fundamental physical constants:2010. Rev. Mod. Phys., 84, 1527–1605.

36. Wood, B.M. and Solve, S. (2009) Areview of Josephson comparison results.Metrologia, 46, R13–R20.

37. Dresselhaus, P.D., Elsbury, M., Olaya, D.,Burroughs, C.J., and Benz, S.P. (2011)10 V programmable Josephson voltagestandard circuits using NbSi-barrierjunctions. IEEE Trans. Appl. Supercond.,21, 693–696.

38. Müller, F., Behr, R., Palafox, L.,Kohlmann, J., Wendisch, R., andKrasnopolin, I. (2007) Improved 10 VSINIS series arrays for applications in

http://www.bipm.org/en/CGPM/db/18/6/

References 101

AC voltage metrology. IEEE Trans. Appl.Supercond., 17, 649–652.

39. Yamamori, H., Yamada, T., Sasaki, H.,and Shoji, A. (2008) 10 V programmableJosephson voltage standard circuit witha maximum output voltage of 20 V.Supercond. Sci. Technol., 21, 105007(6 pp).

40. Müller, F., Scheller, T., Wendisch,R., Behr, R., Kieler, O., Palafox, L.,and Kohlmann, J. (2013) NbSi barrierjunctions tuned for metrological appli-cations up to 70 GHz: 20 V arrays forprogrammable Josephson voltage stan-dards. IEEE Trans. Appl. Supercond., 23,1101005 (5 pp).

41. Kautz, R.L. (1995) Shapiro steps inlarge-area metallic-barrier Josephsonjunctions. J. Appl. Phys., 78, 5811–5819.

42. Benz, S.P. and Hamilton, C.A. (1996) Apulse-driven programmable Josephsonvoltage standard. Appl. Phys. Lett., 68,3171–3173.

43. Monaco, R. (1990) Enhanced AC Joseph-son effect. J. Appl. Phys., 68, 679–687.

44. Williams, J.M., Janssen, T.J.B.M.,Palafox, L., Humphreys, D.A., Behr,R., Kohlmann, J., and Müller, F. (2004)The simulation and measurement of theresponse of Josephson junctions to opto-electronically generated short pulses.Supercond. Sci. Technol., 17, 815–818.

45. Benz, S.P., Dresselhaus, P.D., Rüfenacht,A., Bergren, N.F., Kinard, J.R., andLandim, R.P. (2009) Progress toward a1 V pulse-driven AC Josephson voltagestandard. IEEE Trans. Instrum. Meas.,58, 838–843.

46. Benz, S.P., Dresselhaus, P.D., Burroughs,C.J., and Bergren, N.F. (2007) Precisionmeasurements using a 300 mV Joseph-son arbitrary waveform synthesizer. IEEETrans. Appl. Supercond., 17, 864–869.

47. Jeanneret, B., Rüfenacht, A., Overney,F., van den Brom, H., and Houtzager,E. (2011) High precision comparisonbetween a programmable and a pulse-driven Josephson voltage standard.Metrologia, 48, 311–316.

48. Kieler, O.F., Behr, R., Schleussner, D.,Palafox, L., and Kohlmann, J. (2013)Precision comparison of sine waveformswith pulse-driven Josephson arrays. IEEE

Trans. Appl. Supercond., 23, 1301404(4 pp).

49. Behr, R., Palafox, L., Ramm, G., Moser,H., and Melcher, J. (2007) Direct com-parison of Josephson waveforms usingan AC quantum voltmeter. IEEE Trans.Instrum. Meas., 56, 235–238.

50. Rüfenacht, A., Burroughs, C.J., andBenz, S.P. (2008) Precision samplingmeasurements using AC programmableJosephson voltage standards. Rev. Sci.Instrum., 79, 044704 (9 pp).

51. Waltrip, B.C., Gong, B., Nelson, T.L.,Wang, Y., Burroughs, C.J., Rüfenacht,A., Benz, S.P., and Dresselhaus, P.D.(2009) AC power standard using aprogrammable Josephson voltage stan-dard. IEEE Trans. Instrum. Meas., 58,1041–1048.

52. Ihlenfeld, W.G.K., Mohns, E., Behr, R.,Williams, J., Patel, P., Ramm, G., andBachmair, H. (2005) Characterizationof a high resolution analog-to-digitalconverter with a Josephson AC voltagesource. IEEE Trans. Instrum. Meas., 54,649–652.

53. Palafox, L., Ramm, G., Behr, R.,Ihlenfeld, W.G.K., Müller, F., and Moser,H. (2007) Primary AC power standardbased on programmable Josephson junc-tion arrays. IEEE Trans. Instrum. Meas.,56, 534–537.

54. Behr, R., Williams, J.M., Patel, P.,Janssen, T.J.B.M., Funck, T., and Klonz,M. (2005) Synthesis of precision wave-forms using a SINIS Josephson junctionarray. IEEE Trans. Instrum. Meas., 54,612–615.

55. Lee, J., Schurr, J., Nissilä, J., Palafox,L., and Behr, R. (2010) The Josephsontwo-terminal-pair impedance bridge.Metrologia, 47, 453–459.

56. Hellistö, P., Nissilä, J., Ojasalo, K.,Penttilä, J.S., and Seppä, H. (2003)AC voltage standard based on a pro-grammable SIS array. IEEE Trans.Instrum. Meas., 52, 533–537.

57. Benz, S.P., Pollarolo, A., Qu, J., Rogalla,H., Urano, C., Tew, W.L., Dresselhaus,P.D., and White, D.R. (2011) An elec-tronic measurement of the Boltzmannconstant. Metrologia, 48, 142–153.

58. Lipe, T.E., Kinard, J.R., Tang, Y.-H., Benz, S.P., Burroughs, C.J., and


Dresselhaus, P.D. (2008) Thermalvoltage converter calibrations using aquantum AC standard. Metrologia, 45,275–280.

59. Toonen, R.C. and Benz, S.P. (2009)Nonlinear behavior of electronic com-ponents characterized with precisionmultitones from a Josephson arbitrarywaveform synthesizer. IEEE Trans. Appl.Supercond., 19, 715–718.

60. Jaklevic, R.C., Lambe, J., Silver, A.H.,and Mercereau, J.E. (1964) Quantuminterference effects in Josephson tunnel-ing. Phys. Rev. Lett., 12, 159–160.

61. Gallop, J.C. (1991) SQUIDs, the Joseph-son Effects and SuperconductingElectronics, Adam Hilger, Bristol.

62. Koelle, D., Kleiner, R., Ludwig, F.,Dantsker, E., and Clarke, J. (1999) High-transition-temperature superconductingquantum interference devices. Rev. Mod.Phys., 71, 631–686.

63. Clarke, J. and Braginski, A.I. (eds) (2006)The SQUID Handbook, vol. 1 and 2,Wiley-VCH Verlag GmbH, Berlin.

64. Meissner, W. and Ochsenfeld, R.(1933) Ein neuer Effekt bei Eintritt derSupraleitfähigkeit. Naturwissenschaften,21, 787–788 (in German).

65. Harvey, K. (1972) A precise low tem-perature DC ratio transformer. Rev. Sci.Instr., 43, 1626–1629.

66. Piquemal, F. (2010) in Handbook ofMetrology, vol. 1 (eds M. Gläser and M.Kochsiek), Wiley-VCH Verlag GmbH,Weinheim, pp. 267–314.

67. Drung, D., Götz, M., Pesel, E.,Barthelmess, H.J., and Hinnrichs, C.(2013) Aspects of application andcalibration of a binary compensationunit for cryogenic current comparatorsetups. IEEE Trans. Instrum. Meas., 62,2820–2827.

68. Koch, H. (2004) Recent advances inmagnetocardiography. J. Electrocardiol.,37, 117–122.

69. Cohen, D. and Halgren, E. (2004) Mag-netoencephalography, in Encyclopedia ofNeuroscience, 3rd edn (eds G. Adelmanand B.H. Smith), Elsevier, New York.

70. Weyand, K. (2001) Maintenance anddissemination of the magnetic field unitat PTB. IEEE Trans. Instrum. Meas., 50,470–473.

71. Weyand, K. (1989) An NMR marginaloscillator for measuring magnetic fieldsbelow 50 mT. IEEE Trans. Instrum.Meas., 38, 410–414.

103

5Quantum Hall Effect

The quantum Hall effect (QHE) occurs in two-dimensional electron systemssubjected to a strong magnetic field. It was first observed by K. von Klitzing in1980 when studying the magnetotransport properties of silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) at low temperatures [1]. In1985 K. von Klitzing was awarded the Noble Prize in Physics for his discovery.Already early on, it was realized that the QHE could have tremendous impact onmetrology since it provides quantized values of the electrical resistance. Thesevalues only depend on the elementary charge e, the Planck constant h, and aninteger number. In fact, today, the QHE has revolutionized resistance metrology.Its main application is the DC quantum Hall resistance standard used by nationalmetrology institutes to reproduce and disseminate the ohm. In recent years,the QHE has also been harnessed for AC resistance measurements, that is, forimpedance metrology, and it has been shown that the unit of capacitance, thefarad, can be directly based on the QHE [2, 3].

From a physics point of view, the QHE relates to the fact that in a two-dimensional electron gas in high magnetic fields, noninteracting electrons loseall degrees of freedom of their motion. Their energy spectrum is then discrete,like the energy spectrum of atoms. In this chapter, we focus the discussion ofthe QHE on this aspect since it provides an understanding of the basic physics,even though it does not provide a complete description of the QHE. For a moredetailed discussion, the reader is referred to, for example, Ref. [4–6].

The chapter is organized as follows. First, we repeat some basics of solid-statephysics, which are necessary to understand the basics of the QHE. We then intro-duce the semiconductor structures, in which the QHE is observed. Finally, wediscuss the QHE itself and elaborate on its impact on metrology.

5.1Basic Physics of Three- and Two-Dimensional Semiconductors

We consider crystalline three-dimensional bulk semiconductors with sizes thatare macroscopic on the scale of the de Broglie wavelength or crystalline two-dimensional layered semiconductor structures. For the latter, it is assumed that


104 5 Quantum Hall Effect

the in-plane dimensions of the layers are macroscopic on the scale of the de Brogliewavelength. Thus, we assume that size quantization only occurs in the directionnormal to the layers. Except for this direction, the allowed values of the wave vec-tor 𝐤 are determined by the size of the semiconductor crystal and, hence, are quasicontinuous due to the macroscopic crystal size. The eigenenergies of the electrons(and holes) and their dispersion are represented by the band structure E(𝐤). Theelectrons and holes determine the transport and optical properties of the semi-conductor crystal. More precisely, these properties are governed by the uppermostoccupied band and the lowest unoccupied band, labeled valence band and conduc-tion band, respectively. These bands are separated by the band gap with energy Eg.The dispersion of these bands close to their extrema can often be approximated bythe free-electron relation but with the free-electron mass replaced by the effectivemass m∗. The renormalization of the mass accounts for the influence of the crystallattice in this approximation. In the following, we will only consider “quasi free”electrons, which can be described by the effective-mass approximation. Moreover,we assume that the band structure is isotropic. We start with three-dimensionalsemiconductors and then move on to two-dimensional ones. The effect of a mag-netic field is discussed for both cases.

5.1.1Three-Dimensional Semiconductors

The energy dispersion of quasi free electrons in a three-dimensional isotropicsemiconductor is given by

E(𝐤) = ℏ2|𝐤|22m∗ = ℏ

2

2m∗ (k2x + k2

y + k2z ). (5.1)

Equation (5.1) simply represents the kinetic energy of a free particle with massm∗. If in Hall measurements a magnetic field is applied along the z direction,B = Bz, the electrons will move along cyclotron orbits in the x–y plane providedthat scattering is not too strong. This condition can be expressed by the inequality𝜔C𝜏 ≫ 1, in which 𝜏 is the scattering time and

𝜔C = eBm∗ (5.2)

the (angular) cyclotron frequency. The magnetic field changes the energy disper-sion. In a quantum mechanical treatment, one obtains for a magnetic field alignedalong the z direction

E(kz) =ℏ

2k2z

2m∗ +(

lC + 12

)ℏ𝜔C ± 1

2g𝜇BB. (5.3)

In this equation, lC is an integer number (lC = 0, 1, 2, 3, … ), g the g-factor of theelectron, and 𝜇B = eℏ∕(2me) the Bohr magneton with me being the electron mass.The interpretation of Eq. (5.3) is straightforward. The first term represents thekinetic energy of the “free” motion in z direction. The second term is the energycorresponding to the cyclotron motion in the x–y plane. Quantum mechanically,

5.1 Basic Physics of Three- and Two-Dimensional Semiconductors 105

this motion corresponds to a harmonic oscillator, where lC is the respective quan-tum number. The different bands with quantum numbers lC = 0, 1, 2, 3, … arecalled Landau levels after the Russian physicist Lev Landau. The third term is theZeeman energy of the electron, which has spin components sz = ±1∕2ℏ orientedparallel or antiparallel to the magnetic field B = Bz. Since the Zeeman energy,g𝜇BB∕2, is much smaller than the separation of the Landau levels, ℏ𝜔C, we shallneglect it for the rest of this discussion.

An important quantity for the understanding of the QHE is the density of statesD(E). It specifies the number of states available to electrons within an energyinterval from E to E + dE (per volume in real space). Here, dE is an infinitesi-mal increase in energy. For the occupation of these states, the Pauli principle hasto be considered. From Eq. (5.1), one obtains for quasi free electrons in a three-dimensional semiconductor at zero magnetic field

D3D(E) = 12π2

(2m∗

ℏ2

)3∕2E1∕2. (5.4)

The square-root energy dependence of D3D(E) is illustrated in Figure 5.1.In a magnetic field, the density of states is given by [7]

D1D(E) =ℏ𝜔C(2π)2

(2m∗

ℏ2

)3∕2∑

lC

(E −

(lC + 1

2

)ℏ𝜔C

)−1∕2. (5.5)

Energy

(1,0) (1,1) (1,2) (1,3)

(2,0) (2,1)

0D

2D3D

1D

De

nsity o

f sta

tes

Figure 5.1 Schematic illustration of thedensity of states of three-, two-, one-, andzero-dimensional semiconductors. Theone- and zero-dimensional systems areobtained by applying a magnetic field tothree- and two-dimensional semiconductors,

respectively, and show Landau levels. Onlythe two lowest subbands are sketched forthe two- and one-dimensional case. The dis-crete Landau levels of the zero-dimensionalsemiconductor are labeled by the quantumnumbers (lz , lC).


As noted by the superscript, this is the density of states of a one-dimensionalsemiconductor with its characteristic one-over-square-root energy dependence.D1D(E) is also shown in Figure 5.1 together with D3D(E) and the densities ofstates of zero- and two-dimensional semiconductors, which are discussed inSection 5.1.2. According to Eq. (5.5), D1D(E) has a singularity at the bottom ofeach Landau level. However, in real systems, this singularity is washed out, forexample, due to level broadening that results from scattering. Notwithstandingthat this feature is not included in Eq. (5.5), the equation shows that D1D(E)increases with increasing magnetic field due to the factor ℏ𝜔C.

Concluding this section, we like to emphasize that a magnetic field transformsa three-dimensional semiconductor into a one-dimensional one. If the magneticfield is applied in z direction, the motion of quasi free electrons is quantized withregard to the wave vector components kx and ky and free only with regard to kz.The energy dispersion splits up into Landau levels. The number of electrons, whichcan occupy a Landau level, increases with increasing magnetic field strength.

5.1.2Two-Dimensional Semiconductors

Confinement of the motion of quasi free electrons in one or more dimensionsmodifies their wave function, dispersion, and density of states. As discussed inSection 5.1.1, confinement can be caused by a magnetic field. Confinement canalso be achieved geometrically by the generation of appropriate small structures.The length scale for the occurrence of this so-called size quantization is given bythe de Broglie wavelength of the electron. Confinement in one dimension createsa two-dimensional semiconductor. Let us assume that the confinement is due to arectangular potential well in z direction with infinite barrier height. In this simplecase, the energy dispersion of the electrons is given by

E(kx, ky) = EQW(lz) +ℏ

2

2m∗ (k2x + k2

y ) (5.6)

with the quantization energy

EQW(lz) =ℏ

2

2m∗

π2l2z

L2z

, (5.7)

where Lz is the width of the quantum well and lz the quantum number(lz = 1, 2, … ,∞). Thus, quantized energy levels are obtained resulting from theconfinement in z direction. The quantization energy increases with the square ofthe quantum number lz and with decreasing quantum well width according to1∕L2

z .The confinement changes the density of states from a square-root to a staircase

function

D2D(E) = m∗

πℏ2

∑

lz

Θ(E − Elz), (5.8)

5.1 Basic Physics of Three- and Two-Dimensional Semiconductors 107

where Θ(E) is the Heaviside function (Θ(E < 0) = 0 and Θ(E ≥ 0) = 1). The den-sity of states for the two-dimensional case is also shown in Figure 5.1. The figureillustrates that D2D(E) is finite at the band edge where D3D(E) is zero. This dif-ference has important consequences for many device applications, in particularfor optoelectronic devices [8]. Confinement of electrons by real potential wellswith finite barrier height does not affect the principal features of size quantiza-tion described earlier. Yet, it modifies some important aspects. For a finite barrierheight, the quantization energy for a given quantum state with quantum numberlz is lower and the number of bound quantum states is finite.

In order to move on to a semiconductor system that is capable of showing theQHE, we combine the confinement by a potential well with the effect of a magneticfield, which is discussed in Section 5.1.1. Again, it is assumed that the potentialwell confines the electron motion in the z direction and that the field is appliedalong the z axis, B = Bz. Thus, the field is applied normal to the semiconduc-tor layers used to form a quantum well as will be discussed in more detail inSection 5.2. Combining the previous results, it becomes immediately obvious thatthis arrangement creates a zero-dimensional semiconductor system, in which theelectron motion is confined in all three dimensions. The energy of the electrons isthen given by the sum of the quantization energy EQW(lz) and the energy corre-sponding to the cyclotron motion in the x–y plane. From Eqs (5.3) and (5.7), weobtain (again neglecting the Zeeman term)

E(lz, lC) =ℏ

2

2m∗

π2l2z

L2z

+(

lC + 12

)ℏ𝜔C. (5.9)

The energy does not depend on the wave vector anymore, that is, there is no dis-persion, since the electrons cannot move freely. The energy spectrum consists ofa series of discrete energies characterized by the quantum numbers (lz, lC) sim-ilar the spectrum of an atom. As a consequence, the density of states, D0D(E),shows energy gaps between adjacent Landau levels as illustrated in Figure 5.1. Thequantization energy EQW(lz) is usually larger than the cyclotron energy, which isaccounted for in Figure 5.1. Therefore, it is often sufficient to consider only theLandau levels of the lowest quantum well state with lz = 1. Henceforth, we willuse this description. The number of states per Landau level (and per area), whichcan be occupied by electrons, is given by

D0D = eh

B. (5.10)

Thus, the density of states increases linearly with increasing magnetic field. Asa consequence, the higher Landau levels are gradually depleted as the magneticfield is increased at a constant electron density. At sufficiently high fields, all elec-trons will occupy the lowest Landau level with lC = 0. The possibility to controlthe occupation of the Landau levels with the magnetic field is essential for thedescription of the QHE as will be discussed in Section 5.3.2.


5.2Two-Dimensional Electron Systems in Real Semiconductors

The QHE is observed in two-dimensional electron gases (2DEGs). There-fore, in this section, it will be discussed how a 2DEG can be realized in realsemiconductors. The section starts with an introduction to the properties ofreal semiconductors, like GaAs and AlGaAs, and their heterojunctions andheterostructures. Epitaxial growth techniques will also be treated briefly. TheGaAs/AlGaAs semiconductor system is chosen as example because the majorityof quantum Hall devices used as resistance standards in quantum metrologynowadays are based on this material system. The discussion will show howthe concept of a two-dimensional semiconductor can be made a physicalreality.

The QHE is an electron transport effect. Therefore, free conduction electronswith suitable transport properties must be inserted in a semiconductor het-erostructures by an appropriate doping technique to observe the QHE. A specialdoping technique known as modulation doping is used for this purpose. It resultsin electron gases with a high mobility of the charge carriers as required for theQHE. Modulation doping is discussed at the end of the section.

5.2.1Basic Properties of Semiconductor Heterostructures

Two-dimensional electron systems can be realized by the so-called semicon-ductor heterojunctions. These structures were first suggested by Kroemer [9].H . Kroemer together with Z. I. Alferov received a part of the Noble Prize inPhysics in 2000 for the development of this technology. A heterojunction is theinterface formed between two semiconductors with different band gap energy orbetween a semiconductor and a metal or insulator. A prominent example of asemiconductor/insulator heterojunction is the Si/SiO2 interface in MOSFETs. Aheterostructure is composed of one or more heterojunctions.

The most prominent example of a semiconductor/semiconductor heterojunc-tion is the GaAs/AlGaAs interface. GaAs is a compound III–V semiconductorformed of an element of the third (Ga) and fifth (As) group of the periodic tableof elements. Other prominent III–V semiconductors are, for example, InP, InAs,AlAs, and GaSb. GaAs is a direct-gap semiconductor. The maximum of its upper-most valence band and the minimum of its lowest conduction band are locatedat the center of the Brillouin zone (Γ point). The band gap energy amounts toEg = 1.42 eV at room temperature. In contrast, AlAs has an indirect band gap, thatis, the maximum of the uppermost valence band and the minimum of the lowestconduction band are located at different points of the Brillouin zone. The valenceband maximum is located at the Γ point, as in GaAs. The minimum of the conduc-tion band is found close to the boundary of the Brillouin zone in (1,0,0) direction,that is, close to the X point. At room temperature, the energy of the indirect gap is

5.2 Two-Dimensional Electron Systems in Real Semiconductors 109

Eg = 2.16 eV. Detailed information on the band structure and material parametersof GaAs and AlAs can be found, for example, in Ref. [10].

Besides the binary compounds GaAs and AlAs, a ternary mixed crystal AlGaAscan be grown. In the mixed crystal, the Ga and Al atoms are randomly distributedover the lattice sites of the group III elements. The fraction of the Ga and Alatoms of this ternary III–V compound semiconductor can be continuously var-ied. This property is reflected by the nomenclature AlxGa1−xAs with the aluminummole fraction x varying between 0 and 1. As the aluminum mole fraction x isincreased, the band gap energy varies between 1.42 eV (GaAs, x = 0) and 2.16 eV(AlAs, x = 1). For x < 0.4, AlxGa1−xAs has a direct band gap as GaAs. For largerx, an indirect band gap is obtained as in AlAs. The effective-mass approximationwith an isotropic effective mass m∗, as discussed in Section 5.1, is sufficient todescribe electrons close the Γ point in GaAs and direct-gap AlxGa1−xAs. Since theQHE is observed at low temperatures of a few kelvin and below, we further notethat the band gap energies increase as the temperature is lowered, for example, to1.52 eV in GaAs at 4 K. The principal features of the band structure do not changewith temperature.

With respect to the growth of heterojunctions, semiconductor technology and,in turn, quantum metrology benefit from the fortunate circumstance that GaAsand AlxGa1−xAs exhibit almost the same lattice constant for all values of x. Thisfeature allows the fabrication of GaAs/AlxGa1−xAs heterostructures with almostperfect single crystalline interfaces using epitaxial crystal growth techniques. Thehigh quality of such interfaces is illustrated in Figure 5.2, which shows a transmis-sion electron microscope (TEM) image of a GaAs/AlAs/GaAs heterostructure.The single crystalline structure is continued over the interfaces at which no crystaldefects are observed.

GaAs

GaAs

AIAs

5 nm

Figure 5.2 TEM image of a GaAs/AlAs/GaAs heterostructure. The individual dots representsingle molecular units of GaAs (top, bottom) and AlAs (center). (Courtesy of PTB.)


5.2.2Epitaxial Growth of Semiconductor Heterostructures

High-quality epitaxial crystal growth of GaAs/AlxGa1−xAs heterostructuresis achieved using molecular beam epitaxy (MBE) [11] or metalorganic vaporphase epitaxy (MOVPE), also known as metalorganic chemical vapor deposition(MOCVD) [12]. MBE is performed in an ultrahigh vacuum chamber with abase pressure below 10−10 Pa. This low pressure reflects the ultralow impurityconcentration in the growth chamber. A sketch of an MBE growth chamber isshown in Figure 5.3. Attached to the chamber are effusion cells, which containhigh-purity Ga, Al, and As in the solid state. The effusion cells are heated totemperatures on the order of 1000 ∘C so that the source materials evaporate. Thegaseous source materials are released to the vacuum chamber if shutters in frontof the effusion cells are opened. In the chamber, the source materials condense onthe substrate, where they react with each other. The substrate is often rotated toachieve spatially uniform crystal growth over a large area. The reaction betweenGa, Al, and As is controlled by the temperature of the substrate and the flowrate of the source materials, which can be adjusted by the temperature of theeffusion cells. For example, a substrate temperature above 600 ∘C is requiredfor the growth of a GaAs crystal with perfect stoichiometry. The growth rateis low, typically 1 μm h−1, which corresponds to the growth of one monolayerof GaAs (thickness 0.28 nm) per second. Thanks to the low growth rate, thethickness of the epitaxial layers can be precisely controlled with a resolution ofa single atomic layer. The growth of the layers can be monitored in situ usingreflection high-energy electron diffraction (RHEED). The Si effusion cell shownin Figure 5.3 can be used to add dopants to the GaAs or AlxGa1−xAs layers in acontrolled way. The ultrahigh vacuum of the growth chamber and the high purityof the source materials ensure that the concentration of unwanted impurities isvery low in MBE-grown GaAs/AlxGa1−xAs heterostructures.

In MOVPE, the metallic source materials, Ga and Al, are provided in the formof metalorganic compounds, such as trimethylgallium. The group-V elements aresupplied as hydrides, such as arsine (AsH3). As an alternative, less toxic group-V precursors were also developed and applied, for example, tertiarybutylarsinefor the growth of As compounds [13]. Using a carrier gas (e.g., hydrogen), themetalorganic compounds are transported to the MOVPE reaction chamber. In

As

Si

Al

Ga

Figure 5.3 Schematic representation of anMBE chamber for the epitaxial growth ofGaAs/AlxGa1−xAs heterostructures. The vac-uum pumps needed to achieve a base pres-sure below 10−10 Pa are not shown. (Courtesyof K. Pierz, PTB.)


the chamber, they chemically react with the group-V precursor at the surface ofthe substrate, on which the GaAs/AlxGa1−xAs heterostructure grows epitaxially.In contrast to MBE, MOVPE does not require ultrahigh vacuum, but is carriedout at pressures on the order of 104 Pa.

5.2.3Semiconductor Quantum Wells

Using MBE or MOVPE, a thin layer of GaAs can be sandwiched between twolayers of AlxGa1−xAs. Figure 5.4a depicts the band gap energy of this quantumwell heterostructure versus the growth direction z, that is, the direction normal tothe layers. The aluminum mole fraction is assumed to be x = 0.3, which results inEg = 1.8 eV for AlxGa1−xAs at room temperature. As mentioned earlier, the bandgap energy of GaAs is Eg = 1.42 eV at room temperature.

Most important for the electronic properties of the heterostructure is the bandalignment, which is shown in Figure 5.4b at the Γ point of the Brillouin zone. Theconduction band of GaAs is located below the AlxGa1−xAs conduction band, whilethe opposite order is found for the valence bands. This is referred to as type Ior straddling type band alignment and results in the formation of potential wellsin the conduction and valence band. The potential wells ideally have a rectan-gular shape corresponding to atomically smooth interfaces between GaAs andAlxGa1−xAs, as seen in the TEM image of Figure 5.2. The depth of the wells isgiven by the band edge discontinuities of the conduction band, ΔEC, and valenceband, ΔEV. At the Γ point, their sum must equal the difference between the band

Eg

Ec

Ev

AlGaAs GaAs AlGaAs

1.42 eV

1.42 eV

1.8 eV(x = 0.3)

1.8 eV(x = 0.3)

ΔEc

ΔEv

Growth direction z

(a)

(b)

Figure 5.4 Spatial variation of the band gapenergy Eg of an AlxGa1−xAs/GaAs/AlxGa1−xAs(x = 0.3) quantum well heterostructure (a)and spatial variation of the conduction bandand valence band (b) versus the growthdirection z. The values of Eg refer to roomtemperature. An exchange of charge carri-ers between the layers, which would result inspace charge regions and band bending, isnot considered in the figure.


gap energies of GaAs and AlxGa1−xAs, that is, ΔEg = ΔEC + ΔEV must hold. Howthe band gap difference is split between conduction and valence band depends onthe detailed electronic structure of the interface. For the GaAs/AlGaAs system,we roughly have ΔEC∕ΔEV = 3∕2.

Summarizing this discussion, we note that for sufficiently thin GaAs layers (typi-cally Lz < 100 nm), a rectangular quantum well with finite barrier height is formedin both the conduction and valence band, that is, for electrons and holes. Thus, atwo-dimensional semiconductor is realized, in which a two-dimensional electrongas can be introduced.

5.2.4Modulation Doping

The tremendous technological success of semiconductors rests to a large extenton the fact that the concentration of mobile carriers can be varied by orders ofmagnitude by doping. Doping refers to the replacement of atoms of the hostlattice by atoms with more (donors) or less electrons (acceptors). Doping isrequired to perform electron transport studies in wide-gap semiconductorsat low temperatures T since for kBT ≪ Eg, the density of intrinsic conductionelectrons is negligibly small. Under this condition, donors provide the requiredextra mobile electrons, for example, when studying the QHE. Yet, after theelectrons have been transferred from the donors to the conduction band, thepositively charged donors act as scattering centers. At low temperatures, scatter-ing at ionized donors is the limiting factor of the electron mobility 𝜇 = e𝜏∕m∗

(𝜏 scattering time), which is one of the key parameters for technological appli-cations of semiconductors and also for the QHE. Scattering can be stronglyreduced by modulation doping [14]. In modulation-doped structures, the donorsare spatially separated from the mobile electrons and, thus, scattering at ionizeddonors is greatly reduced. Electron mobilities exceeding 107 cm2 V−1s−1 havebeen achieved at low temperatures [15]. This value is extremely high comparedto, for example, the room temperature electron mobility of 8000 cm2 V−1s−1

in GaAs.The concept of modulation doping can be applied to a quantum well het-

erostructure by introducing Si donors in a thin layer in one of the AlGaAsbarriers. The doped layer must be separated from the GaAs well by an undopedAlGaAs spacer layer. At elevated temperatures, the donors are thermally excitedand their extra electrons are captured in the GaAs well. Thus, a 2DEG withhigh electron mobility forms in the quantum well. At low temperatures, themobile electrons remain in the quantum well and provide a suitable arena forstudies of the QHE. This consideration shows another advantage of modulationdoping, besides the largely increased electron mobility. In modulation-dopedheterostructures, the mobile electrons do not freeze out at low temperatures,in contrast to homogeneously doped semiconductors. Modulation doping hasfound wide application in high-frequency field-effect transistors called MOSFETsor high-electron-mobility transistors (HEMTs).


A 2DEG can also form at a single heterojunction if band bending is taken intoaccount. We consider the modulation-doped structure of Figure 5.5. From bottomto top, the structure consists of a GaAs substrate and buffer layer, the undopedAl0.3Ga0.7As spacer layer, and the Al0.3Ga0.7As[Si] doping layer. A thin GaAs caplayer is required to prevent oxidation of the AlGaAs in real structures. The lowerpart of the figure schematically shows the conduction and valence band profiletogether with the Fermi level EF, which separates unoccupied from occupied elec-tronic states. The Fermi level is constant in thermodynamic equilibrium as shownin the figure. The donors are ionized and the extra electrons of the donors havebeen transferred across the spacer layer to the GaAs, where they are attractedto the interface by the electric field of the ionized donors. The charge transferis accompanied by band bending so that a triangular potential well is formed atthe interface of the GaAs and the Al0.3Ga0.7As spacer layer. The lowest quantizedenergy level of the potential well is located below the Fermi level. Therefore, the

++

+

++

Buffer

Substrate

(a)

(b)

Ec

EF

EV

2DEG

Cap layer

Doping layer

Spacer layer

Semi-insul.GaAs

GaAs

Al0.3

Ga0.7

As

n-Al0.3

Ga0.7

As [Si] = 1 × 1018

cm−3

GaAs

Doping layer

Spacerlayer

Si+

Figure 5.5 Modulation-dopedAl0.3Ga0.7As/GaAs heterostructure. (a) Layersequence showing the GaAs substrateand buffer layer, the undoped Al0.3Ga0.7Asspacer (thickness on the order of 10 nm),the Al0.3Ga0.7As[Si] doping layer (typicalthickness 50 nm, doped with Si donorsat a typical concentration of 1018 cm−3),and the typically 10 nm thick GaAs caplayer. (b) Schematic band profile. In the

conduction band, a triangular potential wellis formed at the interface of the GaAs andthe Al0.3Ga0.7As spacer layer. EF is the Fermilevel. Shown in light gray in the band profileis the lowest quantized energy state of thetriangular potential well. It holds a 2DEG asshown (in light gray) in the layer sequence.(Courtesy of K. Pierz, PTB.) (Please find acolor version of this figure on the color platesection.)


quantized energy level is populated with electrons. As a result, a 2DEG is formedat the interface of the GaAs and the Al0.3Ga0.7As spacer layer. Heterostructuresof the type shown in Figure 5.5 can be grown more easily than rectangular quan-tum wells by MBE or MOVPE. Therefore, they are used in the majority of today’squantum Hall resistors.

