JOSEPHSON CIRCUITS

FOR PROTECTED QUANTUM BITS

by

WEN-SEN LU

A dissertation submitted to the

School of Graduate Studies

Rutgers, The State University of New Jersey

In partial fulfillment of the requirements

For the degree of

Doctor of Philosophy

Graduate Program in Physics and Astronomy

Written under the direction of

Michael Gershenson

And approved by

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

____________________________________________________

New Brunswick, New Jersey

May 2021

ii

ABSTRACT OF THE DISSERTATION

Josephson circuits for protected quantum bits

by WEN-SEN LU

Dissertation Director

Michael Gershenson

Over the past two decades the performance of superconducting quantum bits (qubits)

has been improved: the coherence time of individual qubits has been increased by five

orders of magnitude, from a few nanoseconds to > 100 μs. The much-improved energy

relaxation time, scalability from engineering point of view, and compatibility with

microwave control make superconducting qubits one of the major competitors for quantum

information hardware applications. Despite the progress, experimental realization of

quantum error corrections for logical qubits remains challenging.

One of possible solutions to this problem is the development of so-called protected

qubits whose errors would be suppressed by special symmetries of the underlying

Hamiltonian. The realization of such qubits requires elements not found in the conventional

superconducting circuit toolbox, such as Josephson elements with cos(

) and cos(2)

dependences of the Josephson energy on the phase difference , circuits with a very

large kinetic inductance, and junctions with unusually low .

iii

This thesis focuses on design, fabrication and characterization of circuits based on

low- Josephson junctions (JJs) and superinductors (SIs). By analyzing limitations on the

junction performance imposed by the thermally activated phase slips, we observed a

dramatic reduction of the critical currents in the regime ≤ which was accompanied

by an increase of the zero-bias resistance . The first part of this work provides practical

considerations for the use of such junctions in quantum circuits. With the aim of improving

elements with a very high kinetic inductance, in the second part we developed

superinductors based on the granular Aluminum ( ) films, in which Josephson

junctions are realized between nanoscale grains. The circuits based on such SIs

demonstrate low microwave losses at ultra-low temperatures. Superinductors are an

essential element of a novel qubit that we have developed – the so-called bifluxon. The

qubit consists of a Cooper-pair box (CPB) with low- Josephson junctions shunted by a

superinductor, thus forming a superconducting loop. When the loop is threaded by the

magnetic flux Φ = Φ 2⁄ where Φ is the flux quantum, the qubit offers exponential

suppression of energy decay from charge and flux noises, and dephasing from flux noise.

In the last part of this work, we observed an increase of the energy relaxation time by two

orders of magnitude, up to 100μs, by turning on protection in the bifluxon qubit.

iv

DEDICATION

It has been quite a journey to reach this personal milestone, with all the supports

along the way. I started it with a simple wish that before I was fully trapped by routine

works in the industry, I wanted to spend the last part of my youth learning from and

contributing to frontier research involving quantum engineering and microwave electronics,

I am deeply grateful that today I ended up with much more than I was expecting.

I would like to start with thanking Professor Michael Gershenson (Misha) for his

mentorship throughout this work. As an old mandarin saying goes, he who teaches us for

one day is our father for life. Misha has always been a keen experimentalist, a resourceful

mentor, and a great friend. Indeed, in addition to scientific skills, the most valuable traits I

learnt from the interactions with him is diligently polishing everything. With patience we

polished fabrication recipes, theoretical pictures behind literatures, to scientific writings.

There will be no shine until we spent time really sitting down and polishing down to the

details. Every sentence in this thesis is written under his guidance and influences, one way

or another. I deeply appreciated the opportunity to have him as my mentor.

As another brilliant experimentalist and thinker in our lab, Dr. Konstantin

Kalashnikov always provides me his valuable microwave expertise and extensive data

analysis tools such as python packages and theoretical modeling. I sincerely thank him for

v

his direct contribution and fruitful discussions steering the direction of bifluxon qubits. In

the meanwhile, a special acknowledgement is due to Plamen Kamenov, who diligently kept

our nano-fabrication process consistent and reliable. I admire his attentive attitude, which

later became one of the foundations of this work. Speaking of the foundation, a special

thanks is given to Wenyuan Zhang for her path finding works guiding this thesis. Her

knowledge in microwave electronics and python toolkits are also essential to this work. I

would also like to thank Thomas DiNapoli for encouraging discussions and idea exchange

for this work, which greatly improved my thinking process.

In a separate paragraph I thank Professor Matthew Bell for his pioneering works

from bifluxon qubit experiments to the infrastructure in our lab that I extensively used

during the past years. Nearly all experimentalists would agree that “just keep the tool

running” cruelly simplified the efforts and times from a researcher at the frontline, and

without his solid experience and extensive familiarity to our equipment the experimental

part of this work would never be realized. Even though we did not overlap, his prior efforts

and inputs provide a solid foundation for this work. His influence on this work is ubiquitous.

I would also like to express my appreciations to microwave engineering experience

exchange with energetic scholars Professor Michael Wu, Professor Srivatsan Chakram

Sundar, and Dr. Xiaoyue Jin. They have been invaluable sources of my microwave

vi

knowledgebase. I appreciate the chance to coach and work with young minds during the

past few years including Darren Schachter, Brian Lerner, Shuping Lee, and Mohamed

Zeineldin. By completing various toolkits in the lab we learnt a great deal from each other.

I would also like to thank the supports from Yuriy Streltsov and William Schneider for

their continuous helps from electronic to machining requests from us.

Also, I would like to thank those brilliant minds that I had privilege to work jointly

in the past, in particular Dr. Takane Kobayashi, Dr. Leila Kasaei, Dr. Hussein Hijazi, and

Professor Leonard Feldman on the granular aluminum engineered with helium ion beam

lithography projects. It is quite impressive to see the other end of frontier fabrication

spectrum such as ion beam lithography, and I sincerely hope with the continuous efforts

this technology can soon be leveraged to improve the performance of superconducting

circuits. I thank all members of Eva’s lab, in particular Dr. Junxi Duan, Xinyuan Lai, Dr.

Shaung Wu, Zenyuan Zhang, Nikhil Tilak, Dr. Jinhai Mao, and Professor Eva Andrei, for

being our faithful and resourceful neighbors. From vacuum gaskets to atomic force

microscopy. their proximity and availability were keys to our experimental works. I would

also like to give my deep appreciation to Professor Vitaly Podozorov for providing his

invaluable insights and offering tools during our difficult times.

vii

I would also like to thank my committee members Professor Salur, Professor

Bartynski, Professor Kotliar, and Dr. Chii-Dong Chen, for their precious inputs and

guidance on my thesis writings. I also thanks Jerrell Spotwood, Shirley Hinds, Nancy

Pamula, and Professor Ronald Gilman for their continuous administrative supports and

efforts in the past few years such that I can focus on the scientific part of my study.

I also appreciated the fruitful brainstorms among various occasions with the great

minds in the same building, Dr. Po-Yao Chang, Hsiang-His Kung. Dr. Li-Cheng Tsai,

Ghanashyam Khanal, and Conan Huang. They had provided solid sources for inspirations

which benefit my research.

Finally, I would like to end this important section by echoing the last statement in

the opening paragraph. I pursued my PhD with the aim of extending my engineering

toolbox, and I am deeply grateful that I ended up with growing not only intellectually but

also mentally. By taking up the responsibility with Tammy as being parents, I am truly

thankful to have Willow as our little monster and form a wonderful family. With the on-

going pandemic, the last part of this work can never be done without Tammy’s great

attention and continuous supports in my life.

With all the scientific efforts, dedications, friendships, and loves, here I humbly

present this work to everyone I have the privilege to work with and live with.

viii

Table of Contents

ABSTRACT OF THE DISSERTATION ................................................................ ii

DEDICATION ............................................................................................................... iv

CHAPTERS

Chapter I : Introduction ..................................................................................................1

I.1The Josephson phenomena and modeling of Josephson junctions ......................3

I.2The phase-slip dynamics of a Josephson junction ...............................................6

I.3The current-voltage characteristics of a Josephson junction .............................10

I.4The phase diffusion regime in underdamped junctions .....................................13

I.5Kinetic inductance .............................................................................................15

I.6The non-linear inductance of a Josephson junction ...........................................18

I.7Theory for fluxon-parity protected circuits .......................................................19

I.8Components for fluxon-parity protected circuits...............................................22

I.9Thesis overview .................................................................................................25

Methodology ..............................................................................................27

II.1Fabrication ........................................................................................................27

II.1-1 Lithography ....................................................................................28

II.1-2 Film deposition and junction fabrication .......................................33

II.1-3 Process flows .................................................................................39

II.2Measurement ....................................................................................................42

II.2-1 DC measurement ............................................................................42

II.2-2 MW measurement ..........................................................................47

II.2-3 Temperature control and magnetic fields characterization

using SQUID geometry..................................................................50

II.3Lists of samples ................................................................................................55

II.3-1 Low- junctions ..........................................................................55

II.3-2 Resonators with superinductance ...................................................56

II.3-3 Bifluxon devices ............................................................................56

ix

Thermal effects in low- Josephson junctions .......................................57

III.1Sample design .................................................................................................58

III.1Noise reduction in the measurement setup. ....................................................60

III.2Current-voltage characteristics of low- junctions ......................................62

III.3Josephson junctions with ≈ ...............................................................63

III.4Josephson junction ≈ . and the effect of shunting JJ with

..............................................................................................................67

III.5Discussion: the suppressed switching currents .....................................70

III.6Conclusion and outlook ..................................................................................80

Microresonators fabricated from high-kinetic-inductance

Aluminum films 82

IV.1Introduction.....................................................................................................82

IV.2Experimental details .......................................................................................84

IV.2-1 Design and fabrication ...................................................................84

IV.3Measurement and microwave analysis ...........................................................88

IV.3-2 Microwave setup ............................................................................89

IV.3-3 The procedure of extracting the quality factors and its

analysis ...........................................................................................90

IV.4Discussion .......................................................................................................95

IV.4-1 The resonance frequency analysis .................................................97

IV.4-2 In-depth analysis of () and () fitting ............................99

IV.4-3 The two-tone time-domain measurements and telegraph

noise .............................................................................................101

IV.4-4 Scaling of () .........................................................................104

IV.4-5 Pump-probe measurements of the TLS relaxation time ..............106

IV.5Summary .......................................................................................................107

Fluxon-Parity-Protected Superconducting Qubit .....................................110

V.1Introduction ....................................................................................................110

V.2Suppressing the decoherence .........................................................................111

V.3Experimental setups .......................................................................................115

V.4Transmission measurement ............................................................................117

V.5Time-domain analysis ....................................................................................121

x

V.6The offset charges and mitigation of quasiparticle poisoning .......................125

V.7Conclusion .....................................................................................................128

Conclusion and outlook ...............................................................129

VI.1Junctions with low Josephson energy ...........................................................129

VI.2Superinductors based on granular aluminum thin films ...............................130

VI.3Fluxon-parity protected qubits ......................................................................131

References ....................................................................................133

xi

List of Figures

Figure I-1: Symbolic circuit representation for a Josephson junction ........................ 4

Figure I-2 The tilted cosine potential from Equation I-5 for a junction with

= . ................................................................................................ 6

Figure I-3: The potential and first six quantized energy levels (bands) for

junctions with different charging energy. ................................................. 7

Figure I-4: Schematics of two phase-slip processes ................................................... 9

Figure I-5: The IVCs for junctions with different levels of dissipation. Left:

Q>1 Right: Q<1 ...................................................................................... 11

Figure I-6: (Left) Different regimes for an underdamped junction with small

(dashed lines at 60 nA, 130 nA and 200 nA, respectively). The

corresponding indicate the temperatures above which the

junction enters the UDP regime. The vertical red line separates the

QPS regime from the TAPS which takes place at the crossover

temperature . (Right) Comparison of the UDP dynamics with

the dynamics shown in Figure I-4. (Inset) Equivalent circuit of the

junction with frequency dependent dissipation. Image was adopted

from [24]. ................................................................................................ 14

Figure I-7: Schematics of a fluxon-parity protected circuit, the “bifluxon”. ............ 21

Figure I-8: (a) First two energy levels of the bifluxon qubit as a function of

detuning from degeneracy point. (b) Calculated amplitudes of the

flux (solid lines) and charge (dashed lines) energy dispersion as a

function of qubit parameters. .................................................................. 24

Figure II-1: SEM image for a single junction with in-plane dimension

× ................................................................................. 31

Figure II-2: Standard deviations of normal state resistance of single junction

devices fabricated in our laboratory ........................................................ 31

xii

Figure II-3: Scattering of the junction areas in the chain devices. The data in

blue/orange histograms are before/after the process optimization. ........ 32

Figure II-4: Resistance of the / interface as a function of the Argon

flow rate. ................................................................................................. 35

Figure II-5: The film stacks during the multi-angle deposition and directional

ion milling. The left trench in the cross-sectional view shows a

normal multi-angle deposition, while right trench in the cross-

sectional view shows a narrower trench that suffers from

unexpected film deposition due to the loss of PMMA film

thickness during the directional ion milling step between the 1st

and 2nd deposition. ........................................................................... 35

Figure II-6: The − “dome” for granular aluminum deposited in this

work. The dashed curve corresponds to the literature data for grAl

films deposited at 77K. ........................................................................... 37

Figure II-7: A meandered nanowire made of granular aluminum. The device is

false colored for clarity. The rest of the structure, which was not in

a galvanic contact with the device, was fabricated in order to

reduce the nonuniformity of the nanowire width due to the

proximity effect in the process of e-beam nanolithography. .................. 38

Figure II-8: The sheet resistance of films as a function of the

flow. ........................................................................................................ 38

Figure II-9: Process flow for junction chain project ................................................. 39

Figure II-10: Process flow of microwave resonator project ..................................... 41

Figure II-11: Thermal anchorage points (hand-drawn green blocks) for twisted

pairs and MW cabling ............................................................................. 43

Figure II-12: (left) The OFC tubing filled with copper-powder epoxy at the

end of DC wiring. (right) The detachable sample holder for DC

measurement. .......................................................................................... 44

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Figure II-13 The wiring schematics for DC current source measurements .............. 45

Figure II-14 The wiring schematics for DC voltage source measurements .............. 46

Figure II-15: Characterizing the parasitic capacitance ............................................. 46

Figure II-16 (left) The sample holder and SMA anchoring for MW

measurement. (right) The microwave sample holder. ............................ 47

Figure II-17 The schematics of MW measurements ................................................. 49

Figure II-18: Transmission background check before each cool down .................... 50

Figure II-19: Optimized PID table ............................................................................ 51

Figure II-20: Optimized noise level and thermal fluctuation at different

temperatures. ........................................................................................... 52

Figure II-21 Periodic response of a SQUID to external magnetic field. .................. 53

Figure II-22: Comparison of the magnet currents that correspond to =

for three designs with different area of the SQUID loop. ....................... 54

Figure II-23: Various designs of SQUID used in this work. .................................... 54

Figure II-24: in , in are the normal state resistance and

the area of single junction in the chain. , in are the

single junction Josephson energy ≡ C and charging

energy ≡ , respectively. , , in are the

single junction critical current predicted from Equation I-2 and the

switching current that experimentally measured.

shows how many SQUIDs are in series in the design. .

indicates the source of capacitor shunting the junction where “”

means external capacitor was fabricated, “ ” means the

junction has intrinsically large area parallel plate capacitors to

provide low , and “ ” represents data point that are

measured before external capacitor was fabricated. Finally,

xiv

and ; in are the zero-biased resistance at zero B-field

and B-field at full frustration point, respectively. ................................... 55

Figure III-1: Various designs of SQUIDs. (a) Each SQUID unit cell was

shunted by a large ≈ . the ground. (c) SQUIDs formed

by large JJs with junction area ≈ . .................................. 59

Figure III-2: Device schematics and the circuit diagram for devices with

shunting capacitors and a common ground (opaque blue pad). .............. 60

Figure III-3: The IVCs recorded for a two-unit SQUID device with different

measurement set-ups at = (sample D059B0N1 in

Table III-1). Each SQUID unit consisted of two nominally

identical junctions with an area . × . and resistance

10kΩ ( = . ). Due to a large shunting capacitor to the

ground, this device has ⁄ = ⁄ ≫ .

Without thorough filtering, the IVC was smeared and the IVC

hysteresis, expected for an underdamped junction at low , was

significantly reduced. Filtering of all leads used for the IVC

measuring restores the critical current which is close to , ,

and enables observation of a well-developed hysteresis. The inset

shows that the noise level in our measurements is around . . ....... 61

Figure III-4: (left) The current-voltage characteristic (IVCs) for a chain of 20

SQUIDs with = (sample D059BBN2 in Table III-1).

(right) The enlargement of the region of small currents/voltages.

Note that the resistance is non-zero for all biasing currents. As

soon as the biasing current exceeds = . (indicated by

a cyan arrow), the voltage across the chain rapidly increases, and at

even greater currents approaches the value × ∆/, where

is the number of SQUIDs and ∆ is the sum of superconducting

energy gaps in the electrodes that form a junction. At ≥

× ∆/ dissipation is due to generation of non-equilibrium

quasiparticles. At < the non-zero resistance ≈

is due to thermally activated phase diffusion (section I.4).

