Introduction to Quantum Noise, Measurement and Amplification

A.A. Clerk,1 M.H. Devoret,2 S.M. Girvin,3 Florian Marquardt,4 and R.J. Schoelkopf21Department of Physics, McGill University, 3600 rue UniversityMontreal, QC Canada H3A 2T8∗2Department of Applied Physics, Yale UniversityPO Box 208284, New Haven, CT 06520-82843Department of Physics, Yale UniversityPO Box 208120, New Haven, CT 06520-81204Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics,Ludwig-Maximilians-Universitat MunchenTheresienstr. 37, D-80333 Munchen, Germany

(Dated: Oct. 26, 2008)

The topic of quantum noise has become extremely timely due to the rise of quantum informationphysics and the resulting interchange of ideas between the condensed matter and AMO/quantumoptics communities. This review gives a pedagogical introduction to the physics of quantum noiseand its connections to quantum measurement and quantum amplification. After introducingquantum noise spectra and methods for their detection, we describe the basics of weak continuousmeasurements. Particular attention is given to treating the standard quantum limit on linearamplifiers and position detectors using a general linear-response framework. We show how thisapproach relates to the standard Haus-Caves quantum limit for a bosonic amplifier known in quan-tum optics, and illustrate its application for the case of electrical circuits, including mesoscopicdetectors and resonant cavity detectors.

Contents

I. Introduction 2

II. Basics of Classical and Quantum Noise 5A. Classical noise correlators 5B. Square law detectors and classical spectrum

analyzers 6C. Introduction to quantum noise 7

III. Quantum Spectrum Analyzers 8A. Two-level system as a spectrum analyzer 8B. Harmonic oscillator as a spectrum analyzer 11C. Practical quantum spectrum analyzers 13

1. Filter plus diode 132. Filter plus photomultiplier 143. Double sideband heterodyne power spectrum 154. Single sideband heterodyne power spectrum 15

IV. Quantum Measurements 15A. Weak continuous measurements 17B. Measurement with a parametrically coupled resonant

cavity 181. QND measurement of the state of a qubit using a

resonant cavity 212. Quantum limit relation for QND qubit state

detection 223. Measurement of oscillator position using a

resonant cavity 24

V. General Linear Response Theory 28A. Quantum constraints on noise 28

1. Heuristic weak-measurement noise constraints 282. Generic linear-response detector 293. Quantum constraint on noise 304. Role of noise cross-correlations 32

∗Electronic address: clerk@physics.mcgill.ca

B. Quantum limit on QND detection of a qubit 32

VI. Quantum Limit on Linear Amplifiers andPosition Detectors 33A. Preliminaries on amplification 33B. Standard Haus-Caves derivation of the quantum

limit on a bosonic amplifier 34C. Scattering versus op-amp modes of operation 36D. Linear response description of a position detector 37

1. Detector back-action 372. Total output noise 383. Detector power gain 394. Simplifications for a quantum-ideal detector 405. Quantum limit on added noise and noise

temperature 41E. Quantum limit on the noise temperature of a voltage

amplifier 431. Classical description of a voltage amplifier 432. Linear response description 443. Role of noise cross-correlations 46

F. Near quantum-limited mesoscopic detectors 461. dc SQUID amplifiers 462. Quantum point contact detectors 463. Single-electron transistors and resonant-level

detectors 47G. Back-action evasion and noise-free amplification 47

1. Degenerate parametric amplifier 482. Double-sideband cavity detector 493. Stroboscopic measurements 50

VII. Bosonic Scattering Description of a Two-PortAmplifier 50A. Scattering versus op-amp representations 50

1. Scattering representation 512. Op-amp representation 523. Converting between representations 52

B. Minimal two-port scattering amplifier 531. Scattering versus op-amp quantum limit 532. Why is the op-amp quantum limit not achieved? 55

VIII. Reaching the Quantum Limit in Practice 56

arX

iv:0

810.

4729

v1 [

cond

-mat

.mes

-hal

l] 2

6 O

ct 2

008

2

A. Importance of QND measurements 56B. Power matching versus noise matching 56

IX. Conclusions 57

Acknowledgements 58

A. The Wiener-Khinchin Theorem 58

B. Modes, Transmission Lines and ClassicalInput/Output Theory 591. Transmission lines and classical input-output theory 592. Lagrangian, Hamiltonian, and wave modes for a

transmission line 613. Classical statistical mechanics of a transmission line 624. Amplification with a transmission line and a negative

resistance 64

C. Quantum Modes and Noise of a TransmissionLine 651. Quantization of a transmission line 652. Modes and the windowed Fourier transform 663. Quantum noise from a resistor 68

D. Back Action and Input-Output Theory forDriven Damped Cavities 691. Photon shot noise inside a cavity and back action 692. Input-output theory for a driven cavity 713. Quantum limited position measurement using a

cavity detector 754. Back-action free single-quadrature detection 79

E. Information Theory and Measurement Rate 801. Method I 802. Method II 80

F. Quantum Parametric Amplifiers 811. Non-degenerate case 81

a. Gain and added noise 81b. Bandwidth-gain tradeoff 82c. Effective temperature 83

2. Degenerate case 84

G. Number Phase Uncertainty 84

H. Mach Zehnder Interferometer as a QuantumLimited Detector 851. Basic description of the interferometer 862. Fock state input 86

a. Measurement rate 87b. Dephasing rate 87

3. Coherent state input 89a. Measurement rate 89b. Measurement induced dephasing 89

4. Symmetric coupling 91

I. Using feedback to reach the quantum limit 911. Feedback using mirrors 912. Explicit examples 923. Op-amp with negative voltage feedback 93