5.3The Hall Effect

The understanding of the QHE requires a basic knowledge of the classical Halleffect. Therefore, we give a brief description of the latter before we considerthe QHE.

5.3.1The Classical Hall Effect

5.3.1.1 The Classical Hall Effect in Three Dimensions

The classical Hall effect was discovered by Edwin Herbert Hall in 1879. It refers tothe generation of a voltage in a current-carrying wire placed in an external mag-netic field. As shown in Figure 5.6, the voltage drop occurs perpendicularly to thedirections of the electrical current and the magnetic field.

The Hall effect is the consequence of the Lorentz force acting on moving chargecarriers in a magnetic field. Assuming that only electrons contribute to the cur-rent, the Hall effect results in the accumulation of electrons at the top surface ofthe conductor of Figure 5.6. The charge accumulation gives rise to an electric fieldEy in y direction. Since the current is zero in y direction, the Lorentz force mustbe balanced by the effect of the electric field Ey in the steady state. We obtain forthe total force Fy in y direction

Fy = (−e)Ey − (−e)vxBz = 0, (5.11)

where (−e) is the charge of the electron and vx = ℏkx∕m∗ the electron velocity.Thus, the Hall field is given by Ey = vxBz and the Hall voltage by UH = EyLy =vxBzLy. If the current is expressed as

Ix = jxLyLz = (−e)n3DvxLyLz, (5.12)

x

y

z

IxLy

Lz

B = Bz UH

Figure 5.6 Experimental arrangement forthe observation of the Hall effect in a three-dimensional conductor.

5.3 The Hall Effect 115

where jx is the current density and n3D the electron density in a three-dimensionalconductor, the Hall voltage can be written as

UH = − 1en3D

1Lz

IxBz = RH1Lz

IxBz. (5.13)

The Hall coefficient, defined as RH = −1∕(en3D), is a measure of the carrier density.In fact, the Hall effect is routinely used to determine the carrier density in metalsand semiconductors. For the latter, a more general description can be worked out,which considers electron and hole currents. For the present discussion, the mostimportant quantity is the Hall resistance Rxy defined as

Rxy =UHIx

. (5.14)

The Hall resistance is to be distinguished from the longitudinal resistance Rxx =Ux∕Ix, where Ux is the voltage drop in the direction of the current.

5.3.1.2 The Classical Hall Effect in Two DimensionsThe description of the Hall effect in three dimensions, as given in Section 5.3.1.1,can be extended to a two-dimensional electron gas in a straightforward way. Weconsider a 2DEG in the x–y plane normal to the magnetic field B = Bz. The dimen-sion Lz is then meaningless and the three-dimensional electron density n3D is tobe replaced by a two-dimensional density n2D (number of electrons per area). Intwo dimensions, the Hall voltage becomes

UH = − 1en2D

IxBz. (5.15)

The Hall resistance can be expressed as in Eq. (5.14) with UH taken fromEq. (5.15). The longitudinal resistance Rxx = Ux∕Ix is defined as in the three-dimensional case.

In the literature, often the resistivity 𝜌 is considered rather than the resistanceR since the resistivity characterizes the physical properties of a material or elec-tronic system independent of its size. We note that in two-dimensional space,Rxx = 𝜌xxLx∕Ly. As a consequence, the longitudinal resistivity and resistance havethe same physical dimension. Moreover, the Hall resistance and the Hall resistivityare equal and independent of the size of the two-dimensional conductor:

Rxy = 𝜌xy = − 1en2D

B. (5.16)

The occurrence of components 𝜌xx and 𝜌xy shows that the resistivity 𝛒 is a tensor,which is defined by the relation

(ExEy

)

=(

𝜌xx 𝜌xy−𝜌xy 𝜌xx

)(jxjy

)

. (5.17)

The inverse of the resistivity tensor 𝛒 is the conductivity tensor 𝛔, defined by(

jxjy

)

=(

𝜎xx 𝜎xy−𝜎xy 𝜎xx

)(ExEy

)

. (5.18)


The components of 𝛒 and 𝛔 are related. We explicitly state some of the relationshere since they have implications in the QHE regime:

𝜌xx =𝜎xx

𝜎2xx + 𝜎

2xy

(5.19)

𝜎xx =𝜌xx

𝜌2xx + 𝜌

2xy

(5.20)

5.3.2Physics of the Quantum Hall Effect

For the description of the QHE, we build on the results of Section 5.1, againrestricting the discussion to the Landau levels of the lowest quantum wellstate and neglecting Zeeman splitting. In Section 5.1.2, we have seen that thenumber of electron states per Landau level (and per area) depends linearly on themagnetic field in two-dimensional semiconductors. Let us now consider a 2DEGwith a given electron density n2D at zero temperature. Thus, thermal excitationsbetween different Landaus levels are suppressed. Changing the magnetic field,the density of states D0D = eB∕h can be adjusted such that the Landau levels withlC = 0, 1, … , (i − 1) are completely filled with electrons, while all other Landaulevels (with lC > i − 1) are empty. The filling factor f defined as f = n2D∕D0D isthen given by f = i, that is, by an integer number. An equivalent statement is tosay that the Fermi level is located in the energy gap between the Landau levelswith lC = i − 1 and lC = i. Moreover, the electron density is given by n2D = ieB∕h.If this expression is inserted in Eq. (5.16), we obtain for the absolute value of theHall resistance

Rxy(i) =he2

1i

. (5.21)

For an integer filling factor, the Hall resistance only depends on fundamental con-stants and an integer number.

Moreover, the longitudinal resistance Rxx vanishes, Rxx = 0, since in a com-pletely filled Landau level scattering of electrons is suppressed due to the lack ofempty final states. As a consequence, the longitudinal resistivity is zero, 𝜌xx = 0.Equations (5.19) and (5.20) imply that the longitudinal conductivity is also zero,𝜎xx = 0. Thus, if the Fermi level is located in the energy gap between Landau levels,the current is driven by the Hall voltage.

The simple argument presented so far predicts the particular resistance valuesRxx = 0 and Rxy(i) = h∕(ie2) for singular values of the magnetic field, that is, if (andonly if ) the field corresponds exactly to an integer filling factor. Surprisingly, how-ever, one observes resistance values Rxx = 0 and Rxy(i) = h∕(ie2) over extendedfield ranges around integer filling factors. This finding is illustrated by the experi-mental data of Figure 5.7, obtained from a GaAs/AlGaAs heterostructure at 0.1 K.The Hall resistance shows pronounced plateaus Rxy(i) = h∕(ie2) and Rxx disap-pears over the corresponding field ranges. This experimental result is referred toas the QHE, discovered by K. v. Klitzing when studying 2DEGs in Si MOSFETs in

5.3 The Hall Effect 117

0

0.0

3.0

6.0

9.0

12.0

Rxy (

kΩ

) Rxx (kΩ

)

15.0

2 4

i = 4

i = 3

i = 2

6

B (T)

8 10 12

0.0

0.2

0.4

0.6

0.8

Figure 5.7 Experimentally determined Hallresistance Rxy (left scale) and longitudinalresistance Rxx (right scale) as a function ofmagnetic flux density of a GaAs/AlGaAs

heterostructure at a temperature T = 0.1 K(measurement current 1 μA). Some integerfilling factors are indicated. (Courtesy of F.Ahlers, PTB.)

1980 [1]. The ratio h∕e2 is named von Klitzing constant, RK = h∕e2. Consequently,the quantized Hall resistance can be expressed as

Rxy(i) =RKi

. (5.22)

In order to emphasize the close relation between the QHE and the paradigm ofquantum metrology, that is, the counting of discrete quanta, the filling factor canbe rewritten. We assume that A is the area of the sample and Φ the magnetic fluxthrough it and introduce the flux quantum Φ0 = h∕e (charge e since single elec-trons are considered rather than Cooper pairs with charge 2e as in Chapter 4). Wecan then write

f =n2D

D0D =An2D

AB eh

=NeΦΦ0

=NeNΦ

(5.23)

with Ne and NΦ being the number of electrons and flux quanta in the sample,respectively. Equation (5.23) shows that the filling factor can be interpreted as theratio between the number of electrons and the number of flux quanta.

Considerable theoretical work has been spurred by the experimental result thatthe Hall resistance is quantized over an extended range of the magnetic field orthe filling factor. Theory has been guided by the experimental observation thatthe width of the Hall resistance plateaus depends on the specific properties of theindividual sample. More precisely, the width of the plateaus is found to shrinkfor 2DEGs with very high electron mobility (106 cm2 V−1s−1 and above). At lowtemperatures, when phonon scattering is strongly reduced, the electron mobilityis a measure of disorder-induced scattering. Therefore, the experimental resultsindicate that disorder should be included in the description of the QHE.


Disorder arises from nonideal heterojunctions and from residual impurities andis an intrinsic property of ternary mixed crystals like AlxGa1−xAs. Thus, disorderis present in any real semiconductor mixed crystal heterostructure, where it givesrise to a spatially varying potential. As a consequence, the Landau levels are inho-mogeneously broadened and they are better described as Landau bands [16]. Theelectronic states are localized close to the upper and lower energy boundary of theinhomogeneously broadened Landau bands, as schematically shown in Figure 5.8.Electrons in these localized states are immobile and do not contribute to electronictransport. Only in the center of the Landau bands extended states are found. Elec-trons in extended states can carry current in the usual way. As the magnetic field(or more generally the filling factor) is changed, the Fermi level moves throughthe Landau bands. Yet, as long as the Fermi level moves through localized states,the density of mobile electrons does not change. Since only the mobile electronscontribute to the current, the Hall resistance does not change either and a plateauis observed. Thus, a model involving disorder can qualitatively explain the basicfeature of the QHE.

Despite the progress achieved including disorder in the theoretical descriptionof the QHE, this approach cannot describe all aspects of the QHE. In fact, eventoday, there exists no complete theoretical description of the QHE in real samples.Such a theory should include the effects of finite sample size, finite temperature,and the contacts to the 2DEG.

The finite size of a real semiconductor 2DEG is taken into account in the edgechannel model of the QHE developed by Büttiker [17], which we will briefly outlinein the following. A more detailed summary can be found, for example, in Ref. [18].

The edge channel model considers that the electron density of a 2DEG drops tozero and the Landau levels bend upward at the boundaries of a sample of finitesize. As a consequence, Landau levels, which are completely filled in the interiorof the 2DEG, pierce the Fermi level at points close to the sample edges. At thesepoints, these Landau levels are partially occupied. Therefore, one-dimensionalconducting channels are generated close to the sample edges, one for each pop-ulated Landau level. The classical analog of these edge channels is the skippingorbits of electrons moving in a magnetic field along a boundary. Electronic trans-port in edge channels can be described by the Landauer–Büttiker formalism fortransport in one-dimensional conductors [19–21]. In this approach, the currentis considered as the driving force and the resulting electric field distribution iscalculated. The current is described by transmission and reflection coefficientsand the chemical potential difference over the one-dimensional conductor. If this

E

D(E)

Exte

nded

Exte

nded

Figure 5.8 Schematic representation ofthe density of states of inhomogeneouslybroadened Landau bands. Dotted: localizedstates close to the lower and upper bound-ary of the bands.

5.4 Quantum Hall Resistance Standards 119

approach is applied to a perfect one-dimensional conductor at zero magnetic field,in which no scattering occurs (i.e., ballistic transport), the inverse of the resistance(i.e., the conductance) is found to be quantized in units of e2∕h [22]. For the QHE,edge channels with opposite direction of the current have to be considered, whichare located at opposite edges of the 2DEG. The edge channel model of the QHEthen shows that the Hall resistance is quantized, Rxy(i) = h∕(ie2), if backscatteringof electrons between edge channels of opposite direction is negligible.

The shortcoming of the edge channel model is that it assumes the current to flowonly close to the boundaries of a QHE sample. This assumption contradicts exper-imental observations [6]. Therefore, more sophisticated models of the QHE havebeen developed [6, 23, 24], which are, however, beyond the scope of this introduc-tory text.

Still more theoretical work is needed to develop a complete description of alldetails of the QHE. Nonetheless, the highly accurate quantization of the Hallresistance and its tremendous impact on quantum metrology are undisputed.The metrological impact of the QHE will be the subject of Section 5.4.

At the end of this section, we like to emphasize that we have discussed theinteger QHE, which has to be distinguished from the fractional quantum Halleffect (FQHE). The FQHE occurs in 2DEGs with electron mobilities well above106 cm2 V−1s−1 at very high magnetic fields above 10 T and mK temperatures.Under these conditions, plateaus of the Hall resistance Rxy(f ) = h∕(f e2) areobserved at fractional values of the filling factor f (such as 1/3, 2/3, 2/5, 3/7)[25]. The FQHE is the signature of a new quantum state generated by many-bodyinteraction as first pointed out by Laughlin [26]. In 1998, D. C. Tsui, H. L. Störmer,and R. B. Laughlin received the Noble Prize in Physics for the discovery ofthe FQHE.

5.4Quantum Hall Resistance Standards

The QHE has made tremendous impact on resistance metrology notwithstandingthat a complete theoretical description of the QHE in real semiconductor sampleshas not yet been presented. In this section, we first review the impact of the QHEon measurements of DC resistance. Today, quantum Hall resistance standards areroutinely used by national metrology institutes to reproduce and disseminate theDC ohm. This effort has led to the adoption of a conventional ohm scale. The con-cept is similar to the one of the conventional, Josephson-based volt scale, which istreated in Section 4.1.4.2.

In recent years, AC measurements have also benefitted from the QHE. It hasbeen shown that reproducible quantized impedance values can be obtained if Hallvoltages are measured on specially designed QHE resistors to which an AC cur-rent is applied [3]. The impedances can be directly compared to capacitances sothat a QHE-based representation of the farad is obtained [2]. Quantum Hall mea-surements in the AC regime will be discussed at the end of Section 5.4.


5.4.1DC Quantum Hall Resistance Standards

5.4.1.1 Comparison Between Classical and Quantum-Based Resistance Metrology

The benchmark for the performance of quantized Hall resistors is set by classicalresistance metrology, which is represented schematically in Figure 5.9. The start-ing point for the realization of the SI ohm is a calculable capacitor [27]. Such acapacitor allows an SI value of the capacitance C to be realized traceable to themeter. From the SI farad, SI values of the capacitive reactance, (𝜔C)−1, can bederived. The AC resistance of artifact resistance standards is then linked to thecapacitive reactance with a so-called quadrature bridge at kilohertz frequencies.The artifact resistance standards must have a calculable AC/DC difference [28, 29]so that their DC resistance can be derived to finally obtain a realization of the SIohm. With this approach, the SI ohm can be realized with a relative uncertaintyof a few parts in 108 [30, 31]. Having realized the SI ohm, resistors with unknownvalues can be calibrated against SI resistance standards.

The maintenance and dissemination of the SI ohm with artifact standards suffersfrom their sensitivity to the environmental conditions, such as temperature andpressure. As a result, it is difficult to maintain a resistance scale with a temporaldrift of less than 10−7 per year with artifact standards only [32].

As compared to these benchmark values, the QHE has substantially improvedresistance metrology. Various measurements have established that the QHE is

unknownresistor (DC)

SI ohm (DC)

calculableAC/DC difference

SI ohm (AC)

quadrature bridge

SI farad

calculable capacitor

Figure 5.9 Calibration of an unknown resistor against a cal-culable capacitor, thereby realizing the SI ohm. Not shownin this schematic representation are the various measuringbridges required to upscale or downscale capacitances andresistances.


highly reproducible. Quantized Hall resistances have been found to agree witheach other within an uncertainty of a few parts in 1010 and better independentof the particular sample properties or type of 2DEG if the guidelines for QHEmetrology are followed [33]. For example, QHE measurements of GaAs/AlGaAsheterostructures were found to agree to measurements of Si MOSFETs withinan uncertainty of 3.5 parts in 1010 [34]. More recently, QHE measurements ofGaAs/AlGaAs heterostructures and graphene were found to agree within anuncertainty of 9 parts in 1011 [35, 36]. This result is especially noteworthy sincegraphene is a particular material. Graphene consists of a monolayer of carbonatoms arranged on a hexagonal lattice. Its electronic properties are quite differentfrom those of GaAs/AlGaAs heterostructures or Si MOSFETs, giving rise to ahalf-integer QHE [37–39].

Observation of the guidelines for reliable DC QHE measurements [33] ensuresthat the sample properties and experimental conditions are sufficiently close to theidealized case of the QHE treated in Section 5.3.2. The assumptions made for theidealized case include zero temperature, negligible influence of the contacts tothe 2DEG, and negligible effects due to the finite measuring current. If the guide-lines are observed, the reproducibility of the QHE is better than the uncertaintiesachievable in classical resistance metrology. Therefore, the QHE provides a practi-cal quantum standard of electrical resistance [40]. The QHE links the macroscopicelectrical quantity resistance to the elementary charge and the Planck constant.Therefore, the QHE resistance is independent of space and time as long as thefundamental constants do not vary in time (see the discussion in Section 3.3.4).With respect to the universality of the QHE, we note that theory does not pre-dict the QHE to depend on the gravitational field [41]. Corrections from quantumelectrodynamics are predicted at the level of 1 part in 1020 only [42].

5.4.1.2 The Conventional OhmAlready at the end of the 1980s, it was obvious that the high reproducibility ofthe QHE could harmonize resistance measurements and ensure their worldwidecomparability. To this end, a fixed value of the von Klitzing constant RK had to beagreed upon. In 1987, the General Conference of the Meter Convention instructedthe International Committee for Weights and Measures (CIPM) to recommend avalue of the von Klitzing constant, which should be used for the determination ofthe electrical resistance from QHE measurements [43]. In 1988, the CIPM recom-mended a value that was determined using the best experimental data available atthat time and should be used from 1 January 1990 [44]. This conventional value oragreed-upon value of RK is denoted by RK-90. RK-90 was introduced together withthe conventional value of the Josephson constant KJ-90 (see Section 4.1.4.2) and isgiven by

RK-90 = 25 812.807 Ω. (5.24)

In order to ensure the compatibility of RK-90 and the SI value of RK, RK-90 isassigned a conventional relative uncertainty. The uncertainty was 2 parts in 107

at the time RK-90 was introduced and reduced to 1 part in 107 later. This value is


still a conservative uncertainty margin since the relative deviation between RKand RK-90 is only 1.7 parts in 108 according to the adjustment of the fundamentalconstants in 2010 [45].

In close analogy to the Josephson case of Section 4.1.4.2, we like to emphasizethat the relation

R90 =RK−90

i(5.25)

establishes a new, highly reproducible resistance scale R90. In Eq. (5.25), RK-90 canbe treated as a constant with zero uncertainty since no comparison to SI quan-tities is made. Equation (5.25) provides a representation of the unit ohm, yet nota realization of the ohm according to the definition of the SI. Nonetheless, since1990, the unit ohm, more precisely “ohm90,” is represented, maintained, and dis-seminated using RK-90, thereby taking advantage of the superior reproducibility ofthe QHE. Since then, on-site resistance comparisons between the Bureau Interna-tional des Poids et Mesures (BIPM) and national metrology institutes have shownprimary quantum Hall resistance standards to agree within an uncertainty of a fewparts in 109 [46].

In the new SI, fixed numerical values with zero uncertainty will be assigned tothe elementary charge and the Planck constant. The redefinition will affect thevon Klitzing constant RK in an analogous way as the Josephson constant KJ. Inthe new SI, RK will have a fixed value with zero uncertainty and the QHE can beused to realize the SI ohm. The agreed-upon constant RK-90 will be abrogated.The consequences for resistance measurements are similar to the consequencesfor voltage measurements, which are discussed in Section 4.1.4.2.

5.4.1.3 Technology of DC Quantum Hall Resistance Standards and Resistance ScalingNowadays, GaAs/AlGaAs heterostructures of the type shown in Figure 5.5 aremostly used as quantum Hall resistance standards. The heterostructures arepatterned into Hall bars with typical sizes of several hundred micrometers.Figure 5.10 shows a schematic representation of a Hall bar and Figure 5.11 aphotograph of a quantum Hall resistance standard. The 2DEG of such a quantumHall standard has a typical electron mobility 𝜇 = 5•105 cm2 V−1 s−1 and carrierdensity n2D = 5•1011 cm−2 [46]. The latter corresponds to a magnetic field of10 T for the observation of the i = 2 plateau. Measurements are performed attemperatures around 1 K in liquid helium cryostats.

For metrological purposes, the plateaus with i = 2 and 4 are mostly used.The use of odd filling factors is disadvantageous since, in real structures, the

B

Ux

Ix UH

Ix

Figure 5.10 Schematic representation of atypical Hall bar with two current contactsand six voltage contacts (three on eitherside).


Figure 5.11 Photograph of a GaAs/AlGaAs quantum Hall resistance standard showing twoHall bars mounted in a chip carrier. (Courtesy of PTB.) (Please find a color version of thisfigure on the color plate section.)

uppermost filled Landau level is then separated from the lowest empty Landaulevel only by the small Zeeman energy splitting. The i = 2 and 4 plateausproduce resistance values RK-90∕2 and RK-90∕4 of approximately 12.906 and6.453 kΩ, respectively. Starting from these values, potentiometric methods andcurrent comparator bridges are used to build up the resistance scale, whichextends from milliohms to teraohms [40]. Among the different comparators, thecryogenic current comparator (CCC) [47] is the most accurate instrument. Itallows resistance scaling to be performed with relative uncertainties of 10−9 andbetter (see Section 4.2.3.3). CCCs are used to compare quantum Hall resistancestandards at liquid helium temperatures to secondary resistance standards withdecade values, such as 100Ω, at room temperature. This measurement is the firststep in building up a practical decade resistance scale at room temperature.

The need for resistance values that cover a broad range raises the questionwhether QHE bars, such as the one shown in Figure 5.10, can be connected inseries or in parallel. In principle, a series circuit of m QHE bars should produceaccurate quantized resistance values mRK-90∕i. Likewise, small resistancesRK-90∕(mi) should be reproducible with a parallel circuit.

Regarding this approach, one needs to recall that the quantized Hall resistanceRxy(i) = RK-90∕i is the result of a four-terminal measurement (as is the longitudi-nal resistance Rxx = 0). As shown in Figure 5.10, two separate pairs of contacts areused for current and voltage measurements. Therefore, the resistances of contacts


to the 2DEG do not contribute to the measurement result. In contrast, the con-tact resistances and the resistances of connecting wires affect the measurementif QHE bars are connected in series or parallel circuits. To alleviate this prob-lem, the so-called multiple connection technique was proposed. It reduces thecontribution of contact resistances Rc to approximately (Rc∕Rxy)n, where n − 1 isthe number of additional connections [48]. Since contact resistances have typicalvalues below 1Ω, their effect can be reduced to a negligible level using multipleconnections.

Series and parallel arrays of quantum Hall resistors were fabricated as inte-grated circuits with nominal resistance values from RK-90∕200 to 50 RK-90 [49, 50].Agreement between nominal and measured values was demonstrated within anuncertainty of a few parts in 109 for some arrays [49]. Whether arrays of quan-tum Hall resistors will substantially impact resistance metrology will be seen inthe future.

5.4.1.4 Relation Between the von Klitzing Constant and the Fine-Structure Constant

The von Klitzing constant RK = h∕e2 can be expressed by the fine-structure con-stant α, which is the dimensionless scaling factor of the strength of the electro-magnetic interaction:

RK = he2 =

𝜇0c2α

. (5.26)

In the present SI, the magnetic field constant, 𝜇0, and the velocity of light in vac-uum, c, are constants with zero uncertainty (see Chapter 2). Therefore, an exper-imental determination of the fine-structure constant determines the von Klitzingconstant with the same relative uncertainty and vice versa. The fine-structure con-stant can be derived very precisely from atomic physics measurements, such asthe measurement of the anomalous magnetic moment of the electron. Thanksto atomic physics data, α and RK are known with a relative uncertainty of only3.2 parts in 1010 according to the adjustment of the fundamental constants in2010 [45].

Based on the relation RK = iRxy(i), the QHE provides an alternative method forthe determination of the fine-structure constant. QHE measurements do not relyon quantum electrodynamical calculation, in contrast to the method using theanomalous magnetic moment. Therefore, QHE measurements can serve as anindependent test. In order to derive α from QHE data, the SI value of the quantumHall resistance Rxy(i) must be determined. This measurement can be performedwith the calibration chain of Figure 5.9 with the unknown resistor being a quan-tum Hall resistance standard. Thereby, the SI value of Rxy(i) is traced back to theSI farad. The measurement can be performed with a relative uncertainty of a fewparts in 108 [30, 31]. At this level of uncertainty, agreement is found betweenthe values of the fine-structure constant obtained from atomic physics and theQHE [45].


5.4.2AC Quantum Hall Resistance Standards

In Section 5.4.1, we have assumed that a DC current is applied to a quantum Halldevice. In this section, we assume that an AC current is driven through a Hall barin the QHE regime. The corresponding physics is referred to as the AC quantumHall effect. As a motivation for the use of the AC QHE in metrology, let us considerFigure 5.12, which illustrates how capacitance measurements can be based on theQHE. Reversing the calibration procedure of Figure 5.9, the starting point is a DCquantum Hall resistance standard to represent the ohm. Subsequently, a DC cal-ibration of an artifact resistance standard with calculable AC/DC difference [28,29] is performed. The AC resistance of the artifact standard is derived from theknown AC/DC difference. Using a quadrature bridge, a representation of the faradcan then be obtained so that unknown capacitors can be calibrated against theQHE. Thus, capacitance measurements and, in turn, inductance measurementscan benefit from the high reproducibility of the QHE. Thereby, the QHE is har-nessed for impedance metrology in general. This important calibration procedureis substantially simplified if an AC quantum Hall resistance standard is available,

unknowncapacitor

farad

quadrature bridge

ohm (AC) AC QHE resistor

calculableAC/DC difference

ohm (DC)

DC QHE resistor

Figure 5.12 Calibration of an unknowncapacitor against either a DC quantum Hallresistance standard (dashed box) or againstan AC quantum Hall resistance standard. Not

shown in this schematic representation arethe various measuring bridges required toupscale or downscale capacitances and resis-tances.


as shown in Figure 5.12. In particular, the measurement no longer relies on artifactresistance standards if the AC QHE is utilized.

QHE measurements in the AC regime are to be performed at kilohertz frequen-cies to be compatible with impedance measuring techniques. These AC QHE mea-surements have caused difficulties for many years. In general, the QHE plateauswere not as flat as in DC measurements and, in addition, exhibited an unwantedfrequency and current dependence [51]. These findings were attributed to capac-itive losses in the Hall bar and between the Hall bar and its surroundings [51, 52].

A special double-shielding technique was proposed to solve the problem of thecapacitive loss currents [52]. The basic idea of this technique is to ensure that thecurrent that reaches the current-low terminal of the QHE resistor is exactly equalto the current that generates the Hall voltage. If this condition is met, the Hallresistance is determined properly.

A doubly shielded QHE resistor is illustrated in Figure 5.13. The Hall bar is sur-rounded by two metallic shields separated by a narrow gap. The gap is alignedalong the line of the Hall voltage measurement. The right-hand shield is connectedto the current-low terminal. This connection ensures that the capacitive currentICL, which has generated Hall voltage, reaches the current-low terminal. Next, weconsider the capacitive current ICH, which does not generate Hall voltage since itdoes not cross the Hall voltage line. This current is fed back to its origin by theleft-hand shield and does not reach the current-low terminal, as required.

The double-shielding technique has caused a breakthrough in AC QHE mea-surements. It ensures correct measurements of the Hall resistance, undistorted byAC losses. Flat QHE plateaus are observed using doubly shielded QHE resistors.The residual frequency dependence of the quantized AC Hall resistance is only1.3 parts in 109 kHz−1 in the kilohertz frequency range [3]. Thus, the AC quantumHall resistance standard is as reproducible and reliable as its DC counterpart. InRef. [2], it has been shown that the farad can be traced to the AC QHE with anuncertainty of 6 parts in 109. This result shows that the AC QHE can substantiallyimpact impedance metrology, very much as the DC QHE has impacted resistancemetrology.

To origin

ICH

ICH ICL

ICL

I

UHall

Figure 5.13 Schematic top view of a doubly shielded AC quantum Hall resistance standard.The shields (light gray) are shown as being transparent. The subscripts CL and CH stand forcurrent-low terminal and current-high terminal, respectively.

References 127

References

1. von Klitzing, K., Dorda, G., andPepper, M. (1980) New method forhigh-accuracy determination of thefine-structure constant based on quan-tized hall Resistance. Phys. Rev. Lett., 45,494–497.

2. Schurr, J., Bürkel, V., and Kibble, B.P.(2009) Realizing the farad from two acquantum Hall resistances. Metrologia,46, 619–628.

3. Schurr, J., Kucera, J., Pierz, K., andKibble, B.P. (2011) The quantum Hallimpedance standard. Metrologia, 48,47–57.

4. Prange, R.E. and Girvin, S.M. (eds)(1990) The Quantum Hall Effect,Springer, New York.

5. Janssen, M., Viehweger, O., Fastenrath,U., and Hajdu, J. (eds) (1994) Intro-duction to the Theory of the IntegerQuantum Hall Effect, Wiley-VCH VerlagGmbH, Weinheim.

6. Weis, J. and von Klitzing, K. (2011)Metrology and microscopic picture ofthe integer quantum Hall effect. Phi-los. Trans. R. Soc. London, Ser. A, 369,3954–3974.

7. Madelung, O. (1981) Introduction toSolid State Theory, Springer, Heidelberg,Berlin, New York.

8. Weisbuch, C. and Vinter, B. (eds) (1991)Quantum Semiconductor Structures:Fundamentals and Applications, Aca-demic Press, San Diego, CA.

9. Kroemer, H. (1963) A proposed class ofhetero-junction injection lasers. Proc.IEEE, 51, 1782–1783.

10. Madelung, O. (ed) (1987) Landolt-Börnstein Numerical Data and Func-tional Relationships in Science andTechnology, Group III, vol. 22, Springer,Berlin.

11. Cho, A.Y. and Arthur, J.R. Jr., (1975)Molecular beam epitaxy. Prog. SolidState Chem., 10, 157–191.

12. Stringfellow, G.B. (1999) Organometal-lic Vapor-Phase Epitaxy: Theory andPractice, 2nd edn, Academic Press, SanDiego, CA, London.

13. Stolz, W. (2000) Alternative N-, P- andAs-precursors for III/V-epitaxy. J. Cryst.Growth, 209, 272–278.

14. Dingle, R., Störmer, H.L., Gossard, A.C.,and Wiegmann, W. (1978) Electronmobilities in modulation-doped semi-conductor heterojunction superlattices.Appl. Phys. Lett., 33, 665–667.

15. Umansky, V., de-Picciotto, R., andHeiblum, M. (1997) Extremely high-mobility two dimensional electron gas:evaluation of scattering mechanisms.Appl. Phys. Lett., 71, 683–685.

16. Prange, R.E. (1981) Quantized Hallresistance and the measurement of thefine-structure constant. Phys. Rev. B, 23,4802–4805.

17. Büttiker, M. (1988) Absence of backscat-tering in the quantum Hall effect inmultiprobe conductors. Phys. Rev. B, 38,9375–9389.

18. Haug, R.J. (1993) Edge-state transportand its experimental consequences inhigh magnetic fields. Semicond. Sci.Technol., 8, 131–153.

19. Landauer, R. (1957) Spatial variationof currents and fields due to localizedscatterers in metallic conduction. IBM J.Res. Dev., 1, 223–231.

20. Landauer, R. (1970) Electrical resistanceof disordered one-dimensional lattices.Philos. Mag., 21, 863–867.

21. Büttiker, M. (1986) Four-terminal phase-coherent conductance. Phys. Rev. Lett.,57, 1761–1764.

22. van Wees, B.J., van Houten, H.,Beenakker, C.W.J., Williamson, J.G.,Kouwenhoven, L.P., van der Marel,D., and Foxon, C.T. (1988) Quantizedconductance of point contacts in a two-dimensional electron gas. Phys. Rev.Lett., 60, 848–850.

23. Wei, Y.Y., Weis, J., von Klitzing, K.,and Eberl, K. (1998) Edge strips inthe quantum Hall regime imaged by asingle-electron transistor. Phys. Rev. Lett.,81, 1674–1677.

24. Siddiki, A. and Gerhardts, R.P. (2004)Incompressible strips in dissipative Hallbars as origin of quantized Hall plateaus.Phys. Rev. B, 70, 195335 (12 pp).

25. Tsui, D.C., Störmer, H.L., and Gossard,A.C. (1982) Two-dimensional magneto-transport in the extreme quantum limit.Phys. Rev. Lett., 48, 1559–1562.


26. Laughlin, R.B. (1983) Anomalousquantum Hall effect: an incompress-ible quantum fluid with fractionallycharged excitations. Phys. Rev. Lett., 50,1395–1398.

27. Thompson, A.M. and Lampard, D.G.(1956) A new theorem in electrostat-ics and its application to calculablestandards of capacitance. Nature, 177,888.