Note that for such low- junctions, the switching current is three

xv

orders of magnitude smaller than the Ambegaokar-Baratoff critical

current , ≈ . . ...................................................................... 63

Figure III-5: IVCs of two connected in series SQUIDs with = .

at = (blue curve) and = . (red curve) at base

temperature (sample D059B0N1 in Table III-1). ................................... 64

Figure III-6: (left) IVCs of two connected in series SQUIDs (sample

D059B0N1 in Table III-1) with = . at = . at

different temperature . The order of for each IVC

from top to bottom is from to . .................................. 65

Figure III-7: (a) IVCs of a single SQUID with = . (sample D079N6

in Table III-1) measured at different magnetic fields (we only

marked four selected curves for clarity). A sub-gap voltage plateau

at ≈ appears at > . . (b) The dependence of

on the superconducting solenoid biasing current. (c) The

measured ()/( = ) as a function of

(/). The dash line corresponds to the dependence

∝ . The reason for observed deviations from the

dash line for < . remains unclear. ................................ 66

Figure III-8: (main panel) The IVCs for a chain of 30 SQUIDs (sample

D063BAN6A_bf and D063BAN6_af in Table III-1) based on JJs

with resistance = and = . before (blue) and

after (red) deposition of the electrode that significantly

increased the capacitance across individual SQUIDs. Upper left

inset: The enlargement of the IVC before deposition. Clear

. jumps can be seen that correspond to the voltage drops

∆/ across individual SQUIDs. Lower right inset: the

enlargement of the low- part of the IVCs. Comparison of red and

blue curves shows that Pt deposition significantly increased ,

up to 20 pA and reduced the zero-bias resistance. .................................. 69

Figure III-9: Analytical solutions from [72] for IVCs at different fluctuation

level ≡ . At = a sharp turn was observed around −

× ≈ , indicating a decrease of switching current from

xvi

to . when the noise level increased from ≪ to =

. . ...................................................................................................... 71

Figure III-10: The switching current measured for different devices at

≈ and the magnetic field increasing from = to

B corresponding to = /. The red dash-dotted line

corresponds to the switching current predicted by the IZ theory in

presence of = . ............................................................... 72

Figure III-11: The zero-biased resistance measured for different devices

at ≈ and the magnetic field increasing from =

to B corresponding to = /. The red dash-dotted line

corresponds to predicted by the IZ theory in presence of

= . ................................................................................... 74

Figure III-12: The switching current as a function of measured in

our experiments (grey symbols) and by other experimental groups

(blue dots) [24, 64, 65, 74-86]. All the data have been obtained at

≈ − for Al-AlOx-Al junctions. Note that the

literature data on this plot correspond to samples with different

(the ratio / for a given varies over a wide range).

However, it seems that this is not the main factor that controls

scattering of . For comparison, the blue dashed line

represents , . .............................................................................. 76

Figure III-13: The zero-bias resistance as a function of measured in

our experiments (grey symbols) and by other experimental groups

for Al-AlOx-Al junctions (blue dots) [24, 64, 65, 74-86]. Table

III-1 and Table III-2 summarizes the parameters of these samples,

respectively. All the data have been obtained at the base <

, though the physical temperature of the Josephson circuits

has not been directly measured. .............................................................. 77

Figure IV-1: (a) Microphotograph of a portion of the halfwavelength resonator

capacitively coupled to the coplanar waveguide transmission line.

Light green - Al ground plane and the central conductor of the

transmission line, green - silicon substrate, black - the central strip

xvii

of the resonator made of strongly disordered Al. (b) Several

resonators with different resonance frequencies coupled to the

transmission line ..................................................................................... 87

Figure IV-2: Schematics of the resonator measurement setup ................................. 89

Figure IV-3: Fitting procedure. (a) Blue and red points correspond to the

transmission measured before and after the phase delay is

removed, respectively. After removing the phase delay, the data

form a circle on the IQ-plane with the center at . (b)

Normalized transmission ∗ on the complex plane. The angle

between the center of the ∗ circle and the real axis

corresponds to . (c) The phase versus frequency (blue points)

fitted with = + − [( − )] (red curve). (d,e)

Measured data (blue points) and the fit with Equation IV-3 (red

curve). ..................................................................................................... 92

Figure IV-4 The dependences () at ≈ for the resonators with

different widths. Solid curves represent the theoretical fits of the

quality factor governed by TLS losses [Equation IV-5]. ........................ 93

Figure IV-5: The temperature dependences of resonance frequency shift

()/ (a) and the internal quality factor (b) for the

resonators #2−4 measured at ≈ () and ≫ (∆). The

fitting curves correspond to Equation IV-9 and Equation IV-7,

respectively. ............................................................................................ 97

Figure IV-6: The temperature dependences of for different resonators. ......... 101

Figure IV-7: (a) The dependences of for resonator #1 on the pump tone

power for several values of detuning ∆ between resonance

and pump frequencies. (b) The values of measured versus

detuning ∆ at a fixed number of the pump tone photons in the

resonator ≈ . ......................................................................... 102

Figure IV-8: The time dependence of [] measured at = at

a fixed frequency on the slope of a resonance dip. The microwave

xviii

power corresponds to ≈ . Each point corresponds to the

data averaging over . .................................................................... 104

Figure IV-9: (a) The pulse sequence. (b) The time dependence of

measured at = . . The pump pulse at = +

was applied between = and = . . The pump

tone power corresponds to ¯ ≈ . Each data point was

averaged over 4000 cycles with the same readout delay time. The

inset shows CW measurement of versus with (red) and

without (blue) the pump signal and indicates the position of

used in the relaxation time measurement. The readout power was

at the single photon level for all measurements on this plot. ................ 107

Figure V-1: The tradeoff between the decay and dephasing protection in

superconducting qubits with a single charge or flux degree of

freedom. The band structure (top panels) and wavefunctions

(bottom panels) of a particle in quasiperiodic potentials: (a) the

free-particle regime and (b) the tight-binding regime. The

wavefunction overlap and the energy sensitivity ()/ do

not simultaneously vanish for any point (i). Flux (charge) qubits

correspond to the case in which the control parameter =

, kinetic energy = (), tunneling energy =

(), and |⟩ is a fluxon (charge) basis. .................................... 113

Figure V-2: (a) Simplified circuit scheme of the bifluxon qubit. Charging

energies of the superinductor and CPB are and ,

respectively. The qubit is controlled by the CPB charge and

the magnetic flux . (b) Optical image of the bifluxon qubit,

readout resonator, and the microwave transmission line. The inset

shows the SEM image of its central part: two JJs form the CPB

island (red false color), the long array of larger JJs acts as a

superinductor (blue), the narrow wire (green) forms the closed loop

and couples the qubit to the readout resonator. ..................................... 115

Figure V-3: Spectra of the bifluxon qubit: experimental data for the − |⟩

and − |⟩ transitions (symbols) and the result of exact

diagonalization of the circuit Hamiltonian in Eq. (1) (solid and

xix

dashed lines). (a) Flux dispersion of the transition frequencies

for two values of the CPB charge = , . The inset is an

enlargement of the qubit spectrum near = , displaying the

avoided crossing that characterizes the rate of double phase slips

. (b) Charge dispersion of the transition frequency for

= . ............................................................................................ 118

Figure V-4: (a) Measurements of the bifluxon energy relaxation in the

protected state (red circles) and unprotected state (blue squares).

The sequence of pulses is shown in the inset. The exponential fits

are shown by solid and dashed lines, respectively. Note that the

resonance energy of the qubit in the protected state is

× . (approximately × ), and a nonzero

occupancy of the first excited state [() + ] with

the qubit temperature = = is taken into

account. (b) Demonstration of an absence of qubit excitation by the

gate voltage pulses. ............................................................................... 120

Figure V-5: Energy relaxation time as a function of the flux frustration

(a) and the CPB charge (b). The pale circles represent

all the measured data and the bright circles show the longest

measured for a given operation point. The lines correspond to

fitting to the resistive noise theory. The sharp dip around =

. corresponds to the Purcell decay into the readout resonator...... 122

Figure V-6: The Ramsey fringes measurement. (a) The pulse protocol for

evaluation in the protected state. The protection is turned on for a

fixed time of ; the time delay between two / pulses is

varied in order to record Ramsey fringes. (b) The experimental

data (circles) and the damped-oscillation fitting (the solid line).

Note that the value of = . describes the fringe damping

in the = state. In the protected state (within a time interval

< < ) damping of Ramsey fringes may be caused by

the pulse jitter rather than dephasing (see the text). ..................... 125

Figure V-7: Suppression of quasiparticle poisoning by gap engineering. (a)

Profile of the superconducting gap across the CPB island. The

xx

critical temperature of the thin CPB island is . ÷ . higher

than that in the thicker electrodes. (b)–(d) Spectroscopy of the

readout resonator as a function of for bifluxon qubits: without

gap modulation at (b), and with gap modulation at

(c) and (d). (e) The gap-engineered device at

. The dispersive shift of the readout resonator (color

coded) is measured at a fixed gate voltage over 9 hours. The

shift is converted into using the data of panel (c).

Abrupt jumps reflect the QP events ( = ±), gradual shift

corresponds to a monotonic drift of with a rate of less than

− per minute. .............................................................................. 127

1

CHAPTER I : INTRODUCTION

The superconducting artificial atoms [1, 2] have been an active experimental testbed for

quantum phenomena for more than three decades. As a non-dissipative circuit element

possessing a non-linear inductance, the Josephson junction has found numerous

applications ranging from quantum limited amplifiers [3] to the noisy intermediate-scale

quantum (NISQ) devices [4]. The most notable application in recent years was perhaps the

development of transmon qubits that combined sub-micron Josephson junctions with large

shunting capacitors [5]. Due to the simplicity and robustness of this design, the coherence

time of the transmon qubits has been increased up to 100 s, and the intermediate-scale

circuits comprising ~50 physical qubits have been developed [6].

Several challenges remain, however, for the state-of-art transmon devices. The

reported single- and two-qubit gate fidelities only marginally exceed the threshold for

implementation of the error correction code [7]. The large shunting capacitor, on the one

hand, makes transmon qubit insensitive to charge noise but, on the other hand, reduces the

qubit anharmonicity, and this imposes limitations on the speed of quantum gates [8, 9]. To

scale up the number of physical qubits while maintaining high fidelity, further

improvements of coherence are required.

2

It is currently believed that one of the main sources of decoherence is the two-level

defects, or Two-Level-Systems (TLSs), whose electric dipole moment fluctuates even at

ultra-low temperatures [10, 11]. Such defects usually reside in amorphous oxides that cover

surfaces of superconducting films and substrates. This issue is usually mitigated either by

improving the quality of all surfaces and interfaces in the process of fabrication [12] or by

reduction of qubit-TLSs coupling through proper design of the qubit Hamiltonian [13, 14].

Here, we pursue the second approach and focus on the component-level realization and

verification for protected circuits with preserved fluxon parity. This thesis is built on the

foundation laid out by a series of previous experimental efforts towards developing the

parity protected quantum circuits [15-17], and it reflects our recent work in this field [14,

18, 19].

To optimize the fluxon-parity protected circuit, it is necessary to better understand

the physics of low- Josephson junctions and superinductors based on such junctions.

The first four sections in this chapter review operation of low- superconducting

junctions, whereas in the following two sections we discuss superinductors and their

applications. We will close this chapter with two sections introducing the idea of protective

quantum circuits.

3

I.1 The Josephson phenomena and modeling of Josephson junctions

About five decades after superconductivity was discovered, in 1962, a theoretical

prediction was made by B. D. Josephson that when two pieces of superconductors were

brought in close vicinity and formed a weak link, a zero-voltage tunneling current could

flow through the weak link, or the junction. [20] According to Josephson, the non-

dissipative tunneling current is a non-linear function of the phase differences ≡ 2 −

1 between the phases of wavefunctions of Cooper pair condensates in the

superconducting electrodes:

= C

Equation I-1

The critical current C can be expressed as [21]

C,AB =

2ℎ

2

Equation I-2

where Δ is the superconducting energy gap, is the normal-state resistance of the

junction.

This current-phase non-linearity opens the possibility for using Josephson junction in

various applications including quantum-limited amplifiers and superconducting qubits.

When a current flowing through a junction exceeds the critical current, a non-zero average

4

voltage develops across the junction; this voltage is described by the second Josephson

relation [Equation I-3]:

=

2

Equation I-3

where Φ ≡ ℎ 2⁄ ≈ 20.6 Gμm

Taking into account the capacitance of a junction, , and active losses at high

frequencies, , the equivalent circuit representing the Josephson junction can be

schematically represented in Figure I-1.

Figure I-1: Symbolic circuit representation for a Josephson junction

This is so-called Resistive-and-Capacitive-Shunted-Junction (RCSJ) model of a

Josephson junction. By using the Kirchhoff's law for this circuit, we can write down the

total current as =

+

+ C sin . Or, using the Josephson relations, we can write

=

2

+

2

+ C

Equation I-4

This equation can be rewritten as

5

0 =

2

+

2

1

+ −

C

2 −

C

2

C + .

Equation I-5

This is the equation of motion for a particle with mass

moving in an effective

potential () =C

1 − −

C under a constant force

C

C. The term

represents damping in this system. By considering harmonic oscillations near

the minima of this so-called “washboard” potential, one can introduce the characteristic

frequency of these oscillations, the plasma frequency , and quality factor :

= 2

Φ

Equation I-6

= = 2

Φ ≡

Equation I-7

, where is the so-called Stewart-McCumber parameter[22]. The Stewart-McCumber

parameter and the quality factor have been used to distinguish between the

overdamped limit (, ≪ 1) and the underdamped limit (, ≫ 1) of the junction

dynamics.

The amplitude of the cosine term in the potential is the Josephson energy ≡C

;

the cosine function is tilted with the slope C⁄ , where is the DC current flowing

through the junction. This tilted cosine function (“the tilted washboard potential”) has been

6

used to study the dynamic of Josephson systems. In Figure I-2 we plot the tilted washboard

potential for an // junction with a typical critical current of = 1μA

(corresponding to ≈ 25K) at various biasing current . In Figure I-2 we also marked

the position of the fictitious particle which represents the response of the system at different

.

Figure I-2 The tilted cosine potential from Equation I-5 for a junction with = .

I.2 The phase-slip dynamics of a Josephson junction

To study the dynamic of the system in more detail, let us first assume that the

system is damping-free ( = ∞) and the tilt / = 0. Equation I-5 is then simplified:

Φ

2

φ + ∇− = 0.

7

At the bottom of each well ≈ 2 and, by limiting deviations from the equilibrium

positions, we recover the harmonic oscillator equation

φ + = 0 with the

characteristic frequency =

. The energy difference between the adjacent

oscillator levels ℏ is

ℏ = 2

Equation I-8

where ≡()

is the charging energy associated with the capacitor in Figure I-1. In

Fig. I-3 we draw the first six energy levels for two values of while keeping fixed

[23].

Figure I-3: The potential and first six quantized energy levels (bands) for junctions

with different charging energy.

As we can see in Figure I-3, the energy ratio ⁄ determines whether the system should

be treated classically or quantum mechanically. When ⁄ is large, the ground state

8

wavefunction is localized near the potential minima. With decreasing ⁄ , the

wavefunctions become delocalized over many potential minima due to tunneling.

The non-zero probability of tunneling between adjacent minima of the washboard potential

results in generation of the so-called phase slip, an abrupt 2 change of a phase difference

across the junction and generation of the voltage pulse () such that ∫ ()dt = Φ/2.

The phase slips due to tunneling are known as the quantum phase slips (QPS), they are

characterized by the rate ≈

ℏexp −

.

ℏ [24, 25] where Δ =

8(1 − ⁄ ) and attempt frequency ≈ . However, the phase slips can also

occur due to the over-the-barrier thermal activation at non-zero temperatures. The rate of

the thermally activated phase slips (TAPS) depends exponentially on the temperature:

≈ ⁄ [24, 26]. Figure I-4 schematically shows the phase dynamics

corresponding to the QPS and TASP. At zero tilt, the phase slips with different signs of the

phase change occur with the same probability and, as a result, the average voltage across

the junction is zero. However, when the junction is biased with current , the non-zero tilt

breaks the symmetry and a non-zero average voltage proportional to the phase slip rate is

generated across the junction. The dynamics of the system in this case depends on

dissipation. In the underdamped regime the energy gained by a system in the process of

9

over-the-barrier activation cannot be dissipated and a running-away solution corresponds

to generation of voltage = 2Δ/ across the junction due to breaking of Cooper pairs.

Figure I-4: Schematics of two phase-slip processes

In-situ tunability of the height of potential barriers of the Josephson potential opens

numerous opportunities for basic research and applications. Indeed, soon after B.D.

Josephson discovered the Josephson effect in 1962, Jaklevic et al [27] came up with a split

junction design named Superconducting QUantum Interference Devices (SQUIDs). The

DC SQUID comprises a superconducting loop interrupted with two nominally identical

Josephson junctions. The critical current and for such a system can be modulated by

an external magnetic flux Φ = × threading the superconducting loop :

10

= |cos(Φ Φ⁄ )|

Equation I-9

We will utilize the SQUIDs geometry in this work to investigate the dynamics of phase

slips in the junction as a function of magnetic flux Φ and temperature .