J. Additional Technical Details 941. Proof of Quantum Noise Constraint 942. Simplifications for a Quantum-Limited Detector 953. Derivation of non-equilibrium Langevin equation 964. Linear-Response Formulas for a Two-Port Bosonic

Amplifier 96a. Input and output impedances 97b. Voltage gain and reverse current gain 98

5. Details for Two-port Bosonic Voltage Amplifier withFeedback 98

References 100

I. INTRODUCTION

The physics of classical noise is a topic which is ex-tremely familiar to both physicists and engineers. In thecase of electrical circuits, we usually think of noise as anunavoidable nuisance, learning early on that any dissipa-tive circuit element (i.e. a resistor) at finite temperaturewill inevitably generate Johnson noise. We also knowthat there are spectrum analyzers that can measure thisnoise: roughly speaking, these spectrum analyzers con-sist of a resonant circuit to select a particular frequencyof interest, followed by an amplifier and square law detec-tor (e.g. a diode rectifier) which measures the intensity(mean square amplitude) of the signal at that frequency.

Recently, several advances have led to a renewed inter-est in the quantum mechanical aspects of noise in meso-scopic electrical circuits, detectors and amplifiers. Onemotivation is that such systems can operate simultane-ously at high frequencies and at low temperatures, en-tering the regime where !ω > kBT . As such, quantumzero-point fluctuations will play a more dominant rolein determining their behaviour than the more familiarthermal fluctuations. Recall that in a classical picture,the intensity of Johnson noise from a resistor vanisheslinearly with temperature because thermal fluctuationsof the charge carriers cease at zero temperature. Oneknows from quantum mechanics, however, that there arequantum fluctuations even at zero temperature, due tozero-point motion. How do we describe such zero-pointfluctuations and their consequences in mesoscopic sys-tems? This question will form a central theme of thisreview.

Note that zero-point motion is a notion from quan-tum mechanics that is frequently misunderstood, witheven the most basic of questions leading to confusion.One might wonder, for example, whether it is physicallypossible to use a spectrum analyzer to detect the zero-point motion. As we will discuss extensively, the answeris quite definitely yes, if we use a quantum system asour spectrum analyzer. Consider for example a hydro-gen atom in the 2p excited state lying 3/4 of a Rydbergabove the 1s ground state. We know that this state isunstable and has a lifetime of only about 1 ns before itdecays to the ground state and emits an ultraviolet pho-ton. This spontaneous decay is a natural consequence ofthe zero-point motion of the electromagnetic fields in thevacuum surrounding the atom. In fact, the rate of spon-taneous decay gives a simple way in which to measurethis zero point motion of the vacuum: if one modifies thezero-point noise of the vacuum by, e.g., placing the atomin a resonant cavity, there is a direct change in the atom’sdecay rate (Haroche and Raimond, 2006; Raimond et al.,2001). We will discuss this more in what follows, as wellas provide a thorough discussion of how various differ-ent mesoscopic systems can act as spectrum analyzers of

3

quantum noise.

A second motivation for interest in the quantum noiseof mesoscopic systems comes from the relation betweenquantum noise and quantum measurement. There ex-ists an ever-increasing number of experiments in meso-scopic electronics where one is forced to think about thequantum mechanics of the detection process, and aboutfundamental quantum limits which constrain the perfor-mance of the detector or amplifier used. Noise plays afundamental role in quantum measurement: quantumnoise from the detector acts back on the system beingmeasured to ensure that information about the variableconjugate to the measured variable is destroyed, thus en-forcing the Heisenberg uncertainty principle. A directconsequence of this quantum back-action is that a “phasepreserving” linear amplifier (i.e. an amplifier which am-plifies both quadratures of the input signal by the sameamount 1) necessarily adds a certain minimum amountof noise to the input signal, even if it is otherwise per-fect (Caves, 1982; Haus and Mullen, 1962). There areseveral motivations for understanding in principle, andrealizing in practice, amplifiers whose noise reaches thisminimum quantum limit. Reaching the quantum limiton continuous position detection has been one of thegoals of many recent experiments on quantum electro-mechanical (Cleland et al., 2002; Etaki et al., 2008;Flowers-Jacobs et al., 2007; Knobel and Cleland, 2003;LaHaye et al., 2004; Naik et al., 2006; Poggio et al.,2008; Regal et al., 2008) and opto-mechanical systems(Arcizet et al., 2006; Schliesser et al., 2008) As we willshow, having a near-quantum limited detector would al-low one to continuously monitor the quantum zero-pointfluctuations of a mechanical resonator. Having a quan-tum limited detector is also necessary for such tasks assingle-spin NMR detection (Rugar et al., 2004), as wellas gravitational wave detection (Abramovici et al., 1992).The topic of quantum-limited detection is also directlyrelevant to recent activity exploring feedback control ofquantum systems (Doherty et al., 2000; Geremia et al.,2004; Korotkov, 2001b; Ruskov and Korotkov, 2002);such schemes necessarily need a close-to-quantum-limiteddetector.

In this introductory article, we will discuss both theaforementioned aspects of quantum noise: the descrip-tion and detection of noise in the quantum regime !ω >kBT , and the relation between quantum noise, quantummeasurement and quantum amplification. While someaspects of these topics have been studied in the quan-tum optics and quantum dissipative systems communi-ties and are the subject of several reviews (Braginskyand Khalili, 1992; Gardiner and Zoller, 2000; Haus, 2000;Weiss, 1999), they are somewhat newer to the mesoscopic

1 In the literature this is often referred to by the unfortunate nameof ‘phase insensitive’ amplifier. We prefer the term ‘phase pre-serving’ to avoid any ambiguity.

physics community; moreover, some of the technical ma-chinery developed in these fields is not directly applica-ble to the study of quantum noise in mesoscopic systems.Note also that there are many other interesting aspectsof quantum noise besides those we discuss in this article;we outline some of these in the last, concluding sectionof this article.