28. Gibbings, D.L.H. (1963) A design forresistors of calculable a.c./d.c. resistanceratio. Proc. IEE, 110, 335–347.

29. Kucera, J., Vollmer, E., Schurr, J., andBohacek, J. (2009) Calculable resistors ofcoaxial design. Meas. Sci. Technol., 20,095104 (6 pp).

30. Small, G.W., Rickets, B.W., Coogan,P.C., Pritchard, B.J., and Sovierzoski,M.M.R. (1997) A new determination ofthe quantized Hall resistance in termsof the NML calculable cross capacitor.Metrologia, 34, 241–243.

31. Jeffery, A.M., Elmquist, R.E., Lee, L.H.,Shields, J.Q., and Dziuba, R.F. (1997)NIST comparison of the quantized Hallresistance and the realization of theSI ohm through the calculable capac-itor. IEEE Trans. Instrum. Meas., 46,264–268.

32. Witt, T.J. (1998) Electrical resistancestandards and the quantum Hall effect.Rev. Sci. Instrum., 69, 2823–2843.

33. Delahaye, F. and Jeckelmann, B. (2003)Revised technical guidelines for reliabledc measurements of the quantized Hallresistance. Metrologia, 40, 217–233.

34. Hartland, A., Jones, K., Williams, J.M.,Gallagher, B.L., and Galloway, T. (1991)Direct comparison of the quantizedHall resistance in gallium arsenide andsilicon. Phys. Rev. Lett., 66, 969–973.

35. Janssen, T.J.B.M., Fletcher, N.E., Goebel,R., Williams, J.M., Tzalenchuk, A.,Yakimova, R., Kubatkin, S., Lara-Avila,S., and Falko, V.I. (2011) Graphene,universality of the quantum Hall effectand redefinition of the SI system. New J.Phys., 13, 093026 (6 pp).

36. Janssen, T.J.B.M., Williams, J.M.,Fletcher, N.E., Goebel, R., Tzalenchuk,A., Yakimova, R., Lara-Avila, S.,Kubatkin, S., and Fal’ko, V.I. (2012)Precision comparison of the quantum

Hall effect in graphene and galliumarsenide. Metrologia, 49, 294–306.

37. Novoselov, K.S., Geim, A.K., Mozorov,S.V., Jiang, D., Katsnelson, M.I.,Grigorieva, I.V., Dubonos, S.V., andFirsov, A.A. (2005) Two-dimensional gasof massless Dirac fermions in graphene.Nature, 438, 197–200.

38. Zhang, Y., Tan, Y.-W., Stormer, H.L.,and Kim, P. (2005) Experimental obser-vation of the quantum Hall effect andBerry’s phase in graphene. Nature, 438,201–204.

39. Novoselov, K.S., Jiang, Z., Zhang, Y.,Morozov, S.V., Stormer, H.L., Zeitler, U.,Maan, J.C., Boebinger, G.S., Kim, P., andGeim, A.K. (2007) Room-temperatureQuantum Hall effect in graphene. Sci-ence, 315, 1379.

40. Jeckelmann, B. and Jeanneret, B. (2001)The quantum Hall effect as an electricalresistance standard. Rep. Prog. Phys., 64,1603–1655.

41. Hehl, F., Obukhov, Y.N., and Rosenow,B. (2005) Is the quantum Hall effectinfluenced by the gravitational field?Phys. Rev. Lett., 93, 096804 (4 pp).

42. Penin, A.A. (2009) Quantum Hall effectin quantum electrodynamics. Phys. Rev.B, 79, 113303 (4 pp).

43. Giacomo, P. (1988) News from theBIPM. Metrologia, 25, 115–119(see also Resolution 6 of the 18thMeeting of the CGPM (1987), BIPMhttp://www.bipm.org/en/CGPM/db/18/6/(accessed 22 August 2014)).

44. Quinn, T.J. (1989) News from the BIPM.Metrologia, 26, 69–74.


46. Piquemal, F. (2010) in Handbook ofMetrology, vol. 1 (eds M. Gläser and M.Kochsiek), Wiley-VCH Verlag GmbH,Weinheim, pp. 267–314.

47. Harvey, K. (1972) A precise low tem-perature dc ratio transformer. Rev. Sci.Instrum., 43, 1626–1629.

48. Delahaye, F. (1993) Series and paral-lel connection of multiple quantumHall-effect devices. J. Appl. Phys., 73,7914–7920.

http://www.bipm.org/en/CGPM/db/18/6/

References 129

49. Poirier, W., Bounouh, A., Piquemal, F.,and Andre, J.P. (2004) A new generationof QHARS: discussion about the techni-cal criteria for quantization. Metrologia,41, 285–294.

50. Hein, G., Schumacher, B., and Ahlers,F.J. (2004) Preparation of quantum Halleffect device arrays. Conference onPrecision Electromagnetic Measure-ments Digest 2004, pp. 273–274, ISBN0-7803-8493-8.

51. Ahlers, F.J., Jeanneret, B., Overney, F.,Schurr, J., and Wood, B.M. (2009) Com-pendium for precise ac measurements ofthe quantum Hall resistance. Metrologia,46, R1–11.

52. Kibble, B.P. and Schurr, J. (2008) A noveldouble-shielding technique for ac quan-tum Hall measurement. Metrologia, 45,L25–27.

131

6Single-Charge Transfer Devices and the New Ampere

The paradigm of quantum metrology is the counting of single quanta to linkmacroscopic physical quantities to fundamental constants. This concept becomesparticularly obvious considering the electrical current and the transfer of singlecharges through a conductor. Single-charge transfer entails the controlled manip-ulation of single electrons and single Cooper pairs in normal conductors andsuperconductors, respectively. Its importance for metrology has increased evenmore by the envisaged new definition of the ampere, which links the ampere tothe elementary charge e (see Section 2.2). Single-charge transfer has the potentialto realize the new ampere definition if single charges are transferred in a clockedfashion at a frequency f. This approach yields a quantized current

I = nef = neT , (6.1)

where n is the number of elementary charges transferred per cycle. Equation (6.1)can be viewed as the textbook definition of the current being the charge trans-ferred through a conductor cross section per time interval T = 1∕f . In this sense,quantized current sources provide the most direct realization of the new amperedefinition. As an alternative, the new definition can also be realized using thequantum Hall and the Josephson effect and applying Ohm’s law.

For Eq. (6.1) to be applicable, single charges must be isolated and transferredone by one through a conductor. The basic physics of single-charge transport willbe discussed in Section 6.1. In Section 6.2, we give an overview of quantized cur-rent sources made from normal metals, superconductors, and semiconductors.The basics of superconductors and semiconductors are summarized in Chapters4 and 5, respectively. Detailed reviews of single-charge transfer can be found, forexample, in Refs [1–4]. We conclude Section 6.2 with a discussion of the prospectsof quantum current standards based on single-charge transfer.

An important application of quantum current standards is a fundamental con-sistency test of electrical quantum metrology, which is known as the quantummetrology triangle (QMT), first suggested in [5]. The QMT is described in Section6.3. The consistency test aims at verifying the relation of the Josephson effect, thequantum Hall effect (QHE), and single-charge transport to the fundamental con-stants e and h.


132 6 Single-Charge Transfer Devices and the New Ampere

6.1Basic Physics of Single-Electron Transport

In this section, we discuss the basics of single-charge transfer in normal-metalcircuits. Single-electron transport (SET) rests on two basic physical phenomena,namely, the tunneling of electrons through potential barriers and the so-calledCoulomb blockade. Coulomb blockade occurs in small structures with a largecapacitive charging energy. These phenomena are also the basis of single-chargetransfer in semiconductors and superconductors.

6.1.1Single-Electron Tunneling

Electrons in metals are delocalized. Therefore, the charge on a capacitor C con-nected to a voltage source U by metallic wires can take any value Q = CU eventhough the electron charge (−e) is quantized. This fact, illustrated in Figure 6.1a,raises the question of how single electrons can be manipulated in metals. Afirst clue is obtained noting that a capacitor plate will carry a fixed number ofelectrons and, thus, quantized charge if one of the wires is broken. This situationcan be realized by an open switch as shown in Figure 6.1b. Opening of the switchresults in the localization of the electrons on the capacitor. Of course, breakingthe wire prohibits any further adjustment of the number of electrons on thecapacitor and is not a practical approach. Yet, localization can also be obtainedreplacing the switch by a tunnel element with sufficiently large resistance. In itssimplest form, a tunnel element consists of two metallic contacts separated bya sufficiently thin insulating layer, very similar to the Josephson tunnel junctiondescribed in Chapter 4. The tunnel element (with resistance RT and capacitanceCT) and the capacitor plate form a so-called single-electron quantum box, asshown in Figure 6.1c.

A single-electron quantum box allows single-electron charges to be manipu-lated and can serve as building block of SET devices if two fundamental conditionsare fulfilled. First, the charging energy required to put an extra electron on the

U C Q

U C

UC

(a)

(b)

(c)

n (−e)

n (−e)

RT, CT

Figure 6.1 Comparison of a closed metallic circuit withoutcharge quantization (a), an open metallic circuit with a fixednumber of quantized charges n on the capacitor (b), and asingle-electron quantum box (dashed box in part c) allowingthe manipulation of single electrons.

6.1 Basic Physics of Single-Electron Transport 133

capacitor, E1eC , must be considerably larger than the thermal energy, kBT , to pre-

vent random thermal transfer of electrons:

E1eC = e2

2CΣ≫ kBT , (6.2)

where CΣ is the total capacitance (CΣ = C + CT). Relation (6.2) shows that reli-able SET operation requires very small capacitances (0.1–1 fF), corresponding tostructures with nanometer dimensions. Moreover, in particular for metrologicalapplications, the temperature must be low (often in the millikelvin range).

The second condition relates to quantum fluctuations whose energy, EQF, mustbe much smaller than E1e

C . According to the Heisenberg uncertainty relation, wecan write EQF = ℏ∕𝜏 , where 𝜏 is the RC time constant, 𝜏 = RTCΣ. Thus, we obtainas second condition

RT ≫1𝜋

he2 ≈

RK4

, (6.3)

where RK is the von Klitzing constant. Relation (6.3) expresses that the tunnelresistance must be large enough to localize electrons sufficiently well in the single-electron quantum box.

6.1.2Coulomb Blockade in SET Transistors

Single-electron manipulation can be achieved using a so-called SET transistor ifconditions (6.2) and (6.3) are fulfilled. As shown in Figure 6.2, an SET transistoris a three-terminal device. It consists of two single-electron quantum boxes con-nected such that a small charge island is formed between the tunnel elements.The island is capacitively coupled to a gate voltage UG via the gate capacitance CG.Additionally, a source–drain voltage USD can be applied across the SET transis-tor (shown as being split symmetrically in Figure 6.2). Using SET transistors, clearsignatures of SET phenomena were observed as early as 1987 [6, 7].

In order to understand the operation of an SET transistor, its different energyterms and, in turn, its chemical potential have to be considered. The total electro-static energy of the charge island is given by

Eel−st =(−enexc + Q0)2

2CΣ, (6.4)

where nexc = N − N0 is the number of excess electrons on the island and N thetotal number of electrons. N0 is the number of electrons in equilibrium, that is,

RT,CT RT,CT

−USD/2 USD/2CG

UG

Figure 6.2 Equivalent circuit of an SET transis-tor. The charge island is shown as black dot.


for USD = 0 and UG = 0, which compensate for the positive background charge ofthe island. The gate electrode induces a continuously variable charge Q0 = CGUG.For noninteracting electrons at zero temperature, the total electronic energy of thecharge island, E(N), is obtained if the single-particle energies εi of all N electronsare added to the electrostatic energy:

E(N) =N∑

i=1εi +

(−enexc + CGUG)2

2CΣ. (6.5)

The important quantity for the study of transport phenomena is the chemicalpotential 𝜇, which, by definition, is the energy required to put an extra electronin a system. The chemical potential reflects the mere change of particle number(to be considered also for uncharged particles) as well as changes of the electro-static energy caused thereof (to be considered only for charged particles). If thechemical potential is constant, no net transfer of particles occurs and the currentis zero.

The chemical potential of the charge island, 𝜇C, is calculated subtracting thetotal electronic energy of an island with N − 1 electrons from the correspondingterm for N electrons:

𝜇C(N) ≡ E(N) − E(N − 1) = εN +(nexc − 1∕2

)e2

CΣ− e

CGCΣ

UG. (6.6)

The sum of the last two terms on the right-hand side is the electrostatic potential,−e𝜙N , while the first term is the electrochemical potential, 𝜇elch(N). The electro-static potential, −e𝜙N , can be adjusted by the gate voltage UG. Note that 𝜇C(N),𝜇elch(N), and −e𝜙N have the dimension of an energy even though they are con-ventionally referred to as potentials.

If the number of electrons on the charge island changes by one at a constantgate voltage, the chemical potential changes by Δ𝜇C. We obtain from Eq. (6.6)

Δ𝜇C = εN+1 − εN + e2

CΣ. (6.7)

For a small metallic island with small capacitance, we have εN+1 − εN ≪ e2∕CΣ.Thus, the chemical potential levels of the island are separated by the Coulombenergy e2∕CΣ.

The chemical potential across a metallic SET transistor is plotted in Figure 6.3.The chemical potential of the island is shown for occupation with either N − 1,N , or N + 1 electrons. Furthermore, 𝜇L and 𝜇R are the chemical potential of theelectron source (left) and electron drain (right), respectively, which are related tothe source–drain voltage according to

𝜇L − 𝜇R = eUSD. (6.8)

In Figure 6.3, 𝜇L − 𝜇R is assumed to be smaller than the Coulomb energy e2∕CΣ.Without lack of generality, we can further assume that the charge island is occu-pied by N electrons. Figure 6.3 then illustrates the so-called Coulomb blockade,that is, the suppression of electron transfer due to the Coulomb energy. The figure


Energ

y

μL

μR

μc (N + 1)

μc (N − 1)

μc (N)

e2

CΣ

−eøN

Position across SET transistor

Figure 6.3 Chemical potential across ametallic SET transistor for a fixed gate volt-age UG and a small source–drain voltageUSD. The tunnel barriers (gray) separate themetallic wires on the left (electron source)and right (electron drain) from the chargeisland. The chemical potential of the sourceand drain lead is 𝜇L and 𝜇R, respectively.The chemical potential of the island isshown for occupation with either N − 1, N,or N + 1 electrons.

shows that electrons cannot move from the source lead to the island since 𝜇L islocated below 𝜇C(N + 1). Likewise, electron flow from the island to the drain leadis inhibited since 𝜇C(N) is located below 𝜇R. Thus, the number of electrons on theisland remains constant at N and no current flows. Of course, for this argumentto be valid, we assume that kBT is much smaller than the Coulomb energy.

The Coulomb blockade can be lifted if the gate voltage UG (and thus −e𝜙N ) isadjusted such that

𝜇L > 𝜇C(N + 1) > 𝜇R. (6.9)

Under this condition, electrons can tunnel from the source lead to the island andfurther to the drain lead. The electrons are transferred one by one, that is, single-electron transport occurs. The simultaneous transfer of two or more electrons isnot possible since 𝜇C(N + 2), 𝜇C(N + 3), and so on are still located well above 𝜇L.As a consequence, the number of electrons on the charge island oscillates betweenN and N + 1.

The preceding discussion has yielded the following insight, which is the basis ofclocked single-charge transfer and quantized current sources:

• In an SET transistor, electron transfer occurs either one by one or is inhibitedby Coulomb blockade (for |eUSD| = |𝜇L − 𝜇R| < e2∕CΣ).

• Switching between these two states can be achieved by the adjustment of thegate voltage.

6.1.3Coulomb Blockade Oscillations and Single-Electron Detection

Let us assume again that the source–drain voltage across the SET transistor issmall so that the relation |eUSD| = |𝜇L − 𝜇R| < e2∕CΣ holds. Under this condition,the so-called Coulomb blockade oscillations occur if the gate voltage is contin-uously tuned. The tuning causes the state of the SET transistor to periodicallychange between Coulomb blockade and single-electron transport. The discussionof the physics presented in Section 6.1.2 applies to each period of the Coulombblockade oscillation. Only the electron numbers change from period to period.If one period involves the electron numbers N and N + 1, the following periods


0

1

2

0 1 2−1 −1/2 1/2 3/2

Gate voltage, UG (e/CG)

−1

Excess e

lectr

ons,

n exc

Curr

ent, I S

D

Figure 6.4 Schematic representation of thesource–drain current ISD (top) and the num-ber of excess electrons nexc on the chargeisland (bottom) versus gate voltage UG. Thedot marks the operating point of an SETelectrometer.

involve N + 1, N + 2, and so on. This feature is illustrated in the lower part ofFigure 6.4, where the number of excess electrons on the charge island, nexc, isplotted versus the gate voltage UG. The UG axis is scaled in units of e∕CG, whichis the period length of the Coulomb blockade oscillation. The period length isdetermined by Eq. (6.6), which shows that a gate voltage change of e∕CG shifts thechemical potential of the charge island by e2∕CΣ. The upper part of the figure isa schematic representation of the source–drain current, ISD, versus gate voltage.The current shows peaks with steep slopes (whenever the number of electronson the island oscillates corresponding to the transport of single electrons). Thisfeature is used in applications of SET transistors as electrometers, which have anunprecedented charge resolution on the order of 10−5 e∕

√Hz. For charge detec-

tion, a sensor electrode is coupled to the charge island of an SET transistor andthe operating point is chosen on the flank of a current peak.

Coulomb blockade does not occur for larger source–drain voltages, that is,for |eUSD| = |𝜇L − 𝜇R| ≥ e2∕CΣ. Referring to Figure 6.3, the reason is obvious.For larger source–drain voltages, 𝜇L is located above 𝜇C(N + 1) or 𝜇R lies below𝜇C(N). Either condition precludes Coulomb blockade. The complete dynamicsof the SET transistor can be concisely summarized by the stability diagram ofFigure 6.5, in which the number of excess electrons on the charge island is plottedin the plane of the gate voltage and the source–drain voltage.

0 1

0 10

Gate voltage, UG (e/CG)

−3/2 −1 −1/2 1/2 3/2

e/CΣ

−e/CΣ

Sourc

e-d

rain

vo

lta

ge

, U

SD

−1

(0,1)(−1,0)

(−1,0) (0,1)

Figure 6.5 Stability diagram of an SETtransistor: number of excess electrons nexcon the charge island versus gate voltageUG and source–drain voltage USD. In thegray regions, Coulomb blockade occurs andnexc has a constant value as indicated. Inthe white regions, nexc oscillates betweenthe indicated values corresponding tononzero current flow.


We finally note that the stability diagram also illustrates the behavior of an SETtransistor if USD is tuned at a constant gate voltage UG ≠ (i + 1∕2)e∕CG (i integer).For USD ≪ 0, negative current flow is observed, followed by Coulomb blockadearound USD = 0 and positive current for USD ≫ 0. This behavior is also shown bythe experimental data of Figure 6.9. It is the basis of Coulomb blockade thermom-etry discussed in Chapter 8.

6.1.4Clocked Single-Electron Transfer

Clocked single-electron transfer according to Eq. (6.1) can be realized with single-electron quantum boxes. However, a single normal-metal SET transistor is notsufficient for this purpose. This conclusion can be understood recalling that in theon-state, when the Coulomb blockade is lifted, the source–drain current relies onelectron tunneling, which is a stochastic process. As a consequence, one cannotcontrol the exact number of electrons that tunnel from source to drain in a giventime interval, even though the electrons tunnel one by one.

Controlled clocked transfer of single electrons is feasible if two (or more) SETtransistors are connected in series. Figure 6.6 shows a so-called SET pump withtwo charge islands. The chemical potentials of the islands can be individuallyadjusted by periodic gate voltages UG1 and UG2. The charge islands are separatedfrom each other and from the source and drain leads by three tunnel junctions.The lower part of the figures shows the periodic gate voltages, which are phase-shifted with respect to each other. The phase shift enables the following cycle, inwhich a single electron is transferred from source to drain. First, the chemicalpotential of island 1 is lowered by an increase of UG1 so that an electron cantunnel onto the island from the source. Subsequently, the chemical potential of

−USD/2 USD/2

(a)

(b)

UG2UG1

CG CG

UG2

Time

UG1

Figure 6.6 SET pump consisting of two charge islands and three tunnel junctions (a). Thechemical potentials of the islands are controlled by periodic gate voltages (b).


island 1 is raised again (by lowering UG1), while the chemical potential of island 2is lowered by the increase of UG2. In this phase, the electron tunnels from island1 to island 2. In the last part of the cycle, the decrease of UG2 raises the chemicalpotential of island 2 so that the electron is emitted to the drain lead. This clockedsingle-electron transfer does not require the application of a source–drain biasvoltage (in fact, it is even feasible against a small opposing bias). Therefore, thedevice is referred to as SET pump as opposed to SET turnstile devices whoseoperation relies on a source–drain bias.

The dynamics of SET pumps can be analyzed with a stability diagram, whichshows the number of excess electrons on charge island 1 and 2, (n1, n2), in theplane of the gate voltages UG1 and UG2. Figure 6.7 is a schematic representationof the stability diagram of the SET pump of Figure 6.6. Single-electron pumpingis achieved if the gate voltages are varied such that a triple point is encircled.As shown in the figure, counterclockwise rotation gives rise to clocked single-electron transfer from the source to island 1, island 2, and further to the drain.The stability diagram illustrates in an intuitive way that the direction of the single-electron current is reversed for clockwise rotation. Thus, single-electron pumpingis a reversible process whose direction is determined by the relative phase betweenthe gate voltages.

The first metallic SET pump was demonstrated in 1991 [8] and, since then,metallic SET pumps have considerably impacted metrology. Therefore, the prop-erties of metallic SET pumps are discussed in Section 6.2.1.

Single-electron transport through a metallic SET turnstile device was demon-strated in 1990 [9]. The device consisted of four tunnel junctions separated bythree charge islands. A single alternating gate voltage was applied to the centralisland. In order to realize single-electron transport, a source–drain voltage had tobe applied. This property classifies the device as turnstile, as mentioned earlier. Sofar, SET turnstiles made from normal metals have not reached the accuracy of SETpumps. Therefore, with our focus being on metrology, we will not discuss them inmore detail. The subject of turnstiles, however, will resurface in the context ofsemiconducting and superconducting quantized current sources in Sections 6.2.2and 6.2.3, respectively.

e

e

CGUG2

(0,1)(1,1)

(1,0)(0,0)

CGUG1

Figure 6.7 Schematic representation ofthe stability diagram of an SET pump thatis driven by two gate voltages UG1 andUG2. Shown is the number of excess elec-trons (n1,n2) on the charge island 1 and 2.The closed trajectory corresponds to single-electron pumping from source to drain.

6.2 Quantized Current Sources 139

6.2Quantized Current Sources

In this section, we discuss different implementations of clocked single-electrontransport. The focus is put on the performance of quantized current sources withrespect to their benchmark parameters for the realization of the new ampere: (i)the clock frequency, which determines the magnitude of the quantized currentaccording to Eq. (6.1), and (ii) the accuracy with which the quantized current canbe generated. The latter is not determined by the uncertainty of the frequency,which can be 10−14 and better if derived from atomic clocks (see Chapter 3). Theuncertainty of the quantized current is determined by the transfer error. Thisquantity describes the difference between the number of elementary chargesthat are actually transferred and the intended number n in Eq. (6.1), whichdescribes a perfect source. Mathematically, the transfer error can be expressed as|n − ⟨nS⟩|∕n. Here, ⟨nS⟩ denotes the time-averaged number of elementary chargestransferred per cycle of the clock frequency f . The transfer error and the clockfrequency are often interrelated so that a careful optimization of the overall perfor-mance is required. A comprehensive overview of SET devices is given in Ref. [4].

6.2.1Metallic Single-Electron Pumps

Most metallic SET devices are made of aluminum since it has a stable nativeoxide with good dielectric properties, which can form insulating tunnel barriers.A scanning electron microscopy (SEM) image of an Al/Al oxide SET transistoris shown in Figure 6.8. For SET operation according to the concept outlined inSection 6.1, a weak magnetic field is applied to suppress superconductivity inaluminum. The application of the field yields a normal-metal/insulator system.Pronounced Coulomb blockade can be achieved in Al/Al oxide SET transistors,as illustrated in Figure 6.9.

The clock frequency of metallic SET pumps is determined by the time constant𝜏 = RTCΣ of the tunneling process. This time constant cannot easily be reducedsince the relation RT ≫ RK must hold and the reduction of CΣ requires the fabrica-tion of extremely fine nanostructures. The clock frequency has to fulfill the condi-tion f ≪ (RTCΣ)−1. Otherwise, tunneling events are missed due to the stochasticnature of tunneling. Lowering the frequency increases the probability that tun-neling occurs in each cycle of the drive voltage. For high-accuracy single-electronpumping, the clock frequency must be limited to the 10 MHz range, correspond-ing to picoampere currents.

Other types of transfer errors can occur even at sufficiently low clock frequen-cies. A major source of error is higher-order tunneling, also named cotunneling[12, 13]. Cotunneling refers to the joint tunneling of two or more electrons throughan SET transistor or SET pump in either direction. As an example, consider an SETtransistor with one charge island and two tunnel barriers in the Coulomb block-ade state when single-electron tunneling is energetically forbidden. One type of


500 nm

L = SE1500 nmEHT = 20.0 KV WD = 14 mm MAG = X 50.0 K PHOTO = 0

Al/Al-oxide

Gate

Island

Tunnel junctions

Figure 6.8 SEM image of an Al/AlOx/Al/AlOx/Al SET transistor. Each structure is seen twicedue to the specific fabrication procedure (double-angle shadow evaporation [10, 11]). Cour-tesy of PTB.

−15

−1.5

−1.0

−0.5

0.0

0.5

nA

1.5

−10 −5 0

USD

I SD

5 10 mV

Figure 6.9 Source–drain current versussource–drain voltage of an Al/AlOx/Al/AlOx/Al SET transistor measured at a temper-ature of 25 mK and a magnetic field of 1 T.

The gate voltage was chosen to maximizethe Coulomb blockade and a “Coulomb gap”of 7 meV is observed. (Courtesy of H. Scherer,PTB.)

cotunneling process consists of the simultaneous transfer of one electron from thesource lead to the island and another one from the island to the drain. This processleaves the charge on the island unchanged and does not violate energy conserva-tion, but effectively transfers an electron to the drain. The transfer process canbe viewed as quantum tunneling through a potential barrier resulting from the


Coulomb energy. Obviously, this cotunneling process gives rise to a transfer error.The probability of cotunneling decreases with increasing number of tunnel junc-tions in SET pumps [14].

Besides cotunneling, photon-assisted tunneling needs to be considered whenanalyzing the accuracy of metallic SET pumps [15, 16]. In this process, absorp-tion of photons provides the energy required to lift an electron over the Coulombbarrier. The photon-assisted tunneling rate strongly depends on the shielding ofthe SET device against electromagnetic radiation [17].

Since the accuracy can be enhanced by a larger number of tunnel junction, theNIST fabricated a seven-junction SET pump and experimentally demonstrated atransfer error of 1.5 parts in 108 at a clock frequency of 5.05 MHz [18]. This is anexcellent result achieved by complex technology, which involves the synchronoustuning of six gate voltages. Despite the merits of metallic SET pumps, these resultsalso illustrate their main drawbacks. Their clock frequency and current are lim-ited and a complex multiple-gate setup is required to achieve low uncertainties.The latter issue is aggravated by long-term stability problems due to uncontrolledbackground charges, which change the properties of metallic SET pumps duringoperation. The nature and dynamics of the background charges are not yet com-pletely understood.

An alternative way to suppress cotunneling using less tunnel junctions is toembed the SET device in a high-impedance environment [19]. The first so-calledR-pump with three tunnel junctions and 60 kΩ on-chip resistors in series withthe pump was demonstrated by PTB in 2001 [20]. Three-junction R-pumps havenot reached metrological accuracy. Therefore, five-junction R-pumps were devel-oped, which achieved transfer errors of a few parts in 108 [21]. Seven-junctionSET pumps as well as five-junction R-pumps were employed in fundamental con-sistency tests of electrical quantum metrology, which are discussed in Section 6.3[22, 23].

6.2.2Semiconducting Quantized Current Sources

Clocked single-electron transport in semiconductor structures is governed by thegeneral principles that are outlined in Section 6.1 for metallic SET devices. Liketheir metallic counterparts, semiconducting quantized current sources are builtfrom charge islands and tunnel barriers, sandwiched between a source and drainreservoir. Different driving schemes can be realized, such as turnstile or pumpingoperation.

Yet, there are two important aspects that distinguish semiconductor SETdevices from metallic ones. In semiconductors, the density of free electrons issubstantially smaller than in metals. The smaller density gives rise to a larger deBroglie wavelength, which is of the same order as the size of the charge island.Therefore, size quantization must be taken into account (see Section 5.1). Thequantization energy must then be added to the Coulomb energy e2∕CΣ. The resultis a more complicated potential level structure compared to the equally spaced


levels of a metallic charge island. The charge island in semiconductor SET devicesshould rather be viewed as a quantum dot, that is, a zero-dimensional structurewith an atom-like energy spectrum.

The other important difference concerns the tunnel barriers. In metallic SETdevices, the height and width of the tunnel barriers are fixed, being determinedby the material properties and the thickness of the insulating layer. In contrast,the height and width of the tunnel barriers in semiconductors can be tuned byexternal gate voltages. We will discuss in this section that the tunability of thetunnel barriers is the key to the operation of semiconductor SET devices at higherfrequencies.

6.2.2.1 GaAs Based SET DevicesMost semiconductor quantized current sources have been fabricated either fromGaAs/AlGAs or Si/SiO2. We will first discuss GaAs/AlGAs SET devices. The fab-rication of such devices starts with a high-mobility two-dimensional electron gas(2DEG) in a GaAs/AlxGa1−xAs heterostructure. Further details about the growthand the properties of GaAs/AlxGa1−xAs heterostructures can be found in Section5.2. In order to fabricate a quantum dot, metal electrodes can be deposited on topof the heterostructure. The application of a negative voltage depletes the 2DEGunderneath the electrodes and creates potential barriers. A quantum dot con-nected by tunnel barriers to a source and drain reservoir can be generated usingappropriately shaped electrodes. The first experimental demonstration of clockedsingle-electron transfer in semiconductor structures was achieved with a turnstiledevice fabricated with this method [24].

Alternatively, first a one-dimensional conducting channel is defined by etching,which removes the 2DEG on either side of the channel. The channel is crossedby metallic gate electrodes to define tunnel barriers and, in turn, a quantum dotbetween the barriers. A schematic representation of such an SET device is shownin Figure 6.10. Both an AC and a DC voltage are applied to the left gate electrode

Figure 6.10 Schematic layout of aGaAs/AlGaAs SET pump. Typical parame-ters: width of the one-dimensional channel700 nm, gate width 100 nm, gate separation250 nm. An AC and a DC voltage are applied

to the left gate, while only a DC voltage isapplied to the right gate. (Courtesy of A.Müller, PTB.) (Please find a color version ofthis figure on the color plate section.)


(entrance gate). The potential of the exit gate on the right is adjusted by a DCvoltage only.

The SET device of Figure 6.10 is a nonadiabatic GaAs/AlGaAs SET pump [25].Its operation principle is illustrated in Figure 6.11, which schematically depictsthe different phases of a pumping cycle. Shown is the potential along the con-ducting channel for fixed DC gate voltages and an AC voltage that modulates theentrance tunnel barrier. In (a), the entrance barrier is high and tunneling of elec-trons from the source reservoir into the dot is inhibited. As the entrance barrieris lowered and becomes more transparent, electrons tunnel into the quantum dotif the dot potential levels are located below the source Fermi level (b). The subse-quent increase of the entrance barrier lifts the captured electrons above the Fermilevel (c). As the captured electrons gain energy, they face an increasingly lower andmore transparent exit barrier and, finally, tunnel out of the dot to the drain reser-voir (d). For sufficiently small structures and low temperatures, Coulomb blockadeensures that a small integer number of electrons are transferred per cycle. Theinteger number can be chosen by the adjustment of the DC gate voltages. A sin-gle electron is usually transferred per cycle for high-accuracy operation. Withoutgoing into the details, we note that the parameters of the dynamic quantum dot donot instantaneously follow the clock frequency if it is in the megahertz to gigahertzrange. This behavior classifies the pumping scheme as nonadiabatic. The nona-diabatic behavior is important since in the adiabatic limit, a directional currentcannot be obtained applying a single periodic modulation signal [26].