I.3 The current-voltage characteristics of a Josephson junction

Upon increasing the biasing current and exceeding the critical current , the

non-zero voltage is generated across the junction, which corresponds to “sliding” of a

fictitious mass along the tilted washboard potential. When is reduced, in case of low

dissipation the potential tilt needs to be reduced to nearly zero in order to re-trap the

fictitious particle. A more rigorous calculation can be done for an underdamped junction

as shown in [28]: drops to zero only when becomes less than the retrapping current

≈

. The Q dependence of indicates that the system can be returned to zero

voltage state if only the energy gained from the biasing tilt (as phase advances by 2 from

one potential well to the next one) is dissipated through damping, Q. On the other hand,

in heavily damped junctions (~1) dissipation dominates inertia, and we expect no

hysteresis in the IVC. This corresponds to the case where the fictitious particle slowly

slides down the inclined washboard potential.

11

We illustrate the above scenarios in Figure I-5. In both cases the characteristic scale of

currents is provided by the critical current C,AB. The Ambegaokar - Baratoff relation

Equation I-2 allows us to relate C,AB to the normal-state resistance N , one of important

fabrication parameters.

Figure I-5: The IVCs for junctions with different levels of dissipation. Left: Q>1 Right:

Q<1

Due to non-linear IV characteristics, the Josephson-junction-based elements were

considered attractive for digital applications [29-31]. In these applications, the

underdamped junctions were driven between superconducting and resistive IVC branches

to realize digital zeros and ones. These devices featured ultrafast operation (2 × 10 J/

bit at 770 GHz, [32]) and extremely low energy consumption. They operated in the

“classical” Josephson regimes ≫ , in order to avoid phase slips and maintain

12

the hysteresis in the IVc. Typical parameters are ≈ 2500K and ≈ 4.5mK for

2μm × 2μm niobium junctions which operate at ≈ 4 in the RSFQ circuits [33].

However, due to the requirement of helium refrigeration for operation of these devices and

rapid progress of CMOS industry, superconducting logics found rather limited applications.

With the advancement of superconducting artificial atoms and dilution refrigeration

in the late 1990s Josephson junctions regained attention as non-linear and low-dissipative

elements of superconducting qubits. The Cooper pair box (a small superconducting island

flanked by two JJs) became one of the first solid-state quantum bit (qubit) experimentally

realized in 1998 [34]. Soon after, various superconducting qubits archetypes were

implemented [35] by mapping the logical quantum states to different physical realizations.

The fluxonium qubit, for example [36], is based on a junction with 10 ≳ ⁄ ≳ 1 and

≫ , where is the inductance energy for the superinductor element which will be

addressed below. The latter requirement can be satisfied if the superinductor with

inductance energy = (Φ 2⁄ ) ⁄ has a sufficiently large . Similarly, in the case of

bifluxon [14] it is also required that ≫ to decouple the qubit from flux noises.

Experimentally, tuning of can be achieved either by changing the oxidation parameters

or by varying the junction area (Equation I-2). In this thesis, the low- junctions with

≈ 0.01 ÷ 10K and ≈ 0.001 ÷ 1K have been fabricated by e-beam lithography.

13

Besides and , another characteristic energy scale is provided by the

temperature. Nominally, the base temperature for a standard cryogen-free dilution

refrigerator is around ≈ 0.01 ÷ 0.02K with cooling power in sub-mW range. Any

poor thermal anchoring between sample carrier or wirings to the cold plate could result in

higher sample temperatures which would increase the rate of TAPS process. Depending on

how small is, the temperature could be the dominant parameter that controls

dissipation in the circuit.

I.4 The phase diffusion regime in underdamped junctions

As was discussed in the previous section, the thermally activated phase slips could

harm the performance of low- junctions by driving the system into a resistive state. The

detailed analysis of the effect of non-zero temperature in the underdamped junctions was

provided by Kivioja et. al,[24]. By considering the quality factor at plasma frequency,

, and the energy dissipated between adjacent potential maxima Δ ≈ 8 ⁄ ,

Kivioja et. al showed that the maximum possible power dissipated due to phase diffusion

before switching to a state with ≈ 2∆/ can be expressed as

2

Φ×

Δ

2= ×

14

where = 4 ⁄ is the maximum possible current carried by underdamped junctions

in the Phase Diffusion (UPD) regime. At < , there is non-zero probability for a

fictitious particle to be retrapped after escape from a local minimum. As a result, instead

of a running-away state with = 2Δ ⁄ , the IVC demonstrates a non-zero slope at

< . Such frequency dependent quality factor was also used to explain the IVCs for

underdamped junctions where both hysteresis feature and non-zero have been observed

[37]. The value of therefore provide valuable information regarding the nature of

damping in the junction circuits, and in Chapter III we will use the zero-bias resistance as

one of the main characteristics of low- junctions. The summary of different regimes

considered in [24] is shown in Fig.I-7.

Figure I-6: (Left) Different regimes for an underdamped junction with small

(dashed lines at 60 nA, 130 nA and 200 nA, respectively). The corresponding

indicate the temperatures above which the junction enters the UDP regime. The

vertical red line separates the QPS regime from the TAPS which takes place at the

crossover temperature . (Right) Comparison of the UDP dynamics with the

dynamics shown in Figure I-4. (Inset) Equivalent circuit of the junction with

frequency dependent dissipation. Image was adopted from [24].

15

In this work, we will investigate the regimes of low-frequency transport in low-

underdamped junctions with ≥ ≫ . Two experimental observables, and ,

will be used to classify and characterize the low- junctions used in our quantum circuits.

These results will be presented in Chapter III.

I.5 Kinetic inductance

In the following two sections, we turn our attention to the second components in

the bifluxon circuit, the superinductor, which is a dissipationless element with microwave

impedance greatly exceeding the resistance quantum =

(). In classical linear circuits

an inductor is usually realized as a wound thin wire with an inductance being proportional

to the wire’s length. Such an element unavoidably has a large parasitic capacitance which

drastically reduces the high-frequency impedance of the inductor. As a quantitative

example, consider two parallel wires of radius and distance apart, one end of the

wires is short-circuited. Such device can be modelled as a transmission line where its

specific inductance and specific capacitance can be calculated as = log( ⁄ ) ⁄ and

= log( ⁄ )⁄ , respectively. Even though it is possible to scale up the inductance by

increasing the length of the wire, the impedance remains logarithmically small =

⁄ =

( ⁄ )

. As a result, the inductor has a relatively low cutoff frequency ;

16

above this frequency the circuit response is dominated by parasitic capacitance and the

device no longer works as an inductor. Clever designs and advanced micro-fabrication

techniques, as demonstrated in [38], can still realize the device impedance ≈ 30Ω

greater than R at microwave frequencies, but such device requires a 100μm × 100μm

footprint and sophisticated removal of bulk substrate to reduce the dielectric constant and

minimize the parasitic capacitance. It would be very desirable to find an alternative

candidate for superinductors with a much smaller footprint and less complicated fabrication

methods.

The alternatives do exist, and they are based on the kinetic inductance rather than

on the geometric one. According to the Drude model, the current in a conductor with carrier

density and cross-sectional area under an AC electric field = exp() can be

expressed as

= =

1

1 +

where is the momentum relaxation time. As frequency increases such that ≫ 1,

the response per unit length becomes inductive with

=

2

1

Equation I-10

17

However, this inductance, which is associated with inertia of charge carriers, is usually

masked by dissipation in a normal metal at frequencies lower than

, which is usually in

the THz range. On the other hand, in superconductors due to the zero loss at low

frequencies, one can exploit this inductance at ≪ . Since the energy in such inductors

is stored in the kinetic energy of the Cooper pair condensate with density (Equation

I-10) rather than in the magnetic field, such an inductance is called the “kinetic inductance”.

High kinetic inductance has been reported for several strongly disordered materials

including [39, 40], [41], [42, 43], and granular aluminum ( )[44].

Such materials have been found applications in magnetometers [45] and single-photon

detectors [46].

Still challenges exist for implementing the disordered thin films in the quantum

circuit as SIs. One common use of inductors in quantum circuits is the microwave

resonators for the qubit readout. These resonators have typically a sub-mm length and a

sub- μm width. In order to have predictable specifications and performance, it is required

that the film should have high uniformity at this relatively large length. It is not trivial to

develop the deposition technique that would guarantee uniformity for highly resistive films

approaching the superconductor-insulator transition [47]. Thorough optimization of

18

fabrication processes is required to achieve reproducibility of disordered nanowire

fabrication.

Before we end the discussion for superinductors and make a final remark, let us

briefly review another approach which was also commonly employed. This approach is

based on the use of kinetic inductance of Josephson junctions, with its own pros and cons

which we will address in the next section.

I.6 The non-linear inductance of a Josephson junction

Blessed of being non-linear and non-dissipative simultaneously, the Josephson

junction is also a promising candidate for realizing the SI. The junction non-linearity has

found many applications ranging from Josephson metamaterials [48], superinductors [15,

49, 50], quantum-limited amplifiers [3, 51] to various artificial atoms developed in the

superconducting qubits family [14, 16, 52, 53]. Most of these applications rely on the non-

linear inductance associated with the Josephson current-phase relation Equation I-1,

Equation I-3. Let us rewrite the relations as

= cos ,

2

Φ =

we then have

×

=

cos , and from the definition of inductance =

one expects that the inductance for the Josephson junction can be expressed as:

19

() =Φ

2 cos ≡

cos

where =

is the Josephson inductance.

In order to build a linear superinductor, Josephson junctions with

≫ 1 can be

connected in series to form a chain [15, 49, 50]. The condition

≫ 1 is required to

suppress the phase slips within the chain. The entire chain can therefore be viewed as a

single circuit element with large inductance [54]. Optimization of such superinductors

requires the trade-off between the kinetic inductance of individual junctions (∝ 1/) and

the number of junctions in the chain (the parasitic capacitance of a long chain might be

significant). Still, it is possible to achieve high impedance SIs by reducing the dielectric

constant of the superinductor environment, and a device with > 200Ω has been

reported [55].

I.7 Theory for fluxon-parity protected circuits

The idea of protected superconducting devices for quantum computing could be

dated back to two closely related proposals by Kitaev [56] and Douçot et al [57]. It was

proposed that protection of qubit coherence can be implemented on the hardware level by

designing the proper Hamiltonian:

ℋ, = − cos(2) − cos() +

2∗+

()

2

Equation I-11

20

When () ≪ 1 and ∗ ≫ 1, the ground state of the first three terms corresponds to

a linear combination of the wavefunctions denoted as |⟩ and localized around the

minima of potential at = 2 ⁄ , where and are integers. From the

exponential shift theorem, the effect of cos(2) = + 2⁄ operator in the

first term can be viewed as a shift in the -coordinate by ±2 , which maps |⟩ to

(| + ⟩ + | − ⟩) 2⁄ . As a result, with sufficiently small () the full Hamiltonian ℋ,

projected onto the low-energy subspace becomes

ℋ, = −1

2( + ) +

1

2()(2 ⁄ )

where is the shift operator and |⟩ = | + ⟩ . In the absence of perturbation

() = 0, the ground state wavefunction is -periodic in with -fold degeneracy for

the ground states: Ψ = ∑ |⟩ , = . Such degeneracy decouples the

Hamiltonian from environmental noises in the -space, hence it justified the name “parity

protection”. More advanced fault tolerant gate operations can be achieved with a fast pulse

of () [58]. The parity protection together with fault tolerant gate controls forms a

protective computation scheme for the artificial atoms.

In superconducting artificial atoms, the Josephson Hamiltonian is 2-periodic in

its phase coordinate. It is possible to reconstruct Equation I-11 by utilizing -periodic

Josephson elements (the so-called Josephson rhombi) based on the Aharonov-Bohm effect

21

[16] or 4-periodic Josephson elements based on the Aharonov-Casher effect [17]. In this

work we will be focusing on the realization of components for the latter circuit

implementing the 4-periodicity of Josephson energy.

Figure I-7: Schematics of a fluxon-parity protected circuit, the “bifluxon”.

Figure I-7 provides schematics of a bifluxon circuit [14]. This device consists of a Cooper-

pair box (CPB) with the junction energy scale shunted by a large inductor with the

inductance energy = 2⁄ . The phase coordinates for each element, , , Φ ,

represent the phase across the inductor, the phase across the CPB, and the external flux

threading the superconducting loop, respectively. The associated charge coordinates are ,

, corresponding to the offset charge in the inductor, on the CPB island, and due to

external gate control, respectively (these charges are all normalized by 2). The full circuit

Hamiltonian, as shown in the supplementary material in [14], can be simplified to

22

ℋ± = −4 − ±(),

±() = ± cos( 2⁄ ) +

2 − 2

Φ

Φ

The coordinates [, ], [, ] ∝ are the conjugated pairs [, ] = as in Equation

I-11. Here, the ground state energy in was localized around the minimum near =

2 where is an even or odd integer (2 or 2 + 1) for or , which gives rise

to the aforementioned degeneracy and parity protection for the bifluxon circuit. The

wavefunction can be expressed as Ψ ≈ ( − ) exp− ⁄ 4⁄ and is

associated with a fluxon excitation |2⟩ or |2 + 1⟩.

I.8 Components for fluxon-parity protected circuits

The experimental realization of a bifluxon qubit relies on two factors. Firstly, the

fluxon excitation picture requires level quantization of bound states in the inductor phase

coordinate , which implies the depth of potential wells to be greater but not much

greater than the level spacing ℏ at zero temperature ≳ ℏ . In particular this

requires an accurate characterization of relevant energy scales such as and physical

temperature to correctly describe phase slips phenomena in low- junctions. This

provides motivation for the study of such junctions at ultra-low temperatures.

23

The second requirement is the realization of a superinductor, which we introduced

briefly in section I.5. The SI as a circuit element must satisfy two conditions. Firstly, it

should be non-dissipative over a wide frequency range. Secondly, it should demonstrate

sufficiently small parasitic capacitance such that = ⁄ > , where ≡

ℎ (2)⁄ ≈ 6.5kΩ. The purpose of such superinductor in a bifluxon circuit is to minimize

decoherence induced by the flux noise [14]. The qubit sensitivity to a decoherence

process can be estimated by the amplitude of the charge and flux dispersion of the transition

energy Φ, ,

= Φ = 0, = 0.5 − Φ = π, = 0.5

= Φ = π, = 0 − Φ = π, = 0.5

It has been shown in [14] that a minimized dispersion can be achieved when ⁄ is

sufficiently large and, at the same time, remains small. An exponentially small flux

dispersion can be achieved in the regime where 2 ≪ ℏ exp−2 ⁄ .

24

Figure I-8: (a) First two energy levels of the bifluxon qubit as a function of detuning

from degeneracy point. (b) Calculated amplitudes of the flux (solid lines) and charge

(dashed lines) energy dispersion as a function of qubit parameters.

This requires the implementation of an inductive element with > 30μH and a self-

resonance mode above the resonance frequency of the qubit. Assuming such element is a

part of a qubit operating at 10GHz , in order to have the self-resonance mode =

1 √⁄ = 1 √30 × 10 × ⁄ > 2 × 10 it will require the capacitance to be <

0.3fF or its impedance to be > 300kΩ . The physical realization of such a high

impedance device at microwave frequency with geometrical inductance alone would be

25

impossible, but it can be achieved by using suspended JJ chains [55] or disordered

superconducting nanowires [18, 40, 59].

To implement the fluxon-parity protective scheme, it is therefore necessary to better

understand the ultra-low-temperature characteristics of low- junctions and

superinductors. This is the main focus of this thesis.

I.9 Thesis overview

In this chapter, we briefly reviewed the theoretical background and characteristics

of the elements comprising the bifluxon qubit. Two critical elements, the low- junctions

and a non-dissipative high impedance superinductor, were introduced. In this chapter we

discussed the principles of operation of these elements, their realization will be addressed

in the following chapters. Common methodology used in this work, including nano-

fabrications and DC and MW characterization, will be described in Chapter II. A complete

list of devices studied in this work can also be found at the end of that chapter. In Chapter

III, we characterize the low- junction by reviewing the transport data in DC

measurement for a chain of SQUIDs. By varying the Josephson energy with external

flux, we studied the effect of phase slips in such junctions at ultra-low temperatures. In

Chapter IV we investigate the kinetic inductance of disordered superconducting thin

26

films by fabricating coplanar waveguide resonators. This chapter was adopted from [18].

The works in Chapter V is adopted from the experimental section in [14], which

demonstrates a bifluxon circuit that consist of CPB and the superinductors made from the

Josephson junction chains. In Chapter VI we conclude this thesis by formulating the main

findings and providing an outlook for the future experiments with the bifluxon circuits.

27

METHODOLOGY

In this chapter, we focus on the methods of preparation and characterization of Josephson

junctions and disordered films used in this work. As a book-keeping section, the

methodology unavoidably include many details and become extensive. However, to

benefits the future junior researchers, I will still try to put down comprehensive notes in

this chapter. Modularized process steps will be introduced in the first section, while the

measurement protocols are detailed in the second section.

II.1 Fabrication

Nanofabrication is no doubt one of the fields that require sophisticated training and

experiences in an artisanal setting such as university laboratories. Depending on the scales

of the lab and funding, it is not uncommon that PhD students are the backbone of process

integration and equipment maintenance work force after thorough trainings. Being the

foundation of major discoveries of contemporary electronics, nanofabrication plays a

crucial role as a first step in the study of our quantum circuit. Indeed, nanofabrication is

challenging not only because of our limited understanding of physics behind the scenes in

the uncharted nanoscale realms, but also how robust and repeatable our processes should

be to provide reliable devices with nominally-the-same characteristics from run to run. One

28

common practice in nowadays semiconductor foundries is setting up detail process flow

tables together with the inspection handbook, in order to achieve reproducibility from

devices to devices and between different operators. In this work we follow similar ideas to

maintain our fabrication parameters in project specific flows to keep track of all process

changes. Here in this section, we will first introduce our key process modules with the

corresponding tools and inspection criteria, and then we proceed with the process flows for

different devices in this work.