The remainder of this article is organized as follows.We start in Sec. II by providing a short review of thebasic statistical properties of classical noise; we also dis-cuss the key modifications that arise when one includesquantum mechanics. Next, in Sec. III, we discuss the de-tection of quantum noise using either a two-level systemor harmonic oscillator as a quantum spectrum analyzer;we also discuss the physics of other commonly-used noisedetection schemes. In Sec. IV we turn to quantum mea-surements, and give a basic introduction to weak, con-tinuous measurements. To make things concrete, we dis-cuss heuristically measurements of both a qubit and anoscillator using a simple resonant cavity detector, givingan idea of the origin of the quantum limit in each case.Sec. V is devoted to a more rigorous treatment of quan-tum constraints on noise arising from general quantumlinear response theory. In Sec. VI, we give a thoroughdiscussion of quantum limits on amplification and con-tinuous position detection; we also briefly discuss variousmethods for beating the usual quantum limits on addednoise using back-action evasion techniques. We are care-ful to distinguish two very distinct modes of amplifieroperation (the “scattering” versus “op amp” modes); weexpand on this in Sec. VII, where we discuss both modesof operation in a simple two-port bosonic amplifier. Im-portantly, we show that an amplifier can be quantumlimited in one mode of operation, but fail to be quan-tum limited in the other mode of operation. Finally, inSec. VIII, we highlight a number of practical considera-tions that one must keep in mind when trying to performa quantum limited measurement.

In addition to the above, we have supplemented themain text with several pedagogical appendices whichcover some basic background topics (e.g. the Wiener-Khinchin theorem; the Caldeira-Leggett formalism formodeling a dissipative circuit element; input-output the-ory; etc.), as well as more detailed discussions of specificsystems (e.g. degenerate and non-degenerate paramet-ric amplifiers, the Mach-Zehnder interferometer as detec-tor, etc.). The appendices devote particular attention tothe quantum mechanics of transmission lines and drivenelectromagnetic cavities, topics which are especially rel-evant given recent experiments making use of microwavestripline resonators. The discussion of transmission linesalso gives a very instructive example of many of the moregeneral principles discussed in the main text. Finally,note that while this article is a review, there is consider-able new material presented, especially in our discussionof quantum amplification.

4

TABLE I: Table of symbols and main results.

Symbol Definition / Result

General Definitionsf [ω] Fourier transform of the function or operator f(t), defined via f [ω] =

R∞−∞ dtf(t)eiωt

(Note that for operators, we use the convention f†[ω] =R∞−∞ dtf†(t)eiωt, implying f†[ω] =

“f [−ω]

Ӡ)

SFF [ω] Classical noise spectral density or power spectrum: SFF [ω] =R +∞−∞ dt eiωt〈F (t)F (0)〉

SFF [ω] Quantum noise spectral density: SFF [ω] =R +∞−∞ dt eiωt〈F (t)F (0)〉

SFF [ω] Symmetrized quantum noise spectral density SFF [ω] = 12 (SFF [ω] + SFF [−ω]) = 1

2

R +∞−∞ dt eiωt〈F (t), F (0)〉

χAB(t) General linear response susceptibility describing the response of A to a perturbation which couples to B;in the quantum case, given by the Kubo formula χAB(t) = − i

! θ(t)〈[A(t), B(0)]〉 [Eq. (3.30)]A Coupling constant (dimensionless) between measured system and detector/amplifier,

e.g. V = AF (t)σx, V = AxF , or V = Aσz a†aM, Ω Mass and angular frequency of a mechanical harmonic oscillator.

xZPF Zero point uncertainty of a mechanical oscillator, xZPF =q

!2MΩ .

γ0 Intrinsic damping rate of a mechanical oscillator due to coupling to a bath via V = AxF :

γ0 = A2

2M!Ω (SFF [Ω]− SFF [−Ω]) [Eq. (3.25)]ωc Resonant frequency of a cavityκ, Qc Damping, quality factor of a cavity

Sec. III Quantum spectrum analyzersTeff [ω] Effective temperature at a frequency ω for a given quantum noise spectrum, defined via

SF F [ω]SF F [−ω] = exp

“!ω

kBTeff [ω]

”[Eq. (3.21)]

Fluctuation-dissipation theorem relating the symmetrized noise spectrum to the dissipative partfor an equilibrium bath: SFF [ω] = 1

2 coth( !ω2kBT )(SFF [ω]− SFF [−ω]) [Eq. (3.34)]

Sec. IV Quantum MeasurementsNumber-phase uncertainty relation for a coherent state:

∆N∆θ ≥ 12 [Eq. (4.8), (G12)]

N Photon number flux of a coherent beamδθ Imprecision noise in the measurement of the phase of a coherent beam

Fundamental noise constraint for an ideal coherent beam:SNNSθθ = 1

4 [Eq. (4.16), (G21)]S0

xx(ω) symmetrized spectral density of zero-point position fluctuations of a damped harmonic oscillatorSxx,tot(ω) total output noise spectral density (symmetrized) of a linear position detector, referred back to the oscillatorSxx,add(ω) added noise spectral density (symmetrized) of a linear position detector, referred back to the oscillator

Sec. V: General linear response theoryx Input signalF Fluctuating force from the detector, coupling to x via V = AxFI Detector output signal

General quantum constraint on the detector output noise, backaction noise and gain:

SII [ω]SFF [ω]−˛SIF [ω]

˛2 ≥˛˛ !χIF [ω]

2

˛˛2 “

1 + ∆h

SIF [ω]

!λ[ω]/2

i”[Eq. (5.11)]

where χIF [ω] ≡ χIF [ω]− [χFI [ω]]∗ and ∆[z] = (˛1 + z2

˛−

`1 + |z|2

´)/2.