In the following, we summarize the development and present status ofGaAs/AlGaAs SET pumps. All experimental results were obtained at low tem-peratures in the 100 mK range. The predecessor of the SET pump of Figure 6.10

(a) (b)

(c) (d)

Figure 6.11 Schematic representation ofthe pumping cycle of a GaAs/AlGaAs SETpump. Shown is the temporally varyingpotential of the quantum dot (dark gray),the energy levels of the dot (light gray),and the Fermi level (light gray ribbon). The

transferred electron is shown as dark graydot. The different phases a, b, c, d of thepumping cycle are explained in the text.(Courtesy of A. Müller, PTB.) (Please find acolor version of this figure on the color platesection.)


was a similar GaAs/AlGaAs SET pump, which, however, was driven by twophase-shifted AC gate voltages. With this device, a relative uncertainty of thequantized current of 10−4 was demonstrated at a clock frequency of 547 MHz[27]. Pumping operation could be observed up to 3 GHz. Pumping with a singleAC gate voltage was demonstrated in Ref. [25]. The single-gate pumping schemeconsiderably simplifies the operation and paves the way toward on-chip integra-tion of several components. In Ref. [28], a parallel circuit of three pumps wasrealized to increase the output current. On-chip integration of a GaAs/AlGaAsSET pump and a quantum Hall resistor was shown to yield an all-semiconductorsource of quantized voltages [29]. Furthermore, it has been demonstrated thatsingle-gate GaAs/AlGaAs SET pumps have broad operating margins [30] andthat their accuracy can be improved by the application of a magnetic field [31, 32].The latter effect is not yet fully understood. Nonetheless, applying a field of 14 T,the generation of a quantized current with a relative uncertainty of 1.2 partsin 106 was experimentally demonstrated at 0.95 GHz [33]. In this experiment,the quantized current was directly compared to a reference current, which wastraceable to the Josephson effect and the QHE [33]. The quoted uncertainty isdetermined by the uncertainty of the reference current, and the uncertainty ofthe quantized current may even be lower.

Additional information on the transfer error can be obtained if the quantizedcurrent is analyzed as a function of the DC voltage applied to the exit gate.Figure 6.12 shows a pronounced current plateau obtained at a clock frequencyof 200 MHz. Qualitatively, the mere existence of the plateau proves currentquantization. Moreover, the current–voltage characteristics can quantitativelybe analyzed with the help of a theoretical model of the transport process [34].The theoretical model relates the width of the current plateau to the transfer

−200

0.0

0.5

1.0

1.5

I (ef)

2.0

2.5

3.0

−150 −100

VDC (mV)

−50 0

Figure 6.12 Current in units of ef as a function of the exit gate voltage. Clock frequencyf = 200 MHz, temperature 300 mK, zero magnetic field. (Courtesy of F. Hohls, PTB.)


error. The model predicts a transfer error of 10−8 for the quantized current ofFigure 6.12. This result corroborates that GaAs/AlGaAs SET pumps hold greatpromise to achieve uncertainties as low as the ones demonstrated with metallicSET pumps, yet at frequencies that are 2 orders of magnitude higher.

6.2.2.2 Silicon-Based SET DevicesThe operation principle of Si/SiO2 based SET devices is similar to the one ofGaAs/AlGAs devices. The current through a narrow Si wire is controlled bymetal-oxide-semiconductor field-effect transistors (MOSFETs). In their off-state,the MOSFETs create opaque tunnel barriers and a quantum dot is formedbetween them. The transparency of the barriers can be increased if the MOS-FETs are switched to the on-state. Based on the tunable-barrier concept, SETpumping [35] and SET turnstile operation were demonstrated [36]. In general,silicon technology allows very fine nanostructures to be fabricated, in which theCoulomb energy is increased. As a consequence, SET operation can be observedat elevated temperatures on the order of 20 K, which are substantially higher thanthe operation temperatures of GaAs/AlGAs devices.

In Ref. [35], SET pumping was demonstrated at 1 MHz and a temperature of25 K. The transfer error was on the order of 10−2. Soon after, turnstile operationwas achieved at 20 K, producing a quantized current with similar uncertainty, yetat a considerably higher frequency of 100 MHz [36]. With a similar Si MOSFETdevice as in Ref. [36], single-gate SET pumping was demonstrated in Ref. [37].In this work, the generation of a nanoampere quantized current was achieved,pumping three electrons per cycle of the clock frequency of 2.3 GHz. The transfererror was estimated to be on the order of 1 part in 102. A lower transfer errorwas reported for a quantized current generated with a device that consisted of ametallic NiSi nanowire interrupted by two tunnel barriers [38]. The barriers weredefined by Si MOSFETs and the device was fabricated by industrial silicon-on-insulator (SOI) technology. An uncertainty of the quantized current on the orderof 10−3 could be achieved at a clock frequency of 650 MHz and a temperature of0.5 K. More recently, quantized currents with an uncertainty of a few parts in 105

could be demonstrated at a frequency of 500 MHz with an SET pump fabricatedwith silicon metal-oxide-semiconductor (MOS) technology [39]. The device wasoperated at a temperature of a few 100 mK.

As an outlook, we like to mention that clocked single-electron transfer at giga-hertz frequencies was also observed in a graphene structure [40]. The structureconsisted of two coupled graphene quantum dots and a source and drain leaddefined by lithographic methods. The impact of graphene devices on quantizedcurrent generation will depend on the progress of graphene device technology.

6.2.3Superconducting Quantized Current Sources

In this section, metallic devices are discussed, in which superconductivityplays a role. We begin with hybrid devices, which contain both normal and


superconducting metals. At the end of the section, we briefly introduce devices,in which all metallic elements are in the superconducting state.

Let us consider an SET transistor, similar to the one shown in Figure 6.2, witha normal-metal charge island separated from superconducting source and drainleads by fixed insulating tunnel barriers. This SINIS structure (S superconductor,I insulator, N normal metal) was introduced in Ref. [41] together with the com-plementary NISIN structure. The SINIS circuit shows the better performance [4],and we will restrict the discussion to it. A realistic implementation is, for example,an Al/Al oxide/Cu/Al oxide/Al structure where Al is in the superconducting state.In order to generate a quantized current, a periodic voltage is applied to the gateelectrode of the SINIS circuit and the circuit is biased with a DC source–drainvoltage. Thus, a turnstile device is realized.

In Section 6.1.4, we have discussed that clocked single-electron transfer cannotbe achieved with a single all-normal-metal (NININ) SET transistor. The importantdifference between NININ and SINIS SET transistors is the superconducting gap,which enables clocked single-electron transfer in SINIS turnstiles. The operatingprinciple is schematically illustrated in Figure 6.13. Frame (a) shows the tunnel-ing of an electron from the filled states of the superconducting source lead to thenormal-metal charge island. This process takes place when the filled states of thesource are aligned with the lowest empty level of the charge island. After this tun-neling process, the electron cannot leave the charge island since the empty statesof the drain lead are located at higher energies due to the superconducting gap.Subsequently, the potential levels of the charge island are raised by an appropriateadjustment of the gate voltage as shown in frame (b). Frame (c) illustrates the finalstep of the cycle. The highest filled level of the island has been raised above thesuperconducting gap of the drain lead and the electron can tunnel to empty drainstates. Note that in this phase, the superconducting gap of the source lead inhibitsthe uncontrolled transfer of another electron from the source to the island.

EF Δ

Δ

(a) (b) (c)

Figure 6.13 Operating principle of an SINISturnstile. Shown is the density of states ofthe superconducting source (left) and drain(right) lead and the chemical potential ofthe normal-metal charge island (center ofeach frame). (a) Tunneling of a single elec-tron onto the normal-metal charge island.

(b) Increase of the chemical potential ofthe charge island due to the change of thegate voltage. (c) Tunneling of the electronto the superconducting drain lead. 2Δ is thesuperconducting gap and EF the Fermi level.(Courtesy of A. Müller, PTB.)


Transfer errors can occur due to higher-order tunneling processes, which aretheoretically analyzed in Ref. [42]. Theory predicts that quantized currents of30 pA can be generated with a transfer error of 10−8 with SINIS turnstiles [42]. Forreal structures, the current is expected to be limited to 10 pA at this uncertaintydue to the nonuniformity of the tunnel barriers [4]. Uncertainties of the quantizedcurrent on the order of 10−3 were experimentally demonstrated, limited by theuncertainty of the measuring instrument [43]. The experimental work has alsohighlighted the importance of engineering the on-chip environment of SINISturnstiles to suppress higher-order tunneling processes [44–46].

An important advantage of SINIS turnstiles is their operation by a singleperiodic gate voltage only. As for semiconductor SET pumps (see Section 6.2.2),single-gate operation facilitates the fabrication of parallel circuits with increasedoutput current. In Ref. [47], the parallel operation of ten SINIS turnstile wasdemonstrated. The parallel circuit generated a quantized current of 104 pA ata clock frequency of 65 MHz. The hybrid SINIS turnstile is a promising andversatile concept, which was also implemented with a carbon nanotube as normalconductor [48].

In the following, we discuss all-superconductor devices, which seem to haveseveral conceptual advantages. In such devices, Cooper pairs with charge (−2e)are transferred without dissipation. The dissipationless transport avoids adverseheating effects. Moreover, the doubling of the charge compared to the transportof single electrons doubles the current at a given clock frequency. One may alsoargue that in all-superconductor devices, the transport is coherent and, therefore,better controllable than transport based on stochastic tunneling.

Experimentally, superconducting quantized charge pumps were investigatedthat consisted of several superconducting charge islands separated from eachother and from the superconducting source and drain leads by fixed tunnel barri-ers. The charge on the superconducting islands could be adjusted by gate voltages.Thus, the device design is similar to the one shown in Figure 6.6. Moreover, thepumping concept is similar to the one described in Section 6.1.4 for normal-metalSET pumps. Three-junction and seven-junction superconducting pumps wererealized with this concept [49, 50]. Yet, the transfer error of the quantized currentwas found to be unsatisfactory. This result is ascribed to quasiparticle tunneling,that is, the transfer of single charges e, taking place in addition to Cooper pairtunneling.

Another type of superconducting quantized charge device is the so-calledsuperconducting sluice [51]. The device consists of a single superconductingisland whose charge can be controlled by an AC gate voltage. The charge island isconnected to source and drain leads via superconducting quantum interferencedevices (SQUIDs), which act as switches. Switching between the on- and off-stateof the SQUIDs is realized by magnetic flux pulses, which modulate the criticalcurrent of the SQUIDs (see Section 4.2). The flux pulses are synchronized to theAC gate voltage so that clocked Cooper pair transfer can be realized. Quantizedcurrents of about 1 nA were demonstrated with this concept, yet the uncertaintyof the current could not be improved beyond the 10−2 range [52].


Summarizing these results, the experimental realizations of all-superconductorquantized charge devices have not yet fulfilled the expectations discussed ear-lier assuming an ideal device. One reason is quasiparticle tunneling, which is notaccounted for in the picture of an ideal superconducting circuit.

As an outlook, we like to mention that there are other theoretical conceptsfor quantized current sources based on superconductors. Examples are thephase-locking of Bloch oscillations [5] and quantum phase slip devices [53].These concepts are intriguing since they involve current steps that are the dual ofthe Shapiro steps of Josephson voltage standards. In this approach, the transportof Cooper pairs does not rely on stochastic tunneling, which holds the promise ofhigher accuracy. However, the experimental realization of such quantized currentsources is still at its infancy.

6.2.4A Quantum Standard of Current Based on Single-Electron Transfer

The main difference, which distinguishes an SET-based quantum standard of cur-rent from a quantized current source, is the uncertainty of the output current. Ofcourse, there is some leeway where to draw the line. Yet, it seems reasonable torequire a quantum current standard to realize the ampere according to the newdefinition with an uncertainty that is smaller than the uncertainty of the ampererealization in the present SI. Today, the most accurate realization of the ampereis based on the realization of the SI volt and SI ohm and the use of Ohm’s law.The uncertainty of the SI ampere is limited by the realization of the SI volt, whoseuncertainty is larger than that of the ohm realization. The SI volt and, in turn,the SI ampere can be realized with an uncertainty of a few parts in 107 [54] (seeSection 4.1.4.2). Thus, an SET-based quantum current standard should have anuncertainty of 10−7 or better. Such a standard would realize the redefined amperein a very direct way, that is, according to Eq. (6.1).

The required small transfer error has been achieved with normal-metal seven-junction SET pumps [18] and five-junction R-pumps [21]. Both devices generatecurrents on the order of 1 pA. This small current is sufficient for experiments,in which not the current itself is measured, but the charge that is accumulatedoperating the quantum current standard for a well-defined time [22, 23].

Larger currents are required for precision experiments, in which the cur-rent is measured. Considering the required small transfer error, it seems thatGaAs/AlGaAs or silicon SET pumps have the potential to serve as nanoamperequantum current standards. A parallel circuit of about 100 SINIS turnstiles maybe an alternative.

All these SET devices rely on the stochastic tunneling of single electrons, whichis inherently prone to transfer errors. Therefore, it was proposed to incorporateSET detectors in a series circuit of SET current sources to monitor the transfererrors [55]. Using the information on the transfer errors, the quantized currentof the series circuit can be determined with an uncertainty that is lower than the

6.3 Consistency Tests: Quantum Metrology Triangle 149

uncertainty of the individual SET current sources [55]. Such an error account-ing scheme can be applied if the bandwidth of the SET detectors is larger thanthe error rate [55]. The experimental implementation is feasible since the detectorbandwidth has only to exceed the error rate but not the much larger clock fre-quency. At PTB GaAs/AlGaAs SET pumps were combined with metallic SET elec-trometers in an integrated circuit [56]. A proof-of-principle experiment at a clockfrequency of 30 Hz showed the uncertainty achieved with the integrated circuit tobe 50 times lower than the one achieved with a single SET pump [57]. The exten-sion to higher clock frequencies may allow the realization of accurate quantumcurrent standards based on quantized current sources with limited uncertainty.

6.3Consistency Tests: Quantum Metrology Triangle

The QMT is a consistency test of the three electrical quantum effects, that is, theJosephson effect, the QHE, and single-charge transfer. The QMT aims at verifyingthe relation of the quantized voltage and resistance to the Josephson constantKJ = 2e∕h and the von Klitzing constant RK = h∕e2, respectively. Moreover, theQMT intends to verify that the quantized charge qS, which is transferred throughan SET device, is exactly equal to the elementary charge e, as assumed in Eq. (6.1).The QMT is based on Ohm’s law, which is illustrated by the upper trianglein Figure 6.14. In this section, we briefly discuss the basic idea of the QMTexperiment, its experimental implementations, and their results. Comprehensivereviews can be found in Refs [21, 58].

For convenience, we recall the respective equations of the three quantumeffects:

UJ =nJ fJ

KJ, (6.10)

RQHE =RKi , (6.11)

UJ = RQHE ISET

UJ

ISET

QSET

RQHE

CCR

QSET = CCR UJ

Figure 6.14 The quantum metrology tri-angle in the current version applying Ohm’slaw (upper part, solid lines). As an alterna-tive, the charging of a capacitor is stud-ied in the charge version of the quantummetrology triangle (lower part, dashedlines).


ISET = ⟨nS⟩efS. (6.12)

Equation (6.12) contains the average number of transferred elementary charges⟨nS⟩ to account for the occurrence of transfer errors. Using Ohm’s law, we obtainfrom Eqs (6.10)–(6.12)

nJi⟨nS⟩

fJ

fS= KJRKe = 2. (6.13)

In this equation, nJ and i are known integer step numbers. The frequency ratio fJ∕fSand the average number of transferred charges ⟨nS⟩ must be determined experi-mentally. In order to determine ⟨nS⟩, a single-electron detection scheme must beapplied, such as the error accounting concept discussed in Section 6.2.4 [55]. Ameasurement of the current is not sufficient for this purpose since it would onlydetermine the product of qS and ⟨nS⟩. The proof that Eq. (6.13) is valid is oftenreferred to as the closure of the QMT. Any deviation would cast doubt on thestrict validity of at least one of the equations KJ = 2e∕h, RK = h∕e2, and qS = e.

Different approaches have been followed to close the QMT experimentally. Ina straightforward one, the quantized current generated by an SET device is sup-plied to a quantum Hall resistor and the Hall voltage is measured by comparisonto a Josephson voltage standard. Even if a nanoampere quantized current source isavailable, current amplification is required to achieve an uncertainty on the orderof 10−7. The use of a cryogenic current comparator (CCC; see Section 4.2.3.3) witha high winding ratio was suggested for this purpose [59]. Such an experiment wasrealized with a normal-metal three-junction R-pump as quantized current source[60]. Yet, the experimental setup did not allow ⟨nS⟩ to be measured independently.Therefore, the closure of the QMT could not be investigated. At the time of writ-ing, the closure of a QMT, which directly implements Ohm’s law, has not yet beenreported.

In an indirect approach developed first at the NIST, a cryogenic capacitor withcapacitance CCR is charged by an SET device. The “charge version” of the QMT isillustrated in the lower part of Figure 6.14. A total charge QSET = ⟨nS⟩efSTS is accu-mulated over a well-known time TS. The voltage across the capacitor is measuredby comparison to a Josephson voltage standard. As described in Section 5.4.2, thevalue of CCR can be linked to RQHE via a quadrature bridge. The QMT experimentcan be combined with an electron shuttle measurement, which determines theaverage number of transferred elementary charges ⟨nS⟩. Using a seven-junctionSET pump, the NIST experiment demonstrated the closure of the QMT with arelative standard uncertainty of 0.9 parts in 106 [22, 61]. PTB reported the closureof the QMT with an uncertainty of 1.7 parts in 106 using a five-junction R-pumpto charge the capacitor [23].

Regarding the implications of these results, we first recall that the Josephsoneffect and the QHE are highly reproducible, as discussed in Sections 4.1.4 and 5.4,respectively. Yet, strictly speaking, the high reproducibility only implies that theeffects are universal, but does not give information on how well they are describedby the fundamental constants e and h. More information is obtained from theory,

References 151

which does not predict any appreciable deviation from KJ = 2e∕h and RK = h∕e2

(for a more detailed discussion, see, e.g., [4, 58]). However, theory alone cannotprovide a rigorous proof of the validity of the relations, but experimental evidencehas to be sought. With regard to the QHE, we recall from Section 5.4.1.4 that theSI value of RQHE can be compared to h∕e2 as determined from measurements ofthe fine-structure constant α. No deviation is found at a level of uncertainty of afew parts in 108. This result is corroborated by the adjustment of the fundamen-tal constants [62]. The adjustment of the fundamental constants also verifies thevalidity of the relation KJ = 2e∕h at the level of a few parts in 107. Thus, the presentQMT results mainly support the precision of charge quantization at the level of10−6. The improvement of QMT experiments toward the 10−8 level is desirableeven though the electrical quantum effects already rest on very solid grounds.

References

1. Grabert, H. and Devoret, M.H. (eds)(1992) Single Charge Tunneling, PlenumPress, New York.

2. Likharev, K.K. (1999) Single-electrondevices and their application. Proc. IEEE,87, 606–632.

3. Ono, Y., Fujiwara, A., Nishiguchi, K.,Inokawa, H., and Takahashi, Y. (2005)Manipulation and detection of sin-gle electrons for future informationprocessing. J. Appl. Phys., 97, 031101(19 pp.).

4. Pekola, J.P., Saira, O.-P., Maisi, V.F.,Kemppinnen, A., Möttönen, M., Pashkin,Y.A., and Averin, D.V. (2013) Single-electron current sources: toward arefined definition of the ampere. Rev.Mod. Phys., 85, 1421–1472.

5. Likharev, K.K. and Zorin, A.B. (1985)Theory of the Bloch-wave oscillations insmall Josephson junctions. J. Low Temp.Phys., 59, 347–382.

6. Fulton, T.A. and Dolan, G.J. (1987)Observation of single-electron chargingeffects in small tunnel junctions. Phys.Rev. Lett., 59, 109–112.

7. Kuzmin, L.S. and Likharev, K.K. (1987)Direct experimental observation ofdiscrete correlated single-electron tun-neling. JETP Lett., 45, 495–497.

8. Pothier, H., Lafarge, P., Orfila, P.F.,Urbina, C., Esteve, D., and Devoret,M.H. (1991) Single electron pump fab-ricated with ultrasmall normal tunneljunctions. Physica B, 169, 573–574.

9. Geerligs, L.J., Anderegg, V.F., Holweg,P.A.M., Mooij, J.E., Pothier, H., Esteve,D., Urbina, C., and Devoret, M.H. (1990)Frequency-locked turnstile device forsingle electrons. Phys. Rev. Lett., 64,2691–2694.

10. Niemeyer, J. and Kose, V. (1976) Obser-vation of large dc supercurrents atnonzero voltages in Josephson tunneljunctions. Appl. Phys. Lett., 29, 380–382.

11. Dolan, G.J. (1977) Offset masks for liftoffphotoprocessing. Appl. Phys. Lett., 31,337–339.

12. Averin, D.V. and Odintsov, A.A. (1989)Macroscopic quantum tunneling of theelectric charge in small tunnel junctions.Phys. Lett., A140, 251–257.

13. Geerligs, L.J., Averin, D.V., and Mooij,J.E. (1990) Observation of macroscopicquantum tunneling through the coulombenergy barrier. Phys. Rev. Lett., 65,3037–3040.

14. Jensen, H.D. and Martinis, J.M. (1992)Accuracy of the electron pump. Phys.Rev. B, 46, 13407–13427.

15. Martinis, J.M. and Nahum, M. (1993)Effect of environmental noise on theaccuracy of Coulomb-blockade devices.Phys. Rev. B, 48, 18316–18319.

16. Kautz, R.L., Keller, M.W., and Martinis,J.M. (2000) Noise-induced leakage andcounting errors in the electron pump.Phys. Rev. B, 62, 15888–15902.

17. Kemppinen, A., Lotkhov, S.V., Saira,O.-P., Zorin, A.B., Pekola, J.P., and


Manninen, A.J. (2011) Long hold timesin a two-junction electron trap. Appl.Phys. Lett., 99, 142106 (3 pp).

18. Keller, M.W., Martinis, J.M.,Zimmerman, N.M., and Steinbach,A.H. (1996) Accuracy of electron count-ing using a 7-junction electron pump.Appl. Phys. Lett., 69, 1804–1806.

19. Odintsov, A.A., Bubanja, V., and Schön,G. (1992) Influence of electromagneticfluctuations on electron cotunneling.Phys. Rev. B, 46, 6875–6881.

20. Lotkhov, S.V., Bogoslovsky, S.A., Zorin,A.B., and Niemeyer, J. (2001) Operationof a three-junction single-electron pumpwith on-chip resistors. Appl. Phys. Lett.,78, 946–948.

21. Scherer, H. and Camarota, B. (2012)Quantum metrology triangle exper-iments: a status review. Meas. Sci.Technol., 23, 124010 (13 pp).

22. Keller, M.W., Eichenberger, A.L.,Martinis, J.M., and Zimmerman, N.M.(1999) A capacitance standard basedon counting electrons. Science, 285,1706–1709.

23. Camarota, B., Scherer, H., Keller, M.V.,Lotkhov, S.V., Willenberg, G.-D., andAhlers, F.J. (2012) Electron countingcapacitance standard with an improvedfive-junction R-pump. Metrologia, 49,8–14.

24. Kouwenhoven, L.P., Johnson, A.T., vander Vaart, N.C., Harmans, C.J.P.M., andFoxon, C.T. (1991) Quantized current ina quantum-dot turnstile using oscillat-ing tunnel barriers. Phys. Rev. Lett., 67,1626–1629.

25. Kaestner, B., Kashcheyevs, V., Amakawa,S., Blumenthal, M.D., Li, L., Janssen,T.J.B.M., Hein, G., Pierz, K., Weimann,T., Siegner, U., and Schumacher, H.W.(2008) Single-parameter nonadiabaticquantized charge pumping. Phys. Rev. B,77, 153301, (4 pp).

26. Moskalets, M. and Büttiker, M. (2002)Floquet scattering theory of quantumpumps. Phys. Rev. B, 66, 205320 (10 pp).

27. Blumenthal, M.D., Kaestner, B., Li, L.,Giblin, S., Janssen, T.J.B.M., Pepper, M.,Anderson, D., Jones, G., and Ritchie,D.A. (2007) Gigahertz quantized chargepumping. Nat. Phys., 3, 343–347.

28. Mirovsky, P., Kaestner, B., Leicht, C.,Welker, A.C., Weimann, T., Pierz, K.,and Schumacher, H.W. (2010) Synchro-nized single electron emission fromdynamical quantum dots. Appl. Phys.Lett., 97, 252104 (3 pp).

29. Hohls, F., Welker, A.C., Leicht, C.,Kaestner, B., Mirovsky, P., Müller, A.,Pierz, K., Siegner, U., and Schumacher,H.W. (2012) Semiconductor quantizedvoltage source. Phys. Rev. Lett., 109,056802 (5 pp).

30. Kaestner, B., Kashcheyevs, V., Hein, G.,Pierz, K., Siegner, U., and Schumacher,H.W. (2008) Robust single-parameterquantized charge pumping. Appl. Phys.Lett., 92, 192106 (3 pp).

31. Wright, S.J., Blumenthal, M.D., Gumbs,G., Thorn, A.L., Pepper, M., Janssen,T.J.B.M., Holmes, S.N., Anderson, D.,Jones, G.A.C., Nicoll, C.A., and Ritchie,D.A. (2008) Enhanced current quantiza-tion in high-frequency electron pumpsin a perpendicular magnetic field. Phys.Rev. B, 78, 233311 (4 pp).

32. Kaestner, B., Leicht, C., Kashcheyevs, V.,Pierz, K., Siegner, U., and Schumacher,H.W. (2009) Single-parameter quantizedcharge pumping in high magnetic fields.Appl. Phys. Lett., 94, 012106 (3 pp).

33. Giblin, S.P., Kataoka, M., Fletcher, J.D.,See, P., Janssen, T.J.B.M., Griffiths, J.P.,Jones, G.A.C., Farrer, I., and Ritchie,D.A. (2012) Towards a quantum rep-resentation of the ampere using singleelectron pumps. Nat. Commun., 3, 930(6 pp).

34. Kashcheyevs, V. and Kaestner, B. (2010)Universal decay cascade model fordynamic quantum dot initialization.Phys. Rev. Lett., 104, 186805 (4 pp).

35. Ono, Y. and Takahashi, Y. (2003)Electron pump by a combined single-electron/field-effect transistor structure.Appl. Phys. Lett., 82, 1221–1223.

36. Fujiwara, A., Zimmerman, N.M.,Ono, Y., and Takahashi, Y. (2004)Current quantization due to single-electron transfer in Si-wire charge-coupled devices. Appl. Phys. Lett., 84,1323–1325.

37. Fujiwara, A., Nishiguchi, K., and Ono,Y. (2008) Nanoampere charge pumpby single-electron ratchet using silicon

References 153

nanowire metal-oxide-semiconductorfield-effect transistor. Appl. Phys. Lett.,92, 042102 (3 pp).

38. Jehl, X., Voisin, B., Charron, T., Clapera,P., Ray, S., Roche, B., Sanquer, M.,Djordjevic, S., Devoille, L., Wacquez,R., and Vinet, M. (2013) Hybridmetal–semiconductor electron pumpfor quantum metrology. Phys. Rev. X, 3,021012 (7 pp).

39. Rossi, A., Tanttu, T., Yen Tan, K.,Iisakka, I., Zhao, R., Wai Chan, K.,Tettamanzi, G.C., Rogge, S., Dzurak,A.S., and Möttönen, M. (2014) An accu-rate single-electron pump based on ahighly tunable silicon quantum dot.Nano Lett., 14, 3405–3411.

40. Connolly, M.R., Chiu, K.L., Giblin, S.P.,Kataoka, M., Fletcher, J.D., Chu, C.,Griffith, J.P., Jones, G.A.C., Fal’ko, V.I.,Smith, C.G., and Janssen, T.J.B.M. (2013)Gigahertz quantized charge pump-ing in graphene quantum dots. Nat.Nanotechnol., 8, 417–420.

41. Pekola, J.P., Vartiainen, J.J., Möttönen,M., Saira, O.-P., Meschke, M., andAverin, D.V. (2008) Hybrid single-electron transistor as a source ofquantized electric current. Nat. Phys., 4,120–124.

42. Averin, D.V. and Pekola, J.P. (2008)Nonadiabatic charge pumping in ahybrid single-electron transistor. Phys.Rev. Lett., 101, 066801 (4 pp).

43. Aref, T., Maisi, V.F., Gustafsson, M.V.,Delsing, P., and Pekola, J.P. (2011)Andreev tunneling in charge pump-ing with SINIS turnstiles. Europhys.Lett., 96, 37008 (6 pp).

44. Lotkhov, S.V., Kemppinen, A., Kafanov,S., Pekola, J.P., and Zorin, A.B. (2009)Pumping properties of the hybridsingle-electron transistor in dissipa-tive environment. Appl. Phys. Lett., 95,112507 (3 pp).

45. Pekola, J.P., Maisi, V.F., Kafanov, S.,Chekurov, N., Kemppinen, A., Pashkin,Y.A., Saira, O.-P., Möttönen, M., andTsai, J.S. (2010) Environment-assistedtunneling as an origin of the dynesdensity of states. Phys. Rev. Lett., 105,026803 (4 pp).

46. Saira, O.-P., Möttönen, M., Maisi, V.F.,and Pekola, J.P. (2010) Environmentally

activated tunneling events in a hybridsingle-electron box. Phys. Rev. B, 82,155443 (6 pp).

47. Maisi, V.F., Pashkin, Y.A., Kafanov, S.,Tsai, J.S., and Pekola, J.P. (2009) Parallelpumping of electrons. New J. Phys., 11,113057 (9 pp).

48. Siegle, V., Liang, C.-W., Kaestner, B.,Schumacher, H.W., Jessen, F., Koelle,D., Kleiner, R., and Roth, S. (2010) Amolecular quantized charge pump. NanoLett., 10, 3841–3845.

49. Geerligs, L.J., Verbrugh, S.M., Hadley,P., Mooij, J.E., Pothier, H., Lafarge, P.,Urbina, C., Esteve, D., and Devoret,M.H. (1991) Single Cooper pair pump.Z. Phys. B, 85, 349–355.

50. Aumentado, J., Keller, M.W., andMartinis, J.M. (2003) A seven-junctionCooper pair pump. Physica E, 18,37–38.

51. Niskanen, A.O., Pekola, J.P., and Seppä,H. (2003) Fast and accurate single-islandcharge pump: implementation of aCooper pair pump. Phys. Rev. Lett., 91,177003 (4 pp).

52. Vartiainen, J.J., Möttönen, M., Pekola,J.P., and Kemppinen, A. (2007) Nanoam-pere pumping of Cooper pairs. Appl.Phys. Lett., 90, 082102 (3 pp).

53. Mooij, J.E. and Nazarov, Y.V. (2006)Superconducting nanowires as quan-tum phase-slip junctions. Nat. Phys., 2,169–172.

54. Flowers, J. (2004) The route to atomicand quantum standards. Sciene, 306,1324–1330.

55. Wulf, M. (2013) Error accounting algo-rithm for electron counting experiments.Phys. Rev. B, 87, 035312 (5 pp).

56. Fricke, L., Wulf, M., Kaestner, B.,Kashcheyevs, V., Timoshenko, J.,Nazarov, P., Hohls, F., Mirovsky, P.,Mackrodt, B., Dolata, R., Weimann, T.,Pierz, K., and Schumacher, H.W. (2013)Counting statistics for electron capturein a dynamic quantum dot. Phys. Rev.Lett., 110, 126803 (5 pp).

57. Fricke, L., Wulf, M., Kaestner, B., Hohls,F., Mirovsky, P., Mackrodt, B., Dolata, R.,Weimann, T., Pierz, K., Siegner, U., andSchumacher, H.W. (2014) Self-referencedsingle-electron quantized current source.Phys. Rev. Lett., 112, 226803 (6 pp).


58. Keller, M.W. (2008) Current status of thequantum metrology triangle. Metrologia,45, 102–109.

59. Piquemal, F. and Geneves, G. (2000)Argument for a direct realization ofthe quantum metrological triangle.Metrologia, 37, 207–211.

60. Devoille, L., Feltin, N., Steck, B.,Chenaud, B., Sassine, S., Djordevic,S., Seron, O., and Piquemal, F. (2012)Quantum metrological triangle exper-iment at LNE: measurements on athree-junction R-pump using a 20 000:1

winding ratio cryogenic current com-parator. Meas. Sci. Technol., 23, 124011,(11 pp).

61. Zimmerman, N.M. and Eichenberger,A.L. (2007) Uncertainty budget forthe NIST electron counting capaci-tance standard, ECCS-1. Metrologia, 44,505–512.


155

7The Planck Constant, the New Kilogram, and the Mole

Mass is a difficult quantity. This becomes obvious when trying to explain in a fewwords what mass is about. Macroscopically, it is related to matter, though matteris not a clearly defined concept in physics. Mass (gravitational mass) is propor-tional to the weight of a body, which is the gravitational force imposed by thegravitational field of a second mass, for example, the earth. More strictly speak-ing, weight is any force which affects the free fall of a body. Mass (inertial mass)reflects the resistance of a body to change its velocity, and the force needed tochange its velocity is proportional to its mass. According to Einstein’s equivalenceprinciple, inertial mass and gravitational mass are identical. This has been con-firmed by experiments on the level of 10−12. Furthermore, mass is equivalent toenergy according to Einstein’s famous formula

E = m0c2 (7.1)

as reflected in the so-called mass deficit corresponding to the binding energy of acomposed system. In this equation, m0 is the mass of a body at rest. According tospecial relativity, the inertial mass depends on its velocity, v:

m(v) =m0

√

1 − v2

c2

. (7.2)

Macroscopic masses come about by the sum of the mass of the constitutingelementary particles reduced by the mass deficit. Yet, how elementary particlesreceive their mass has long been a puzzle. Within the standard model of particlephysics, elementary particles receive their mass through the interaction with theso-called Higgs field. It was named after Peter Higgs who together with FrancoisEnglert received the 2013 Nobel Prize in Physics after the discovery of a particle atthe European Organization for Nuclear Research (CERN) Large Hadron Colliderwhich might be the long-searched Higgs boson.