The processes involved throughout the work are modularized into two sub-groups:

lithography and film formation. By combining above modules to form a process layer for

a specific circuit element such as a junction, meander, or capacitor, we can study the

devices connected to various electromagnetic environments.

II.1-1 Lithography

Let us begin with the lithography module. In this work the lithography works are

performed using the NPGS/SEM ebeam writing bundle with FEI Sirion XL30 30keV

SEM from Thermal Fisher Scientific. After a user defines the layout for a specific project

in DesignCAD 2000 LT, a runfile containing the pattern information and sequential

instructions is assembled in the Nanometer Pattern Generation System (NPGS) maintained

29

by Dr. Joe Nabity under Windows XP. Typically, two types of instructions, beam

movement and beam dosage, are generated in the NPGS and sent to different part of the

ebeam control PC. The NPGS system will translate and send the pre-defined movement

commands to a local computer to control the beam movements, and then convert the beam

dosage to the electric signal dwell time on a beam blanker (Scanservice model 880) based

on the ratio between the nominal dosage and the beam current user provided in the runfile

before beginning of each process. The alignment between beam blanker, aperture, and

condensing coil is essential to the optimum ebeam writing and maximized beam current

outputs; it is in general recommended to perform the vacuum bakeout and full

mechanical/software alignments on annual basis.

Although the recent advancement in ebeam writer enables reproducible -scale

patterning, high uniformity of such pattern across a device such as long chains of junctions

remains challenging in a laboratory setting. For designs involved in this work such as

junction chains requiring reproducible junction areas across a sub-millimeter size window,

longer working distance and rather small aperture size (size 4, 30μm in diameter) are used

to maximize the depth of focus to accommodate the differences in heights across the chip.

The typical beam size is about 5 − 10nm while the maximum field of view is 1mm.

For devices larger than the field of view we stitched several write windows and verified

30

the stitching results through a series of stitching markers. To ensure the uniform thermal

distribution during the preparation of ebeam-resist, the chip was prebaked on the surface

of a thick copper slab heated by the hotplate and monitored with a standalone thermometer.

All lithography parameters are finetuned based on the specific pattern density and are

recorded in the process flow tables listed at the end of this section. We would like to point

out that a shortened baking time is adopted during the resist pre-bake step in order to

minimize the thermal budgets applied to junctions in the case that another lithography layer

is required after junction formation. Because of that, it is necessary to pay extra attention

to the development condition such as the MIBK solution temperature to stay consistent

between batches. The solution temperature was monitored at 23 ± 0.5 by a K-type

thermal couple.

The junctions in this work are fabricated by the Manhattan pattern technique with

angle evaporation and differential development in bilayer e-beam resists, and the details

can be found in chapter 3 in [60]. The in-situ oxidation process performed between

deposition of the bottom and top aluminum electrodes is tuned to produce the required

value of . Typically we used the dry Oxygen partial pressure 1 − 100 torr and

oxidized the structures for 5 − 15 minutes. With optimization of this process in our lab,

the standard deviations for the normal state resistance for a nominally sub-μm-wide

31

single junction devices across the 7mm × 7mm chip [Figure II-1] did not exceed 2.4%.

[Figure II-2]. Albeit the sampling was taken at a relatively small scale of fabrication batch,

such standard deviation is comparable to the state of art < 3.5% as reported in [61].

Figure II-1: SEM image for a single junction with in-plane dimension ×

Figure II-2: Standard deviations of normal state resistance of single junction devices

fabricated in our laboratory

32

For junction array devices with an overall sub-millimeter length, however, the

standard deviation of junction areas across the chain went up to 40% with this lithography

recipe before further fine-tuning (blue histogram in Figure II-3). This was mostly due to

shorter working distances used in fabrication of the single junction devices, which resulted

in a smaller depth of focus and therefore greater in-plane variations. By increasing working

distance from 8mm to 15mm, we reduced the junction area variation to less than 10%

across a 200μm-long chain (orange histogram in Figure II-3).

Figure II-3: Scattering of the junction areas in the chain devices. The data in

blue/orange histograms are before/after the process optimization.

33

Small scattering in and are crucial for proper operation of our SQUID

based circuit: minimizing deviations from the nominal values of and is important

for realizing the maximum superinductance in the junction-based superinductors where

simultaneously frustration of all SQUID unit cells is required. In this work we typically

observe a 90%-reduction of critical current for the SQUID structure at the magnetic field

corresponding to full frustration Φ = Φ/2, where Φ is the magnetic flux through the

SQUID loop.

II.1-2 Film deposition and junction fabrication

Three project-specific process modules are included in this section: uniform

junction oxidation during ebeam evaporation, ion milling integration, and granular

aluminum AC sputtering. Other standard process parameters can be found in the process

flow tables listed at the end of the fabrication section.

Junction fabrication

The junction oxidation non-uniformity has been a major bottleneck in developing

junction technologies for decades. Although aluminum films form robust and pin-free

tunneling barriers, the barrier transparency is extremely sensitive to residual moisture

presented in the chamber or oxygen source. In this work a liquid nitrogen cold trap was

34

inserted between the sample space and oxygen source tank to remove the potential moisture

before entering the chamber, and the chamber was evacuated down to < 5 × 10 torr

before oxygen was introduced for oxidation.

Ion milling for hybrid device integration

To integrate superconducting elements fabricated in different chambers and ensure

good superconducting contact between different layers, we used the 10mm DC cathode

KRI Argon ion source for in-situ ion milling of the film surface before deposition of top

layers. One common issue in the process of -ion milling of aluminum oxide with

PMMA mask is weak selectivity of milling. We were able to perform deeper etching by

dividing a long etching run into several short etching sequences to allow the PMMA cool

down between individual steps. In this work the milling recipe was tested by measuring

the critical current of the resulting joint and ensuring that this current is two to three order

of magnitude greater than the critical current of all junctions in the devices (Figure II-4).

The removal rate of baked PMMA is ~3A/s at acceleration voltage 90V, and this should

be taken into account in the design of the junction fabricated by multi-angle deposition to

avoid unexpected film deposited into the undercut as shown in Figure II-5.

35

Figure II-4: Resistance of the / interface as a function of the Argon flow rate.

Figure II-5: The film stacks during the multi-angle deposition and directional ion

milling. The left trench in the cross-sectional view shows a normal multi-angle

deposition, while right trench in the cross-sectional view shows a narrower trench

that suffers from unexpected film deposition due to the loss of PMMA film thickness

during the directional ion milling step between the 1st and 2nd deposition.

36

Granular aluminum sputtering

One typical method to fabricate granular aluminum () films is to evaporate

aluminum target in a reduced oxygen environment. Unfortunately, this significantly

shortened the filament lifetime in our e-gun evaporation source. For this reason, we adopted

an alternative method: the reactive magnetron sputtering of pure Al in reduced Ar+O2

atmosphere. In this work the films were prepared in an AC sputtering chamber with

RF power 150watt, DC bias −150 at a rate 4A/s in the atmosphere of and .

The process starts with the base pressure < 10 torr and the pure target was pre-

sputtered in pure plasma at a rate of 0.5A/s for 5 minutes. After pre-cleaning of the

target, we turned off the plasma, and introduced and in the chamber using two

feedback-controlled mass flow meters (Sierra MicroTrak 100 and MicroTrak 101) to

maintain the partial pressure at 5 × 10 torr and 4 × 10 torr, respectively. Only

after this chamber preparation we ignited the plasma to deposit films. We would

like to point out that in our process no substrate cooling is used and the grain size in our

film (~4) is greater than that for the films deposited at 77K (~3 in [62]).

The dependence of on the normal-state resistivity for such films is shown in

[Figure II-6]. By controlling the deposition rate and the partial pressure, we can tune

the resistivity of superconducting grAl films between 10 Ωcm and 8 × 10 Ωcm

37

(Figure II-8) which provides the sheet kinetic inductance in a range between 2.5 fH sq⁄ to

2 nH sq⁄ for 40nm thick films. The films are then patterned into various geometries with

nanolithography. Figure II-7 shows one of such structures - a meandered nanowire with

sub-μm width and few thousands of squares total, which provides the kinetic inductances

up to a few μH.

Figure II-6: The − “dome” for granular aluminum deposited in this work. The

dashed curve corresponds to the literature data for grAl films deposited at 77K.

38

Figure II-7: A meandered nanowire made of granular aluminum. The device is false

colored for clarity. The rest of the structure, which was not in a galvanic contact with

the device, was fabricated in order to reduce the nonuniformity of the nanowire width

due to the proximity effect in the process of e-beam nanolithography.

Figure II-8: The sheet resistance of films as a function of the flow.

39

II.1-3 Process flows

Figure II-9: Process flow for junction chain project

40

41

Figure II-10: Process flow of microwave resonator project

42

II.2 Measurement

II.2-1 DC measurement

All experiments presented in this work were performed using a BlueFors SD250

dilution refrigerator rated 250 μW at 100 mK and the base temperature around 25 mK.

The detailed working principle of dilution refrigeration could be found in previous thesis

works from our lab, for example chapter 3 and 4 in [63]. The external magnetic field used

in this work was same order of magnitude as Earth’s magnetic field, and to reduce stray

magnetic fields the entire cryostat is enclosed with a customized μ-metal shield which

enabled attenuation of the Earth’s field by two orders of magnitude. Two customized sets

of wiring and instrument modules labeled as DC and MW have been developed and

installed in the cryostat.

The wiring for DC setup inside the cryostat consists of 12 twisted pairs made of

resistive alloys : (5: 1) with multiple thermal anchoring points to isolate the

sample from room temperature and to reduce the thermal loads as shown in Figure II-11.

Near the cold finger which supported the sample holder, twisted pairs are carefully winded

on an Oxygen-Free Copper (OFC) tubing filled with copper-powder epoxy (Figure II-12

left), which serves as a cryogenic lowpass filter and a final thermal anchoring point before

connecting to the sample holder. On the detachable sample holder, 100kΩ surface mount

43

metal film resistors are installed in each channel with non-magnetic silver epoxy on a PTFE

laminated PCB (Arlon AD1000) which has high dielectric constant = 10.2 and

thickness 500 μm (Figure II-12 right). The Device-Under-Test (DUT) was mounted

inside a 7-by-7 housing on the sample holder by cryogenic grease (Apiezon N) .

We used aluminum wire bonds to connect the DUT and copper fingers on the PCB.

Figure II-11: Thermal anchorage points (hand-drawn green blocks) for twisted pairs

and MW cabling

44

Figure II-12: (left) The OFC tubing filled with copper-powder epoxy at the end of DC

wiring. (right) The detachable sample holder for DC measurement.

The epoxy filter, twisted pairs and thermal anchoring together provides channels

with minimal noise level for common transport measurements. To measure low-

junctions, however, due to the non-linear IVC response and small critical currents

(typically within the fA - nA range) a careful design of the measurement circuitry is

required as reported in [37]. Additional filtering of the current source in order to prevent

fluctuations in biasing circuitry is necessary to properly record the IVc for junctions with

sub-kelvin (see chapter 4 in [64]). Indeed, measurements on low- junctions are

challenging in part due to the low critical currents which is sensitive to electronic noises

and thermal activations, and in part due to the high zero bias resistance in certain cases (>

45

20MΩ as reported in [65], for example). In this work we try to mitigate the first concern

by installing cascaded low pass filters at the source end and tackle the second concern by

employing a voltage preamplifier (DL Instrument 1201) with input impedance as high as

few GΩ when being DC-coupled. The comparison of filtering effects on the IVcs will be

elaborated in section III.1.

The wiring and instrument schematic outside of the fridge is shown in Figure II-13

and Figure II-14 for current and voltage biasing schemes, respectively. A commercial LC

low pass filter (BLP 1.9+, DC − 1.9MHz) and a homemade RC filter (DC − 8Hz) box with

variable biasing resistors up to 1GΩ are inserted between the source Tektronix AFG3252

(or Keithley 6200) and DUT. Two types of biasing boxes are made and used depends on

the biasing schemes. The voltage drops across DUT are amplified with a voltage preamp

DL1201 and measured by HP 34401A digital multimeter with PC automation scripts such

as LabVIEW and Python module QCoDeS.

Figure II-13 The wiring schematics for DC current source measurements

46

Figure II-14 The wiring schematics for DC voltage source measurements

As a final remark before we turn our attention to MW setup, in this work we

characterized the parasitic capacitances with AH 2500A capacitance bridge (Andeen-

Hagerling). Indeed, it is difficult to directly measure the parasitic, but still we managed to

estimate the value by extending the wiring stage by stage. Here the results are attached for

future references (Figure II-15).

Figure II-15: Characterizing the parasitic capacitance

47

II.2-2 MW measurement

The MW setup was developed in previous experiments in our lab [66]. Commercial

semi-rigid SMA biaxial cables are thermally anchored with SMA sockets screwed

at each flange. A photon-leakage-free sample holder with two microwave launch ports

made from OFC are shown in Figure II-16 left. The launch ports are vertically mounted

with silver epoxy at a AD1000 PCB which has precut copper traces with trace width and

spacing satisfying 50Ω impedance matching condition for microwave applications (Figure

II-16 right). The fabricated DUT is then connected to the copper fingers on the PCB with

aluminum wire bonds.

Figure II-16 (left) The sample holder and SMA anchoring for MW measurement.

(right) The microwave sample holder.

48

With this setup (Figure II-17) we have performed the transmission measurement

over a wide range of MW power including testing of the MW resonators in the single-

photon population regime. The probe signal at and the pump signal at , generated by

two microwave synthesizers, were coupled to the input of the cryostat through directional

couplers. Depending on the experiment performed, the pump signal could be pulsed using

an internal RF switch of the microwave synthesizer. Attenuators and low-pass filters were

installed in the microwave input line to prevent leakage of thermal radiation into the

resonator. The signal, after passing the sample, was amplified by a cryogenic high-electron

mobility transistor (HEMT amplifier Caltech CITCRYO 1-12, 35 dB gain between 1 ÷

12 GHz) and two 30dB room-temperature amplifier. Two cryogenic Pamtech isolators

(each provides 18dB isolation between 3 ÷ 12 GHz ) were anchored at the base

temperature to reduce the 5K noise from the HEMT amplifier. The amplified signal was

downconverted to the intermediate frequency (IF) = | − | ≈ 30MHz using mixer

M1 with the local oscillator signal . The IF signal was digitized using the card

AlazarTech ATS 9870 at 1GS/s. The magnitude and phase of the signal was obtained

by digital demodulation as

= ⟨() (2)⟩ + ⟨() (2)⟩

= (⟨() (2)⟩ ⟨() (2)⟩⁄ )

Equation II-1

49

where ⟨⋯ ⟩ stands for the time averaging over integer number of periods, typically 10.

The reference phase was provided by mixer M2. The entire instrumentation and data

acquisition is also preformed via PC automation scripts such as LabVIEW and Python

module QCoDeS.

Figure II-17 The schematics of MW measurements

Before each MW measurement run, the transmission of the sample holder and DUT

assembly was recorded over a wide frequency range and logged to compare with previous

50

measurements. The purpose of such comparison is to ensure no spurious resonances or

losses exist in the entire MW setup due to material fatigue or loose connections (Figure

II-18).

Figure II-18: Transmission background check before each cool down

II.2-3 Temperature control and magnetic fields characterization using

SQUID geometry

To closely monitor the temperature of the DUT a thin film NTC

thermometer was attached to the sample holder, and three brass braids directly connected

the holder to the coldplate for enhanced thermal anchoring. At the cryogenic temperature,

51

an optimized PID close-controlled loop for the heater is required to maintain the device at

stable temperature for duration up to an hour and beyond.

In this work the optimized PID table was obtained by monitoring the stability of

the resistance of underdamped junctions at the phase diffusion branch of the IVC, as we

discussed in I.4. The calibrated PID table for different temperatures and the corresponding

noise level for the system are shown in Figs. II-19 and II-20.

Figure II-19: Optimized PID table

52

Figure II-20: Optimized noise level and thermal fluctuation at different temperatures.

53

Taking advantage of the periodicity of the cosine dependence of () in

Equation I-9, we can also monitor the performance of our superconducting magnet. This

periodicity of the resistance of a SQUID biased at > is shown in Figure II-21.

Figure II-21 Periodic response of a SQUID to external magnetic field.

54

We did observe flux focusing effects for different designs with different metal density.

Flux focusing is due to flux repulsion from nearby superconducting pads, it results in an

increase of the magnetic field in the SQUID loop (Figure II-22). In Figure II-23 we listed

design variants as we used in this work.

Figure II-22: Comparison of the magnet currents that correspond to = for

three designs with different area of the SQUID loop.

Figure II-23: Various designs of SQUID used in this work.

55

II.3 Lists of samples

II.3-1 Low- junctions

Below we listed low- devices we studied in Chapter III.