[Note: 1 + ∆[z] ≥ 0 and ∆ = 0 in most cases of relevance, see discussion around Eq. (5.16)]α Complex proportionality constant characterizing a quantum-ideal detector:

|α|2 = SII/SFF and sin (arg α[ω]) = !|λ[ω]|/2√SII [ω]SF F [ω]

[Eqs. (5.17,J15)]

Γmeas Measurement rate (for a QND qubit measurement) [Eq. 5.23]Γϕ Dephasing rate (due to measurement back-action) [Eqs. (4.44),(5.18)]

Constraint on weak, continuous QND qubit state detection :η = Γmeas

Γϕ≤ 1 [Eq. (5.24)]

Sec. VI: Quantum Limit on Linear Amplifiers and Position DetectorsG Photon number (power) gain, e.g. in Eq. (6.7b)

Input-output relation for a bosonic scattering amplifier: b† =√

Ga† + F† [Eq.(6.7b)](∆a)2 Symmetrized field operator uncertainty for the scattering description of a bosonic amplifier:

(∆a)2 ≡ 12

˙a, a†

¸− |〈a〉|2

5

TABLE I: Table of symbols and main results.

Symbol Definition / Result

Standard quantum limit for the noise added by a phase-preserving bosonic scattering amplifierin the high-gain limit, G ( 1, where 〈(∆a)2〉ZPF = 1

2 :(∆b)2

G ≥ (∆a)2 + 12 [Eq. (6.10)]

GP [ω] Dimensionless power gain of a linear position detector or voltage amplifier(maximum ratio of the power delivered by the detector output to a load, vs. the power fed into signal source):

GP [ω] = |χIF [ω]|24Im χF F [ω]·Im χII [ω]] [Eq. (6.25)]

For a quantum-ideal detector, in the high-gain limit: GP )h

Im α|α|

4kBTeff!ω

i2[Eq. (6.29)]

Sxx,eq[ω, T ] Intrinsic equilibrium noise Sxx,eq[ω, T ] = ! coth“

!ω2kBT

”[−Im χxx[ω]] [Eq. (6.31)]

Aopt Optimal coupling strength of a linear position detector which minimizes the added noise at frequency ω:

A4opt[ω] = SII [ω]

|λ[ω]χxx[ω]|2SF F [ω][Eq. (6.36)]

γ[Aopt] Detector-induced damping of a quantum-limited linear position detector at optimal coupling, fulfillsγ[Aopt]

γ0+γ[Aopt]=

˛Im α

α

˛1√

GP [Ω]= !Ω

4kBTeff* 1 [Eq. (6.41)]

Standard quantum limit for the added noise spectral density of a linear position detector (valid at each frequency ω):Sxx,add[ω] ≥ limT→0 Sxx,eq[w, T ] [Eq. (6.34)]

Effective increase in oscillator temperature due to coupling to the detector back action,for an ideal detector, with !Ω/kB * Tbath * Teff :Tosc ≡ γ·Teff+γ0·Tbath

γ+γ0→ !Ω

4kB+ Tbath [Eq. (6.42)]

Zin, Zout Input and output impedances of a linear voltage amplifierZs Impedance of signal source attached to input of a voltage amplifierλV Voltage gain of a linear voltage amplifierV (t) Voltage noise of a linear voltage amplifier

(Proportional to the intrinsic output noise of the generic linear-response detector [Eq. (6.53)] )I(t) Current noise of a linear voltage amplifier

(Related to the back-action force noise of the generic linear-response detector [Eqs. (6.52)] )TN Noise temperature of an amplifier [defined in Eq. (6.46)]ZN Noise impedance of a linear voltage amplifier [Eq. 6.49)]

Standard quantum limit on the noise temperature of a linear voltage amplifier:kBTN[ω] ≥ !ω

2 [Eq.(6.61)]

Sec. VII: Bosonic Scattering Description of a Two-Port AmplifierVa(Vb) Voltage at the input (output) of the amplifier

Relation to bosonic mode operators: Eq. (7.2a)Ia(Ib) Current drawn at the input (leaving the output) of the amplifier

Relation to bosonic mode operators: Eq. (7.2b)λ′I Reverse current gain of the amplifiers[ω] Input-output 2× 2 scattering matrix of the amplifier [Eq. (7.3)]

Relation to op-amp parameters λV , λ′I , Zin, Zout: Eqs. (7.7)ˆV ( ˆI) Voltage (current) noise operators of the amplifierFa[ω], Fb[ω] Input (output) port noise operators in the scattering description [Eq. (7.3)]

Relation to op-amp noise operators ˆV, ˆI: Eq. (7.9)

II. BASICS OF CLASSICAL AND QUANTUM NOISE

A. Classical noise correlators

Consider a classical random voltage signal V (t). Thesignal is characterized by zero mean 〈V (t)〉 = 0, andautocorrelation function

GV V (t− t′) = 〈V (t)V (t′)〉 (2.1)

whose sign and magnitude tells us whether the voltagefluctuations at time t and time t′ are correlated, anti-correlated or statistically independent. We assume that

the noise process is stationary (i.e., the statistical proper-ties are time translation invariant) so that GV V dependsonly on the time difference. If V (t) is Gaussian dis-tributed, then the mean and autocorrelation completelyspecify the statistical properties and the probability dis-tribution. We will assume here that the noise is due tothe sum of a very large number of fluctuating chargesso that by the central limit theorem, it is Gaussian dis-tributed. We also assume that GV V decays (sufficientlyrapidly) to zero on some characteristic correlation timescale τc which is finite.