In spite of all these complications with the quantity mass, the definition of itsunit, kilogram, in the present SI system was apparently straightforward and sim-ple: it relates any mass to the mass of the international kilogram prototype (seeSection 2.2.3). The mass of the kilogram prototype was defined to be equal to themass of a cubic decimeter of pure water at the temperature of its highest densityof about 4 ∘C. It has been realized in the Kilogramme des Archives of the French


156 7 The Planck Constant, the New Kilogram, and the Mole

Academy of Sciences (for further reading of the history of the kilogram definitionand prototype, see [1, 2]). The choice of the inert Pt/Ir alloy as the material of theprototype should ensure a stable standard provided the appropriate handling andcleaning procedure would be applied. At the time of the first Conférence Généraledes Poids et Mesures (CGPM) in 1889, 30 Pt/Ir copies of the prototype had beenproduced and distributed among the 17 signature countries of the Meter Con-vention (Convention du Mètre) as their national mass standard and the BureauInternational des Poids et Mesures (BIPM). Countries joining the meter conven-tion at a later time (at the time of writing, there are 55 signatories of the MeterConvention) were also entitled to receive a Pt/Ir copy of the prototype. Subsequentcomparisons of the national prototypes with the international kilogram prototypeperformed in 1950 and 1990 revealed a problem with the apparently straightfor-ward and simple definition of the mass unit kilogram: obviously, there has been adrift between the mass of the international prototype and its copies by an averageof about 30 μg over 100 years with a trend toward an increase of the mass in themajority of national prototypes (see Figure 7.1).

It should be noted, however, that the measurements shown in Figure 7.1 cannotexclude a mass drift of the international prototype instead.

To overcome the obvious weakness of the present SI definition, it has beensuggested by the International Committee for Weights and Measures (CIPM) tonewly define the kilogram in terms of the Planck constant (see Section 2.2.3). This,at a first glance, might sound strange. However, it is fully in line with the intentionto base the SI units on constants of nature. The Planck constant, conventionallylabeled h, is definitely one of the fundamental constants of nature. It was intro-duced originally by Max Planck in 1900 when developing a theoretical descriptionof the emission spectrum of a so-called blackbody radiator [3]. Its consequencethat the energy of a harmonic oscillator has to be quantized in terms of E = hν(𝜈 being the frequency) laid the base for the quantum theory.

−1001880 1900 1920 1940

Year

1960 1980 2000

−80

−60

−40

−20

0

Δm/μ

g 20

40

60

80

100

Figure 7.1 Mass difference of different national kilogram prototypes (black) and BIPMworking standards (gray) and the international kilogram prototype defining the horizontalline at Δm= 0.

7 The Planck Constant, the New Kilogram, and the Mole 157

Actually, the strategy for linking the kilogram to the Planck constant is the sameas for the present definition of the meter which is linked to the speed of light. Abasic requirement besides those already mentioned in Section 2.2 is that the con-stant chosen (defining constant) includes in its unit the unit to be defined. Thisis true for the speed of light with the unit meter per second as well as for thePlanck constant, which has the unit of action m2 kg s−1 equal to J s. The questionon how fundamental a constant really is is not always easy to answer (see, e.g.,[4, 5]). Without question, however, the speed of light, c, and the Planck constant,h, are fundamental in the theory of relativity and quantum physics, respectively.Indeed, as first pointed out by Planck, c and h together with Newton’s gravita-tional constant, G, set up universal units for length, time, and mass (Planck units),which, however, turned out to be unpractical (see, e.g., [6]). Of course, as dis-cussed in Section 2.2, in practice, there are further requirements for the choiceof the respective constant. Most important, before definition, its value must bemeasured with the required uncertainty, and the link between the unit and therespective defining constant must be feasible experimentally in order to realizethe unit. However, it should be noted, that the definitions in the new SI leave roomfor different realizations. For the kilogram, the Consultative Committee for Massand Related Quantities (CCM) of the CIPM had required a relative uncertaintyfor the Planck constant of at least 2× 10−8 [7]. Furthermore, the CCM requiredthat at least three independent experiments yield consistent values for the Planckconstant with relative standard uncertainties not larger than 5 parts in 108.

Several experimental approaches have or had been pursued to provide thelink between a macroscopic mass and the Planck constant, for example, voltagebalance, superconducting magnetic levitation, the Avogadro experiment, and thewatt balance. The latter two will be described in a little more detail in Sections 7.1and 7.2.

In a voltage balance [8–10], the force between the electrodes of a capacitor witha known and traceable voltage applied to it is compared to the weight of a cali-brated mass. Though these experiments were initially performed to realize the voltor to determine the Josephson constant (see Section 4.1.2), they could also link amass to the Planck constant if the capacitance is traced back to the von Klitzingconstant (see Section 5.4.2). However, to our knowledge, these experiments havenot been pursued since the best relative uncertainty achieved was at the 10−7 level.A dynamic version of a voltage balance for measuring inertial mass and relatingit to the Planck constant has been developed at the Chinese metrology institute,the National Institute of Metrology (NIM) [11] (see Section 7.2). In the super-conducting magnetic levitation experiments [12–15], use is made of the idealdiamagnetic property of a superconductor (see Section 4.2.1.1): a superconduct-ing material with a calibrated mass is levitated in the magnetic field created by acurrent driven coil. Change of the current results in the levitation of the supercon-ducting mass at different heights. Measuring the current in terms of the Josephsonand quantum Hall effect then provides the link between mass and the Planck con-stant. Even though fractional uncertainties of order 10−6 had been achieved, theseexperiments also have not been further continued.


7.1The Avogadro Experiment

The Avogadro experiment [16, 17], sometimes also called the “X-ray crystal den-sity (XRCD)” experiment, is aimed at the determination of the Avogadro constant,NA, by counting the number of atoms in a mole of a high-purity Si single crystal.However, as will be shown later (see Equation 7.5), it also provides an indepen-dent approach for a precise determination of the Planck constant. It had originallybeen pursued to provide an alternative definition of the kilogram by tracing it toan exactly defined atomic mass, such as 12C or 28Si, or the mass of an elementaryparticle [18]. Since the relative masses of atoms and elementary particles such asthe electron can be determined very accurately using Penning traps, one basicallywould have been free in the choice of the reference mass.

Here, we would like to mention another experiment which also would providea direct link of a macroscopic to an atomic mass, the so-called ion accumulationexperiment [19]. Its idea has been to use a modified mass spectrometer where ionsof a specific element are generated and finally collected in a Faraday cup connectedto a balance. The moving ions correspond to a current, which can be measuredby the Josephson and quantum Hall effect (see Chapters 4 and 5) and integratedover the entire accumulation time, thereby providing the number of ions imping-ing onto the Faraday cup. The monoisotopic elements 197Au and 209Bi had beenused as ion sources. A proof of principle had been demonstrated by accumulat-ing 38 mg of Bi. The atomic mass unit determined from this experiment agreedwith the Committee on Data for Science and Technology (CODATA) value within9× 10−4 [20]. Nevertheless, foreseeing the difficulties encountered to reduce theuncertainty by more than 4 orders of magnitude, the experiment has not beenpursued further.

Coming back to the Avogadro experiment, the Avogadro constant is the numberof specified entities in the amount of substance of 1 mol in a pure substance, forexample, the number of atoms in 12 g of the carbon isotope 12C. It is a scalingfactor that links atomic and macroscopic properties. According to the definitionof the mole, we have, for example,

M(12C) = NAm(12C), (7.3)

where M(12C) = 12 g mol−1 is the molar mass of 12C and m(12C) is its atomic mass.Note that according to the proposed new definition of the mole (see Sections 2.2.6and 7.3), the exact equation M(12C) = 12 g mol−1 is no longer valid, but M(12C) hasto be determined experimentally.

For a perfect pure silicon single crystal, the Avogadro experiment relates itsmass, m, to the number of Si atoms contained in the crystal, NSi, and the massof the Si atom, mSi:

m = NSimSi = NSiMSiNA

, (7.4)

where MSi is the molar mass of Si. For a perfect single crystal, the number of atomscontained is given by its volume, V , divided by the volume occupied by one atom,

7.1 The Avogadro Experiment 159

V Si; thus,

NSi =V

VSi= 8V

a30

, (7.5)

where a0 is the lattice parameter (lattice constant) and thus a30 is the volume of

the Si crystal unit cell. The factor 8 accounts for the fact that in a perfect Si singlecrystal, the unit cell contains 8 Si atoms. Combining Eqs (7.5) and (7.4) yields

m = 8Va3

0

MSiNA

. (7.6)

As Si has three stable isotopes, 28Si, 29Si, and 30Si, the molar mass, MSi, is given bythe sum of the molar mass of the isotopes weighted by their (amount of substance)abundances, fi:

MSi =∑

ifiMi

Si =∑

ifiAi

rMu (7.7)

with Air being the relative atomic mass and Mu being the molar mass unit. For

natural Si, the abundances are about f28 = 0.922, f29 = 0.047, and f30 = 0.031. Ifwe now finally consider that the molar Planck constant, NAh, is given by

NAh = α2c2R∞

Aer Mu, (7.8)

we end up with the basic relation linking a macroscopic mass to the Planck con-stant [21]

m = 8Va3

0

2R∞hcα2

∑

i

fiAir

Aer

, (7.9)

where R∞ is the Rydberg constant, c the velocity of light in vacuum, 𝛼 thefine-structure constant, and Ae

r the relative atomic mass of the electron. Notethat the Avogadro constant does not enter explicitly, yet, the name “Avogadroexperiment” is still kept. For the realization of the kilogram according to Eq.(7.9), the volume of the single crystal, the lattice parameter, and the isotopiccomposition have to be measured. Of course, also the mass of the sphere has tobe measured with the required uncertainty for the determination of the Planckconstant. The relative atomic masses are measured by comparing cyclotronfrequencies in Penning traps. Since none of these quantities requires traceabilityto the unit of mass, the Avogadro experiment represents a primary realization ofthe newly defined kilogram. The constants appearing additionally in Eq. (7.9) areknown with sufficiently small uncertainty: the speed of light in vacuum is definedand thus exact. The Rydberg constant, fine-structure constant, and relativeatomic mass of the electron are known with relative standard uncertainties of5× 10−12, 3.2× 10−10, and 4× 10−10 [22], respectively, and thus, their uncertaintycontribution can be neglected at the required 10−8 level.

We shall now discuss the individual measurements a little more in detail,considering also additional constrains. Silicon has become the material of choicedue to its use in microelectronics where large-sized high-purity and almost


perfect crystals can be synthesized. However, in view of the small fractionaluncertainty of the order 10−8 required for the realization of the new kilogram,crystal perfection and purity (i.e., defect (vacancy) and impurity concentration)have to be investigated. The major impurities to be considered, after growthand purification by multiple float zone crystallization, are interstitial oxygenand substitutional carbon and boron. Their concentration can be determinedby infrared spectroscopy [23]. An estimate of the vacancy concentration can beobtained by positron annihilation experiments [24].

In view of the most precise determination of the volume of the macroscopicSi single crystal, a sphere has been chosen (see Figure 7.2). The volume then canbe determined by a series of diameter measurements scanning the entire surface.The diameters are measured by optical interferometry [25, 26]. The layout of aspecially constructed spherical Fizeau interferometer is schematically shown inFigure 7.3. The central part consists of a temperature-controlled vacuum chambercontaining the sphere and the Fizeau optics. The two arms of the interferome-ter are illuminated by plane wave light coming from tunable diode lasers throughmultimode optical fibers. The Fizeau objectives are carefully adjusted to have theirfocal point in the center of the sphere. First, the diameter of the empty etalon, D,and subsequently the distances between the sphere surface and the reference sur-faces, d1 and d2, are measured. The diameter of the sphere, d, is then obtainedby subtracting d1 and d2 from D. With this technique, some 10 000 diameterscan be measured simultaneously depending on the resolution of the camera sys-tem. The sphere can be rotated around the horizontal and vertical axis to coverit completely by overlapping diameter measurements. The obtainable uncertaintydepends critically on how well the shape of the sphere matches the wave frontof the interferometer light. Having high-quality objectives and production of analmost perfect sphere are thus the most critical issues. Furthermore, the surface

Figure 7.2 Photo of a single crystal Si sphere used in the Avogadro experiment. The diam-eter and mass, respectively, are about 10 cm and 1 kg. (Courtesy of PTB.) (Please find a colorversion of this figure on the color plate section.)


Vacuum casingmK controlled

Camera

Collimator

Multi-modefibre

Beamsplitter

Diode laser

d = D − d1 - d

2

d d2

Dd

1

Fizeauobjective

Figure 7.3 Schematic drawing of the spherical Fizeau interferometer constructed at PTB.(Courtesy of A. Nicolaus, PTB.)

of the sphere generally is covered by different surface layers, in particular siliconoxide, which not only have to be considered for the mass correction but also inevaluating the interferometry results, due to their different index of refraction andresulting phase shifts.

The measured diameter topography of a silicon sphere is shown in Figure 7.4, forexample. Peak-to-valley deviations from a perfect sphere are of the order of some

Figure 7.4 Diameter variations of a single crystal Si sphere. (Courtesy of A. Nicolaus, PTB.)(Please find a color version of this figure on the color plate section.)


10 nm, resulting in uncertainty of the volume determination of presently about2× 10−8 [27, 28].

As mentioned before, the composition and thickness of the surface layer of theSi spheres must be determined for both the mass correction and volume determi-nation. It is important to note that for the mass correction, only the relative massof the respective elements with respect to Si enters, and thus, the traceability toa mass standard is not required. The standard methods applied for thickness andoptical constants measurement are X-ray reflectometry (XRR) and optical spectralellipsometry (SE). However, since the surface layer not only may contain differentsilicon oxides (SiOx) but possibly also chemisorbed water and other contaminants,analytical methods, in particular X-ray photoelectron spectroscopy (XPS), X-rayfluorescence (XRF), and near edge X-ray absorption fine structure (NEXAFS), areapplied. Combining the results of the individual experiments enabled to developa detailed model of the surface layer [29] and to estimate its contribution to theoverall fractional uncertainty of the experiment to presently about 10−8.

For the measurement of the lattice parameter, a combined optical and X-rayinterferometer is used [30], as schematically shown in Figure 7.5. The X-ray inter-ferometer follows the design by Bonse and Hart [31]. It consists of three parallelsingle crystal Si plates, each of about 1 mm thickness and separated by the sameamount. The surface of these plates is orthogonal to the lattice plans to be mea-sured. In the case of Si, the spacing d220 of the {2 2 0} planes is measured becauseof their low absorption. The lattice parameter a0 is then obtained according toa0 =

√8d220.

The operation principle of the X-ray interferometer is illustrated in Figure 7.6.The first plate (labeled S) acts as a beam splitter for the incident X-ray due to Braggreflection at the crystal planes. The two other plates (M and A) act as transmissionoptics, where the two plane waves generated by plate S are recombined by plateM at the position of the analyzer (plate A). Moving the analyzer orthogonal to

Mo Kα

Opticalinterferometer

Multianodedetector

Displacement

PhotodiodeNd:YAG

Fixed crystal Analyzer

Figure 7.5 Schematic layout of the combined optical and X-ray interferometer for measur-ing the lattice constant of crystalline Si. (Courtesy of E. Massa, G. Mana, INRIM.) (Please finda color version of this figure on the color plate section.)


S

M

A

D

d

Figure 7.6 Operation principle of aBonse–Hart X-ray interferometer. The threesingle crystal plates labeled S, M, and A act,respectively, as beam splitter and transmis-sion optics for the incident X-ray. The crystal

planes are indicated by the black dots rep-resenting the atom position (not to scale).The crystal plates must have equal thickness,and the spacing between them must be thesame.

Figure 7.7 Photo of the central part ofthe INRIM X-ray interferometer showingthe Si crystal plates right in the center. Theexperiment is performed in a temperature-stabilized vacuum chamber. A Mo K

𝛼X-ray

source and an iodide-stabilized single-mode

He–Ne laser are used for the X-ray and opti-cal interferometer, respectively. (Courtesyof E. Massa, G. Mana, INRIM.) (Please find acolor version of this figure on the color platesection.)

the direction of the lattice planes causes a periodic modulation of the transmittedand diffracted beams (Moiré effect) with a period of the lattice spacing, d, andindependent of the X-ray wavelength. The central part of the X-ray interferome-ter of the Italian metrology institute INRIM, Torino, [30] is shown in Figure 7.7.Considering the effect of point defects on the lattice parameter, the average latticeconstant of a macroscopic Si sphere can be determined presently with a fractionaluncertainty of the order of 10−9 [30] corresponding to attometer resolution.


Finally, the ratio of the atomic masses of silicon with its specific isotope compo-sition and the electron (factor

∑i

fiAir∕Ae

rin Eq. (7.9)) has to be determined, which

is basically a molar mass determination of the Si crystal (c.f. Eq. (7.7)). The stan-dard technique for molar mass determination is by gas mass spectrometry, whichof course requires the crystal to be dissolved and transferred to a gaseous com-pound by chemical reactions [32, 33]. Actually, it turned out in the frame of thedetermination of the Avogadro constant that the uncertainty of the molar massdetermination of natural Si of about 3× 10−7 limited the achievable total uncer-tainty to about 10−7 [34]. This was the start of an international research effort toproduce a high-purity single crystal with highly enriched 28Si [35] where 29Si and30Si make only a small correction to the molar mass. In fact, for a crystal withf 28 ∼ 0.9999, the uncertainty of the molar mass could in principle be reduced byseveral orders of magnitude as compared to natural Si [36].

The production of the high-quality 28Si single crystal proceeded in several stepsstarting with the enrichment of SiF4 gas by centrifugation at the Science and Tech-nical Center (Centrotech) in St Petersburg, Russia, from which, after conversioninto SiH4, a polycrystal was grown at the Institute of Chemistry of High-PuritySubstances of the Russian Academy of Sciences (ICHPS RAS) in Nizhny Nov-gorod. Finally, the polycrystal was transformed into a 5 kg 28Si single crystal byfloating zone (FZ) single crystal growth at the Leibniz Institute for Crystal Growth(IKZ) in Berlin, Germany. Two precise spheres were produced from the singlecrystal rod at the CSIRO, Australia [37], for the subsequent determination of theAvogadro constant [27, 28].

Nevertheless, the molar mass of the enriched 28Si had to be measured. This wasdone by a modified isotope dilution mass spectrometry (IDMS) in combinationwith a multicollector inductively coupled plasma (ICP) mass spectrometer [38].In IDMS, a spike with an isotope of the substance to be determined is added tothe sample to be analyzed. Since the chemical behavior of the isotopically markedsubstance and the nonmarked substance is identical, their peak ratio of the massspectrometer signal reflects the mass ratio of both. As the mass of the spike canbe measured before adding it, this serves as a calibration, and the unknown massfraction can be determined. The basic idea of the IDMS molar mass determinationof the enriched 28Si crystal was to treat the sum of 29Si and 30Si as a virtual ele-ment in the sample. Preparing gravimetrically a blend with a ratio R(30Si/29Si)∼ 1for calibration and adding a spike of a highly enriched 30Si crystal to the samplethen enabled to determine the mass fraction of all three isotopes by measuring theamount ratio R(30Si/29Si) in the original and spiked sample [39–41]. This proce-dure avoided to explicitly measure the very small amount ratio R(29Si/28Si) andR(30Si/28Si), which could hardly be measured with the required precision [39].With this new approach, the molar mass of the enriched Si crystal has been deter-mined with fractional uncertainty of 8.2× 10−9 [39].

On the basis of the present results, the Avogadro experiment with an enriched28Si single crystal definitely has the potential for a primary realization of the newdefinition of the kilogram with an uncertainty of ≤2× 10−8, thus fulfilling therequirement of the Consultative Committee for Units (CCU) and CIPM. It is

7.2 The Watt Balance Experiment 165

worth mentioning that the production of high-quality enriched 28Si crystals notonly has given a boost to future mass metrology but also to other areas in science,such as quantum information technology [42].

7.2The Watt Balance Experiment

The watt balance experiment [43–46] also provides a direct link between a macro-scopic mass and the Planck constant. It compares mechanical and electrical power.The basic idea for this experiment was first proposed by Kibble [47] and realizedby Kibble et al. at the NPL, UK [48], and by Olson et al. at the NBS/NIST, Gaithers-burg, USA [49, 50].

The watt balance experiment is performed in two phases. The principle of theexperiment is illustrated by means of Figure 7.8. Consider two coils: one (coil 1)carrying current, I1, is fixed, and the other one (coil 2) carrying current, I2, ismovable in the vertical direction (upper left part of Figure 7.8). The vertical (z)component of the force imposed on the movable coil, Fz, then is given by

Fz = I2∂Φ12∂z

, (7.10)

where ∂Φ12∕∂z is the vertical gradient of the magnetic flux generated by the cur-rent thru coil 1. This force can be balanced (force mode) by connecting coil 2 to abalance loaded with an appropriate mass, such that mg = −Fz (g is the local grav-itational acceleration). Thus,

mg = −I2∂Φ12∂z

. (7.11)

In the second phase (velocity mode), the second coil is an open circuit, and theinduced voltage is measured when moving it vertically with a constant velocity,vz. The induced voltage then is given by

U2 = −∂Φ12∂t

= −∂Φ12∂z

∂z∂t

= −∂Φ12∂z

vz. (7.12)

Combining Eqs (7.11) and (7.12) then yields

mgvz = I2U2. (7.13)

(The subscript 2 will be abolished in the following.) This equation equals mechan-ical and electrical power which is reflected in the label “watt balance.”

The same result is obtained for a geometry as indicated in the right-hand partof Figure 7.8 where a coil with wire length L is placed in a horizontal, purely radialmagnetic field Br. ∂Φ12∕∂z in Eqs (7.10)–(7.12) is then to be replaced by −BrL.

Since with the exception of the BIPM experiment (see later text) the experimentis split into these two phases, it is actually a virtual comparison of the electrical andmechanical watt. Note, however, that Eq. (7.10) actually is only one component of avector equation. Neglecting the other components implies tremendous constrainson the alignment of the experiment.


I2

UU

Br

Br

I2

I1

I1

Fz

vz

vz

Fz

‘‘Force mode’’

‘‘Velocity mode’’

Figure 7.8 Principle of the watt balance experiment.

If the voltage, U , is measured against a Josephson voltage standard, it can beexpressed as (see Chapter 4)

U = C1UJ,1 = C1ifJ,1K−1J = C1ifJ,1

(h2e

)

, (7.14)

where i is an integer number (Shapiro step number), fJ,1 the Josephson frequency,KJ the Josephson constant, and C1 the calibration factor. The current, I, can bemeasured as the voltage drop across a resistor, R. Measuring voltage and resistancein terms of the Josephson and von Klitzing constant, respectively, then yields

I = UR

=C2UJ,2

C31n

RK=

C2jfJ,2

(h2e

)

C31n

(he2

) =C2C3

jn e2

fJ,2, (7.15)

where j and n are integer numbers, j denoting again the respective Shapiro stepand n labeling the quantum Hall plateau (filling factor), and RK is the von Klitzingconstant. Combining Eqs (7.13)–(7.15) yields

m = C4

fJ,1fJ,2h

gvz, (7.16)

where C is a combination of the different calibration factors multiplied withthe integer numbers i, j, and n. Equation (7.16) is the fundamental watt balanceequation relating a macroscopic mass to the Planck constant correspondingto Eq. (7.9) in the Avogadro experiment. Since none of the quantities on theright-hand side of Eq. (7.16) require traceability to a mass standard, the wattbalance experiment is also a primary realization of the kilogram. However, itmust be noted that it has been implied that the Josephson constant, K J, andthe von Klitzing constant, RK, respectively, are exactly 2e∕h and h∕e2, which once

7.2 The Watt Balance Experiment 167

again points out the importance of the metrological triangle experiment (seeSection 6.3). Thus, the quantities to be measured in the watt balance experimentare the Josephson frequencies fJ,1 and fJ,2; the gravitational acceleration, g; andthe velocity, vz, given the numbers of the Shapiro step i and j and the filling factorn used in the calibration. For the determination of the Planck constant the masshas to be measured as well.

The essential ingredients of a watt balance experiment then are [44]:

• A suitable balance also allowing the required alignment (a detailed descriptionof the alignment procedure can be found in Ref. [51]).

• A magnet providing the magnetic flux which could be a permanent magnet, anelectromagnet (mostly superconducting solenoids), or a combination of both.

• A setup for the velocity measurement. For this, the movement of the coil isdetected by an interferometer, usually operated in vacuum to avoid uncertain-ties due to the refractive index of air.

• A Josephson and quantum Hall standard for measuring the current in the forcemode and the voltage induced in the coil in the velocity mode.

• A gravimeter to measure the gravitational acceleration and its spatial profile.

In addition, sensors and actors are required to monitor and control the align-ment.

At the time of writing, seven laboratories worldwide had a watt balance inoperation or under construction, namely, the National Institute of Standards andTechnology (NIST), the National Research Council (NRC), the Federal Instituteof Metrology (METAS), the French Metrology Institute (LNE), the MeasurementStandards Laboratory (MSL), the Korea Research Institute of Standards andScience (KRISS), and the BIPM. Though based on the same underlying principledescribed earlier, they differ in their specific design as described in detail inrecent reviews [44–46]. An example is the schematic drawing of the NIST wattbalance (NIST-3) shown in Figure 7.9 [53, 54, 52]. In the present version, asuperconducting magnet providing a radial magnetic field with a flux density of0.1 T is installed, and a wheel balance with a test mass of 1 kg is used. The originalNPL watt balance used a beam balance and permanent magnet [48]. Followingthe decision of NPL to stop the watt balance project, the latest version of theNPL Mark II watt balance was transferred to the NRC, and the first results werereported in 2012 [55].

The specific features of the METAS watt balance were to separate the force andvelocity mode and to use a 100 g test mass instead of 1 kg. Furthermore, a paral-lel and homogeneous horizontal magnetic field generated by two flat poles of aSmCo permanent magnet was used [44]. The first results have been reported byEichenberger et al. [56].

The LNE watt balance experiment [57, 58] uses atomic interferometry forgravimetry [59], a special guiding stage to ensure motion of the coil along the ver-tical axis, and a programmable Josephson array associated with a programmablebias source as voltage reference [60].


Knife edge Balance wheel

Multi-filamentband

Countermass

Velocity modemotor

Moving coil

Interferometer(1 of 3)

NorthTrimcoil

West

z

1 m

Uppersuperconductingsolenoid

Spider

Mass

Interfero-meter

Trimcoil

Stationarycoils

Lowersuper-conductingsolenoid

Figure 7.9 Schematic drawing of the NISTwatt balance. The magnetic field is createdby two superconducting solenoids wired inseries opposition creating a magnetic fluxdensity at the moving coil radius of about

0.1 T. The trim coil is used to achieve a 1/rdependence of the field [52]. (Courtesy ofNIST.) (Please find a color version of thisfigure on the color plate section.)

The specific approach of the BIPM watt balance is to carry out simultaneouslythe force mode and velocity mode [61, 62]. A major challenge in this scheme is toseparate the induced voltage due to the motion of the coil from the resistive volt-age drop due to the simultaneously flowing balance current. One possible way toovercome this could be to employ a superconducting moving coil [63]. The balancecurrent driven through the coil in the force mode would then not cause a voltagedrop, and the measured voltage solely would be due to the induced voltage of thevelocity mode. The present activities at BIPM, however, are focused on the devel-opment of a room-temperature version which enables both the simultaneous (onephase) and the conventional two-phase operations [45, 62].

At this end, METAS, NPL, NIST, and NRC have reported values of the Planckconstant with uncertainties in the range of 1.9× 10−8 (NRC) [64] to 2.9× 10−7

(METAS) [56]. However, the individual results are, at present, not fully consis-tent. Further experiments definitely will resolve this, and then the watt balancealso has the potential for a primary realization of the new kilogram.

For the sake of completeness, the aforementioned experiment (called Joulebalance) at the NIM, China, has to be included here, although it differs from

7.3 The Mole: Unit of Amount of Substance 169

the watt balance experiments discussed so far. It follows the design of anelectrodynamometer and operates in the force mode only equating the magneticenergy difference and gravitational potential energy difference between twoknown vertical positions of a coil [65]. The electromagnetic force to compensatethe weight of the test mass is created by two coils aligned parallel to each other,which requires the determination of a mutual inductance between the two coils.An uncertainty for the Planck constant of 8.9 ppm has been achieved so far, yet,further improvement seems possible [66].

To summarize, there are presently at least two experiments which have thepotential to realize the kilogram according to its new definition, the watt balanceand the Avogadro experiment. Both, however, are large-scale experiments andprobably not in a position to provide day-to-day calibrations of secondarystandards. Therefore, the CGPM in 2011 encouraged the BIPM to develop a poolof reference standards to facilitate the dissemination of the unit of mass afterredefinition [67]. This pool will include Pt/Ir as well as Si sphere reference massesstored under special conditions. From these reference masses, a mean mass ofthe ensemble will be calculated, giving each of them a statistical weight reflectingits stability. Traceability according to the new definition will be maintained bycalibrating one or more of the reference masses by a primary realization of thekilograms. The ensemble will then be used for the worldwide dissemination ofthe kilogram and, thus, in some sense will replace the old international kilogramprototype.

7.3The Mole: Unit of Amount of Substance

The mole is the unit of the quantity amount of substance and one of the base unitsof the SI, sometimes called the “SI unit of chemists” even though there is an ongo-ing dispute among chemists on the usefulness of the mole. It is used to quantifyan ensemble of entities in a thermodynamic sense (like in the ideal gas equationpV = nRT) and to quantify entities in stoichiometric chemical reactions [68]. Thepresent definition is based on a fixed value of the molar mass of 12C, M(12C):

M(12C) = Ar(12C)Mu = 0.012 kg mol−1 (7.17)

with Ar(12C) and Mu = 10−3 kg mol−1 being the relative atomic mass of 12C and themolar mass unit, respectively. This definition links the mole to the kilogram. In theproposed new definition, the magnitude of the mole is set by fixing the numericalvalue of the Avogadro constant, NA. A mole then is the amount of substance ofa system that contains NA-specified entities. Thus, the dependence on the kilo-gram definition is abandoned. Yet, as a consequence, the molar mass of 12C is nolonger exact but has an uncertainty equal to the uncertainty of the molar massunit Mu. The uncertainty of Mu can be analyzed on the basis of Eq. (7.8) whichcan be rewritten:

Mu =2NAh

cR∞

α2Aer

. (7.18)


Since NA, h, and c will be exact in the new SI, the uncertainty will be given bythe second factor, and since me = 2hR∞∕cα2, it is equal to the uncertainty of themass of the electron, me, which according to the 2010 CODATA evaluation [22] is4.4× 10−8. This, in general, will add only a minor contribution to the molar massuncertainty of any atom or molecule X

M(X) = Ar(X)Mu (7.19)

as well as to the most widely used method to determine the amount of substance,n, through weighing according to

n = mAr(X)Mu

. (7.20)

The realization of the mole will still be achieved via a primary direct method asdescribed, for example, in [69]. Thus, in essence, the daily life of an analyticalchemist will not be changed, yet the new definition would add to the consistencyof the new SI and make clear the distinction between the quantities of amount ofsubstance and mass.

References

1. Quinn, T. (2012) From Artifacts toAtoms, Oxford University Press, Oxford,New York.

2. Davis, R. (2003) The SI unit of mass.Metrologia, 40, 299–305.

3. Planck, M. (1900) Zur Theorie desGesetzes der Energieverteilung im Nor-malspektrum. Verh. Dtsch. Phys. Ges., 2,237–245 (in German).

4. Flowers, J. and Petley, B. (2004) inAstrophysics, Clocks and FundamentalConstants (eds S.G. Karshenboim and E.Peik), Springer, Berlin, Heidelberg, pp.75–93.

5. Lévy-Leblond, J.-M. (1979) in Problemsin the Foundations of Physics; Proceed-ings of the International School of Physics“Enrico Fermi” Course LXXXII (ed G.Toraldo di Francia), North Holland,Amsterdam, pp. 237–263.

6. Okun, L.B. (2004) in Astrophysics, Clocksand Fundamental Constants (eds S.G.Karshenboim and E. Peik), Springer,Berlin, Heidelberg, pp. 57–74.

7. Resolution CCM/13-31ahttp://www.bipm.org/cc/CCM/Allowed/14/31a_Recommendation_CCM_G1%282013%29.pdf (accessed 15November 2014).

8. Clothier, W.K., Sloggett, G.J.,Bairnsfather, H., Currey, M.F., andBenjamin, D.J. (1989) A determination ofthe volt. Metrologia, 26, 9–46.

9. Bego, V., Butorac, J., and Illic, D. (1999)Realization of the kilogram by measur-ing at 100 kV with the voltage balanceETF. IEEE Trans. Instrum. Meas., 48,212–215.

10. Funck, T. and Sienknecht, V. (1991)Determination of the volt with theimproved PTB voltage balance. IEEETrans. Instrum. Meas., 40, 158–161.

11. Li, S., Zhang, Z., He, Q., Li, Z., Lan,J., Han, B., Lu, Y., and Xu, J. (2013)A proposal for absolute determina-tion of inertial mass by measuringoscillation periods based on the quasi-elastic electrostatic force. Metrologia, 50,9–14.