Figure II-24: in , in are the normal state resistance and the area

of single junction in the chain. , in are the single junction Josephson

energy ≡ C ⁄ and charging energy ≡ ⁄ , respectively. ,,

in are the single junction critical current predicted from Equation I-2 and the

switching current that experimentally measured. shows how many SQUIDs

are in series in the design. . indicates the source of capacitor shunting the

junction where “” means external capacitor was fabricated, “ ” means the

junction has intrinsically large area parallel plate capacitors to provide low , and

“ ” represents data point that are measured before external capacitor was

fabricated. Finally, and ; in are the zero-biased resistance at zero B-

field and B-field at full frustration point, respectively.

56

II.3-2 Resonators with superinductance

Below we listed four resonators in Chapter IV designed with different , and .

II.3-3 Bifluxon devices

In Chapter V we demonstrate a prototype bifluxon circuit, and here we list the parameters

for each component (SI and CPB) in the circuit.

With the reliable and consistent methodologies ranging from fabrication, measurement, to

experimental control knobs such as temperature and magnetic fields, we are ready to move

on to the investigation and realization of the circuit elements for bifluxon qubits.

57

THERMAL EFFECTS IN LOW- JOSEPHSON JUNCTIONS

Josephson junctions with the Josephson energy less than 0.5 have already been

employed as non-linear elements of superconducting qubits [36]. Though of these

junctions remains much greater than the physical temperature of qubits (10 ÷ 50), a

non-zero rate of thermally activated phase slips in these junctions might limit the qubit

coherence. Indeed, the coherence time of superconducting qubits approaches 1 [67],

and even rare events at such a time scale might become significant.

In the past strong experimental and theoretical effort was aimed at better

understanding of a crossover from the classical Josephson behavior (well-defined phase

difference, very large quantum fluctuations of charge) to the Coulomb-blockade regime

(localized charges, very large quantum fluctuations of phase). This crossover is commonly

attributed to the decrease of the parameter /. The coherent phase slip processes (the

so-called quantum phase slips, or QPS), whose rate exponentially increases with

approaching / ≈ 1, do not contribute to dissipation, though they might affect the

dephasing time in qubits.

In this thesis we are mostly concerned with a different regime Δ ≫ ≥ ≫ ,

which is more relevant to operation of protected superconducting circuits. In order to

explore the dynamics of low- junctions at ultra-low temperatures, we designed JJs with

58

a very low transparency of the tunneling oxide and the values of Josephson energy

between 0.1 ÷ 10. The charging energy =()

of these junctions remained below

10 due to shunting the junctions with external capacitors. This chapter reviews the

results of the low-frequency transport measurements with these structures.

III.1 Sample design

All the samples studied in this Chapter have been implemented as SQUIDs, in order

to be able to in-situ tune by applying the external magnetic field. Figure III-1

schematically shows the design of a chain of SQUIDs formed by small junctions

(0.2 × 0.2 ). The area of the SQUID loop varied between 6.8 and 49. Our

experiments were focused on the JJs with 1 > ≫ : this regime is relevant to the

quantum circuits in which JJs are shunted with large external capacitors (such as the

transmon qubit). Large / ratio would also significantly reduce the rate of quantum

phase slips ∝ −2

[68]. Typical specific capacitance of the junction tunneling

barrier is about 50 fF/ , and in order to reduce down to 20 the

junctions should either have relatively large in-plane dimensions ( > 4 ) or be

shunted with external capacitors ( > 200fF). We have used both methods in different

structures (Figure III-1). In the external capacitors approach, several designs of the

59

shunting capacitors have been implemented. In the approach where we introduced

relatively large JJs, in order to keep below 1 the oxidation recipes were fine-tuned

for the growth of low-transparency tunneling barrier. Figure III-2 shows that that

each SQUID unit cell is flanked by two large metal pads, which are used as shunting

capacitors to the common ground when the entire chain was covered by an additional

top electrode (sputtered film). A few nm native oxide grown at the atmospheric

pressure serves as a pinhole-free dielectric for this parallel-plate with a typical

capacitance around 500fF for 50 pad area. Such corresponds to a charging

energy per each cell as low as =()

= 8mK.

The chains of SQUIDs were designed to provide access to individual SQUIDs or

pairs of SQUIDs within a chain. We did not expect to observe strong effect of the

environment on the IVCs of SQUIDs with ≫ , and our observations are in line with

this expectation.

Figure III-1: Various designs of SQUIDs. (a) Each SQUID unit cell was shunted by a

large ≈ . the ground. (c) SQUIDs formed by large JJs with junction area

≈ . .

60

Figure III-2: Device schematics and the circuit diagram for devices with shunting

capacitors and a common ground (opaque blue pad).

III.1 Noise reduction in the measurement setup.

The noise reduction was our primary concern in characterization of low-

junctions. Most of our measurements have been performed in the constant current mode.

The value of the critical current at = 0 for an // JJ with = 1 is

, = 30 according to the Ambegaokar-Baratoff formula (Equation I-2). With

further reduction of and proliferation of phase slips at elevated temperatures , the

current range well below 1 becomes relevant.

Figure III-3 illustrates the importance of proper filtering of noises in both the

current supply part and the voltage recording part of the measuring setup. By using the

61

combination of cascaded low-pass filters and 100 resistors on the sample holder, we

were able to record switching currents in the range (Figure III-3).

Figure III-3: The IVCs recorded for a two-unit SQUID device with different

measurement set-ups at = (sample D059B0N1 in Table III-1). Each

SQUID unit consisted of two nominally identical junctions with an area

. × . and resistance 10kΩ ( = . ). Due to a large shunting

capacitor to the ground, this device has ⁄ = ⁄ ≫ . Without

thorough filtering, the IVC was smeared and the IVC hysteresis, expected for an

underdamped junction at low , was significantly reduced. Filtering of all leads used

for the IVC measuring restores the critical current which is close to , , and

enables observation of a well-developed hysteresis. The inset shows that the noise level

in our measurements is around . .

62

III.2 Current-voltage characteristics of low- junctions

Below we focus on the results of measurements at < 200 – in this

temperature range one can neglect transport of the thermally-excited quasiparticles in -

based superconducting circuits. We are mainly interested in two features of the IVCs: the

switching current and the zero-bias resistance = ⟨⟩/ measured at small DC

voltages ⟨⟩ ≪ 2∆/ and currents ≪ . Figure III-4 shows how these two quantities

have been determined in the experiments on a chain of 20 SQUIDs with = 40 at

= 25 (sample D059BBN2 in Table III-1).

63

Figure III-4: (left) The current-voltage characteristic (IVCs) for a chain of 20 SQUIDs

with = (sample D059BBN2 in Table III-1). (right) The enlargement of the

region of small currents/voltages. Note that the resistance is non-zero for all biasing

currents. As soon as the biasing current exceeds = . (indicated by a cyan

arrow), the voltage across the chain rapidly increases, and at even greater currents

approaches the value × ∆/, where is the number of SQUIDs and ∆ is the

sum of superconducting energy gaps in the electrodes that form a junction. At ≥

× ∆/ dissipation is due to generation of non-equilibrium quasiparticles. At <

the non-zero resistance ≈ is due to thermally activated phase

diffusion (section I.4). Note that for such low- junctions, the switching current is

three orders of magnitude smaller than the Ambegaokar-Baratoff critical current

, ≈ . .

For convenience, below we separately discuss the data for junctions with ≈ 1

and ultra-low- junctions.

III.3 Josephson junctions with ≈

In the regime ≫ , and < 200 one may expect to observe the

“classical” behavior of Josephson junctions: ≈ , and a well-developed IVC

hysteresis typical for underdamped junctions with the McCumber parameter ≫ 1

(section I.3 and [22]). Indeed, in zero magnetic field, the IVCs shown in Figure II-5 meet

64

these expectations. The value of = 18 is close to , = 32, and, within the

accuracy of our measurements ≈ 10 ÷ 10 , depending on the magnitude of , we

could not detect a non-zero .

Figure III-5: IVCs of two connected in series SQUIDs with = . at =

(blue curve) and = . (red curve) at base temperature (sample D059B0N1 in

Table III-1).

At full frustration Φ = 0.5Φ where we expect to be minimized (Equation

I-9), this device demonstrated behavior that resembled the Coulomb-blockade regime (red

curve in Figure II-5). Note that for most of the studied samples in this regime is of an

order of the base temperature. The zero-bias resistance at full frustration (Φ =

0.5Φ, ) becomes strongly dependent on temperature at > 0.2 (Figure III-6) where

the concentration of thermally generated quasiparticles becomes significant. The drop of

with temperature at > 0.25 (Figure III-6) can be approximated by the Arrhenius

65

dependence (Φ = 0.5Φ, ) ∝

with ≈ 1.63 . Assuming that the

quasiparticle current at a given voltage is proportional to the concentration of quasiparticles

where () = () × exp

, this number is comparable to the value =

2.40 extracted from the plot of the density of thermal quasiparticles for aluminum

(Figure 2.1 in [69]). For < 0.25 a weak decrease of with decreasing has been

observed.

Figure III-6: (left) IVCs of two connected in series SQUIDs (sample D059B0N1 in

Table III-1) with = . at = . at different temperature . The

order of for each IVC from top to bottom is from to .

66

Figure III-7: (a) IVCs of a single SQUID with = . (sample D079N6 in Table

III-1) measured at different magnetic fields (we only marked four selected curves for

clarity). A sub-gap voltage plateau at ≈ appears at > . . (b) The

dependence of on the superconducting solenoid biasing current. (c) The

measured ()/( = ) as a function of (/) . The dash line

corresponds to the dependence ∝

. The reason for observed

deviations from the dash line for

< . remains unclear.

Even in the “classical” regime ≫ , we obtained several unexpected results.

Firstly, we noticed that the dependence of (Φ) for some samples could significantly

deviate from the expected dependence (Φ) = (Φ = 0) × cos (Φ/Φ) in the B-

67

fields approaching the full frustration cos

≈ 0 (Figure III-7). Secondly, we have

observed sub-gap ( < 2∆/) voltage steps on the IVCs (Figure III-7 a), which

significantly reduced the accuracy of extraction of and close to full frustration.

We speculated that one reason for appearance of sub-gap steps might be the Fiske

resonances [70, 71]. However, we could not identify the circuit elements that would be

responsible for the corresponding resonance frequencies at = (75μeV) ℎ⁄ ≈ 18GHz

(for the silicon substrate this frequency corresponds to a length scale ≈ 5). This issue

requires further investigation.

As we will discuss below, for the devices with ( = 0) < 1 further deviations

from the “classical” Josephson behavior have been observed. With decreasing ,

rapidly drops well below the , value (it becomes several orders of magnitude smaller

than , for devices with ≈ 0.1), and dramatic increase of the zero-bias resistance

is observed.

III.4 Josephson junction ≈ . and the effect of shunting JJ with

As it was expected, we observed the charging effects in the low- JJs without

external capacitive shunts. Figure III-8 shows the IVC for a chain of 30 SQUIDs based on

JJs with resistance = 40Ω and = 0.17K. The IVCs have been measured for the

68

same device before and after the deposition of top capacitor pad. This top electrode,

being isolated from the rest of the circuit by the native oxide, significantly increased

the capacitance connected in parallel with individual SQUIDs (from 2 fF to 500 fF) and

decreased from 1.5K to 8mK (

from 0.13 to 25 before and after Pt deposition,

respectively).

Deposition of the electrode did not significantly affect the IVCs at voltages

> × 2∆ - the normal state resistance of the chain decreased by about 2.5% after an

additional lithographic cycle and deposition, from 610Ω to 595Ω. However, the

IVCs have been significantly modified at low currents/voltages: the lower inset in Figure

III-8 shows that Pt deposition significantly increased , up to 20 pA and reduced the

zero-bias resistance R . We attribute these findings to suppression of the charging

effects and associated with them quantum phase slips with the rate ∝

exp−2 ⁄ by an increase of the shunting capacitance.

69

Figure III-8: (main panel) The IVCs for a chain of 30 SQUIDs (sample

D063BAN6A_bf and D063BAN6_af in Table III-1) based on JJs with resistance

= and = . before (blue) and after (red) deposition of the

electrode that significantly increased the capacitance across individual SQUIDs.

Upper left inset: The enlargement of the IVC before deposition. Clear .

jumps can be seen that correspond to the voltage drops ∆/ across individual

SQUIDs. Lower right inset: the enlargement of the low- part of the IVCs.

Comparison of red and blue curves shows that Pt deposition significantly increased

, up to 20 pA and reduced the zero-bias resistance.

70

III.5 Discussion: the suppressed switching currents

In this section we include all our junction devices in the regime ≪ ≤ and

provide brief discussions for the suppressed switching currents and potential source

of dissipation.

It is expected that the main source of dissipation in the regime ≪ ≤ ≪ ∆

is a relatively high rate of incoherent thermally activated phase slips (TAPS). The DC

transport in the regime ≪ has been theoretically considered by Ivanchenko and

Zilberman [72] (the “IZ theory”). This theory considers a stochastic noise V() as a

driving force for switching the underdamped junctions in the resistive state. The equation

of motion for a classical Josephson junction (Equation I-4) can be rewritten as:

+ ()

=

2

+

2

+ C

By solving the corresponding Fokker-Planck equation, the DC Josephson current

( = V + V) can be parameterized by the ratio of the potential energy Ω and the

noise fluctuation (Θ) , ≡ Ω ⁄ where Ω = 2eIR ℏ⁄ and D = ΘR(2 ℏ⁄ ) . Note

that the source of noise fluctuations can be either thermal Johnson-Nyquist noise [73] or

electronic noises, and in the analysis all noises appear as an emf, (k ⁄ , , … ).

The resulting IVCs obtained in [72] are shown in Figure III-9. At z = 2 a sharp turn

around 10 × ⁄ ≈ 5 indicates a decrease of switching current from , to

71

0.02, when the noise level increased from Θ ≪ Ω to Θ = 0.5Ω, and an increase of a

zero-bias resistance .

.

Figure III-9: Analytical solutions from [72] for IVCs at different fluctuation level ≡

⁄ . At = a sharp turn was observed around × ⁄ ≈ , indicating a

decrease of switching current from to . when the noise level increased

from ≪ to = . .

The IZ theory predicts that ∝ at small [74]. In Figure III-10, we

plotted the predictions of according to the IZ theory for = 20. This

noise corresponds to the Johnson-Nyquist noise = 4Δ generated at =

50 by the 100 resistors connected in series with the device in Figure II-13. The

bandwidth was estimated as Δ ≈ 2⁄ , where 2⁄ = 0.5 ÷ 5GHz is the range of

plasma frequency of the shunted JJs.

72

Figure III-10: The switching current measured for different devices at ≈

and the magnetic field increasing from = to B corresponding to

= /. The red dash-dotted line corresponds to the switching current predicted

by the IZ theory in presence of = .

Most of the data points in Figure III-10 are 1-2 orders of magnitude smaller

than predicted by the IZ theory. A possible explanation for this discrepancy might be

more complex phase dynamics in the devices with a very high TAPS rate, outside of the

limits of applicability of the IZ theory. Another possibility is an exponentially strong

73

sensitivity of the TAPS rate to the noise level in the setup and the physical temperature of

a device, the parameters that are not easy to control in all experiments.

Figure III-10 also included the data for five devices where we tuned

(Equation I-9) by varying the external magnetic flux threading the SQUIDs loop. The

effective for these devices was calculated as = 2 cos 2

. By tuning

over an order of magnitude, we observed rather complicated dependences that

varied between and .

74

Figure III-11: The zero-biased resistance measured for different devices at ≈

and the magnetic field increasing from = to B corresponding to

= /. The red dash-dotted line corresponds to predicted by the IZ theory

in presence of = .

The zero-bias resistance in Figure III-11 follows a similar trend. Being

unmeasurably low at > 10, rapidly increases at < 1, and becomes much

greater than the normal-state resistance at < 0.1 . Instead of a well-defined

“superconductor-to-insulator” transition at a certain value of / , a broad crossover

75

between these two limiting regimes is observed. Note that different JJ samples (single

junctions and arrays) demonstrate similar values of though their charging energies

could vary over a wide range (see Table III-1).

Our findings are in line with the prior experiments with low- junctions [24, 64,

65, 74-86]. Despite large scattering of the data in Figure III-12 and Figure III-13, a very

rapid drop of and increase of has been observed in most of the experiments as

soon as becomes significantly less than 1K. Figure III-12 shows that for typical

experimental conditions, the crossover between the “classical” behavior ∝ to the

behavior controlled by the thermal diffusion of phase occurs at ≈ 1. Note that the

literature data in Figs. III-12 and III-13 correspond to samples with different values of the

ratio / . However, large scattering range of and hides possible effect of

charging. For the same reason, it is unclear if the impedance of the environment plays any

significant role in these experiments: similar values of could be observed for single

JJ in a highly-resistive environment (> 100 as in [80] and our setup), single JJ in a

low-impedance environment [82], and chains of SQUIDs frustrated by the magnetic field

[75, 78].

76

Figure III-12: The switching current as a function of measured in our

experiments (grey symbols) and by other experimental groups (blue dots) [24, 64, 65,

74-86]. All the data have been obtained at ≈ − for Al-AlOx-Al

junctions. Note that the literature data on this plot correspond to samples with

different (the ratio / for a given varies over a wide range). However, it

seems that this is not the main factor that controls scattering of . For comparison,

the blue dashed line represents ,.

77

Figure III-13: The zero-bias resistance as a function of measured in our

experiments (grey symbols) and by other experimental groups for Al-AlOx-Al

junctions (blue dots) [24, 64, 65, 74-86]. Table III-1 and Table III-2 summarizes the

parameters of these samples, respectively. All the data have been obtained at the

base < , though the physical temperature of the Josephson circuits has not

been directly measured.