6

The spectral density of the noise as measured by aspectrum analyzer is a measure of the intensity of thesignal at different frequencies. In order to understandthe spectral density of a random signal, it is useful todefine its ‘windowed’ Fourier transform as follows:

VT [ω] =1√T

∫ +T/2

−T/2dt eiωtV (t), (2.2)

where T is the sampling time. In the limit T % τc theintegral is a sum of a large number N ≈ T

τcof random

uncorrelated terms. We can think of the value of theintegral as the end point of a random walk in the complexplane which starts at the origin. Because the distancetraveled will scale with

√T , our choice of normalization

makes the statistical properties of V [ω] independent ofthe sampling time T (for sufficiently large T ). Noticethat VT [ω] has the peculiar units of volts

√secs which is

usually denoted volts/√

Hz.The spectral density (or ‘power spectrum’) of the noise

is defined to be the ensemble averaged quantity

SV V [ω] ≡ limT→∞

〈|VT [ω]|2〉 = limT→∞

〈VT [ω]VT [−ω]〉 (2.3)

The second equality follows from the fact that the v(t)is real valued. The Wiener-Khinchin theorem (derived inAppendix A) tells us that the spectral density is equal tothe Fourier transform of the autocorrelation function

SV V [ω] =∫ +∞

−∞dt eiωtGV V (t). (2.4)

The inverse transform relates the autocorrelation func-tion to the power spectrum

GV V (t) =∫ +∞

−∞

dω

2πe−iωtSV V [ω]. (2.5)

We thus see that a short auto-correlation time impliesa spectral density which is non-zero over a wide range offrequencies. In the limit of ‘white noise’

GV V (t) = σ2δ(t) (2.6)

the spectrum is flat (independent of frequency)

SV V [ω] = σ2 (2.7)

In the opposite limit of a long autocorrelation time, thesignal is changing slowly so it can only be made up outof a narrow range of frequencies (not necessarily centeredon zero).

Because V (t) is a real-valued classical variable, it natu-rally follows that GV V (t) is always real. Since V (t) is nota quantum operator, it commutes with its value at othertimes and thus, 〈V (t)V (t′)〉 = 〈V (t′)V (t)〉. From this itfollows that GV V (t) is always symmetric in time and thepower spectrum is always symmetric in frequency

SV V [ω] = SV V [−ω]. (2.8)

As a prototypical example of these ideas, let us con-sider a simple harmonic oscillator of mass M and fre-quency Ω. The oscillator is maintained in equilibriumwith a large heat bath at temperature T via some in-finitesimal coupling which we will ignore in consideringthe dynamics. The solution of Hamilton’s equations ofmotion are

x(t) = x(0) cos(Ωt) + p(0)1

MΩsin(Ωt)

p(t) = p(0) cos(Ωt)− x(0)MΩ2 sin(Ωt), (2.9)

where x(0) and p(0) are the (random) values of the po-sition and momentum at time t = 0. It follows that theposition autocorrelation function is

Gxx(t) = 〈x(t)x(0)〉 (2.10)

= 〈x(0)x(0)〉 cos(Ωt) + 〈p(0)x(0)〉 1MΩ

sin(Ωt).

Classically in equilibrium there are no correlations be-tween position and momentum. Hence the second termvanishes. Using the equipartition theorem 1

2MΩ2〈x2〉 =12kBT , we arrive at

Gxx(t) =kBT

MΩ2cos(Ωt) (2.11)

which leads to the spectral density

Sxx[ω] = πkBT

MΩ2[δ(ω − Ω) + δ(ω + Ω)] (2.12)

which is indeed symmetric in frequency.

B. Square law detectors and classical spectrum analyzers

Now that we understand the basics of classical noise,we can consider how one experimentally measures a clas-sical noise spectral density. With modern high speeddigital sampling techniques it is perfectly feasible to di-rectly measure the random noise signal as a function oftime and then directly compute the autocorrelation func-tion in Eq. (2.1). This is typically done by first per-forming an analog-to-digital conversion of the noise sig-nal, and then numerically computing the autocorrelationfunction. One can then use Eq. (2.4) to calculate thenoise spectral density via a numerical Fourier transform.Note that while Eq. (2.4) seems to require an ensembleaverage, in practice this is not explicitly done. Instead,one uses a sufficiently long averaging time T (i.e. muchlonger than the correlation time of the noise) such thata single time-average is equivalent to an ensemble aver-age. This approach of measuring a noise spectral densitydirectly from its autocorrelation function is most appro-priate for signals at RF frequencies well below 1 MHz.

For microwave signals with frequencies well above 1GHz, a very different approach is usually taken. Here, thestandard route to obtain a noise spectral density involves

7

first shifting the signal to a lower intermediate frequencyvia a technique known as heterodyning (we discuss thismore in Sec. III.C.3). This intermediate-frequency signalis then sent to a filter which selects a narrow frequencyrange of interest, the so-called ‘resolution bandwidth’.Finally, this filtered signal is sent to a square-law detector(e.g. a diode), and the resulting output is averaged overa certain time-interval (the inverse of the so-called ‘videobandwidth’). It is this final output which is then takento be a measure of the noise spectral density.

It helps to put the above into equations. Ignoring forsimplicity the initial heterodyning step, let

Vf [ω] = f [ω]V [ω] (2.13)

be the voltage at the output of the filter and the inputof the square law detector. Here, f [ω] is the (ampli-tude) transmission coefficient of the filter and V [ω] is theFourier transform of the noisy signal we are measuring.From Eq. (2.5) it follows that the output of the squarelaw detector is proportional to

〈I〉 =∫ +∞

−∞

dω

2π|f [ω]|2SV V [ω]. (2.14)

Approximating the narrow band filter centered on fre-quency ±ω0 as2

|f [ω]|2 = δ(ω − ω0) + δ(ω + ω0) (2.15)

we obtain

〈I〉 = SV V (−ω0) + SV V (ω0) (2.16)

showing as expected that the classical square law detectormeasures the symmetrized noise power.