12. Frantsuz, E.T., Gorchakov, Y.D., andKhavinson, V.M. (1992) Measurementsof the magnetic flux quantum, Planckconstant, and elementary charge atVNIIM. IEEE Trans. Instrum. Meas., 41,482–485.

13. Frantsuz, E.T., Khavinson, V.M.,Genevès, G., and Piquemal, F. (1996)A proposed superconducting magneticlevitation system intended to monitor

http://www.bipm.org/cc/CCM/Allowed/

References 171

stability of the unit of mass. Metrologia,33, 189–196.

14. Shiota, F. and Hara, K. (1987) A study ofa superconducting magnetic levitationsystem for an absolute determination ofthe magnetic flux quantum. IEEE Trans.Instrum. Meas., 36, 271–274.

15. Shiota, F., Miki, Y., Fujii, Y., Morokuma,T., and Nezu, Y. (2000) Evaluation ofequilibrium trajectory of superconduct-ing magnetic levitation system for thefuture kg unit of mass. IEEE Trans.Instrum. Meas., 49, 1117–1121.

16. Becker, P. (2001) History and progressin the accurate determination of theAvogadro constant. Rep. Prog. Phys., 64,1945–2008.

17. Becker, P. and Bettin, H. (2011) TheAvogadro constant: determining thenumber of atoms in a single-crystal 28Sisphere. Philos. Trans. R. Soc. London,Ser. A, 369, 3925–3935.

18. Becker, P. (2003) Tracing the defini-tion of the kilogram to the Avogadroconstant using a silicon single crystal.Metrologia, 40, 366–375.

19. Gläser, M. (2003) Tracing the atomicmass unit to the kilogram by ion accu-mulation. Metrologia, 40, 376–386.

20. Schlegel, C., Scholz, F., Gläser, M.,Mecke, M., and Bethke, G. (2007) Accu-mulation of 38 mg of bismuth in acylindrical collector from a 2.5 mA ionbeam. Metrologia, 44, 24–28.

21. Stenger, J. and Göbel, E.O. (2012) Thesilicon route to a primary realizationof the new kilogram. Metrologia, 49,L25–L27.


23. Zakel, S., Wundrack, S., Niemann,H., Rienitz, O., and Schiel, D. (2011)Infrared spectrometric measurementof impurities in highly enriched 28Si.Metrologia, 48, S14–19.

24. Gebauer, J., Rudolf, F., Polity, A.,Krause-Rehberg, R., Martin, J., andBecker, P. (1999) On the sensitivity limitof positron annihilation: detection ofvacancies in as-grown silicon. Appl.Phys. A, 68, 411–416.

25. Kuramoto, N., Fujii, K., and Yamazawa,K. (2011) Volume measurement of 28Sispheres using an interferometer with aflat etalon to determine the Avogadroconstant. Metrologia, 48, S83–95.

26. Bartl, G., Bettin, H., Krystek, M., Mai,T., Nicolaus, A., and Peter, A. (2011)Volume determination of the Avo-gadro spheres of highly enriched 28Siwith a spherical Fizeau interferometer.Metrologis, 48, S96–103.

27. Andreas, B., Azuma, Y., Bartl, G., Becker,P., Bettin, H., Borys, M., Busch, I., Gray,M., Fuchs, P., Fujii, K., Fujimoto, H.,Kessler, E., Krumrey, M., Kuetgens, U.,Kuramoto, N., Mana, G., Manson, P.,Massa, E., Mizushima, S., Nicolaus, A.,Picard, A., Pramann, A., Rienitz, O.,Schiel, D., Valkiers, S., and Waseda,A. (2011) Determination of the Avo-gadro constant by counting the atomsin a 28 Si crystal. Phys. Rev. Lett., 106,030801-1–030801-4.

28. Andreas, B., Azuma, Y., Bartl, G., Becker,P., Bettin, H., Borys, M., Busch, I., Fuchs,P., Fujii, K., Fujimoto, H., Kessler, E.,Krumrey, M., Kuetgens, U., Kuramoto,N., Mana, G., Massa, E., Mizushima,S., Nicolaus, A., Picard, A., Pramann,A., Rienitz, O., Schiel, D., Valkiers, S.,Waseda, A., and Zakel, S. (2011) Count-ing the atoms in a 28Si crystal for anew kilogram definition. Metrologia, 48,S1–13.

29. Busch, I., Azuma, Y., Bettin, H., Cibik,L., Fuchs, P., Fujii, K., Krumrey, M.,Kuetgens, U., Kuramoto, N., andMizushima, S. (2011) Surface layerdetermination for the Si spheres ofthe Avogadro project. Metrologia, 48,S62–82.

30. Massa, E., Mana, G., Kuetgens, U., andFerroglio, L. (2011) Measurement of the{220} lattice-plane spacing of a 28Si x-rayinterferometer. Metrologia, 48, S37–43.

31. (a) Bonse, U. and Hart, M. (1965) Anx-ray interferometer. Appl. Phys. Lett.,6, 155–156; (b) Bonse, U. and Hart, M.(1965) Principles and design of Laue-case x-ray interferometer. Z. Phys., 188,154–164.

32. De Bièvre, P., Lenaers, G., Murphy,T.J., Peiser, H.S., and Valkiers, S. (1995)


The chemical preparation and charac-terization of specimens for "absolute"measurements of the molar mass of anelement, exemplified by silicon, for rede-terminations of the Avogadro constant.Metrologia, 32, 103–110.

33. Bulska, E., Drozdov, M.N., Mana, G.,Pramann, A., Rienitz, O., Sennikov, P.,and Valkiers, S. (2011) The isotopic com-position of enriched Si: a data analysis.Metrologia, 48, S32–36.

34. Becker, P., Bettin, H., Danzebrink, H.-U.,Gläser, M., Kuetgens, U., Nicolaus, A.,Schiel, D., DeBievere, P., Valkiers, S., andTaylor, P. (2003) Determination of theAvogadro constant via the silicon route.Metrologia, 40, 271–287.

35. Becker, P., Schiel, D., Pohl, H.-J.,Kaliteevski, A.K., Godisov, O.N.,Churbanov, M.F., Devyatykh, G.G.,Gusev, A.V., Bulanov, A.D., Adamchik,S.A., Gavva, V.A., Kovalev, I.D.,Abrosimov, N.V., Hallmann-Seiffert,B., Riemann, H., Valkiers, S., Taylor,P., DeBievre, P., and Dianov, E.M.(2006) Large-scale production of highlyenriched 28Si for the precise determina-tion of the Avogadro constant. Meas. Sci.Technol., 17, 1854–1860.

36. Becker, P., Friedrich, H., Fujii, K.,Giardini, W., Mana, G., Picard, A.,Pohl, H.-J., Riemann, H., and Valkiers, S.(2009) The Avogadro constant determi-nation via enriched silicon-28. Meas. Sci.Technol., 20, 092002-1–092002-20.

37. Leistner, A. and Zosi, G. (1987) Polish-ing a 1 kg silicon sphere for a densitystandard. Appl. Opt., 26, 600–601.

38. Rienitz, O., Pramann, A., and Schiel,D. (2010) Novel concept for the massspectrometric determination of absoluteisotopic abundances with improved mea-surement uncertainty: part I. theoreticalderivation and feasibility study. Int. J.Mass Spectrom., 289, 47–53.

39. Pramann, A., Rienitz, O., Schiel, D.,Schlote, J., Güttler, B., and Valkiers,S. (2011) Molar mass of silicon highlyenriched in 28Si determined by IDMS.Metrologia, 48, S20–25.

40. Mana, G. and Rienitz, O. (2010) Thecalibration of Si isotope-ratio mea-surements. Int. J. Mass Spectrom., 291,55–60.

41. Mana, G., Rienitz, O., and Pramann,A. (2010) Measurement equations forthe determination of the Si molar massby isotope dilution mass spectrometry.Metrologia, 47, 460–463.

42. Saeedi, K., Simmons, S., Salvali, J.Z.,Dluhy, P., Riemann, H., Abrosimov,N.V., Becker, P., Pohl, H.-J., Morton, J.L.,and Thewalt, M.L.W. (2013) Room-temperature quantum bit storageexceeding 39 minutes using ionizeddonors in Silicon-28. Science, 342,830–833.

43. Eichenberger, A., Jeckelmann, B., andRichard, P. (2003) Tracing Planck’sconstant to the kilogram by elec-tromechanical methods. Metrologia,40, 356–365.

44. Eichenberger, A., Genevès, G., andGournay, P. (2009) Determination ofthe Planck constant by means of a wattbalance. Eur. Phys. J. Spec. Top., 172,363–383.

45. Stock, M. (2011) The watt balance:determination of the Planck constantand redefinition of the kilogram. Phi-los. Trans. R. Soc. London, Ser. A, 369,3936–3953.

46. Special issue Watt and joule balances,the Planck constant and the kilogram,(Robinson, I.A. (Ed.)) (2014) Metrologia,51 (2).

47. Kibble, B.P. (1976) in Atomic Massesand Fundamental Constants, vol. 5 (edsJ.H. Sanders and A.H. Wapstra), PlenumPress, New York, pp. 545–551.

48. Kibble, B.P., Robinson, I.A., and Bellis,J.H. (1990) A realization of the SIwatt by the NPL moving-coil balance.Metrologia, 27, 173–192.

49. Olsen, P.T., Bower, V.E., Phillips, W.D.,Williams, E.R., and Jones, G.R. (1985)The NBS absolute ampere experi-ment. IEEE. Trans. Instrum. Meas.,34, 175–181.

50. Olsen, P.T., Elmquist, R.E., Phillips, W.D.,Williams, E.R., Jones, G.R., and Bower,V.E. (1989) A measurement of the NBSelectrical watt in SI units. IEEE Trans.Instrum. Meas., 38, 238–244.

51. Robinson, I.A. and Kibble, B.P. (2007)An initial measurement of Planck’sconstant using the NPL Mark II wattbalance. Metrologia, 44, 427–440.

References 173

52. Schlamminger, S., Haddad, D., Seifert, F.,Chao, L., Newell, D.B., Liu, R., Steiner,R.L., and Pratt, J.R. (2014) Determina-tion of the Planck constant using a wattbalance with a superconducting magnetsystem at the National Institute of Stan-dards and Technology. Metrologia, 51,S15–S24.

53. Steiner, R., Newell, D., and Williams, E.(2005) Details of the 1998 watt balanceexperiment determining the Planck con-stant. J. Res. Natl. Inst. Stand. Technol.,110, 1–26.

54. Steiner, R.L., Williams, E.R., Newell,D.B., and Liu, R. (2005) Towards anelectronic kilogram: an improved mea-surement of the Planck constant andelectron mass. Metrologia, 42, 431–441.

55. Steele, A.G., Meija, J., Sanchez, C.A.,Yang, L., Wood, B.M., Sturgeon, R.E.,Mester, Z., and Inglis, A.D. (2012) Rec-onciling Planck constant determinationsvia watt balance and enriched-siliconmeasurements at NRC Canada. Metrolo-gia, 49, L8–10.

56. Eichenberger, A., Baumann, H.,Jeanneret, B., Jeckelmann, B., Richard,P., and Beer, W. (2011) Determinationof the Planck constant with the METASwatt balance. Metrologia, 48, 133–141.

57. Geneves, G., Gournay, P., Gosset, A.,Lecollinet, M., Villar, F., Pinot, P.,Juncar, P., Clairon, A., Landragin, A.,Holleville, D., Dos Santos, F.P., David, J.,Besbes, M., Alves, F., Chassagne, L., andTopcu, S. (2005) The BNM Watt balanceproject. IEEE Trans. Instrum. Meas., 54,850–853.

58. Gournay, P., Geneves, G., Alves, F.,Besbes, M., Villar, F., and David, J.(2005) Magnetic circuit design for theBNM Watt balance experiment. IEEETrans. Instrum. Meas., 54, 742–745.

59. Pereira dos Santos, F., Le Gouet, J.,Mehlstäubler, T., Merlet, S., Holleville,D., Clairon, A., and Landragin, A. (2008)Gravimètre à atoms froids. Rev. Fr.Métrol., 13, 33–40, (in French).

60. Maletras, F.-X., Gournay, P., Robinson,I.A., and Geneves, G. (2007) A biassource for dynamic voltage measure-ments with a programmable Josephsonjunction array. IEEE Trans. Instrum.Meas., 56, 495–499.

61. Picard, A., Bradley, M.P., Fang, H., Kiss,A., de Mirandes, E., Parker, B., Solve,S., and Stock, M. (2011) The BIPM wattbalance: improvements and develop-ments. IEEE Trans. Instrum. Meas., 60,2378–2386.

62. Fang, H., Kiss, A., Picard, A., and Stock,M. (2014) A watt balance based on asimultaneous measurement scheme.Metrologia, 51, S80–S87.

63. de Mirandes, E., Zeggah, A., Bradley,M.P., Picard, A., and Stock, M. (2014)Superconducting moving coil system tostudy the behaviour of superconductingcoils for a BIPM cryogenic watt balance.Metrologia, 51, S123–S131.

64. Sanchez, C.A., Wood, B.M., Green,R.G., Liard, J.O., and Inglis, D. (2014) Adetermination of Planck’s constant usingthe NRC watt balance. Metrologia, 51,S5–S14.

65. Zhang, Z., He, Q., Li, Z., Lu, Y., Zhao, J.,Han, B., Fu, Y., Li, C., and Li, S. (2011)Recent development on the joule balanceat NIM. IEEE Trans. Instrum. Meas., 60,2533–2538.

66. Zhang, Z., He, Q., Lu, Y., Lan, J., Li, C.,Li, S., Xu, J., Wang, N., Wang, G., andGong, H. (2014) The joule balance inNIM of China. Metrologia, 51, S25–S31.

67. BIPM www.bipm.org/en/scientific/mass/pool_artefacts/ (accessed 15 November2014).

68. Milton, M.J.T. and Mills, I.M. (2009)Amount of substance and the proposednew definition of the mole. Metrologia,46, 332–338.

69. Milton, M.J.T. and Quinn, T.J. (2001)Primary methods for the measurementof amount of substance. Metrologia, 38,289–296.

http://www.bipm.org/en/scientific/mass/

175

8Boltzmann Constant and the New Kelvin

As already pointed out in Section 2.2, the present definition of the unit of thermo-dynamic temperature, kelvin, is based on a material artifact, namely, the triple-point-of-water (TPW) temperature. The TPW is the temperature (273.16 K) andpressure (611.73 Pa) where all three phases of water, that is, liquid, solid (ice), andvapor, coexist. Though ideally the TPW can be considered a constant of nature, inreality, its precise temperature depends on many parameters, like isotopic com-position, purity, and so on, which are often difficult to quantify precisely. Never-theless, according to the present definition of the kelvin, the temperature of theTPW is always exactly 273.16 K. The effect of isotope composition has been con-sidered by defining the isotope composition of the water to be used [1]. Keeping inmind that the determination of isotope ratios also exhibits uncertainties and thatpurity and its temporal variation are very difficult to specify absolutely, it is obvi-ous that we are at a situation which is not so different from the kilogram discussedin Chapter 7. Today, the realization of the TPW is possible with an uncertainty ofa few parts in 107.

In the new SI, the definition of the kelvin by a material artifact will be replacedby tracing it back to a fundamental constant consistent with the other envisagedchanges of the SI base units. Since, in the laws of physics, temperature oftenappears as thermal energy, kBT , it seemed only natural to take the Boltzmannconstant, kB, as the reference, as proposed by the International Committee forWeights and Measures (CIPM) and noted by the 24th General Conference onWeights and Measures (CGPM) in 2011 [2]. But again, to make the transitionfrom the present unit to the newly defined one as smooth as possible, theBoltzmann constant has to be determined with the highest possible precision atthe TPW temperature. Actually, the Comité consultatif de thermométrie (CCT)of the CIPM has specified [3] that an uncertainty of the order of 1× 10−6 has tobe achieved, including at least two fundamentally different methods of primarythermometry.

The new definition of the kelvin links the unit of temperature to the unit ofenergy, the joule (1 J= 1 kg m2 s−2). The unit of temperature is then independentof a particular temperature other than in the present definition. The latter is ofparticular advantage when scaling the unit to very high and low temperatures.


176 8 Boltzmann Constant and the New Kelvin

8.1Primary Thermometers

For primary thermometers, the relation between the measurand and thethermodynamic temperature is explicitly known or calculable with the necessaryuncertainty and does not contain any other temperature-dependent quantitiesand constants. The most common primary thermometers relevant for the deter-mination of the Boltzmann constant and the realization of the new kelvin arebased on thermal equations of state, such as the constant volume gas thermome-ter (CVGT), the refractive index gas thermometer (RIGT), and the more recentlydeveloped dielectric constant gas thermometer (DCGT). Another gas ther-mometer based on measuring the speed of sound, the acoustic gas thermometer(AGT), has also received great attention in view of a precise determination ofthe Boltzmann constant [4]. We shall briefly describe DCGT and AGT as wellas radiation thermometers (total radiation as well as spectral radiation). In theframe of quantum metrology, noise thermometers and thermometers based onmolecule absorption spectroscopy (Doppler broadening thermometry (DBT))as well as Coulomb blockade thermometers (CBTs) will be considered finally.Other primary thermometers, which are presently of less relevance for thedetermination of the Boltzmann constant, like magnetic thermometers, will notbe discussed here, but the reader is referred to the respective literature [5, 6].

The base for many primary thermometers is given by the thermal state equationfor ideal gases:

pV = nRT = NkBT , (8.1)

where p, V , and T are the state variables for pressure, volume, and temperature,respectively; n is the amount of substance; N is the number of particles; andR is the universal gas constant, R = kBNA (NA is the Avogadro constant; seeChapter 7). Yet, even though noble gases, in particular, behave approximately likeideal gases at the TPW temperature, for a precise determination of the Boltzmannconstant, even the smallest deviations from the ideal gas behavior must be takeninto account. This is usually accomplished by determining experimentally at aconstant temperature the dependence of the measurand (e.g., pressure) on thedensity of the gas. These isotherms are then fitted according to a virial expansion,

pV = nRT(1 + B(T)∕Vm + C(T)∕V 2m + · · ·), (8.2)

and extrapolated for zero density. B(T) and C(T) in Eq. (8.2) are, respectively, thesecond and third density virial coefficients, and V m is the molar volume, Vm = V∕n.For absolute isotherm CVGT, a constant volume is subsequently filled with differ-ent amounts of gas at constant temperature to obtain different pressures. From aplot of (pV)/n versus 1/V m, the product RT is then directly obtained according toEq. (8.2). However, we will not discuss CVGT any further because the achievableuncertainty seems to be limited to a few parts in 105 [4].

8.1 Primary Thermometers 177

8.1.1Dielectric Constant Gas Thermometry

DCGT is based on the Clausius–Mossotti relation relating the relative dielectricconstant of a gas, 𝜀r, to its static electric polarizability, α0, according to

𝜀r − 1𝜀r + 2

= NV

α03𝜀0

, (8.3)

where 𝜀0 is the electric constant, which is exactly defined in the present SI.Replacing the number density (N/V ) in Eq. (8.3) using the state equation ofthe ideal gas (Eq. (8.1)), taking into account that 𝜀r𝜀0 = 𝜀, and approximating𝜀r + 2 ≈ 3 as valid for ideal gases yield

p =kT(𝜀 − 𝜀0)

α0. (8.4)

The dielectric constant 𝜀 is determined by capacity measurements. Accordingly,in the DCGT experiment, the pressure dependence of the capacitance of acapacitor containing the measuring gas at a constant temperature has to bemeasured. However, in addition, the polarizability, α0, has to be known withthe required uncertainty. This actually is fulfilled for 4He, where ab initio QEDcalculations for α0 have meanwhile achieved an uncertainty well below 10−6 [7].As already said earlier, for a precise determination of kB, deviations from the idealgas behavior must be considered. Combining the virial expansion (Eq. (8.2)) withthe Clausius–Mossotti relation (8.3) then yields [4]

p ≈𝜒

3A𝜀

RT+ 𝜅eff

[

1 + B (T)3A

𝜀

𝜒 + C(T)(3A

𝜀)2 + · · ·

]

, (8.5)

where 𝜒 =(𝜀∕𝜀0 − 1

)is the dielectric susceptibility, A

𝜀= NA

α0∕3𝜀0 is the molarpolarizability, and 𝜅eff is the effective compressibility of the capacitor used tomeasure 𝜒 , taking into account changes of its dimension with pressure [8]. Therelative change of the capacitance

C(p) − C(0)C(0)

= 𝜒 + 𝜀

𝜀0𝜅effp (8.6)

where C(p) and C(0) are, respectively, the capacitance of the gas-filled and evac-uated capacitor is then measured at constant temperature. From a polynomial fitof the plot of p versus C(p)−C(0)∕C(0) at the TPW, 3A

𝜀∕RTTPW and thus the Boltzmannconstant kB are obtained. However, because of the very small susceptibility ofgases (e.g., for He at the TPW and 0.1 MPa 𝜒 ≈ 7 × 10−5), these measurementsare extremely demanding as they require audio-frequency capacitance bridgesproviding uncertainties of a few parts in 109 [9] and pressure measurement upto 7 MPa. The most recent value of the Boltzmann constant obtained in thePhysikalisch–Technische Bundesanstalt dielectric constant gas thermometer(PTB DCGT) experiment at the TPW has an estimated relative standard uncer-tainty of 4.3× 10−6 [10]. Within this uncertainty, the result is in agreement with


the 2010 Committee on Data for Science and Technology (CODATA) value[11]. Since further improvement of the DCGT experiment seems feasible, theDCGT method bears the potential for a primary realization of the new kelvindefinition.

8.1.2Acoustic Gas Thermometry

Acoustic gas thermometry (AGT) [12] applies a resonance method to measurethe low-pressure speed of sound. It presently bears the potential for the lowestuncertainty determination of the Boltzmann constant. It is based on the tworelations

12

mv2rms =

32

kBT (8.7)

and

v2rms =

3γ0

u20. (8.8)

In Eq. (8.7), m is the mass of the atom and v2rms the root-mean-square velocity of

an atom. Equation (8.7) thus relates the kinetic energy of an atom to its thermalenergy kBT . Equation (8.8) then relates v2

rms to the zero-frequency speed of soundof the gas, u0, where γ0 = cp∕cv is the zero-pressure limit of the ratio of specificheat capacitance at constant pressure (cp) and constant volume (cv). CombiningEqs (8.7) and (8.8) and replacing the mass of an atom by the molar mass, M, of thegas divided by the Avogadro constant, m = M∕NA, yield

kB =Mu2

0γ0NAT

. (8.9)

For a monatomic gas, γ0 = 3∕5. The Avogadro constant, NA, presently is knownwith an uncertainty of 4.4× 10−8 according to CODATA [11]. When the experi-ment is performed at the TPW temperature, the molar mass, M, and the speed ofsound, u0, have to be measured to determine kB.

Argon is usually employed in the AGT experiments. Since Ar has three stableisotopes, 40Ar, 36Ar, and 38Ar (in total, 23 isotopes of Ar are known), the chal-lenge for the molar mass determination is to quantify its isotopic compositionand purity (chemical composition). A detailed study for the molar mass deter-mination can be found in, for example, [13]. The speed of sound is measured inan acoustic resonator. Today, spherical or almost (quasi)spherical resonators (see,e.g., [13, 14]) and cylindrical resonators [15] are used. While in the early high-precision experiment [14] the resonator was filled with high-purity mercury toestimate its geometric dimensions (volume), microwave resonances of the sameresonator are now frequently used to determine its dimension [16]. Quasisphericalresonators are advantageous as compared to perfect spherical resonators becausethe otherwise degenerate microwave modes are then partly resolved, allowing a


Figure 8.1 Photo of the assembled National Physics Laboratory (NPL) 1 l copper AGTresonator. (Courtesy of NPL.) (Please find a color version of this figure on the color platesection.)

better determination of the dimensions and their thermal variation [17]. Togetherwith the measured frequencies of the acoustic resonances, the speed of sound isthen derived. An AGT resonator is shown in Figure 8.1.

Though AGT had already been applied in the late 1970s to determine theuniversal gas constant [18], the first high-precision determination of theBoltzmann constant using AGT was reported by Moldover et al. [14] using astainless steel spherical resonator. The uncertainty of the value of the Boltzmannconstant reported in their experiment was 1.8× 10−6. More recent experimentshave confirmed that uncertainties at the 10−6 level can readily be obtained[13, 19], including experiments with He gas instead of Ar [20–22] and the useof cylindrical instead of spherical resonators [15, 23]. However, at the time ofwriting, a significant difference exists between the results with the smallestuncertainty [13, 19], which certainly will be resolved as research is continued.

So, after the new definition of the kelvin by fixing the value of the Boltzmannconstant, AGT definitely seems to be a promising technique to realize the kelvin.


8.1.3Radiation Thermometry

For absolute radiation thermometry (RT), primary sources of radiation arerequired. Again, primary means that a physical law relates the radiance of thissource to a temperature without referring to parameters which themselvesdepend on temperature or radiation power. The two major sources used todayfor absolute RT are blackbodies and synchrotrons. While synchrotron radiationindeed allows for a precise realization of the primary scale of spectral radiancefrom the visible to the X-ray regime [24], it does not allow a direct determinationof the Boltzmann constant.

Absolute thermometers based on blackbody radiometers measure the spectralradiance as a function of frequency, Lν(ν,T), the frequency integrated total radi-ance, L(T). The temperature dependence of the spectral radiance of a blackbodyradiator is described by the Planck law [25]:

Lν(ν,T) = 2hc2

0ν3[

exp(

hνkBT

)

− 1]−1

. (8.10)

The total radiance is given by the Stefan–Boltzmann law:

L(T) =∫

∞

0Lν(ν,T)dν = 𝜎

𝜋

T4 (8.11)

with the Stefan–Boltzmann constant:

𝜎 =2𝜋5k4

B

15c20h3

. (8.12)

A blackbody absorbs all incoming radiation regardless of its frequency and angleof incidence. Its absorptivity and emissivity equal 1, and the emission spectrum asdescribed by Eq. (8.10) is the maximum possible radiance at a given frequencyand temperature. The challenge for the construction of blackbody radiators isto meet the ideal conditions mentioned earlier as close as possible. Blackbod-ies usually consist of a cavity with a small hole. The inner surface of this cav-ity is covered with a suitable material. The choice of the material depends onthe temperature and frequency considered. The blackbody radiators developed atthe Physikalisch–Technische Reichsanstalt (PTR) by Lummer and Kurlbaum [26]allowed at that time an unprecedented precise determination of their emissionspectrum and led the path for the discovery of the Planck law.

Precise absolute RT became possible only through the development of cryo-genic radiometers [27, 28]. Cryogenic radiometers are electrical substitutioncalorimeters where the heating by the incoming radiation is compared to an equalelectrical heating. Operating the radiometer at cryogenic temperatures (usuallyusing liquid He), as compared to room temperature, results in a considerableincrease in sensitivity and accuracy (e.g., due to the much smaller heat capacityand hence increased thermal diffusivity of the used material (e.g., Cooper)) as wellas smaller radiation losses and background radiation [27]. In practice, RT does


not detect the total radiation emitted into a complete hemisphere but only theradiation passing through a suitable aperture system. Consequently, additionalsources of error relate to its temperature, geometry, and diffraction effects [27].

Spectral radiometry, as compared to total radiometry, has the advantage that itcan select and restrict the frequency to the maximum of the emission spectrumof the blackbody, considering that for the determination of kB with the smallestpossible uncertainty, the operating temperature should be at the TPW or closeto it.

With all these restrictions, RT is not able to determine the Boltzmann constantas precisely as the other techniques described here; however, spectral radiometryin particular will probably be the favored method for dissemination of the high-temperature scale after the new definition of the kelvin based on kB.

8.1.4Doppler Broadening Thermometry

Doppler broadening of spectral emission or absorption lines of atomic or molec-ular gases has already been discussed in Chapter 3. The Doppler effect for elec-tromagnetic waves accounts for the frequency change when source and detectormove relative to each other. For an atom or molecule with a resonance frequencyν0 moving with velocity (speed), s, toward a tunable laser source at rest, absorptionwill take place when the frequency of the laser, ν′, is shifted to the red accordingto (neglecting the second-order Doppler effect)

ν′ = ν0

(1 − s

c

). (8.13)

Considering next a gas at low pressure in thermal equilibrium at a given temper-ature, the velocity distribution of the atoms or molecules then will be describedby a Maxwell–Boltzmann distribution proportional to exp[−(s∕s0)2] wheres2

0 = 2kBT∕m with m being the mass of the atom or molecule. This transformsinto a Gaussian absorption profile with e-fold half width, ΔνD:

ΔνDν0

=(2kBT

mc2

)1∕2. (8.14)

The relative mass of the respective ion can be measured with small uncertainty(of order 10−9 or better) using Penning traps. Conversion to absolute massesrequires knowledge of the Avogadro constant (present uncertainty according toCODATA: 4.4× 10−8 [11]), or alternatively an independent measure of the massof the molecule is needed. If the experiment is then performed at a known tem-perature, for example, at the TPW, the Boltzmann constant can be determinedby measuring frequencies. This proposal was first made by Bordé [29], and anexperimental proof of principle was demonstrated by Daussy et al. [30].

A laser spectroscopy setup as used for Doppler broadening thermometry (DBT)is schematically shown in Figure 8.2. Key features of the setup are (i) the stabilized,frequency tunable laser system whose frequency is traced back to the SI unit, for


Frequencytuning

Thermostat

Absorption cell

Absorption spectrum

Laser frequency

Δ

ν0

Detector

Frequencystabilization

Intensitystabilization

Iin Iout

Iout

Laser

Figure 8.2 Schematic of the laser spectrometer setup for DBT.

example, by employing a femtosecond frequency comb (see Section 3.3.1); (ii) thetemperature-stabilized absorption cell containing the molecular gas; and finally(iii) the detection system. The laser system of choice of course depends on the gasand the spectral feature used for the experiment. A rovibrational absorption lineof the ammonia molecule 14NH3 at the frequency of 28 953 694 MHz and a CO2laser stabilized to an absorption line of OsO4 were used in the initial demonstra-tion of DBT [30]. The absolute laser frequency (linewidth< 10 Hz) was measuredby comparison with a Cs fountain clock (see Section 3.2). Tunability of the lasersource had been achieved by electro-optic sideband modulation.

A component (R12) of the ν1 + 2ν02 + ν3 combination band of CO2 (ν1 and ν3 are

the symmetric and antisymmetric stretching mode and ν02 is the bending mode)

and an external-cavity diode laser emitting at 2.006 μm (linewidth∼ 1 MHz) wereused by Casa et al. [31, 32]. Further, a rovibrational absorption line in 13C2H2 wasinvestigated by Koichi et al. using a tunable diode laser locked to a frequency comb[33], and a line at 1.39 μm in the ν1 +ν3 band of H2

18O has been measured usinga pair of offset-frequency locked external-cavity diode lasers [34].

The DBT experiment naturally requires the best achievable control of the gastemperature. To keep the uncertainty of the temperature measurement as smallas possible, the temperature of the TPW or close to it has been used. The Frenchgroup, for example, has used an ice water-filled thermostat, keeping the temper-ature at 213.15 K [30, 35, 36]. A gas cell included in a temperature-controlledthermal shield, surrounded by a cooled enclosure under vacuum, is used in theexperiments of the Italian group [31, 32], allowing a variation of the gas tempera-ture between 270 and 330 K.

When fitting the measured absorption profile to determine the Doppler broad-ening, other line broadening (or narrowing) mechanisms must be considered


like the contribution of the Lorentzian-shaped homogeneous linewidth (seeSection 3.3), the second-order Doppler effect, the Lamb–Dicke narrowing, orthe possible overlap with neighboring absorption lines, and so on (for a detailedlineshape analysis, see [37, 38]).

The smallest uncertainty of the value of the Boltzmann constant measured byDBT today is of the order of some 10 ppm [34, 35], with further improvementspossible. DBT thus will also have the potential for a primary realization of thenew kelvin.

8.1.5Johnson Noise Thermometry

Johnson noise thermometry (JNT) is based on the Nyquist relation [39]

⟨U2⟩ = 4kBTRΔf (8.15)

relating the mean square noise voltage of a resistor, ⟨U2⟩, to its resistance, R,the Boltzmann constant, and temperature. Δf is the frequency bandwidth.Equation (8.15) is a high-temperature approximation valid for frequenciesf ≪ kBT∕h. As the noise voltage is generated by the thermal motion of theelectrons in the resistor, the statistical nature of this mechanism requires suffi-ciently long measuring times, t, depending on the uncertainty required. For thedetermination of a certain temperature, this is quantified by the relation

ΔTT

≈ 2.5√

tΔf. (8.16)

According to Eq. (8.16), for example, at a bandwidth of 20 kHz, a measurementtime of several weeks is required to obtain an uncertainty of order 10−5. Given thevery small noise voltages, this long measurement time naturally is causing con-siderable problems such as the stability of the electronic devices and extra noisesources (e.g., amplifiers, leads). Thus, the determination of an unknown temper-ature is usually done by comparing the mean square noise voltage of the resistorwith a resistor at a known reference temperature, T0, at equal bandwidth. Thesedays, the switched-input digital correlator technique is frequently used for thesemeasurements [40, 41].