78

Table III-1: List for the SQUIDs chain devices included in this chapter. in ,

in are the normal state resistance and the area of single junction in the

chain. , in are the single junction Josephson energy ≡C

and

charging energy ≡

, respectively. ,, in are the single junction

critical current predicted from Equation I-2 and the switching current that

experimentally measured. shows how many SQUIDs are in series in the

design. . indicates the source of capacitor shunting the junction where

“” means external capacitor was fabricated, “ ” means the junction has

intrinsically large area parallel plate capacitors to provide low , and “ ”

represents data point that are measured before external capacitor was fabricated.

Finally, and ; in are the zero-biased resistance at zero B-field and B-

field at full frustration point, respectively.

79

Table III-2: List for the SQUIDs devices from literatures [24, 64, 65, 74-86], the

source can be found by following the Ref. # in Chapter VIII.

Our observations are in line with an expected strong dependence of the TAPS rate

on the sample parameters in the regime ≪ ≤ ≪ ∆. Indeed, one can estimate the

TAPS rate as Γ = −

, where is the plasma frequency (or an attempt rate)

and −

is the probability of the over-the-barrier excitation. For example, at

= 0.25 and

= 1.32 , the rate decreases from 3 × 10 to 0.1 if

80

the physical temperature decreases from 50 to 20 . This also explains why the

experimental results might be so sensitive to the noise level in the experimental setup.

III.6 Conclusion and outlook

Phase slips in JJs have been actively studied over the last three decades in different

types of Josephson circuits (single JJs, JJ arrays, etc.) over wide ranges of and . In

our work we focused on the thermally activated phase slips (TAPS), which, in contrast to

the quantum (a.k.a. coherent) phase slips, result in dissipation. At sufficiently low

temperatures ≪ ∆, where the concentration of quasiparticles becomes negligibly low,

the TAPS are expected to be the only source of dissipation.

We observed that in all studied devices with < 1 the switching current

is significantly suppressed with respect to . At the same time, we observed a very

rapid growth of with decreasing Josephson coupling below ≈ 1. Large scattering

of the data might reflect a steep dependence of the rate of incoherent phase slips on the

physical temperature and noise level in different experimental setups. Our observations are

in line with most of the data reported in literature.

The observed enhanced dissipation in Josephson circuits with < 1 imposes

limitations on the performance of superconducting qubits based on low- junctions. This

81

important issue requires further theoretical and experimental studies. Especially important

direction would be measurements of the coherence time in the qubits with systematically

varied Josephson energy over the range = 0.1 − 1. One of the signatures of TAPS-

induced decoherence is an observation of a steep temperature dependence of the coherence

time at < 100 [87].

82

MICRORESONATORS FABRICATED FROM HIGH-KINETIC-

INDUCTANCE ALUMINUM FILMS

This chapter is based on a joint work with W. Zhang, K. Kalashnikov, P. Kamenov, T.

DiNapoli and M.E. Gershenson (Rutgers University) [18]. My main contribution to this

work was the development of fabrications, microwave characterization techniques, and

data analysis. In this chapter, we will be investigating the MW response of disordered

superconducting thin films by characterizing the coplanar waveguide resonators made from

granular () films.

IV.1 Introduction

The development of novel quantum circuits for information processing requires the

implementation of ultra-low-loss microwave resonators with small dimensions [88].

Superconducting resonators have become ubiquitous parts of high-performance

superconducting qubits [89, 90] and kinetic-inductance photon detectors [39]. An

important resource for resonator miniaturization is the kinetic inductance of

superconductors, , which can exceed the magnetic ("geometrical") inductance by orders

of magnitude in narrow and thin superconducting films [39]. High kinetic inductance

translates into a high impedance of the MW elements, slow propagation of

83

electromagnetic waves, and small dimensions of the MW resonators. Ultra-narrow wires

and thin films of and [39, 40], [41], [42, 43], and granular

[44] were studied recently as candidates for high- applications. Research in high-

elements also have an important fundamental aspect. According to the Mattis-Bardeen

(MB) theory [91], the kinetic inductance of a thin superconducting film ( = 0) is

proportional to the resistance of the film in the normal state, , and thus increases with

disorder. This theory, however, cannot be directly applied to strongly disordered

superconductors near the disorder-driven superconductor-to-insulator transition (SIT).

Recent theories predict a rapid decrease of the superfluid density near the SIT and the

emergence of sub-gap delocalized modes that would result in enhanced dissipation at

microwave frequencies [47, 92]. Thus, the study of the electrodynamics of strongly

disordered superconductors may also contribute to a better understanding of the disorder-

driven SIT.

In this Chapter, we present a detailed characterization of the half-wavelength

microwave resonators fabricated from disordered Aluminum films. Our interest in high-

films was stimulated by the possibility of fabrication of superinductors (dissipationless

elements with microwave impedance greatly exceeding the resistance quantum =

()

[15, 93, 94]), and the development of SI-based protected qubits [17]. We have fabricated

84

resonators with an impedance as high as 5kΩ, ultra-small dimensions and relatively

low losses. The study of the temperature dependences of the resonance frequency and

intrinsic quality factor at different MW excitation levels allowed us to identify

resonator coupling to two-level systems (TLS) in the environment as the primary

dissipation mechanism at ≲ 250mK; at higher temperatures the losses can be attributed

to thermally excited quasiparticles.

IV.2 Experimental details

IV.2-1 Design and fabrication

Preparations of disordered films

The standard method for the fabrication of disordered films is the deposition

of at a reduced oxygen pressure [95, 96]. Such films consist of nanoscale grains (3 ÷

4 in diameter) partially covered by . We have fabricated the films by DC

magnetron sputtering of an 6N-purity target in the atmosphere of and .

Typically, the partial pressures of and were 5 × 10mbar and (3 ÷ 7) ×

10 mbar, respectively. The films were deposited onto the intrinsic substrates at

room temperature. By controlling the deposition rate and pressure, the resistivity of the

studied films can be tuned between 10Ωcm and 10Ωcm . In order to improve

85

reproducibility, prior to the disordered deposition the target was pre-cleaned in a pure

plasma by sputtering at a rate of 0.6 nm/s for more than 2 minutes.

Critical currents of narrow disordered films

One important consideration when designing superconducting devices out of thin

film is the critical current based on the dimension of the devices. To calculate the Ginzburg-

Landau depairing current (0) for strongly disordered films at ≪ , we used the

equation for the critical supercurrent density =

√

ℏ

[28]. The concentration of

Cooper pairs can be found either from the measured kinetic inductance per square

⊡, or from the result of the Mattis-Bardeen theory ⊡ =

=

ℏ⊡

∆ where is the

film thickness. The supercurrent density is uniform over the cross section of a

superconducting film provided that the film width < ⁄ , where is the London

penetration length. This condition is satisfied for all studied films. Thus, one can estimate

as

(0) = ∙ () =1

3√3

∆

⊡(0) ≈ 1.07

⊡(0)

Equation IV-1

The coherence length (0) can be found from the data on the upper critical

magnetic field for granular films, ≈ 4 [97, 98]. This yields an estimate (0) =

Φ (2)⁄ ≈ 10 nm. The data in Table IV-1 show that the values of the microwave

86

current ∗ = 2∗ ⁄ , which corresponds to the onset of strong nonlinearity of the

resonator response, are of the same order of magnitude as the current (0) ⁄

corresponding to the bifurcation threshold.

Table IV-1: Summary of predicted and measured depairing currents

Fabrication of microwave resonators

The hybrid microcircuits containing the CPW half-wavelength resonators coupled

to a CPW transmission line (TL) have been fabricated using e-beam lithography. [Figure

IV-1 (b)] As the first step, a 50Ω TL was fabricated by the e-gun deposition of a 140nm-

thick film of pure on a pre-patterned substrate and successive lift-off. The use of pure

TL facilitated the impedance matching with the MW set-up and reduced the number of

spurious resonances (a large number of these resonances is observed if high- films are

used for both the TL and resonator fabrication). After the second e-beam lithography,

several half-wavelength disordered resonators were fabricated in the openings in the

ground plane. Before each metal deposition, reactive ion etching with 75mbar

plasma at a power of 30 watts for 30 seconds was used to remove the e-beam resist

87

residue from the substrate surface. The width of the central strip of the resonators varied

between 0.5 ÷ 10µm, and the strip-ground distance was fixed at 4µm with lithography

alignment precision < 0.5μm. [Figure IV-1 (a)]

Figure IV-1: (a) Microphotograph of a portion of the halfwavelength resonator

capacitively coupled to the coplanar waveguide transmission line. Light green - Al

ground plane and the central conductor of the transmission line, green - silicon

substrate, black - the central strip of the resonator made of strongly disordered Al.

(b) Several resonators with different resonance frequencies coupled to the

transmission line

For the resonator characterization at ultra-low temperatures, we used a microwave

setup developed for the study of superconducting qubits [15]. The resonators were designed

with the resonance frequencies ≈ 2 ÷ 4GHz, which allowed us to probe the first three

harmonics of the resonators within the setup frequency range 2 ÷ 12GHz . Different

resonance frequencies of the resonators enabled multiplexing in the transmission

measurements. In order to ensure accurate extraction of the internal quality factor , the

88

resonators were designed with a coupling quality factor of the same order of

magnitude as .

IV.3 Measurement and microwave analysis

The resonators in this work were characterized using a wide range of MW power

, two-tone (pump-probe) measurements, and time domain measurements. The

resonator parameters , , and were found from the simultaneous measurements of

the amplitude and the phase of the transmitted signal () using the procedure

described in [89, 90]. The kinetic inductance of the central conductor of the resonators,

which exceeded the magnetic inductance by several orders of magnitude, was calculated

as = 14

(the capacitance between the resonator strip and the ground was

obtained from the Sonnet simulations). The parameters of several representative resonators

are listed in Table IV-2.

Table IV-2: Summary of the measured parameters of resonators

89

IV.3-2 Microwave setup

Figure IV-2: Schematics of the resonator measurement setup

All measurements were performed in the BlueFors™ BF-SD250 dilution

refrigerator with a base temperature of ≈ 25mK. To reduce stray magnetic fields, a µ-

metal shield was installed outside of the cryostat. We used the microwave measurement

setup [Figure IV-2] developed for the research in superconducting qubits; it was described

in Chapter II. The setup enabled the resonator testing over a wide range of MW power,

90

including the single-photon population regime, the two-tone (pump-probe) and time

domain measurements. DC setup

On the same resonator chip, we also patterned Hall bars to characterize critical

currents for the disordered films. The critical currents were measured using an

Arbitrary Waveform Generator (Tektronix™ AFG3252) and HP™ 34401A multimeter

(see II.2-1 DC measurement).

IV.3-3 The procedure of extracting the quality factors and its analysis

In this section we will describe the detailed steps to extract quality factors in the

microwave analysis. The magnitude and phase of the transmitted signal have been

used to extract the quality factors , , and and the resonance frequency .

Typically, an asymmetry in the coupling of a resonator to the input and output ports results

in deviation of the resonator response from a symmetric Lorentzian function [99]. If the

coupling between the resonator and the transmission line is weak, the frequency

dependence () near the resonance frequency is described by the following

equation [99, 100]:

() = 1 − ||⁄

1 + 2( ⁄ − 1)

Equation IV-2

91

The phase delay can be found from the value of

[()] measured over

a range of away from the resonance. All other parameters in [Equation IV-2] have been

determined similar to the iteration procedure described in [100]. We first selected the initial

values of unknown parameters in [Equation IV-2] and ran a multi-variable nonlinear fitting

procedure for the entire model. The output of the nonlinear fit was used to obtain the final

values of unknown parameters and the error bars. The parameter initialization procedure

was as follows. After elimination of the phase delay , the data () formed a

circle on the -plane [Figure IV-3 (a)]. The prefactor corresponds to the center of

this circle. For the normalized circle

∗ = () ⁄

Equation IV-3

the angle between the off-resonance points and the I-axis corresponds to , and the circle

diameter corre ponds to the ratio of ||⁄ [Figure IV-3 (b)]. Next, we translated ∗

so that the circle center coincided with the origin. can then be obtained from fitting the

phase of the translated ∗ , , versus frequency with = + tan[2(1 − ⁄ )]

(see Figure IV-3 (c)). Figure IV-3 (d,e) show the experimental data with the result of fitting.

92

Figure IV-3: Fitting procedure. (a) Blue and red points correspond to the

transmission measured before and after the phase delay is removed, respectively.

After removing the phase delay, the data form a circle on the IQ-plane with the center

at . (b) Normalized transmission ∗ on the complex plane. The angle between

the center of the ∗ circle and the real axis corresponds to . (c) The phase versus

frequency (blue points) fitted with = + [( − ⁄ )] (red curve). (d,e)

Measured data (blue points) and the fit with Equation IV-3 (red curve).

With the extracted quality factors, we next proceed with the analyses of losses. We

observed the enhancement of the internal quality factor with increasing the average

number of photons in the resonators, = 2Q (ℎ

)⁄ [101], where =

(Q + Q

) is the loaded quality factor. The dependences () for three resonators

93

with different measured at the base temperature ≈ 25mK are shown in Figure IV-4.

Similar behavior of () have been observed for many types of CPW superconducting

resonators (see, e.g. [28, 102] and references therein), including the resonators based on

disordered films [44, 103]. Note that the increase of with the input MW power

is limited by the resonance distortion by bifurcation at > ∗. For the resonators

with ≳ 10 the onset of bifurcation is observed for the microwave currents ∗ =

2∗ ⁄ which scale approximately as ⁄ [104], where is the Ginzburg-

Landau depairing current in the central strip in the previous section (Equation IV-1).

Figure IV-4 The dependences () at ≈ for the resonators with different

widths. Solid curves represent the theoretical fits of the quality factor governed by

TLS losses [Equation IV-5].

The power-dependent intrinsic losses can be attributed to the resonator coupling to

the TLS with the Lorentzian shaped distribution

94

() ≈1

( − ℎ) + (ℏ ⁄ )

Equation IV-4

where is the energy of TLS and is its dephasing time [105]. Once the MW power

reaches some characteristic level and the Rabi frequency of the driven TLS

Ω ≈ exceeds the relaxation rate 1 √⁄ , the population of the excited TLS

increases, and the amount of energy that the TLS with ≈ can absorb from the

resonator decreases. Thus, the high "burns the hole" in the density of states (DoS)

of dissipative TLS. The width of the "hole" is 2⁄ , the power-dependent factor can be

written as

= 1 +

Equation IV-5

where and correspond to and , respectively. Note that the exponent is

known to be dependent on the electric field distribution in a resonator [106], and the

characteristic power increases with temperature by orders of magnitude due to a strong

-dependence of and [107]. Taking into account the TLS saturation at high

temperature, the power dependence of the TLS-related part of the loss tangent can be

expressed as follows [12]:

(, ) =

ℎ

ℎ

2

Equation IV-6

95

By fitting the experimental data with Equation IV-6 we found and , the

obtained parameters are listed in Table IV-3. We found that larger values of correspond

to wide strips, and the extracted scales as the square of the electric field on the surface

of the resonator. The experimental dependences () measured for resonators #2 − 4 at

≅ 1 and ≫ 1 [Equation IV-8(b)] are well described by the sum of the TLS

contribution [Equation IV-6] and the MB term = () ()⁄ [43]:

() = , , , + (, (0))

Equation IV-7

The agreement of measured with the prediction of Equation IV-7 over the whole

measured temperature range proves that the losses in the developed resonators are limited

by the sum of TLS and MB terms.

Table IV-3: Summary of the fitting parameters

IV.4 Discussion

The measured sheet kinetic inductance ⊡ = 2 nH ⊡⁄ is similar to that reported

for granular films in [103] and in [108], and exceeds by a factor-of-2 ⊡

realized for ultra-thin disordered films of [42, 109]. For the disordered films

96

with < 10mΩ · cm, ⊡ is in good agreement with the result of the MB theory [42],

⊡( = 0) = ℏ⊡ ∆(0)⁄ , where ∆(0) is the BCS energy gap at = 0K. Very large

values of ⊡ allowed us to realize the characteristic impedance = ⁄ as high as

5 kΩ for the resonators with narrow ( = 0.7µ) central strips. The speed of propagation

of the electromagnetic waves in such resonators does not exceed 1% of the speed of light

in free space; accordingly, their length is two orders of magnitude smaller than that for the

conventional CPW resonators with the impedance = 50 Ω.

To identify the physical mechanisms of losses in the resonators, we measured the

dependences of and on the temperature ( = 25 ÷ 450 mK) and the microwave

power . Below we show that in the case of moderately disordered films (resonators

#2 − 4), both the dissipation and dispersion at < 0.25K can be attributed to the resonator

coupling to the TLS in the environment, whereas at higher temperatures they are controlled

by the T dependence of the complex conductivity of superconductors, () = () −

() [42].

97

IV.4-1 The resonance frequency analysis

Figure IV-5: The temperature dependences of resonance frequency shift

()/

(a) and the internal quality factor (b) for the resonators #2−4 measured at

≈ () and ≫ (∆) . The fitting curves correspond to Equation IV-9 and

Equation IV-7, respectively.