We thus have two very different basic approaches forthe measurement of classical noise spectral densities: forlow RF frequencies, one can directly measure the noiseautocorrelation, whereas for high microwave frequencies,one uses a filter and a square law detector. For noisesignals in intermediate frequency ranges, a combinationof different methods is generally used. The whole storybecomes even more complicated, as at very high frequen-cies (e.g. in the far infrared), devices such as the so-called ‘Fourier Transform spectrometer’ are in fact basedon a direct measurement of the equivalent of an auto-correlation function of the signal. In the infrared, visibleand ultraviolet, noise spectrometers use gratings followedby a slit acting as a filter.

2 A linear passive filter performs a convolution Vout(t) =R +∞−∞ dt′ F (t − t′)Vin(t′) where F is a real-valued (and causal)

function. Hence it follows that f [ω], which is the Fourier trans-form of F , obeys f [−ω] = f∗[ω] and hence |f [ω]|2 is symmetricin frequency.

C. Introduction to quantum noise

Based on our review of classical noise, one expects thatthe study of quantum noise involves spectral densities ofthe form

Sxx[ω] =∫ +∞

−∞dt eiωt〈x(t)x(0)〉 (2.17)

where x is a quantum operator (in the Heisenberg repre-sentation) and the angular brackets indicate the quantumstatistical average evaluated using the quantum densitymatrix. Note that we will use S[ω] throughout this re-view to denote the spectral density of a classical noise,while S[ω] will denote a quantum noise spectral density.As a simple example of the important differences fromthe classical limit, consider the same harmonic oscillatorproblem as above. The solutions of the Heisenberg equa-tions of motion are the same as for the classical case butwith the initial position and momentum replaced by thecorresponding quantum operators:

x(t) = x(0) cos(Ωt) + p(0)1

MΩsin(Ωt)

p(t) = p(0) cos(Ωt)− x(0)MΩ sin(Ωt). (2.18)

Just as before, it follows that the position autocorrelationfunction is

Gxx(t) = 〈x(t)x(0)〉 (2.19)

= 〈x(0)x(0)〉 cos(Ωt) + 〈p(0)x(0)〉 1MΩ

sin(Ωt).

Classically the second term on the RHS vanishes becausein thermal equilibrium the position and momentum areuncorrelated random variables. As we will see shortly be-low for the quantum case, the symmetrized (sometimescalled the ‘classical’) correlator vanishes in thermal equi-librium

〈xp + px〉 = 0, (2.20)

just as it does classically. Notice however that

[x(0), p(0)] = i! (2.21)

which implies that there must inevitably be some corre-lations between position and momentum in the quantumcase since 〈x(0)p(0)〉 − 〈p(0)x(0)〉 = i!.

We can evaluate these correlations using the represen-tation of the operators in terms of the harmonic oscillatorladder operators

x = xZPF(a† + a)

p =i!

2xZPF(a† − a) (2.22)

where xZPF is the RMS zero-point uncertainty of x in thequantum ground state

x2ZPF ≡ 〈0|x2|0〉 =

!2MΩ

. (2.23)

8

-5 0 5 10-10

0

2

4

6

8

10

> 0absorptionby reservoir

< 0emission

by reservoir

h /kBT

SVV [ ]/2RkBT

FIG. 1 Quantum noise spectral density of voltage fluctuationsacross a resistor (resistance R) as a function of frequency atzero temperature (dashed line) and finite temperature (solidline).

For the case of thermal equilibrium, we obtain

〈p(0)x(0)〉 = −i!2

〈x(0)p(0)〉 = +i!2

(2.24)

Not only are the position and momentum correlated, buttheir correlator is imaginary!3 This means that, despitethe fact that the position is an hermitian observable withreal eigenvalues, its autocorrelation function is complexand given from Eq. (2.20) by:

Gxx(t) = x2ZPF

nB(!Ω)e+iΩt + [nB(!Ω) + 1]e−iΩt

,

(2.25)where nB is the Bose-Einstein occupation factor. Thecomplex nature of the autocorrelation function followsfrom the fact that the operator x does not commute withitself at different times.

Because the correlator is complex it follows that thespectral density is no longer symmetric in frequency

Sxx[ω] = 2πx2ZPF (2.26)

× nB(!Ω)δ(ω + Ω) + [nB(!Ω) + 1]δ(ω − Ω)

The Bose-Einstein factors suggest that the positive fre-quency part of the spectral density has to do with stimu-lated emission of energy into the oscillator and the nega-tive frequency part of the spectral density has to do withemission of energy by the oscillator. That is, the posi-tive frequency part of the spectral density is a measureof the ability of the oscillator to absorb energy, while thenegative frequency part is a measure of the ability of the

3 Notice that this occurs because the product of two non-commuting hermitian operators is not itself an hermitian op-erator.

oscillator to emit energy. Fig. 1 illustrates this for thecase of the voltage noise spectral density of a resistor(see Appendix C.3 for more details).

The qualitative picture described above will be con-firmed when we consider in detail how it is that quantumspectrum analyzers work. Given that x(t) and x(0) donot commute, it is not possible to experimentally mea-sure the complex autocorrelation function Gxx(t) as theexpectation value of some Hermitian observable. Nev-ertheless we will see below that it is possible to deter-mine its Fourier transform Sxx[ω] for both positive andnegative frequencies by means of certain non-equilibriummeasurements (Aguado and Kouwenhoven, 2000; Gavishet al., 2000; Lesovik and Loosen, 1997; Schoelkopf et al.,2003).