In this correlator, the signals stemming from the two channels are digitized, andthe required operations (averaging, multiplication) are done by software. Thus,amplifier noise, noise of the leads, and drift are eliminated. For relative tempera-ture measurement (Figure 8.3a), the unknown temperature, TS, is then obtainedaccording to

TS =⟨U(TS)2⟩

⟨U(T0)2⟩

R(T0)R(TS)

T0. (8.17)

Absolute temperature measurements and hence the determination of theBoltzmann constant, however, require the reference resistor to be replaced by avoltage standard. This has been implemented in a cooperation led by the National


T R(T ) R(T)

Vr

A1S S

4 K

(b)

e− T

e−e−

e−

e−

A2

A1

A2(a)

T0 R(T0)

Figure 8.3 Schematic block diagram for the switched-input noise correlator. (a) shows theconventional relative method and (b) the absolute method employing a quantum voltagenoise source (V r) as the reference. (From [5], with kind permission from Wiley-VCH.)

Institute of Standards and Technology (NIST) by employing pulse-driven ACJosephson standards (see Section 4.1.4.4) as quantized voltage noise sources(QVNSs) [42–45]. The Josephson QVNS is a Σ–Δ digital-to-analog converterproducing a sequence of pulses with quantized pulse area

∫U(t)dt = nK−1

J , (8.18)

where K J is the Josephson constant, K J = 2e/h (see Chapter 4). An M-bit long dig-ital code is then used to synthesize a waveform composed of a series of harmonicsof the pulse pattern repetition frequency with equal amplitude but random phase.This results in a pseudorandom noise waveform with calculable power spectraldensity and thus pseudonoise voltage spectral density, UQVNS:

UQVNS = K−1J Qm(MfS)1∕2 (8.19)

with m being the number of Josephson junctions, M the number of bits that deter-mine the length of the digital waveform, and f S the clock frequency. Note that M isproportional to the clock frequency f S (and to the inverse of the spacing betweenthe harmonic tones that make up the noise waveform) [46] so that the voltagegenerated by the pulse-driven Josephson array is proportional to the clock fre-quency, as discussed in Section 4.1.4.4. Q is a dimensionless amplitude factor [46].Comparing the mean square noise voltage spectral density of the thermal resistor⟨U2(T)∕Δf ⟩ to the mean square pseudonoise voltage spectral density using againa switched-input digital correlator (see Figure 8.3b) yields

⟨U2(T)∕Δf ⟩⟨U2

QVNS⟩=

4kBTRK−2

J Q2m2MfS. (8.20)

Measuring the resistance in units of the von Klitzing constant, R=XRRK, the abso-lute temperature is obtained according to

T =⟨U2(T)∕Δf ⟩⟨U2

QVNS⟩

hQ2m2MfS16kBXR

. (8.21)

Performing the measurement at the TPW allows the determination of the Boltz-mann constant, as demonstrated by Benz et al. [46]. The relative uncertainty intheir experiment was determined to be 12 parts in 106. In view of the recently


reported results by a NIST–NIM cooperation [47], an uncertainty of the order of1 ppm seems feasible, and thus, absolute JNT is a promising alternative to realizeand disseminate the temperature scale after the new definition of the kelvin [48].

8.1.6Coulomb Blockade Thermometry

Primary Coulomb Blockade Thermometry (CBT) in the low-temperatureregime (T < 4 K) is based on the current–voltage characteristic of metallicsingle-electron transport (SET) transistors fabricated, for example, on the basisof (Al/AlOx) tunneling junctions (see Section 6.1.2). As a result of the Coulombblockade, the differential conductance, dISD/dUSD, exhibits a dip around zerosource–drain voltage (see Figure 8.4). The characteristic parameter for theoccurrence of this dip is the ratio of the Coulomb energy:

Ec =e2

CΣ, (8.22)

where CΣ is the total capacitance of the SET transistor to the thermal energykBT . The Coulomb blockade is very pronounced for EC ≫ kBT , resulting in awide and deep dip in the differential conductance. With increasing temperature,this dip broadens and its depth decreases. In the temperature regime, where ECis comparable to or even smaller than kBT (weak Coulomb blockade regime), thehalf width of the conductance dip depends solely on the temperature. For an SET

I

USD

2

USD

RT,C

1.00

(a)

(b)

0.98

0.96

0.94

−4.0 −2.0 0.0

T1

T2 > T1

Voltage (meV)

2.0 4.0

RT,C

2− +

dI S

DdU

SD

(a. u)

Figure 8.4 Schematic illustration of the differential conductance (b) of an SET transistor (a)in the weak Coulomb blockade regime.


array with N tunnel junctions, the half width (in first order) is given by [49]

ΔV 1∕2 ≅ 5.44 NkBT

e. (8.23)

For a single SET transistor with a metallic island and two tunnel junctions (oneon either side of the island), the half width is then equal to 10.88 kBT/e. Oneof the limiting factors for the accuracy is the inevitable spread of the junctionparameters [50]. To reduce this, many sophisticated junction arrays have beenrealized [51–53]. The uncertainties which have been achieved so far are of theorder 10−4 –10−3 [51, 52].

Finally, we mention another low-temperature primary electronic thermometerutilizing tunnel elements and measuring the voltage generated in these elementsin the shot noise regime (shot noise thermometer (SNT)) [54, 55].

8.2Realization and Dissemination of the New Kelvin

Presently, AGT enables the determination of the Boltzmann constant with thesmallest uncertainty. It would therefore be the favored method for the realizationof the new kelvin and the determination of the TPW temperature. At considerablyhigher and lower temperatures, however, other methods will be superior. RT andDBT seem to be advantageous at very high temperatures. In contrast, dielectricconstant gas thermometry (DCGT) as well as JNT, possibly complemented byCoulomb blockade and shot noise thermometry, will be employed at low tem-peratures.

Even though the agreed international temperature scale (ITS) could be aban-doned after the new definition, it will probably be kept for practical reasons. How-ever, the values of the fixed points pinning up the ITS will be determined withsmaller uncertainties using the most suitable primary method.

References

1. BIPM http://www.bipm.org/en/publications/si-brochure/kelvin.html(accessed 15 November 2014).

2. resolution 1, 24th CGPM 2011;http://www.bipm.org/utils/common/pdf/24_CGPM_Resolutions (accessed 15November 2014).

3. Consultative Committee for Ther-mometry http://www.bipm.org/cc/CCT/Allowed/Summary_reports/RECOMMENDATION_web_version.pdf(accessed 15 November 2014).

4. for a review see e.g.: Fellmuth, B.,Gaiser, C., and Fischer, J. (2006)

Determination of the Boltzmannconstant—status and prospects. Meas.Sci. Technol., 17, R145–R159.

5. Fischer, J. (2010) in Handbook ofMetrology, vol. 1 (eds M. Gläser andM. Kochsiek), Wiley-VCH Verlag GmbH,Weinheim, pp. 349–381.

6. Fischer, J. and Fellmuth, B. (2005) Tem-perature metrology. Rep. Prog. Phys., 68,1043–1094.

7. Lach, G., Jeziorski, B., and Szalewicz, K.(2004) Radiative corrections to thepolarizability of helium. Phys. Rev. Lett.,92, 233001-1-4.


http://www.bipm.org/utils/common/

http://www.bipm.org/

References 187

8. Zandt, T., Gaiser, C., Fellmuth, B., Haft,N., Thiele-Krivoi, B., and Kuhn, A.(2013) Temperature: Its Measurementand Control in Science and Industry,vol. 8, AIP Conference Proceedings, vol.1552, AIP, pp. 130–135.

9. Fellmuth, B., Bothe, H., Haft, N., andMelcher, J. (2011) High-precision capac-itance bridge for dielectric-constantgas thermometry. IEEE Trans. Instrum.Meas., 60, 2522–2526.

10. Gaiser, C., Zandt, T., Fellmuth, B.,Fischer, J., Jusko, O., and Sabuga, W.(2013) Improved determination ofthe Boltzmann constant by dielectric-constant gas thermometry. Metrologia,50, L7–L11.


12. for a recent review see: Moldover, M.R.,Gavioso, R.M., Mehl, J.B., Pitre, L., dePodesta, M., and Zhang, J.T. (2014)Acoustic gas thermometry. Metrologia,51, R1–R19.

13. de Podesta, M., Underwood, R., Sutton,G., Morantz, P., Harris, P., Mark, D.F.,Stuart, F.M., Vargha, G., and Machin, G.(2013) A low-uncertainty measurementof the Boltzmann constant. Metrologia,50, 354–376.

14. Moldover, M.R., Trusler, J.P.M., Edwards,T.J., Mehl, J.B., and Davis, R.S. (1988)Measurement of the universal gasconstant R using a spherical acousticresonator. J. Res. Natl. Bur. Stand., 93,85–144.

15. Zhang, J.T., Lin, H., Feng, X.J., Sun, J.P.,Gillis, K.A., Moldover, M.R., and Duan,Y.Y. (2011) Progress towards redetermi-nation of the Boltzmann constant with afixed-path-length cylindrical resonator.Int. J. Thermophys., 32, 1297–1329.

16. Mehl, J.B. and Moldower, M.R. (1986)Measurement of the ratio of the speedof sound to the speed of light. Phys. Rev.A, 34, 3341–3344.

17. Mehl, J.B., Moldover, M.R., andPitre, L. (2004) Designing quasi-sphericalresonators for acoustic thermometry.Metrologia, 41, 295–304.

18. Colclough, A.R., Quinn, T.J., andChandler, T.R.D. (1979) An acoustic

determination of the gas constant. Proc.R. Soc. Lond., A, 368, 125–139.

19. Pitre, L., Sparasci, F., Truong, D.,Guillou, A., Risegari, L., and Himbert,M.E. (2011) Measurement of theBoltzmann constant kB using a quasi-spherical acoustic resonator. Int. J.Thermophys., 32, 1825–1886.

20. Gavioso, R.M., Benedetto, G.,Giuliano Albo, P.A., Madonna Ripa, D.,Merlone, A., Guianvarc’h, C., Moro, F.,and Cuccaro, R. (2010) A determinationof the Boltzmann constant from speedof sound measurements in helium at asingle thermodynamic state. Metrologia,47, 387–409.

21. Gavioso, R.M., Benedetto, G.,Madonna Ripa, D., Giuliano Albo,P.A., Guianvarc’h, C., Merlone, A.,Pitre, L., Truong, D., Moro, F., andCuccaro, R. (2011) Progress in INRIMexperiment for the determination ofthe Boltzmann constant with a quasi-spherical resonator. Int. J. Thermophys.,32, 1339–1354.

22. Segovia, J.J., Vega-Maza, D., Martín,M.C., Gómez, E., Tabacaru, C., and delCampo, D. (2010) An apparatus basedon a spherical resonator for measuringthe speed of sound in gases and fordetermining the Boltzmann constant.Int. J. Thermophys., 31, 1294–1309.

23. Zhang, J., Lin, H., Feng, X.J., Gillis, K.A.,Moldover, M.R., Zhang, J.T., Sun, J.P.,and Duan, Y.Y. (2013) Improved deter-mination of the Boltzmann constantusing a single, fixed-length cylindricalcavity. Metrologia, 50, 417–432.

24. see e.g. Ulm, G. (2003) Radiometry withsynchrotron radiation. Metrologia, 40,S101–S106.

25. Planck, M. (1900) Zur Theorie derEnergieverteilung im Normalspektrum.Verh. Deutsch. Phys. Ges., 2, 237–245.

26. Lummer, O. and Kurlbaum, F. (1898)Der elektrisch geglühte absolut schwarzeKörper und seine Temperaturmessung.Verh. Deut. Phys. Ges., 17, 106–111.

27. Quinn, T.J. and Martin, J.E. (1985)A radiometric determination of theStefan–Boltzmann constant. Proc. R.Soc. Lond. A, 316, 85–189.

28. Martin, J.E., Fox, N.P., and Key, P.J.(1985) A cryogenic radiometer for


absolute radiometric measurements.Metrologia, 21, 147–55.

29. (a) Bordé, C.J. (2002) Atomic clocksand inertial sensors. Metrologia, 39,435–463; (b) Bordé, C. (2005) Base unitsof the SI, fundamental constants andmodern quantum physics. Philos. Trans.R. Soc. A, 363, 2177–2201.

30. Daussy, C., Guinet, M., Amy-Klein, A.,Djerroud, K., Hermier, Y., Briaudeau, S.,Bordé, C.J., and Chardonnet, C. (2007)Direct determination of the Boltzmannconstant by an optical method. Phys.Rev. Lett., 98, 250801-1–250801-4.

31. Casa, C., Castrillo, G., Galzerano, G.,Wehr, R., Merlone, A., Di Serafino, D.,Laporta, P., and Gianfrani, L. (2008)Primary gas thermometry by means oflaser-absorption spectroscopy: determi-nation of the Boltzmann constant. Phys.Rev. Lett., 100, 200801-1–200801-4.

32. Castrillo, A., Casa, G., Merlone, A.,Galzerano, G., Laporta, P., and Gianfrani,L. (2009) On the determination of theBoltzmann constant by means of pre-cision molecular spectroscopy in thenear–infrared. C. R. Phys., 10, 894–906.

33. Koichi, M.T., Yamada, K.M.T., Onae,A., Hong, F.-L., Inaba, H., and Shimizu,T. (2009) Precise determination of theDoppler width of a rovibrational absorp-tion line using a comb-locked diodelaser. C.R. Phys., 10, 907–915.

34. Moretti, L., Castrillo, A., Fasci, E.,De Vizia, M.D., Casa, G., Galzerano, G.,Merlone, A., Laporta, P., and Gianfrani,L. (2013) Determination of the Boltz-mann constant by means of precisionmeasurements of H2

18O line shapes at1.39 μm. Phys. Rev. Lett., 111, 060803-1–060803-5.

35. Lemarchand, C., Djerroud, K.,Darquié, B., Lopez, O., Amy-Klein, A.,Chardonnet, C., Bordé, C.J., Briaudeau,S., and Daussy, C. (2010) Determi-nation of the Boltzmann constant bylaser spectroscopy as a basis for futuremeasurements of the thermodynamictemperature. Int. J. Thermophys., 31,1347–1359.

36. Djerroud, K., Lemarchand, C.,Gauguet, A., Daussy, C., Briaudeau,S., Darquié, B., Lopez, O., Amy-Klein,A., Chardonnet, C., and Bordé, C.J.

(2009) Measurement of the Boltzmannconstant by the Doppler broadeningtechnique at a 3.8 x 10−5 level. C. R.Phys., 10, 883–893.

37. Bordé, C.J. (2009) On the theory of lin-ear absorption line shapes in gases. C. R.Phys., 10, 866–882.

38. Lemarchand, C., Triki, M., Darquié, B.,Bordé, C.J., Chardonnet, C., and Daussy,D. (2011) Progress towards an accuratedetermination of the Boltzmann con-stant by Doppler spectroscopy. New J.Phys., 13, 1–22.

39. Nyquist, H. (1928) Thermal agitation ofelectronic charge in conductors. Phys.Rev., 32, 110–113.

40. Brixi, H., Hecker, R., Oehmen, J.,Rittinghaus, K.F., Setiawan, W., andZimmermann, E. (1992) in Tempera-ture and its Measurement and Controlin Science and Industry, vol. 6 (ed. J.F.Schooley), American Institute of Physics,New York, pp. 993–996.

41. Edler, F., Kühne, M., and Tegeler, E.(2004) Noise temperature measurementsfor the determination of the thermody-namic temperature of the melting pointof palladium. Metrologia, 41, 47–55.

42. Benz, S.P., Martinis, J.M., Nam, S.W.,Tew, W.L., and White, D.R. (2002) inProceedings TEMPMEKO 2001 Interna-tional Symposium on Temperature andThermal Measurements in Industry andScience, vol. 8 (eds B. Fellmuth, J. Seidel,and G. Scholz), VDE, Berlin, pp. 37–44.

43. Tew, W.L., Benz, S.P., Dresselhaus, P.D.,Coakley, K.J., Rogalla, H., White, D.R.,and Labenski, J.R. (2010) Progress innoise thermometry at 505 K and 693 Kusing quantized voltage noise ratio spec-tra. Int. J. Thermophys., 31, 1719–1738.

44. Benz, S., White, D.R., Qu, J.F., Rogalla,H., and Tew, W. (2009) Electronic mea-surement of the Boltzmann constantwith a quantum-voltage-calibrated John-son noise thermometer. C.R. Phys., 10,849–858.

45. White, D.R., Benz, S.P., Labenski, J.R.,Nam, S.W., Qu, J.F., Rogalla, H., andTew, W.L. (2008) Measurement time andstatistics for a noise thermometer with asynthetic-noise reference. Metrologia, 45,395–405.

References 189

46. Benz, S.P., Pollarolo, A., Qu, J., Rogalla,H., Urano, C., Tew, W.L., Dresselhaus,P.D., and White, D.R. (2011) An elec-tronic measurement of the Boltzmannconstant. Metrologia, 48, 142–153.

47. Qu, J. (2014) Improvements in the Boltz-mann constant measurement with noisethermometry at NIM. Conference onPrecision Electromagnetic Measurements(CPEM), Rio de Janeiro, Brazil.

48. see also Engert, J., Beyer, J., Drung, D.,Kirste, A., Heyer, D., Fleischmann, A.,Enss, C., and Barthelmess, H.-J. (2009)Practical noise thermometers for lowtemperatures. J. Phys.: Conf. Ser., 150,012012.

49. Pekola, J.P., Hirvi, K.P., Kauppinen, J.,and Paalanen, M.A. (1994) Thermometryby arrays of tunnel junctions. Phys. Rev.Lett., 73, 2903–2906.

50. Hirvi, K.P., Kauppinen, J.P., Korotkov,A.N., Paalanen, M.A., and Pekola, J.P.(1995) Arrays of normal metal tunneljunctions in weak Coulomb blockaderegime. Appl. Phys. Lett., 67, 2096–2098.

51. Begsten, T., Claeson, T., and Delsing,P. (1999) Coulomb blockade thermom-etry using a two-dimensional array oftunnel junctions. J. Appl. Phys., 86,3844–3847.

52. Pekola, J.P., Holmqvist, T., and Meschke,M. (2008) Primary tunnel junctionthermometry. Phys. Rev. Lett., 101,206801-1–206801-4.

53. Feschchenko, A.V., Meschke, M.,Gunnarson, D., Prunnila, M., Roschier,L., Penttilä, J.S., and Pekola, J.P. (2013)Primary thermometry in the interme-diate Coulomb blockade regime. J. LowTemp. Phys., 173, 36–44.

54. Spitz, L., Lehnert, K.W., Siddigi, I., andSchoelkopf, R.J. (2003) Primary elec-tronic thermometry using the shotnoise of a tunnel junction. Science, 300,1929–1932.

55. Spitz, L., Schoelkopf, R.J., and Pari, P.(2006) Shot noise thermometry downto 10 mK. Appl. Phys. Lett., 89, 183123-1–183123-3.

191

9Single-Photon Metrology and Quantum Radiometry

Photons are the massless bosonic quanta of electromagnetic radiation carryingthe energy E = h𝜈, where 𝜈 is the light frequency. The concept of photons goesback to Planck [1] and Einstein [2] at the beginning of the twentieth century eventhough the name photon (from the Greek word phos meaning light) was createdonly in 1926 by Lewis [3, 4].

Max Planck, when developing his famous radiation formula, postulated thatthe energy exchange between electromagnetic radiation and the wall of a blackbody can take place only in discrete quanta of energy h𝜈. Albert Einstein then usedthe concept of light quanta to explain the photoelectric effect, for which he wasawarded the 1921 Nobel Prize in physics.

The concept of the photon, however, became much more prominent with thedevelopment of quantum optics showing that there is more to quantify the natureof light than the wavelike quantities like, for example, frequency, intensity, andpolarization. These are, in particular, the coherence properties of light as describedby correlation functions for the field, intensity, and photon number [5, 6].

The first-order (field-) correlation function

g(1)(r1, t1; r2, t2) =⟨

E∗ (r1, t1)

E(r2, t2)⟩

[⟨|||E(r1, t1)

|||

2⟩⟨|||E(

r2, t2)|||

2⟩]1∕2(9.1)

describes the spectral properties. The pointed brackets ⟨ ⟩ denote an ensembleaverage. The Fourier transform of g(1)(r1, t1; r2, t2) is the spectrum of the radiationsource and accounts for the contrast (visibility) of interference patterns of the elec-tromagnetic field reflecting the phase correlation of the light field. For plane wavesand stationary fields where the ensemble average can be replaced by a time averageand neglecting further spatial dependencies, Eq. (9.1) simplifies to

g(1)(𝜏) =⟨E∗(t)E(t + 𝜏)⟩

⟨|E(t)|2⟩. (9.2)

The second-order correlation function g(2)(r1, t1; r2, t2)describes the intensity cor-relations:

g(2)(r1, t1; r2, t2) =⟨

E∗ (r1, t1)

E∗(r2, t2)E(r1, t1)E(r2, t2)⟩

⟨|||E(r1, t1)

|||

2⟩⟨|||E(

r2, t2)|||

2⟩ . (9.3)


192 9 Single-Photon Metrology and Quantum Radiometry

For plane waves and stationary classical fields, this can be written in terms of inten-sities, I:

g(2)(𝜏) =⟨I(t + 𝜏)I(t)⟩

⟨I(t)⟩2 . (9.4)

More general, in terms of photon creation and annihilation operators, respec-tively, a†(t) and a(t), the second-order correlation function for stationary fieldsneglecting again any spatial dependencies is given by

g(2)(𝜏) =⟨

a† (t)a†(t + 𝜏)a(t)a(t + 𝜏)⟩

⟨a† (t)a(t)⟩2 , (9.5)

where a†a gives the photon number, n, of the respective mode.While the first-order correlation function can be measured by means of, for

example, a standard Michelson interferometer, measurement of the second-ordercorrelation function is generally carried out using a Hanbury Brown and Twissinterferometer [7] (Figure 9.1). To further prove that the photons are indistin-guishable, two-photon interference (Hong–Ou–Mandel experiment [8]) must beperformed. For the following, we shall considerg(2)(𝜏) only.

For any state of light g(2)(𝜏 → ∞) = 1, because photon emission is uncorrelatedfor large delay times (Figure 9.2). For 𝜏 = 0, the probability for simultaneousdetection of two photons can be increased, unaltered, or decreased with respectto g(2)(𝜏 → ∞). For thermal light, g(2)(0) = 2; for coherent light (coherent insecond order), g(2)(0) = 1; and for nonclassical light, g(2)(0) < 1. For a “true”single-photon emitter (single-photon Fock state), g(2)(0) = 0. While g(2)(0) = 2corresponds to bunching of photons, g(2)(0) < 1 reflects anti-bunching. In termsof photon-counting statistics, g(2)(0) = 2 corresponds to Bose–Einstein statistics,while g(2)(𝜏) ≡ 1 reflects Poisson statistics [9]. g(2)(0) < 1 is associated withsub-Poissonian statistics even though it does not need to occur together withphoton anti-bunching [10].

Bunching of thermal light is simply a manifestation of fluctuating electromag-netic fields and Bose statistics: more photons are emitted when the instantaneouslight intensity is higher than the mean intensity. Consequently, the probability to

Detector

Correlation

50/50 Beamsplitter

Coherentlight source

Detector

Figure 9.1 Setup of the Hanbury Brown and Twiss interferometer.

9.1 Single-Photon Sources 193

2.0

1.5

1.0

g(2

) (τ)

0.5

0.00 1 2

Thermal state

Coherent state

Nonclassical Fock state

τ/τcoh

3 4

Figure 9.2 Second-order correlation function g(2)(𝜏) versus delay time 𝜏 normalized to the

coherence time 𝜏coh.

detect another photon is increased. In contrast, anti-bunching of a single two-levelemitter is due to the fact that when a photon is emitted, the emitter returns intothe ground state and a second photon cannot be emitted simultaneously.

Single-photon emitters are of considerable interest for quantum informationapplications like quantum cryptography and quantum computing [11]. In quan-tum metrology, another promising application could be in the field of radiome-try and photometry, where they could provide quantum standards for (spectral)radiative power and luminous flux on the base of counting photons from a single-photon emitter. Knowing the photon energy h𝜈, the (spectral) radiative power ofa single-photon emitter, Φ, is given by the number of photons emitted per timeinterval, r, multiplied by hν ∶ Φ = r•hν (note the analogy to single electron tunnel-ing; Eq. (6.1)). However, due to the small energy of a single photon, for example,a photon at a wavelength of 500 nm carries energy of about 4× 10−19 J, high repe-tition rate single-photon sources are needed to bridge the many orders of magni-tude in power relevant for practical applications.

9.1Single-Photon Sources

Even though “quasi” single photons may be generated by strongly attenuating acoherent light source, for example, a laser, the photon statistics will be unaltered,that is, it remains Poissonian. Therefore, here, we have to consider a differentapproach for sources that emit nonclassical light, showing g(2)(0) < 1 (preferablyg(2)(0) = 0) and thus obeying a sub-Poissonian statistics.

The fundamental element of a single-photon source is an optical transition(a two-level system in the simplest case) of an individual radiation sourcepreferably with high quantum efficiency. This could be single neutral atoms,single ions, single molecules, single-color centers, or semiconductor quantum


dots. A single photon can be emitted at any arbitrary time or also triggered by theuser, thus being a deterministic source. The single-photon emitter then often willbe coupled to a resonant cavity that causes the radiation to be emitted into a well-defined spatial mode with high collection efficiency. Furthermore, the cavity canenhance the spontaneous emission rate (Purcell effect) and narrow the spectralbandwidth of the emission. For the sake of completeness, also probabilistic single-photon sources should be mentioned. These sources are based on parametricdown-conversion or four-wave mixing producing always pairs of photons whereone photon can be used to herald the creation of the other photon (so-called“heralded single photon”). For further reading, see for example, [12–15].

Photon anti-bunching has been first observed in the resonance fluorescence ofNa atoms continuously excited by a dye laser by Kimble et al. [16]. Dietrich andWalther did observe anti-bunching emission of a laser-cooled single ion storedin a Paul radio-frequency trap [17]. In molecule fluorescence, anti-bunchingwas first reported by De Martini et al. [18], Kitson et al. [19], and Brunel et al.[20]. Recently, a new single-photon source based on Rydberg excitations in anRb gas held in a linear optical lattice has been demonstrated [21–23]. In viewof potential applications, however, solid-state single-photon emitters might bemore promising even though these may require in some cases cooling to lowtemperatures. Here, two systems have gained considerable interest recently,namely, color centers in diamond, in particular the nitrogen-vacancy (NV) andthe silicon-vacancy (SiV) centers, and semiconductor quantum dots.

9.1.1(NV) Color Centers in Diamond

A (NV) color center is formed by a substitutional nitrogen atom and an adjacentvacancy in the diamond lattice. The (NV) centers are prepared in type 1 syntheticdiamond that usually contains homogeneously dispersed nitrogen impurities.Vacancies are created by electron or neutron irradiation. Subsequent annealingat about 900 C results in the formation of the (NV) centers (a small number of(NV) centers actually are already present without extra annealing). The (NV)center exhibits two charge states, electrically neutral and negatively charged. Asimplified energy level structure of the (NV) center is shown in Figure 9.3.

Electron transitions between the 3A ground and the 3E excited state (the labelingof the energy levels is according to the C3V symmetry group), separated by 1.945 eV(637 nm), produce absorption and luminescence. The 3A state and the 3 E state aresplit into the states with spin quantum number mS =±1 and mS = 0 by ∼5.6 μeV[24, 25] and ∼2.9 μeV [26], respectively, due to the magnetic interaction of theunpaired electrons at the NV− center. The mS =±1 states are additionally splitdue to hyperfine interaction, that is, interaction between the electron and nuclearspins. The metastable singlet state 1A that acts as a nonradiative trap state for theexcitation is also indicated. However, its energetic position is not known exactly.


2.9 μeV

5.6 μeV

1.9

45 e

V

±1

±1

0

3E

1A

3A

0

Figure 9.3 Energy level structure (not to scale)of the (NV)− center.

The room temperature photoluminescence of the (NV)−centers exhibits a zero-phonon emission line at 637 nm accompanied by a broad (≈120 nm) phonon-assisted recombination band [27, 28]. The (NV)− emission shows high quantumefficiency close to one and short recombination lifetime (≈11 ns) [27]. Individual(NV) centers can be addressed using microscope imaging technique. Figure 9.4ashows a confocal microscopy raster scan of a part of a diamond sample with (NV)color centers. The bright regions show the emission from (NV)−centers in nanodi-amonds [29]. The second-order correlation function of the emission of an individ-ual center is shown in Figure 9.4b. The pronounced dip at 𝜏 = 0 clearly shows thequantum nature of the emission. We note that g(2) (𝜏) is larger than one for delaytimes which are larger than the radiative recombination lifetime. This fact relatesto the presence of the 1A state to which the excited state can relax.

Besides the emission from the NV center with photon rates up to 1 MHz [29],also other defect-related emission of diamond [30] like the emission from SiV cen-ters in diamond nanocrystals was investigated [31]. These centers exhibit emissionbetween approximately 730 and 750 nm, depending on the local stress in the nan-odiamonds with zero-phonon linewidth in the order of 0.7–2 nm and photon ratesof up to 6 MHz [31–33]. Other defect centers under investigation in diamond arethe nickel-related color center (NE8) [34], the chromium-related center [35, 36],and the interstitial carbon-related color center, TR 12, emitting at 470 nm. In caseof the latter, it has been shown that single-color centers can be created selectivelyusing focused ion beams [37].

As a further promising step toward practical applications, electrical excitation ofthe (NV)0 color center in diamond at room temperature has been realized recently[38]. This has been achieved by fabricating a more or less standard LED struc-ture, a pin diode with p- and n-doped diamond, and sandwiching an intrinsic dia-mond layer that contains the (NV) center. Besides its potential for single-photonemitters, color center defects in diamond are of particular interest for single-spinmanipulation [39]. Further, nanoscale nuclear magnetic resonance (NMR) spectrahave been recorded using the (NV) center [40, 41].


2.0

(a)

(b)

1.5

1.0

g2

(τ)

0.5

0.0−100n −50n 50n 100n0

Delay time, τ (s)

Figure 9.4 Confocal microscopy raster scanof a part of a diamond sample with (NV)color centers. The bright regions show theemission of the (NV) centers in nanodia-monds (a) and second-order correlation func-tion g(2)(𝜏) (b) of spectrally filtered emission

from an individual (NV)− center (some aremarked by circles in the upper frame). Thepronounced dip at 𝜏 = 0 clearly shows thequantum nature of the emission. (Courtesy ofS. Kütt, PTB.)

9.1.2Semiconductor Quantum Dots

The characteristic feature of semiconductor quantum dots is their discrete elec-tronic states, similar to atoms (this is why semiconductor quantum dots are oftencalled artificial atoms) due to size quantization in all three spatial dimensions(i.e., L< 100 nm (see semiconductor quantum wells in Chapter 5)). For furtherreading, see [42–44]. One way to fabricate semiconductor quantum dots is tostart with a two-dimensional electron gas (2DEG) formed in semiconductorheterostructures like GaAs/AlGaAs or InGaAs/GaAs (see Section 5.2). Quantum


T = 2.3 K In0.4Ga0.6As/GaAs

p-Shell

PL(μW)

0.24

0.20

0.15

0.11

0.08

PL inte

nsity (

a. u)

0.06

0.05

2X

1X

0.04

0.03

0.02

0.015

1280 1290

Energy (meV)

1300 1310

s-Shell

λL = 632.8 nm

Figure 9.5 Photoluminescence spectra(T = 2.3 K) of a single InGaAs/GaAs quantumdot for different excitation intensities, show-ing at low excitation a single line due to thelowest state (s-shell) exciton recombination

(1X). At higher excitation also, biexciton (2X)recombination as well as emission from thenext excited state (p-shell) of the quantumdot is observed. (From [51], with kind per-mission from Elsevier.)

dots then can be formed by (electron beam) lithography and subsequent chemicaletching (see e.g., [45]). However, the optical quality of free-standing quantum dotsis poor even when overgrown with a larger band gap material. Semiconductorquantum dots have also been fabricated by laser-induced interdiffusion [46].Most fabrication techniques, however, rely on self-assembled quantum dotsformed during epitaxial growth of slightly lattice-mismatched semiconductors(Stranski–Krastanov growth modus [47]). III–V (e.g., InP in GaAs and InGaAsin GaAs) as well as II–VI heterostructures (e.g., CdSe in ZnS) have been mostlystudied so far. Single-photon emitters in the blue spectral regime have also beenrealized with InGaN/GaN quantum dots [48]. The photoluminescence at low


T = 4 K

Exciton 1X

Biexciton 2X

M

Energy (eV)

Inte

nsity (

a.

u)

1.32 1.34

1.5

1.0

0.5

Time τ (ns)

1X

CW-excitation

g(2

) (τ)

0.0−10 0 10

Figure 9.6 CW-photoluminescence of an InAs/GaAs quantum dot and second-order corre-lation function of the exciton (1X) emission. (Courtesy of P. Michler, University Stuttgart.)

temperatures and weak excitation originate from neutral and charged excitons(and biexcitons), that is, Coulomb-bound electron-hole pair recombination [49,50]. A photoluminescence spectrum of a single InGaAs/GaAs quantum dot isshown in Figure 9.5, for example, [51].