We start the data analysis with the temperature dependence of the relative shift of

the resonance frequency () ⁄ ≡ [() − (25mK)] (25mK)⁄ . Figure IV-5 (a)

shows the dependences () ⁄ measured for three resonators (#2 − 4) with different

width . The low-temperature part of () ⁄ is governed by the -dependent TLS

contribution to the imaginary part of the complex dielectric permittivity () = () +

(). It should be noted that, in contrast to the TLS-related losses, the frequency shift

() is expected to be weakly power-dependent [43]. Indeed, the temperature

dependences measured for the different values of almost coincide; this simplifies the

98

analysis and reduces the number of fitting parameters. The low-temperature part of

() is well described by the following equation [39]:

()

=

ℜ

1

2+

1

2

ℎ

−

ℎ

Equation IV-8

Here Ψℜ() is the real part of the complex digamma function, the TLS

participation ratio, is the energy stored in the TLS-occupied volume normalized by the

total energy in the resonator, and the loss tangent characterizes the TLS-induced

microwave loss in weak electric fields at low temperatures ≪ ℎ . The product

is the only fitting parameter, its values are listed in Table IV-3. The obtained values of

are close to that found for -based [43] and -based resonators [108, 110].

Note that resonator #4 demonstrates the most pronounced increase of () with

temperature due to the stronger electric fields and a larger participation ratio characteristic

of the high- resonators [12].

At > 0.25 K, rapidly drops due to the decrease of the superfluid density. The

dependences () over the whole studied range can be described as

() ⁄ = () ⁄ +

() ⁄

Equation IV-9

where

99

()

=

1

2() − (25)

(25)

Equation IV-10

is the resonance shift due to the -induced break of Cooper pairs and subsequent increase

of the kinetic inductance, calculated in the thin film limit [43]. The only free parameter in

() ⁄ is the gap energy ∆(0), which can be found by fitting of the high- portion

of () ⁄ [Equation IV-9]; the measured ratio ∆(0) = is about 10% greater

than the BCS value of 1.76 , which is consistent with previously reported data [62].

IV.4-2 In-depth analysis of () and () fitting

To identify the dominant mechanisms of losses in the studied resonators, we have

analyzed the experimental dependences () and () on the basis of the theory of

two-level systems [111] and the Mattis-Bardeen theory of the complex impedance of

superconductors [91].

The losses due to the real part of the complex impedance of superconductors, =

− , can be estimated using the Mattis-Bardeen theory. In the thin film limit [112]:

() = () ()⁄ ; where

() =

ℎ [() − ( + ℎ)] ×

( + + ℎ)

√ − ( + ℎ) −

() =

ℎ [1 − 2( + ℎ)] ×

( + + ℎ)

√ − ( + ℎ) −

Equation IV-11

100

() is Fermi-Dirac distribution function, Δ(0) is the energy gap. The temperature-

dependent shift of the resonance frequency is given in the previous section by Equation

IV-10. Since the frequency shift () does not depend on the MW power, the fitting

procedure included the following steps:

• fitting the () dependence with only two free parameters: Δ(0) (which controls

the behavior of () term at > 300mK), and the product of the participation ratio

and the material loss tangent, (which governs the rising part of ());

• finding the index from the linear part of () at = 25 plotted on the

double-log scale.

• knowing , , Δ(0), one can find the low temperature characteristic power (i.e.

the number of photons (0)) using () measured at the base temperature = 25mK;

• finding () by fitting () data for both low and high values of the input power.

Significant change in the population of the ground and excited TLS states due to

Rabi oscillations is expected at the average number of photons in the resonator > .

The characteristic value ≈ 1 ⁄ depends on the TLS relaxation time () and

the dephasing time (). In agreement with [113] where the TLS relaxation time was

shown to be , ≈ 1 + , we found that the extracted characteristic values of

depend on the temperature as () = (0) + , ≈ 2 [Figure IV-6].

101

Figure IV-6: The temperature dependences of for different resonators.

IV.4-3 The two-tone time-domain measurements and telegraph noise

We obtained an additional information on the TLS related dissipation by

performing the pump-probe experiments in which was measured at a low-power ( ≈

1) probe signal while the power of the pump signal at the frequency was varied

over a wide range. Figure IV-7 (a) shows the dependences () measured at different

detuning values ∆ = − = 0, ±1, ±10MHz. Note that we have not observed any

changes in when the pump signal was applied at the second and third harmonics of the

resonator. Also, was -independent when we monitored the second harmonic and

applied the pump signal at the first harmonic.

102

Figure IV-7: (a) The dependences of for resonator #1 on the pump tone power

for several values of detuning ∆ between resonance and pump frequencies. (b)

The values of measured versus detuning ∆ at a fixed number of the pump tone

photons in the resonator ≈ .

Since the resonator coupling to the pump signal varies by several orders of

magnitude within the detuning range 0 ÷ 10, it is more informative to analyze

as a function of the average number of the "pump" photons in the resonator, =

1 −

− ()

ℎ , where and = 1 − are the

transmission and reflection amplitudes at the pump frequency, respectively. The

dependence on the detuning ∆ for a fixed ≈ 1000 is depicted in Figure IV-7

(b). The resonance behavior of (∆) is expected since only a narrow TLS band

[Equation IV-4] contributes to dissipation: the "hole" extension in the DoS is limited by

≈ ⁄ around the pump frequency. Indeed, using the approach developed in [114], one

can obtain the following expression:

(∆) = 1 +( 2⁄ )

∆ + (1 2⁄ )

103

where is the off-resonance quality factor and introduced by Equation IV-5 factor κ

might be calculated as = ⁄ . The dephasing time is the only fitting parameter,

and it is found to be ≈ 60ns. This result agrees with the measurements of the dephasing

time for individual TLS in amorphous tunnel barrier in Josephson junctions [10].

By application of the MW pulses at the pump frequency, we observed that the

characteristic time at which varies with does not exceed 36 . For several

resonators we have observed the telegraph noise in the resonance frequency on the time

scale of 1 ÷ 10. This noise can be attributed to interactions of the resonators with a small

number of strongly coupled TLS.

Interactions between the high-frequency (coherent, > ) TLS with the low-

frequency (thermal, < ) fluctuators result in the TLS spectral diffusion as well as

the flicker noise. The telegraph noise in the resonance frequency is expected if some

of the TLS with ≈ are strongly coupled to a resonator. Typical TLS densities for

/ junctions are ≈ 1 (GHz · µm) −1 [11]. Interestingly, the number of

strongly coupled TLS for our resonators (assuming that the strongly coupled TLS are in

the oxide layer of the resonator) is of the order of unity (1 (GHz · µm) × 0.1MHz ×

10µm ). To study the telegraphy noise, we repetitively measured at a fixed

frequency on a slope of the resonance dip for a few minutes. Figure IV-8 shows an example

104

of the measured telegraph noise in Re[] . The characteristic time scale of random

switching between two Re[] levels is 1 ÷ 10s.

Figure IV-8: The time dependence of [] measured at = at a fixed

frequency on the slope of a resonance dip. The microwave power corresponds to

⟨⟩ ≈ . Each point corresponds to the data averaging over .

IV.4-4 Scaling of ()

The ground and excited states of TLS become equally populated when the Rabi

oscillation frequency Ω = · ℏ⁄ exceeds the rate 1 ⁄ , or, equivalently, when the

electric field in the TLS-occupied volume exceeds the critical value ≈ ℏ ⁄ .

In order to understand the variation of the observed characteristic power for different

resonators, we considered the dependence of the maximum electric field near the surface

on the resonator parameters.

105

The standard way to evaluate the characteristics of CPW resonators is by the

Schwarz-Cristoffel (SC) mapping of the coplanar topology to the trivial parallel-plate

capacitor geometry. Let us consider a zero-thickness CPW with a central strip width 2

and a ground-to-ground distance 2. The transfer function for the mapping of the upper

half-plane to the rectangle is given by

() =

( − )( + )( − )( + )

here is an integration constant, chosen to be = 1. The half-width of the equivalent

capacitor is calculated as

= () =1

(1 − )(1 − )

≡1

()

() is also known as the complete elliptic integral of the first kind, = ⁄ . Similarly,

the height of the capacitor is

=1

(1 − )(1 − )

⁄

≡1

1 −

the electric field in the -plane for the given voltage across the capacitor is uniform

and can be easily obtained as

=

=

√1 −

the corresponding field in the -plane scales with the factor () = ⁄ which is

() =1

( − )( − )

thus, for example, the field strength at the center of microstrip is

106

|( = 0)| = · ( = 0) =

√1 −

accordingly, the power in the CPW can be written as

≈

≈

1 −

therefore, we expect that the characteristic power scales as ≈ (()) ⁄ , which is

in agreement with the experimental data.

IV.4-5 Pump-probe measurements of the TLS relaxation time

We have performed the time domain measurements of the TLS relaxation time for

resonator #1 using the pulse sequence shown in Figure IV-9 (a). A 0.5s-long pump pulse

was applied to the resonator at the beginning of each duty cycle. A readout pulse at the

single-photon power level lasting for 36ms followed the pump pulse and was digitized to

obtain . The readout delay time was varied between 0s and 1s. Figure IV-9 (b) shows

the result of the experiment at the readout frequency = 2.4258 GHz and the pump

frequency = + 1 MHz. The change in |()| at = 0.5s is consistent with CW

measurements at the same readout frequency and power level when a pump tone was turned

on and off. This indicates that an upper limit of the TLS relaxation time for our sample is

much less than 36 ms.

107

Figure IV-9: (a) The pulse sequence. (b) The time dependence of || measured at

= . . The pump pulse at = + was applied between =

and = . . The pump tone power corresponds to ¯ ≈ . Each data

point was averaged over 4000 cycles with the same readout delay time. The inset

shows CW measurement of versus with (red) and without (blue) the pump

signal and indicates the position of used in the relaxation time measurement. The

readout power was at the single photon level for all measurements on this plot.

IV.5 Summary

In conclusion, we have fabricated CPW half wavelength resonators made of

strongly disordered films. Because of the very high kinetic inductance of these films,

we were able to significantly reduce the length of these resonators, down to ≈ 1% of that

of conventional CPW resonators with a 50Ω impedance. Due to ultra-small dimensions

108

and relatively low losses at mK temperatures, these resonators are promising for the use

in quantum superconducting circuits operating at ultralow temperatures, especially for the

applications that require numerous resonators, such as multi-pixel MKIDs [28, 104]. The

high impedance = ⁄ of the narrow resonators can be used for effective coupling

of spin qubits [115, 116]. The high resonator impedance imposes limitations on the strength

of resonator coupling to the transmission line. For the studied CPW resonators with ≈

5 kΩ, the strongest realized coupling (when half of the resonator length was used as the

element of capacitive coupling to the transmission line) corresponded to ≈ 10. On the

other hand, for many applications, such as large MKID arrays that require a high loaded

factor, this should not be a limitation.

We have shown that the main source of losses in these resonators at ≪ is the

coupling to the resonant TLS. A comparison of our results with those of the other groups

shows that the obtained values, increasing from (1 ÷ 2) × 10 in the single-photon

regime to 3 × 10 at high microwave power, are typical for the CPW superconducting

resonators with similar TLS participation ratios. This implies that the disorder in films

does not introduce any additional, anomalous losses. Most likely, the relevant TLS are

located near the edges of the central resonator strip either in the native oxide on the

substrate surface or in the amorphous oxide covering the films. Further increase of can

109

be achieved by the methods aimed at the reduction of surface participation, such as

substrate trenching (see [117] and references within) and increasing the gap between the

center conductor and the ground plane [106]. The evidence for that was provided by the

results of [103] obtained for the modified three-dimensional microstrip structures based on

disordered films. It is also worth mentioning that the losses can be reduced using TLS

saturation by the microwave signal outside of the resonator bandwidth but within the TLS

spectral diffusion range. A fundamental issue pertinent to all strongly disordered

superconductors is the development of a better understanding of the impedance of

superconductors near the disorder-driven SIT. This issue requires further research, and the

microwave experiments with the resonators made of disordered and other disordered

materials demonstrating the SIT may shed light on the nature of this quantum transition.

110

FLUXON-PARITY-PROTECTED SUPERCONDUCTING QUBIT

This chapter is based on joint work with K. Kalashnikov, P. Kamenov, and M.E.

Gershenson (Rutgers University), W.T. Hsieh and M.Bell (University of Massachusetts

Boston), and A. Di Paolo and A. Blais (Universite de Sherbrooke) [14]. My main

contribution to this work was the development of fabrications and microwave

characterization techniques.

V.1 Introduction

Over the past two decades superconducting qubits have emerged as one of the most

promising platforms for quantum computing, and the coherence of these qubits has been

improved by five orders of magnitude [88, 118]. Even with this spectacular progress,

implementation of error correction codes remains challenging [119]. Further improvement

in coherence will require the development of new approaches for mitigating harmful effects

due to uncontrollable microscopic degrees of freedom, such as two-level systems (TLSs)

in the qubit environment [11]. This route is provided by the improvement of materials for

fabrication of superconducting qubits, which can lead to the reduction of the TLS density.

A complementary approach, which we consider in this work, is based on the reduction of

the qubit-TLS coupling by qubit design.

111

V.2 Suppressing the decoherence

Qubit coherence is characterized by the energy relaxation (decay) time and the

dephasing time . The decay rate Γ ≡ 1 ⁄ due to coupling to a fluctuating parameter

is proportional to the transition amplitude |⟨|ℋ|⟩| , where ℋ is the coupling

Hamiltonian and |⟩ and |⟩ are the qubit’s ground state and first excited state,

respectively. Since the external noise couples to local operators, decreasing of the overlap

of |⟩ and |⟩ wavefunctions can significantly reduce Γ. This strategy is exploited by

several qubit designs in which localization of the logical-state wavefunctions occurs within

distinct and well-separated minima of the qubit potential, such as the “heavy fluxonium”

qubit [120, 121].

On the other hand, a small dephasing rate Γ ≡ 1 ⁄ requires the qubit transition

frequency to be insensitive to fluctuations of . The first-order decoupling of a qubit

from noise has been achieved at the so-called “sweet spot” , where ⁄ |= 0

[122]. However, the coherence times achieved with this approach are insufficient for the

implementation of the error correction codes, even if the drifts of the qubit operating point

are eliminated over the timescale of operations. To remedy this, a “sweet-spot-everywhere”

approach has been realized in the transmon qubit [5, 123]: an exponentially strong

112

suppression of the qubit sensitivity to noise has been achieved by delocalization of the

qubit wavefunctions in charge space.

It is, however, worth noting that the two approaches of and protection by

qubit design come into conflict in the case of devices with a single degree of freedom in

the qubit Hamiltonian (which we refer to as 1D qubits). For instance, at the dephasing

sweet spot of the “heavy fluxonium” [120, 121] wavefunctions become delocalized due to

its hybridization, which limits the decay time (Figure V-1 (a), = 1 ), whereas

protection can be realized only at the slope of the dispersion curve where is small

(Figure V-1 (a), = 2 ). In turn, the charge insensitivity of the transmon qubit is

accompanied with strong dipole matrix elements that limit (Figure V-1 (b), = 3, 4).

Additionally, the flatness of the transmon-qubit bands results in a strong reduction of the

spectrum anharmonicity, potentially leading to a leakage of information outside of the

computational subspace [124]. These examples suggest that a qubit Hamiltonian with full

noise protection against relaxation and dephasing, i.e., exponentially large and ,

cannot be implemented in a single-mode superconducting quantum device. This conflict,

however, can be reconciled by the so-called “few body” qubits [13] that incorporate more

than one degree of freedom in the qubit Hamiltonian (the dimensionality > 1) [16, 125-

127].

113

Figure V-1: The tradeoff between the decay and dephasing protection in

superconducting qubits with a single charge or flux degree of freedom. The band

structure (top panels) and wavefunctions (bottom panels) of a particle in

quasiperiodic potentials: (a) the free-particle regime and (b) the tight-binding regime.

The wavefunction overlap and the energy sensitivity ()

/ do not simultaneously

vanish for any point (i). Flux (charge) qubits correspond to the case in which the

control parameter = , kinetic energy = (), tunneling energy =

(), and |⟩ is a fluxon (charge) basis.

Another concept of qubit protection exploits symmetries of Hamiltonians with

> 1 [58], an example being the qubit based on Josephson rhombi arrays [128],

experimentally realized in [16]. In a single rhombus threaded by half of the magnetic flux

quantum, the transport of individual Cooper pairs (CPs) is suppressed due to destructive

Aharonov-Bohm interference, such that the rhombi chain supports correlated transport of

CPs (i.e., acts as a cos(2) Josephson element). The dephasing time of the qubit can be

114

enhanced by delocalization of wavefunctions over the states with the same CP parity,

which does not compromise . Importantly, this qubit design enables on-demand

switching of the qubit coupling to the environment (including the readout) on and off,

which facilitates qubit manipulations. This also provides a route to fault-tolerant gates

immune to noises in the control lines [129]. An improved version can be built by parallel

connection of several rhombi chains [130]

Here we focus on the implementation of a complementary circuit preserving the

parity of fluxon in a superconducting loop, which consists of a split Cooper-pair box (CPB)

and a superinductor (SI), and is depicted in Figure V-2 (a). The probability of single-fluxon

tunneling in and out of the loop can be tuned by the CPB charge of the CPB island

(hereafter we refer to CPB charge modulo 2 due to periodicity). At = 1 (where

is the electron charge) Aharonov-Casher interference results in a 4-periodic potential (a

cos(/2) Josephson element), which preserves the fluxon parity in the loop [17, 43, 131].