In closing we note that in the high temperature limitkBT % !Ω we have

nB(!Ω) ∼ nB(!Ω) + 1 ∼ kBT

!Ω. (2.27)

Substitution of this approximation into Eq. (2.27) repro-duces the classical expression in Eq. (2.12).

The results presented above can be extended to thecase of a bath of many harmonic oscillators. As describedin Appendix C a resistor can be modeled as an infiniteset of harmonic oscillators and from this model the John-son/Nyquist noise of a resistor can be derived.

III. QUANTUM SPECTRUM ANALYZERS

A. Two-level system as a spectrum analyzer

Consider a quantum system (atom or electrical circuit)which has its two lowest energy levels ε0 and ε1 separatedby energy E01 = !ω01. We suppose for simplicity thatall the other levels are far away in energy and can beignored. The resulting quantum two-level can be used asa spectrometer of quantum noise (Aguado and Kouwen-hoven, 2000; Schoelkopf et al., 2003); we discuss this indetail in what follows.

The states of any two-level system (here after abbrevi-ated TLS) can be mapped onto the states of a fictitiousspin-1/2 particle since such a spin also has only two statesin its Hilbert space. With spin down representing theground state (|g〉) and spin up representing the excitedstate (|e〉) [Made sign in front of σz consistent with latersections], the Hamiltonian is (taking the zero of energyto be the center of gravity of the two levels)

H0 =!ω01

2σz. (3.1)

In keeping with our discussion in the introduction, ourgoal is to now see how the rate of ‘spin-flip’ transitionsinduced by an external noise source can be used to an-alyze the spectrum of that noise. Suppose for examplethat there is a noise source with amplitude f(t) which

9

can cause transitions via the perturbation4

V = AF (t)σx, (3.2)

where A is a coupling constant. The variable F (t) rep-resents the noise source. We can temporarily pretendthat F is a classical variable, although its quantum op-erator properties will be forced upon us very soon. Fornow, only our two-level spectrum analyzer will be treatedquantum mechanically.

We assume that the coupling A is under our controland can be made small enough that the noise can betreated in lowest order perturbation theory. We take thestate of the two-level system to be

|ψ(t)〉 =(

αg(t)αe(t)

). (3.3)

In the interaction representation, first-order time-dependent perturbation theory gives

|ψI(t)〉 = |ψ(0)〉 − i

!

∫ t

0dτ V (τ)|ψ(0)〉. (3.4)

If we initially prepare the two-level system in its groundstate, the amplitude to find it in its excited state at timet is from Eq. (3.4)

αe = − iA

!

∫ t

0dτ 〈e|σx(τ)|g〉F (τ),

= − iA

!

∫ t

0dτ eiω01τF (τ). (3.5)

Since the integrand in Eq. (3.5) is random, αe is a sumof a large number of random terms; i.e. its value is theendpoint of a random walk in the complex plane (as dis-cussed above in defining the spectral density of classicalnoise). As a result, for times exceeding the autocorre-lation time τc of the noise, the integral will not growlinearly with time but rather only as the square root oftime, as expected for a random walk. We can now com-pute the probability

pe(t) ≡ |αe|2 =A2

!2

∫ t

0

∫ t

0dτ1dτ2 e−iω01(τ1−τ2)F (τ1)F (τ2)

(3.6)which we expect to grow quadratically for short timest < τc, but linearly for long times t > τc. Ensembleaveraging the probability over the random noise yields

pe(t) =A2

!2

∫ t

0

∫ t

0dτ1dτ2 e−iω01(τ1−τ2) 〈F (τ1)F (τ2)〉

(3.7)

4 The most general perturbation would also couple to σy but weassume that (as is often, though not always, the case) a spincoordinate system can be chosen so that the perturbation onlycouples to σx. Noise coupled to σz commutes with the Hamil-tonian but is nevertheless important in dephasing coherent su-perpositions of the two states. We will discuss such processeslater.

Introducing the noise spectral density

SFF (ω) =∫ +∞

−∞dτ eiωτ 〈F (τ)F (0)〉, (3.8)

and utilizing the Fourier transform defined in Eq. (2.2)and the Wiener-Khinchin theorem from Appendix A, wefind that the probability to be in the excited state indeedincreases linearly with time at long times,5

pe(t) = tA2

!2SFF (−ω01) (3.9)

The time derivative of the probability gives the transitionrate from ground to excited states

Γ↑ =A2

!2SFF (−ω01) (3.10)

Note that we are taking in this last expression the spec-tral density on the negative frequency side. If F were astrictly classical noise source, 〈F (τ)F (0)〉 would be real,and SFF (−ω01) = SFF (+ω01). However, because as wediscuss below F is actually an operator acting on the en-vironmental degrees of freedom,

[F (τ), F (0)

]*= 0 and

SFF (−ω01) *= SFF (+ω01).Another possible experiment is to prepare the two-level

system in its excited state and look at the rate of decayinto the ground state. The algebra is identical to thatabove except that the sign of the frequency is reversed:

Γ↓ =A2

!2SFF (+ω01). (3.11)

We now see that our two-level system does indeed act as aquantum spectrum analyzer for the noise. Operationally,we prepare the system either in its ground state or in itsexcited state, weakly couple it to the noise source, andafter an appropriate interval of time (satisfying the aboveinequalities) simply measure whether the system is nowin its excited state or ground state. Repeating this pro-tocol over and over again, we can find the probability ofmaking a transition, and thereby infer the rate and hencethe noise spectral density at positive and negative fre-quencies. Note that in contrast with a classical spectrumanalyzer, we can separate the noise spectral density atpositive and negative frequencies from each other sincewe can separately measure the downward and upwardtransition rates. Negative frequency noise transfers en-ergy from the noise source to the spectrometer. That is,

5 Note that for very long times, where there is a significant de-pletion of the probability of being in the initial state, first-orderperturbation theory becomes invalid. However, for sufficientlysmall A, there is a wide range of times τc " t " 1/Γ for whichEq. 3.9 is valid. Eqs. 3.10 and 3.11 then yield well-defined rateswhich can be used in a master equation to describe the full dy-namics including long times.