Anti-bunching of the exciton emission has been demonstrated in quantum dotsembedded in resonant microcavities [52–54]. The CW-photoluminescence spec-trum of an InAs/GaAs quantum dot embedded in a microdisk resonator struc-ture is shown in Figure 9.6 together with the second-order correlation function(inset). The photoluminescence clearly exhibits the exciton and biexciton emis-sion together with some spurious background emission (M), which couples into awhispering gallery mode of the microdisk. The second-order correlation functionof the spectrally filtered exciton emission shows clearly the nonclassical behaviorof the emission.

Electrical excitation of single-photon quantum dot emission has also beenachieved [55–58]. In either case, optical or electrical excitation, it must beassured that the radiative transition is only excited once at a time to ensuresingle-photon emission. In the case of optical excitation, this occurs throughabsorption saturation together with the combined effect of an anharmonicmultiexciton spectrum and slow relaxation of highly excited quantum dots [53,54] in the electrical case through Coulomb blockade (see Section 6.1.2) [56, 58].

9.2Single-Photon Detectors

Single-photon detectors are required to test the fidelity of single-photon sources.Photon detectors usually convert an incoming photon into an electrical signal,which is then further processed (e.g., amplified) electronically. Single-photon

9.2 Single-Photon Detectors 199

detectors sometimes are classified into nonphoton-number-resolving andphoton-number-resolving detectors, respectively, even though this distinction isnot always strict. For a detailed listing and comparison of single-photon detectorssee [14]. Nonphoton-number-resolving detectors can only distinguish betweenzero and more than zero photons while photon-number-resolving detectors arecapable to count the numbers of incoming photons (within a certain uncertainty).

A detailed overview about the current state of the art is given in, for example,[14, 59].

9.2.1Nonphoton-Number Resolving Detectors

Most familiar nonphoton-number-resolving detectors are photomultiplier tubes(PMTs) and avalanche photodiodes (APDs). While the detection efficiency of SiAPDs (up to 80% for InGaAs APDs; for the near-IR spectral regime, the quantumefficiency is lower) is higher than for PMTs (typically 25%, up to 40%), the darkcount rate of APDs is higher, which often requires cooling below room temper-ature. Further, since APDs for single-photon detection (single-photon avalanchediodes, SPADs) [60] are usually operated in the so-called Geiger mode with a biasvoltage greater than the breakdown voltage of the diode, the avalanche currentdoes not terminate by itself after an incoming photon pulse but instead must beturned off by lowering the bias voltage. As a result, dead times of SPADs generallyare larger than for PMTs, depending of course crucially on the detector electron-ics. The dead time limit of SPADs can be overcome partly by using a multiplexdetector array consisting of fiber splitters and an array of detectors addressableindividually by optical switches [61].

9.2.2Photon-Number-Resolving Detectors

Photon-number-resolving detectors often are based on superconducting mate-rials with sharp superconducting to normal metal transitions. Most promisingdevices to date are superconducting transition-edge sensors (TESs) [62] due totheir high efficiency and low dark counts. TES basically are microcalorimetersthat measure the energy of the absorbed photons. The operation principle of aTES is illustrated in Figure 9.7.

The thermal sensor of TES is made out of a thin film of superconductingmaterial deposited on an isolating substrate. TES have been made with tungstenand aluminum [62, 63], titanium [64], and hafnium [65]. Also bilayers of a super-conductor and a normal metal (Ti/Au and Ti/Pd) [66] and trilayers of Ti/Au/Ti[67] have been used, which enables to vary the superconductor transitiontemperature due to the proximity effect. The superconducting film is structuredby standard lithography techniques and contacted by superconducting wires,mostly Al.


R

TcT

ΔT = εEυ/Ce

υ

Figure 9.7 Operation principle of aTES depicting the resistance, R, versustemperature T close to the superconductortransition temperature TC. ΔT is the increaseof the temperature due to the absorption

of a photon (𝜀 is the detection efficiency,Eν = hν is the energy of the photon, and Cethe electronic heat capacity). (Please find acolor version of this figure on the color platesection.)

A constant bias voltage of the superconducting film provides an electrothermalfeedback (ETF) such that the temperature is maintained [62]. The constant biasvoltage source can also be realized by a constant current source together with abias resistor with much lower resistance than the TES resistance. The reductionof the current flowing through the sensor due to absorption of photons is read outwith a DC superconducting quantum interference device (SQUID) operated in aflux-locked loop (see Section 4.2.3.1) [66] (see Figure 9.8).

Figure of merit of TES single-photon detectors are the detection efficiency, 𝜀,(ratio of the detected energy to the incident energy) and rise and fall time of thesensor. While the detection efficiency in fiber-coupled devices can be quite high(up to 98% [64]), fall times usually are moderate (some hundred nanoseconds toseveral microseconds).

Ibias

Rbias

Ites

Lin SQUID

Vout

Rfb

Lfb

TES

Figure 9.8 ETF–TES bias circuit with a DC SQUID read-out.

9.3 Metrological Challenge 201

9.3Metrological Challenge

The major challenge for the use of single-photon emitters as quantum standardsfor radiative power or luminous flux is to provide the link between optical powermeasurements in the milliwatts regime to the level of single-photon emitters. Thiswould require either single-photon sources operating at extremely high repetitionrate or absolutely linear detectors traced to a primary standard (e.g., a cryora-diometer).

To calibrate SPADs operating in the few photon regime, that is, below about106 photons per second, many orders in intensity have to be bridged to transferthe SI scale down to these very small intensities, considering the limited linearityof SPADs. Thus, careful attenuation of a calibrated light source, for example, alaser, is required, using two or more neutral density filters with high attenuationsubsequently measured in situ (see Figure 9.9a [68]). Alternatively a synchrotron

Microscopeobjective

Si-Diode

Si-SPAD(a)

(b)

Beamsplitter

Laser770 nm

Monitordetector

Aperture

Synchrotron radiationϕMLS (𝜆)

Trap detectorStrap (𝜆)

PRIhigh

PRIlow

Detectors

SPADQE*SPAD (𝜆)

Filter @ 651.34 nmF (𝜆)

Lens Focus

Filter 3 Filter 2

Figure 9.9 Schematic of two setups usedto calibrate single-photon avalanche diodes(SPADs). (a) Using in situ calibration of neu-tral density filters and the known radiativepower of a stabilized laser. The Si diode iscalibrated, for example, by the use of a cry-oradiometer. The subsequent calibration ofthe individual filters is required because oftheir very high attenuation factor. For thecalibration of the SPAD, then both filters areapplied together. (After [68].) (b) Calibration

using synchrotron radiation. The photon rate(PR Ihigh) in the high ring current range ismeasured in the focus of the spectrally fil-tered synchrotron radiation by a calibratedtrap detector using its known responsivity,Strap(𝜆). In a second step, the count rate ofthe SPAD is measured in the low ring cur-rent range. The quantum efficiency of theSPAD, QE∗

SPAD, then can be calculated using

the synchrotron current ratio in the high-and low-current mode. (After [69].)


radiation source, where the emitted radiation power is proportional to the numberof stored electrons, has been used. Since the number of stored electrons can bewidely varied from one up to more than 1011, a SPAD can be calibrated in thesingle-photon regime without using attenuators [69]. The Figure 9.9b depicts theprinciple of this experiment.

Finally, we note that heralded single photons as generated by parametric downconversion can be used also for absolute calibrations of APDs in the few photonregime [15, 70, 71].

Presently, a new definition of radiometric or photometric quantities based onsingle-photon sources is not considered because quantum radiometry (or pho-tometry) is not yet mature for practical applications. However, keeping in mind theprogress that has been made in scaling up the current of single-electron devices(see Chapter 6) and the improvement in the collection efficiency of single-photonemitters reported recently [72], it might be feasible in the near future.

References

1. Planck, M. (1900) Zur Theorie desGesetzes der Energieverteilung im Nor-malspektrum. Verh. Deutsch. Phys. Ges.,2, 237–245 (in German).

2. Einstein, A. (1905) Über einen dieErzeugung und Verwandlung des Lichtsbetreffenden heuristischen Gesicht-spunkt. Ann. Phys., 17, 132–149. (inGerman).

3. Lewis, G.N. (1926) The conservation ofphotons. Nature, 118, 874–875.

4. A critical discussion of the conception“photon” can be found in: Lamb, W.E.(1995) Anti-photon. Appl. Phys. B, 60,77–84.

5. (a) Glauber, R.J. (1963) The quantumtheory of optical coherence. Phys.Rev., 130, 2529–2539; (b) Glauber,R.J. (1963) Coherent and incoherentstates of the radiation field. Phys. Rev.,131, 2766–2788.

6. for further reading see e.g: Mandel,L. and Wolf, E. (1995) Optical Coher-ence and Quantum Optics, CambridgeUniversity Press, New York.

7. Hanbury Brown, R. and Twiss, R.Q.(1956) Correlation between photons intwo coherent beams of light. Nature,177, 27–29.

8. Hong, C.K., Ou, Z.Y., and Mandel, L.(1987) Measurement of subpicosecondtime intervals between two photons

by interference. Phys. Rev. Lett., 59,2044–2046.

9. see e.g. Martinelli, M. and Martelli, P.(2008) Laguerre mathematics in opticalcommunications. Opt. Photonics News,19, 30–35.

10. Zou, X.T. and Mandel, L. (1990) Photon-antibunching and sub-Poissonian photonstatistics. Phys. Rev. A, 41, 475–476.

11. Boumester, D., Ekert, A., and Zeilinger,A. (eds) (2000) The Physics of QuantumInformation, Springer, Berlin.

12. Lounis, B. and Orrit, M. (2005) Singlephoton sources. Rep. Prog. Phys., 68,1129–1179.

13. Grangier, P. (2005) Experiments withsingle photons. Semin. Poincare, 2,1–26.

14. Eisaman, M.D., Fan, J., Migdal, A., andPolyakov, S.V. (2011) Single-photon-sources and detectors. Rev. Sci. Instrum.,82, 071101-1–071101-25.

15. Sergienko, A.V. (2001) in Proceedingsof the International School of Physics“Enrico Fermi” Course CXLVI (eds T.J.Quinn, S. Leschiutta, and P. Tavella), IOSPress, Amsterdam, pp. 715–746.

16. Kimble, H.J., Dagenais, M., and Mandel,L. (1977) Photon antibunching in reso-nance fluorescence. Phys. Rev. Lett., 39,691–695.

References 203

17. Diedrich, F. and Walther, H. (1987) Non-classical radiation of a single stored ion.Phys. Rev. Lett., 58, 203–206.

18. De Martini, F., Giuseppe, G., andMarrocco, M. (1996) Single-mode gen-eration of quantum photon states byexcited single molecules in a microcavitytrap. Phys. Rev. Lett., 76, 900–903.

19. Kitson, S., Jonsson, P., Rarity, J., andTapster, P. (1998) Intensity fluctuationspectroscopy of small numbers of dyemolecules in a microcavity. Phys. Rev. A,58, 620–627.

20. Brunel, C., Lounis, B., Tamarat, P., andOrrit, M. (1999) Triggered source of sin-gle photons based on controlled singlemolecule fluorescence. Phys. Rev. Lett.,83, 2722–2725.

21. Urban, E., Johnson, T.A., Hanage, T.,Isenhower, L., Yavuz, D.D., Walker, T.G.,and Saffman, M. (2009) Observation ofRydberg blockade between two atoms.Nat. Phys., 5, 110–114.

22. Gaetan, A., Miroshnychenko, Y.,Wilk, T., Chotia, A., Viteau, M.,Comparat, D., Pillet, P., Browaeys, A.,and Grangier, P. (2009) Observation ofcollective excitation of two individualatoms in the Rydberg blockade regime.Nat. Phys., 5, 115–118.

23. Dudin, Y.O. and Kuzmich, A. (2012)Strongly interacting Rydberg excita-tions of a cold atom gas. Science, 336,887–889.

24. Loubser, J.H.N. and van Wyk, J.A. (1977)Electron spin resonance in annealed type1b diamond. Diamond Res., 11, 4–7.

25. Loubser, J.H.N. and van Wyk, J.A. (1978)Electron spin resonance in the studyof diamond. Rep. Prog. Phys., 41 (8),1201–1249.

26. Fuchs, G.D., Dobrovitski, V.V.,Hanson, R., Batra, A., Weis, C.D.,Schenkel, T., and Awschalom,D.D. (2008) Excited-state spec-troscopy using single spinmanipulation in diamond. Phys.Rev. Lett., 101, 117601-1–117601-4.

27. Kurtsiefer, C., Mayer, S., Zarda, P., andWeinfurter, H. (2000) Stable solid-statesource of single photons. Phys. Rev. Lett.,85, 290–293.

28. Brouri, R., Beveratos, A., Poizat, J.-P.,and Grangier, P. (2000) Photon anti-bunching in the fluorescence ofindividual color centers in diamond.Opt. Lett., 25, 1294–1296.

29. Schmunk, W., Rodenberger, M.,Peters, S., Hofer, H., and Kück, S. (2011)Radiometric calibration of single pho-ton detectors by a single photon sourcebased on NV-centers in diamond. J.Mod. Opt., 58, 1252.

30. Pezzagna, S., Rogalla, D., Wildanger,D., Meijer, J., and Zaitsev, A. (2011)Creation and nature of optical cen-tres in diamond for single-photonemission—overview and critical remarks.New J. Phys., 13, 035024-1–035024-28.

31. Neu, E., Steinmetz, D., Riedrich-Möller,J., Gsell, S., Fischer, M., Schreck, M.,and Becher, C. (2011) Single photonemission from silicon-vacancy colourcentres in chemical vapour depositionnano-diamonds on iridium. New J. Phys.,13, 025012-1–025012-21.

32. Riedrich-Möller, J., Kipfstuhl, L.,Hepp, C., Neu, E., Pauly, C., Mücklich,F., Baur, A., Wandt, M., Wolff, S.,Fischer, M., Gsell, S., Schreck, M.,and Becher, C. (2012) One- andtwo-dimensional photonic crystal micro-cavities in single crystal diamond. Nat.Nanotechnol., 7, 69.

33. Neu, E., Fischer, M., Gsell, S., Schreck,M., and Becher, C. (2011) Fluorescenceand polarization spectroscopy of singlesilicon vacancy centers in heteroepitaxialnanodiamonds on iridium. Phys. Rev. B,84, 205211-1–205211-8.

34. Marshall, G.D., Gaebel, T., Matthews,J.C.F., Enderlein, J., O’Brian, J.L., andRabeau, J.R. (2011) Coherence propertiesof a single dipole emitter in diamond.New J. Phys., 13, 055016.

35. Aharonovich, I., Castelletto, S.,Simpson, D.A., Starey, A., McCallum, J.,Greentree, A.D., and Prawer, S. (2009)Two-level ultrabright single photonemission from diamond nanocrystals.Nano Lett., 9, 3191–3195.

36. Aharonovich, I., Castelletto, S., Simpson,D.A., Greentree, A.D., and Prawer, S.(2010) Photophysics of chromium-related diamond single-photon emitters.Phys. Rev. A, 81, 043813-1–043813-7.


37. Naydenov, B., Kolesov, R., Batalov, A.,Meijer, J., Pezzanga, S., Rogalla, D.,Jelezko, F., and Wrachtrup, J. (2009)Engineering single photon emitters byion implantation in diamond. Appl. Phys.Lett., 95, 181109-1–181109-3.

38. Mizuochi, N., Makino, T., Kato, H.,Takeuchi, D., Ogura, M., Okushi, H.,Nothaft, M., Neumann, P., Gali, A.,Jelezko, F., Wrachtrup, J., and Yamasaki,S. (2012) Electrically driven single pho-ton source at room temperature indiamond. Nat. Photonics, 6, 299–303.

39. Jelezko, F. and Wrachtrup, J. (2012)Focus on diamond-based photonics andspintronics. New J. Phys., 14, 105024-1–105024-3.

40. Staudacher, T., Shi, F.S., Pezzagna, S.,Meijer, J., Du, J., Meriles, C.A., Reinhard,F., and Wrachtrup, J. (2013) Nuclearmagnetic resonance spectroscopy on a(5-Nanometer)3 sample volume. Science,339, 561–563.

41. Mamin, H.J., Kim, M., Sherwood, M.H.,Rettner, C.T., Ohno, K., Awschalom,D.D., and Rugar, D. (2013) Nanoscalenuclear magnetic resonance with anitrogen-vacancy spin sensor. Science,339, 557–560.

42. Bimberg, D., Grundmann, M.,Ledentsov, N.N. (1998) Quantum DotHeterostructures, John Wiley & Sons,Inc., Chichester (1999)

43. Hawrylak, P. and Wojs, A. (1998) Quan-tum Dots, Springer, Berlin.

44. Shields, A.J. (2007) Semiconductor quan-tum light sources. Nat. Photonics, 1,215–223.

45. Steffen, R., Forchel, A., Reinecke, T.,Koch, T., Albrecht, M., Oshinowo, J.,and Faller, F. (1996) Single quantum dotsas local probes of electronic propertiesof semiconductors. Phys. Rev. B, 54,1510–1513.

46. Brunner, K., Bockelmann, U., Abstreiter,G., Walther, M., Böhm, G., Tränkle, G.,and Weimann, G. (1992) Photolumi-nescence from a single GaAs/AlGaAsquantum dot. Phys. Rev. Lett., 69,3216–3219.

47. Stranski, I.N. and Krastanov, L. (1938)Zur Theorie der orientierten Ausschei-dung von Ionenkristallen aufeinander.

Sitzungsber. Akad. Wiss. Wien Math.-Naturwiss., 146, 797–810.

48. Jarjour, A.F., Taylor, R.A., Oliver, R.A.,Kappers, M.J., Humphreys, C.J., andTahraoui, A. (2007) Cavity-enhancedblue single-photon emission from asingle InGaN/GaN quantum dot. Appl.Phys. Lett., 91, 052101-1–052101-3.

49. Dekel, E., Gershoni, D., Ehrenfeld,E., Garcia, J.M., and Petroff, P. (2000)Cascade evolution and radiative recom-bination of quantum dot multiexcitonsstudied by time-resolved spectroscopy.Phys. Rev. B, 62, 11038–11045.

50. Finley, J.J., Fry, P.W., Ashmore, A.D.,Lemaitre, A., Tartakovskii, A.I.,Oulton, R., Mowbray, D.J., Skolnick,M.S., Hopkinson, H., Buckle, P.D., andMaksym, P.A. (2001) Observation ofmulticharged excitons and biexcitons ina single InGaAs quantum dot. Phys. Rev.B, 63, 161305-1–161305-4.

51. Findeis, F., Zrenner, A., Böhm, G., andAbstreiter, G. (2000) Optical spec-troscopy on a single InGaAs/GaAsquantum dot in the few-exciton limit.Solid State Commun., 114, 227–230.

52. Press, D., Götzinger, S., Reitzenstein, S.,Hofmann, C., Löffler, A., Kamp, M.,Forchel, A., and Yamamoto, Y. (2007)Photon antibunching from a singlequantum-dot-microcavity system in thestrong coupling regime. Phys. Rev. Lett.,98, 117402-1–117402-5.

53. Michler, P., Imamoglu, A., Mason, M.D.,Carson, P.J., Strouse, G.F., and Buratto,S.K. (2000) Quantum correlation amongphotons from a single quantum dotat room temperature. Nature, 406,968–970.

54. Michler, P., Kiraz, A., Becher, C.,Schoenfeld, W.V., Petroff, P.M., Zhang,L., Hu, E., and Imamoglu, A. (2000) Aquantum dot single-photon turnstiledevice. Science, 290, 2282–2285.

55. Ward, M.B., Farrow, T., See, P., Yuan,Z.L., Karimov, O.Z., Bennet, A.J.,Shields, A.J., Atkinson, P., Cooper,K., and Ritchie, D.A. (2007) Electricallydriven telecommunication wavelengthsingle-photon source. Appl. Phys. Lett.,90, 063512-1–063512-3.

References 205

56. Imamoglu, A. and Yamamoto, Y. (1994)Turnstile device for heralded single pho-tons: coulomb blockade of electron andhole tunneling in quantum confinedp-i-n heterojunctions. Phys. Rev. Lett.,72, 210–213.

57. Benson, O., Satori, C., Pelton, M., andYamamoto, Y. (2000) Regulated andentangled photons from a single quan-tum dot. Phys. Rev. Lett., 84, 2513–2516.

58. Kim, J., Benson, O., Kan, H., andYamamoto, Y. (1999) A single-photonturnstile device. Nature, 397, 500–503.

59. Hadfield, R.H. (2009) Single-photondetectors for optical quantum infor-mation applications. Nat. Photonics, 3,696–705.

60. Sappa, F., Cova, S., Ghioni, M., Lacaita,A., Samori, C., and Zappa, F. (1996)Avalanche photodiodes and quenchingcircuits for single photon detection.Appl. Opt., 35, 1956–1976.

61. Castelletto, S.A., De Giovanni, I.P.,Schettini, V., and Migdall, A.L. (2007)Reduced deadtime and higher ratephoton-counting detection using a mul-tiplexed detector array. J. Mod. Opt., 54,337–352.

62. (a) Irwin, K.D. (1995) An application ofelectrothermal feedback for high resolu-tion cryogenic particle detection. Appl.Phys. Lett., 66, 1998–2000; (b) Irwin,K.D., Nam, S.W., Cabrera, B., Chugg, B.,and Young, B. (1995) A quasiparticle-trap-assisted transition-edge sensor forphonon-mediated particle detection. Rev.Sci. Instrum., 66, 5322–5326.

63. Lita, A.E., Miller, A.J., and Nam, S.W.(2008) Counting near-infrared single-photons with 95% efficiency. Opt.Express, 16, 3032–3040.

64. Fukuda, D., Fujii, G., Numata, T.,Yoshizawa, A., Tsuchida, H., Fujino,H., Ishii, H., Itatani, T., Inoue, S., andZama, T. (2009) Photon number resolv-ing detection with high speed and highquantum efficiency. Metrologia, 46,S288–292.

65. Lita, A.E., Calkins, B., Pellochoud,L.A., Miller, A.J., and Nam, S. (2009)

High-efficiency photon-number-resolvingdetectors based on hafnium transition-edge sensors. AIP Conf. Proc., 1185,351–354.

66. Rajteri, M., Taralli, E., Portesi, C.,Monticone, E., and Beyer, J. (2009)Photon-number discriminating super-conducting transition-edge sensors.Metrologia, 46, S283–S287.

67. Taralli, E., Portesi, C., Lolli, L.,Monticone, E., Rajteri, M., Novikov,I., and Beyer, J. (2010) Impedance mea-surements on a fast transition-edgesensor for optical and near-infraredrange. Supercond. Sci. Technol., 23,105012-1–105012-5.

68. Kück, S., Hofer, H., Peters, S., andLopez, M. (2014) Detection efficiencycalibration of silicon single photonavalanche diodes traceable to a nationalstandard. 12th International Conferenceon New Developments and Applicationsin Optical Radiometry (NEWRAD 2014),Espoo, Finland, June 24–27, 2014, p. 93,http://newrad2014.aalto.fi/Newrad2014_Proceedings.pdf (accessed)

69. Müller, I., Klein, R., Hollandt, J., Ulm,G., and Werner, L. (2012) Traceablecalibration of Si avalanche photodiodesusing synchrotron radiation. Metrologia,49, S152–S155.

70. Brida, G., Genovese, M., and Gramenga,M. (2006) Twin-photon techniques forphoto-detector calibration. Laser Phys.Lett., 3, 115–123.

71. Polyakov, S.V., Ware, M., and Migdall,A. (2006) High accuracy calibration ofphoton-counting detectors. Proc. SPIE,6372, 63720J.

72. (a) Lee, K.G., Chen, X.W., Eghlidi,H., Kukura, P., Lettow, R., Renn, A.,Sandoghdar, V., and Götzinger, S. (2011)A planar dielectric antenna for direc-tional single-photon emission andnear-unity collection efficiency. Nat.Photonics, 5, 166–169; (b) Chen, X.W.,Götzinger, S., and Sandoghdar, V. (2011)99% efficiency in collecting photonsfrom a single emitter. Opt. Lett., 36,3545–3547.

http://newrad2014.aalto.fi/Newrad2014_

207

10Outlook

As we have tried to outline in our book, modern quantum science and technologyhas progressed considerably and paved the way to a new system of units basedon constants of nature. In particular, the kilogram, the ampere, the kelvin, andthe mole should be defined with respect to fixed numerical values of, respec-tively, the Planck constant h, the elementary charge e, the Boltzmann constant kB,and the Avogadro constant NA. Together with the present definition of the second,the meter, and the candela based on fixed numerical values of, respectively, theground-state hyperfine splitting frequency of the cesium 133 atom 𝜈(133Cs)hfs; thespeed of light in vacuum, c; and the spectral luminous efficacy, K cd, of monochro-matic radiation of frequency 540× 1012 Hz, the International System of Units, theSI, then will be based completely on constants of nature. All artifacts will be abol-ished. Measurements in all areas like science, production, trade, and protectionof human health and environment then will be based on solid grounds and areexpected to be more stable in space and time.

At the time of writing, it seems that a consensus is emerging on the criteria setfor the adoption of the new definitions, and the experimental results are approach-ing agreement among each other. Considering the presently ongoing efforts tofurther improve the experiments, it seems possible to adopt the new SI at the nextGeneral Conference on Weights and Measures (CGPM) in 2018 [1].

In addition, present scientific progress in optical frequency standards is pro-ceeding toward a new definition of the second. The new definition will be basedon the frequency of an electronic transition of an atom or ion in the optical fre-quency regime instead of the presently used microwave transition in the 133Csatom. These new frequency standards will contribute also to the investigation ofbasic physics questions related to, for example, relativity and quantum gravity. Inany case, as our understanding of the universe proceeds, the SI has to adapt to theprogress in science and technology as it always has done in the past.

A remaining challenge for future metrology is to bring the intrinsic advantageof the new SI to the “workbench,” that is, to make its potential fully accessibleto all kind of users, including industrial production, health care, environmen-tal protection, and finally regulators and standardization bodies. The first stepsin this direction have been taken, for example, in Josephson voltage metrology.Due to the maturity of the Josephson technology, an AC quantum voltmeter (see


208 10 Outlook

Section 4.1.4.5) could be developed that is suitable for use by industrial calibrationlaboratories [2]. In general, electrical quantum standards will find wider applica-tion if they can be operated at higher temperatures. Further progress in materialscience may enable this development.

References

1. Milton, M.J.T., Davis, R., and Fletcher, N.(2014) Towards a new SI: a review ofprogress made since 2011. Metrologia, 51,R21–30.

2. Lee, J., Behr, R., Palafox, L., Katkov, A.,Schubert, M., Starkloff, M., and

Böck, A.C. (2013) An ac quantum volt-meter based on a 10 V programmableJosephson array. Metrologia, 50,612–622.

209

Index

aacoustic resonator 178

AlAs 108

AlGaAs 109

Allan variance 24

antibunching of photons 192, 194, 198

arrays of QHE resistors 123

avalanche photodiodes (APD) 199

bband structure 104

BCS theory 62

black body radiometer 180

Bloch oscillation 148

Bordé–Ramsey atom interferometry 38, 44

bunching of photons 192

cchemical potential 134

Clausius–Mosotti relation 177

clock comparison 25, 51

coherence properties of light 191

conductivity tensor 115

constant volume gas thermometer (CVGT)

176

Cooper pair 62

coordinated universal time (UTC) 11

coplanar waveguides 70

correlation function 191

– first-order 191

– second-order 191, 195, 198

cotunneling 139

Coulomb blockade 132, 134, 135, 185

Coulomb energy 134

cryogenic current comparator 94, 123, 150

cryogenic radiometer 180

Cs-atomic clock 11

Cs-fountain clock 32

current biasing of SQUID 91

current, critical 64, 70

cyclotron frequency 104

ddefining constants 21, 157

density of states 105–107

deterministic single-photon source 194

disorder 118

doping 112

Doppler effect 12, 25, 26, 37, 181

eedge channel model 118

effective electron mass 104

Einstein’s equivalence principle (EEP) 49,

155

electrochemical potential 134

electron shelving technique 44

electrostatic potential 134

energy dispersion 104, 106

ephemeris second 11

Er-doped fiber laser 42

error accounting 149, 150

exciton 198

ffilling factor 116, 117

fine-structure constant 49, 124

Fizeau interferometer 160

flux quantum 64, 65, 77, 81, 117

flux transformer 92

flux-locked loop 92, 200

fractional quantum Hall effect 119

frequency chain 39

frequency stability 23

frequency, characteristic 69, 76, 77

QuantumMetrology: Foundation of Units and Measurements, First Edition.Ernst O. Göbel and Uwe Siegner.© 2015 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2015 by Wiley-VCH Verlag GmbH & Co. KGaA.

210 Index

gGaAs 108

GaAs/AlGaAs heterostructure 109, 121,

142, 196

gauge block 15

General Conference of Weights and

Measures 1

graphene 121, 145

guidelines for QHE metrology 121

GUM 5

hHall

– coefficient 115

– field 114

– resistance 115

– voltage 114

Hall bar 122

Hanbury Brown and Twiss interferometer

192

Heisenberg uncertainty relation 6, 12, 133

heterojunction 108, 113

high-temperature superconductors 62

highly enriched 28Si 164

hydrogen maser 35

iimpedance metrology based on the QHE

125

international kilogram prototype 15, 155,

169

international temperature scale (ITS) 18,

186

ion accumulation experiment 158

isotope dilution mass spectrometry 164

jJohnson noise thermometry 81, 183

Josephson arbitrary waveform synthesizer

78

Josephson constant 64, 149, 166

Josephson equations 64

Josephson impedance bridge 80

Josephson junction 63

– overdamped 67

– SINIS 68, 69, 74, 75

– SIS 68, 69

– SNS 68, 69, 74, 75, 77

– underdamped 67

Joule balance 168

kKilogramme des Archives 155

lLamb–Dicke regime 31

Landau level 105, 107

Landauer–Büttiker formalism 118

leap second 11

linewidth, homogeneous 26

localized electronic states 118

London penetration depth 83

mmagic wavelength 45

magnetic flux quantization 84

magnetic moment, measurement of 94

magnetically shielded room 97

magneto-optical trap 27

magneto-transport 103

magnetocardiography 96

magnetoencephalography 96

magnetometer 90, 92

McCumber parameter 67

measurement uncertainty 5

Meissner–Ochsenfeld effect 82

metal organic chemical vapor deposition

110

metal organic vapor phase epitaxy 110

metal-oxide-semiconductor field-effect

transistor (MOSFET) 103, 108, 121, 145

Meter Convention 1

Mètre des Archives 1

metrology 1

microstrip line 70

mobility of electrons 112

mode-locking 39

molar mass of Si 158, 164

molar mass unit 169

molecular beam epitaxy 110

nneutral atom clocks 43

noise

– 1/f 9

– quantum 8

– shot 9

– thermal 8

noise power spectral density 7

nomenclature of atomic states 10

nuclear magnetic resonance 98

Nyquist relation 8, 183

ooctupole transition 46

optical lattice clock 45

optical molasses 26

Index 211

pPaul trap 30

Penning trap 30

phase relaxation time 36

photomultiplier tubes (PMT) 199

photon-assisted tunneling 141

Planck law 180

probabilistic single photon source 194

qquantization energy 106

quantized current 131

quantized voltage noise sources 183

quantum dot, self-assembled 197

quantum interference 87, 89

quantum jump fluorescence detection 46

quantum logic spectroscopy 47

quantum optics 191

quantum phase slip 148

quantum voltmeter 80

quantum well 106, 111

quantum-based electrical power standard

80

rR-pump 141

Rabbi frequency 13

Ramsey technique 12, 32

RCSJ model 66, 91

realization of the SI ohm 120

recoil limit 29

recoil momentum 26, 38

representation of the ohm 122

representation of the volt 73

resistance bridge 95

resistance metrology 119

resistivity 115

Rydberg frequency 50

ssaturated absorption spectroscopy 37

scaling of resistance values 123

Schrödinger equation 63

Shapiro steps 65

Si single crystal 158

single ion frequency standards 46

single-electron

– electrometer 136

– pump 137, 139, 143

– quantum box 132

– transistor 133

– turnstile 138, 146

single-photon emitter 193

SINIS structure 146

Sisyphus cooling 28

size quantization 104, 106, 141, 196

spectral radiance 180

speed of sound 178

superconducting quantum interference

device (SQUID) 81

SQUID gradiometer

– first-order 93

– second-order 94

stability diagramm

– SET pump 138

– SET transistor 136

Stefan–Boltzmann law 180

superconducting magnetic levitation 157

superconducting quantized charge pump

147

superconducting sluice 147

superconducting transition edge sensor

199

supercontinuum 42

supercurrent 63

sympathetic cooling 48

synchrotron radiation 180, 202

tthermal converter 73, 80, 81

thermal state equation 176

titanium–sapphire laser 41

total radiance 180

transfer error 139, 145

triple point of water 18, 175

tunnel element 132

two-dimensional electron gas (2DEG) 103,

112, 142, 196

uuniversality of the QHE 121

vvirial expansion 176

voltage balance 72, 157

voltage, characteristic 70, 75

von Klitzing constant 117, 133, 149, 166

xx-ray crystal density (XRCD) 158

x-ray interferometer 162

zZeeman energy 105

Zeeman slower 26

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