In the case of perfectly symmetric CPB junctions, the two degenerate logical states with

different fluxon-number parity reside in disjoint regions of the Hilbert space, forbidding

qubit decay. It is moreover possible to delocalize the wavefunction within each parity state

via double fluxon tunneling in order to provide protection against pure dephasing by flux

noise. Below we refer to such an element as a “bifluxon” qubit.

115

Figure V-2: (a) Simplified circuit scheme of the bifluxon qubit. Charging energies of

the superinductor and CPB are and , respectively. The qubit is controlled by

the CPB charge and the magnetic flux . (b) Optical image of the bifluxon

qubit, readout resonator, and the microwave transmission line. The inset shows the

SEM image of its central part: two JJs form the CPB island (red false color), the long

array of larger JJs acts as a superinductor (blue), the narrow wire (green) forms the

closed loop and couples the qubit to the readout resonator.

V.3 Experimental setups

In this work, the bifluxon qubit is realized as a split junction CPB (a

superconducting island flanked by two small and nominally identical JJs with Josephson

116

and charging energies and , respectively; see Figure V-2 (b)] shunted by a SI,

which is implemented as an array of = 122 larger JJs with corresponding energies

and . The sizes of small ( 0.11 × 0.16 ) and large ( 0.21 × 0.30 )

junctions are chosen in order to allow phase-slip events across the CPB junctions

(/ ≈ 1) but suppress the phase slips in the array (/ ≈ 1). As long as the

inductive energy of the SI chain = / is much smaller than , the phase across

the SI is close to an integer number of 2. The stray capacitance of the superconducting

islands to the ground in combination with the junction capacitances results in charging

energies and of the CPB and the SI, respectively. The self-resonant mode of the

SI with frequency , determined by the SI inductance and its stray capacitance to the

ground , should remain well above the qubit transition frequency (usually a few GHz)

in order to avoid qubit coupling to this mode.

The bifluxon qubit is controlled by the magnetic flux in the loop Φ and the

offset charge , induced by applying the dc bias voltage to the coupling capacitor

between the microstrip line and the CPB island. In order to perform the dispersive

measurements of the bifluxon qubit, the device is inductively coupled to a lumped element

readout resonator with capacitance = 120 and inductance = 4 . For the

coupling, a portion of the bifluxon superconducting loop with kinetic inductance =

117

0.4 is shared with the readout resonator. The qubit-resonator coupling constant for the

device described in this paper is /2 = 52MHz. The full list of experimental parameters

is provided below:

Table V-1: The bifluxon qubit parameters estimated from a test structure: Josephson

and charging energies, areas of the junctions in the CPB and SI array, and the

number of junctions in the array and its total inductance.

Table V-2: The resonator parameters: inductance, capacitance, shunting inductance,

and loaded and intrinsic quality factors.

Table V-3: The fitting parameters: the CPB junction Josephson energy, charging

energies of CPB and SI, the SI inductive energy, the CPB junctions’ asymmetry, and

noise factors of SI and CPB modes.

V.4 Transmission measurement

In the transmission measurements, the microwave signals travel along the

microstrip line that is coupled to the readout resonators of up to five different bifluxon

qubits measured in the same cooldown. The qubits are individually addressed due to

118

different resonant frequencies of the readout resonators. The bifluxon qubit, readout

resonator, and microstrip transmission lines are fabricated in a single multiangle electron-

beam deposition of aluminum through a liftoff mask as we introduced in II.1.

Figure V-3: Spectra of the bifluxon qubit: experimental data for the |⟩ − |⟩ and

|⟩ − |⟩ transitions (symbols) and the result of exact diagonalization of the circuit

Hamiltonian in Eq. (1) (solid and dashed lines). (a) Flux dispersion of the transition

frequencies for two values of the CPB charge = ,

. The inset is an

enlargement of the qubit spectrum near = , displaying the avoided crossing

that characterizes the rate of double phase slips . (b) Charge dispersion of the

transition frequency for = .

119

The pump tone induces the |0⟩ − |⟩ transitions at the resonance frequencies

= ( − )/ℎ. The measurement tone probes the dispersive shift of the coupled

readout resonator. Although the dispersive measurements in the protected regime are

complicated by significantly reduced qubit-readout coupling, the signal-to-noise ratio in

the spectroscopic measurements is sufficiently high to identify the resonances even in the

protected regime. The flux dependencies of the resonance frequencies () and

at = 0,

are shown in Figure V-3(a). The obtained spectra are in good

agreement with the results of diagonalization of the circuit Hamiltonian [solid lines in

Figure V-3(a)], with fitting parameters /ℎ = 27.2GHz , /ℎ = 7.7GHz , /ℎ =

0.94GHz, and /ℎ = 10GHz, and asymmetry between the CPB junctions Δ ≡ −

= ℎ × 6GHz.

The extracted values are consistent with the expected JJ parameters. The normal-

state resistance of the CPB junctions extracted from using the Ambegaokar-Baratoff

relation agrees within 20% with the normal state resistance of test junctions

fabricated on the same chip. Both CPB and SI charging energies agree well with the typical

aluminum-based junction capacitance 50fF/μm and specific capacitance of micron-size

islands on silicon substrates 0.085fF/μm [132].

120

We also observe an additional resonance at 13.9GHz, whose position does not

depend on and . We attribute this resonance to the lowest-frequency mode of the

superinductor, which corresponds to characteristic impedance of the SI = 14Ω.

Figure V-4: (a) Measurements of the bifluxon energy relaxation in the protected state

(red circles) and unprotected state (blue squares). The sequence of pulses is shown in

the inset. The exponential fits are shown by solid and dashed lines, respectively. Note

that the resonance energy of the qubit in the protected

state is × . (approximately × ), and a nonzero occupancy of the

first excited state [( ⁄ ) + ]⁄ with the qubit temperature = =

is taken into account. (b) Demonstration of an absence of qubit excitation by

the gate voltage pulses.

121

V.5 Time-domain analysis

In the time-domain experiments the signal-to-noise ratio, reduced by weak qubit-

readout coupling, is too low to employ conventional pulse protocols (decay, Rabi

oscillations, and Ramsey fringes). For this reason, we designed special pulse sequences for

and measurements in the protected regime. The pulse sequence used for probing

the decay is shown in Figure V-4(a). Initially the qubit is prepared in the ground

unprotected state ( = 0). A microwave pulse at the resonant frequency ()

excited

the qubit, and then the protection is turned on by applying a pulse of the gate voltage

corresponding to the offset charge =

. We have used pulses with the rise and drop

times approximately 30ns ≫ 1/(()

− ( ⁄ )

)) , which is sufficiently long to ensure

adiabatic evolution of the qubit between protected and unprotected states. After time Δ,

the protection is removed by setting = 0 and the qubit state is measured. As a control

experiment, we apply the gate voltage pulses alone, without a pulse; the absence of

qubit excitation proved the adiabaticity of gate manipulations; see Figure V-4(b).

122

Figure V-5: Energy relaxation time as a function of the flux frustration (a)

and the CPB charge (b). The pale circles represent all the measured data and the

bright circles show the longest measured for a given operation point. The lines

correspond to fitting to the resistive noise theory. The sharp dip around = .

corresponds to the Purcell decay into the readout resonator.

The main result of this work — the dependence of on the qubit control

parameters and — is presented in Figure V-5. Dashed lines represent fits to the

model that take into account resistive losses in the capacitively coupled environment and

readout resonator (Purcell effect is pronounced near /2 = 0.3). An increase of

123

in the protected regime by an order of magnitude provides evidence for the qubit’s dipole

moment suppression. The longest decay time of greater than 100 is measured at full

flux frustration = , which corresponds to a minimum qubit energy ( ⁄ )

=

0.4GHz.

Direct measurements of the decoherence time in the protected regime, by either

Rabi or Ramsey techniques, are not feasible because of vanishing coupling of the qubit o

microwave pulses. For this reason, we modified the measurements of Ramsey fringes by

analogy with the aforementioned measurements. The pulse sequence is shown in

Figure V-6(a). Both 30ns long /2 microwave pulses detuned from the qubit transition

frequency by 4MHz are applied in the unprotected state ( = 0 ), and the qubit is

measured after the end of the second pulse. Between the /2 pulses, while the qubit

undergoes free precession, the qubit’s protected state is restored by applying a gate voltage

pulse ( = 1 2⁄ ). After averaging over 1000 cycles, the Ramsey fringes are recorded by

varying the delay between the end of the gate pulse and the second /2 pulse.

Ramsey fringes measured according to this procedure for one of the flux “sweet

spots” at = 0 are shown in Figure V-6(b); the pulse for these measurements

is 0.27μs ong. The difference between the amplitudes of Ramsey ringes at moments Δ =

0 and 0.27μs may provide information on dephasing in the protected state if this is the

124

only source of dephasing. However, the accuracy of this technique is limited by the

pulse jitter. Indeed, in the rotating frame of the unprotected state, the qubit’s state vector

rotates in the equatorial plane of the Bloch sphere as soon as the protection is turned on.

The angular velocity of these rotations, = ()

− ( ⁄ )

ℏ⁄ , is large ( > 2 × 1GHz)

at both flux sweet spots = 0, ; and even a small jitter can result in a significant error

in the position of the qubit’s state vector at the end of the pulse. According to the

specification, the jitter time of the pulse generator used in our experiments could be as large

as 0.3ns. This jitter-induced phase uncertainty alone, without invoking any dephasing in

the protected state, is sufficient to explain the reduced amplitude of Ramsey fringes at

Δ = 0.28μs. Thus, these measurements can impose only the lower limit on , which is

close to 1μs for the data in Figure V-6(b). Future experiments with better controlled

pulses of different lengths may provide more detailed information on at both sweet

spots.

125

Figure V-6: The Ramsey fringes measurement. (a) The pulse protocol for

evaluation in the protected state. The protection is turned on for a fixed time of

; the time delay between two / pulses is varied in order to record Ramsey

fringes. (b) The experimental data (circles) and the damped-oscillation fitting (the

solid line). Note that the value of = . describes the fringe damping in the

= state. In the protected state (within a time interval < < )

damping of Ramsey fringes may be caused by the pulse jitter rather than

dephasing (see the text).

V.6 The offset charges and mitigation of quasiparticle poisoning

Quasiparticle poisoning (QP) presents a problem for charge-sensitive quantum

superconducting devices [133, 134]. In particular, for a bifluxon qubit in a protected state,

126

tunneling of a nonequilibrium quasiparticle into or out of the CPB island would remove

protection. To minimize QP, we use so-called gap engineering [135, 136].

In Figure V-7(a) we show the superconducting gap in the CPB island and the outer

electrodes that form the CPB Josephson junctions. Because of the dependence of the

critical temperature of films on their thickness, the gap in the thin (20 ) CPB island

is greater than that in thicker (60 ) outer electrodes. This difference , which we

estimate to be approximately (0.3 ÷ 0.4) (see [137]), is sufficiently large to block

tunneling of nonequilibrium quasiparticles with energies greater than onto the CPB

island at sufficiently low temperatures. The efficiency of this technique is demonstrated in

Figure V-7(b) to Figure V-7(d). If both the CPB island and outer electrodes are thick (Δ ≈

0), we observe a characteristic “eye” pattern [135] in the spectroscopic measurements,

which reflects rapid ± jumps of the CPB charge on the timescale of a single scan of the

resonance of the readout resonator; see Figure V-7(b). This pattern vanishes if the gap

engineering is employed and reappears only at higher temperatures, where the

quasiparticles are thermally excited in the CPB island (compare panels (c) and (d) of Figure

V-7). Gap engineering and careful infrared and magnetic shielding of the device allow us

to increase the time intervals between the QP events up to 30mins. In Figure V-7(e) we

show that, in addition to rare QP events, in the gap-engineered device we observe slow

127

monotonic drift of whose origin remains unclear. Because of this drift, we have to

measure (and, if necessary, readjust) before each time-domain measurement.

Figure V-7: Suppression of quasiparticle poisoning by gap engineering. (a) Profile of

the superconducting gap across the CPB island. The critical temperature of the thin

CPB island is . ÷ . higher than that in the thicker electrodes. (b)–(d)

Spectroscopy of the readout resonator as a function of for bifluxon qubits:

without gap modulation at (b), and with gap modulation at (c) and

(d). (e) The gap-engineered device at . The dispersive shift of

the readout resonator (color coded) is measured at a fixed gate voltage over 9

hours. The shift is converted into using the data of panel (c). Abrupt

jumps reflect the QP events ( = ± ⁄ ), gradual shift corresponds to a monotonic

drift of with a rate of less than per minute.

128

V.7 Conclusion

In this work, we develop and characterize a symmetry-protected superconducting

qubit that offers simultaneous exponential suppression of energy decay from charge and

flux noises, and dephasing from flux noise. Provided the offset charge on the CPB island

is an odd number of electrons, the qubit potential corresponds to that of a (/2)

Josephson element, preserving the parity of fluxons in the loop via Aharonov-Casher

interference. In this regime, the logical-state wavefunctions reside in disjoint regions of

Hilbert space, thereby ensuring protection against energy decay. By switching the

protection on, we observe a tenfold increase of the decay time, reaching up to 100 .

Though the qubit is sensitive to charge noise, the sensitivity is much reduced in comparison

with the charge qubit, and the charge-noise-induced dephasing time of the current device

exceeds 1 . Implementation of full dephasing protection can be achieved in the next-

generation devices by combining several (/2) Josephson elements in a small array.

129

CONCLUSION AND OUTLOOK

In this work we developed and characterized quantum superconducting circuits that

serves as a platform for the realization of protected qubits with simultaneous exponential

suppression of energy decay from charge and flux noise. This chapter summarizes the

results and provides outlook for future work.

VI.1 Junctions with low Josephson energy

In Chapter III, we address the transition to a dissipative transport in low-

junctions driven by the incoherent phase slips. We observed that the switching current,

which corresponds to an abrupt increase of the voltage across the JJ to V = 2Δ ⁄ and

generation of quasiparticles, is strongly reduced in such junctions with respect to the

Ambegaokar-Baratoff’s prediction for the critical current. The premature switching ≪

; is accompanied by an increase of the zero-bias resistance . We attributed this

behavior to the temperature activated phase slips whose rate exponentially increases with

decreasing .The dissipation associated with TAPS might limit the performance of qubits

based on low- junctions [13, 14, 36]. Steep temperature dependence of the coherence

time at temperatures could be a signature of TAPS-induced decoherence. This

130

dependence, observed in the experiments with the “heavy” fluxonium [87], requires further

investigations.

VI.2 Superinductors based on granular aluminum thin films

In Chapter IV, we fabricated and characterized CPW half wavelength resonators

made of strongly disordered films. Due to a very high kinetic inductance of these

films, we were able to significantly reduce the footprint of the CPW resonators from

size down to tens of . We realized a superinductor with impedance ≈ 5Ω

approaching the resistance quantum . High kinetic inductance in combination with low

microwave losses make films promising for a wide range of microwave applications

which include kinetic inductance photon detectors and superconducting quantum circuits.

In a more recent work [19], by meandering the nanowires, we were able to fabricate

several lumped-element superinductors with impedance 10 ÷ 30Ω and

inductance 0.1 ÷ 1.3.

To fabricate films reliably, further material characterization is required. One

of the common concerns in fabricating thin films is that disorder and grain

connectivity are not well controlled, which underscores the need for better control of the

critical film formation parameters. Another promising direction of research is the use of

131

for mitigating the quasiparticle poisoning as demonstrated in Chapter V. By

introducing disorder in Al films (Figure II-6), it is possible to increase from 1.3 to

2. Such enhancement opens possibilities for suppression of quasiparticle poisoning by

for gap engineering. Wea re currently working on fabricating the quasiparticle barriers by

in-situ modification of the structure of Al films with the focus ion beam technology.

VI.3 Fluxon-parity protected qubits

In Chapter V we developed and characterized a symmetry-protected

superconducting qubit that offers simultaneous exponential suppression of energy decay

from charge and flux noises, and dephasing from flux noise. The qubit consists of a Cooper-

pair box (CPB) shunted by a superinductor, forming a superconducting loop. Provided the

offset charge on the CPB island is an odd number of electrons, the qubit potential

corresponds to that of a cos (/2) Josephson element, preserving the parity of fluxons in

the loop via Aharonov-Casher interference. In this regime, the logic-state wavefunctions

reside in disjoint regions of phase space, thereby ensuring the protection against energy

decay. By switching the protection on, we observed a ten-fold increase of the decay time,

reaching up to 100 s. Though the qubit is sensitive to charge noise, the sensitivity is much

reduced in comparison with the charge qubit, and the charge-noise-induced dephasing time

132

of the current device exceeds 1 s. Implementation of the full dephasing protection can be

achieved in the next-generation devices by combining several cos( /2 ) Josephson

elements in a small array.

Ideally the lowest-energy states of a fully symmetric bifluxon are exactly

degenerate at /2 = = 1 2⁄ . However, slight deviations from this point would

open a gap in the spectrum and lead to decoherence. As being discussed in [14], in order

to mitigate dephasing due to both charge and flux noises, one strategy could be to combine

an increase of / with strong reduction of the inductive energy . As we mentioned

in Chapter V, an exponentially small flux dispersion can be achieved in the regime ≫

2 . Fulfilling this condition requires the implementation of an ultrahigh-impedance

superconductor with > 30μH and self-resonance frequencies greater than 1GHz. It is

therefore a primary interest in the future to pursue superinductors with even greater

presented in -based superinductors as in [19].

133

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