10

it represents energy emitted by the noise source. Positivefrequency noise transfers energy from the spectrometer tothe noise source.6 Naively one imagines that a spectrom-eters measures the noise spectrum by extracting a smallamount of the signal energy from the noise source andanalyzes it. This is not the case however. There mustbe energy flowing in both directions if the noise is to befully characterized.

We now rigorously treat the quantity F (τ) as a quan-tum Heisenberg operator which acts in the Hilbert spaceof the noise source. The previous derivation is unchanged(the ordering of F (τ1)F (τ2) having been chosen cor-rectly in anticipation of the quantum treatment), andEqs. (3.10,3.11) are still valid provided that we interpretthe angular brackets in Eq. (3.7,3.8) as representing aquantum expectation value (evaluated in the absence ofthe coupling to the spectrometer):

SFF (ω) =∫ +∞

−∞dτ eiωτ

∑

α,γ

ραα 〈α|F (τ)|γ〉〈γ|F (0)|α〉.

(3.12)Here, we have assumed a stationary situation, wherethe density matrix ρ of the noise source is diagonal inthe energy eigenbasis (in the absence of the coupling tothe spectrometer). However, we do not necessarily as-sume that it is given by the equilibrium expression. Thisyields the standard quantum mechanical expression forthe spectral density:

SFF (ω) =∫ +∞

−∞dτ eiωτ

∑

α,γ

ραα ei! (εα−εγ)τ |〈α|F |γ〉|2

= 2π!∑

α,γ

ραα |〈α|F |γ〉|2δ(εγ − εα − !ω).(3.13)

Substituting this expression into Eqs. (3.10,3.11), we de-rive the familiar Fermi Golden Rule expressions for thetwo transition rates.

In standard courses, one is not normally taught thatthe transition rate of a discrete state into a continuumas described by Fermi’s Golden Rule can (and indeedshould!) be viewed as resulting from the continuum act-ing as a quantum noise source which causes the am-plitudes of the different components of the wave func-tion to undergo random walks. The derivation presented

6 Unfortunately, there are several conventions in existence fordescribing the noise spectral density. It is common in engi-neering contexts to use the phrase ‘spectral density’ to meanSFF (+ω) + SFF [−ω]. This is convenient in classical problemswhere the two are equal. In quantum contexts, one sometimessees the asymmetric part of the noise SFF (+ω) − SFF (−ω) re-ferred to as the ‘quantum noise.’ We feel it is simpler and clearerto discuss the spectral density for positive and negative frequen-cies separately, since they each have simple physical interpreta-tions and directly relate to measurable quantities. This conven-tion is especially useful in non-equilibrium situations where thereis no simple relation between the spectral densities at positiveand negative frequencies.

here hopefully provides a motivation for this interpreta-tion. In particular, thinking of the perturbation (i.e. thecoupling to the continuum) as quantum noise with asmall but finite autocorrelation time (inversely relatedto the bandwidth of the continuum) neatly explains whythe transition probability increases quadratically for veryshort times, but linearly for very long times. We find thispicture to be considerably superior to the tortured argu-ments about time scales and order of limits invoked inthe usual derivation of Fermi’s Golden Rule.

It it is important to keep in mind that our expressionsfor the transition rates are only valid if the autocorrela-tion time of our noise is much shorter that the typicaltime we are interested in; this typical time is simply theinverse of the transition rate. The requirement of a shortautocorrelation time in turn implies that our noise sourcemust have a large bandwidth (i.e. there must be largenumber of available photon frequencies in the vacuum)and must not be coupled too strongly to our system. Thisis true despite the fact that our final expressions for thetransition rates only depend on the spectral density atthe transition frequency (a consequence of energy con-servation).

One standard model for the continuum is an infinitecollection of harmonic oscillators. The electromagneticcontinuum in the hydrogen atom case mentioned above isa prototypical example. The vacuum electric field noisecoupling to the hydrogen atom has an extremely shortautocorrelation time because the range of mode frequen-cies ωα (over which the dipole matrix element couplingthe atom to the mode electric field +Eα is significant) isextremely large, ranging from many times smaller thanthe transition frequency to many times larger. Thus, theautocorrelation time of the vacuum electric field noise isconsiderably less than 10−15s, whereas the decay time ofthe hydrogen 2p state is about 10−9s. Hence the inequal-ities needed for the validity of our expressions are veryeasily satisfied.

To close our discussion of the TLS quantum spectrom-eter, let us consider the special case where our noisesource is in thermodynamic equilibrium. In this case, thetransition rates of the TLS must obey detailed balanceΓ↓/Γ↑ = eβ!ω01 in order to give the correct equilibriumoccupancies of the two states of the spectrometer. Thisin turn implies that the spectral densities obey 7

SFF (+ω01) = eβ!ω01SFF (−ω01). (3.14)

Without the crucial distinction between positive and neg-ative frequencies, and the resulting difference in rates,one would always find that our two level system is com-pletely unpolarized (i.e. there is an equal probability

7 One can of course prove this detailed balance relation rigorouslyby calculating the quantum noise for a noise source in a thermalstate. Consider for example Eq. (2.25) and use the identity (1 +nB)/nB = exp β!ω.

11

to be in either of the two states). Equivalently, a clas-sical noise source (where SFF (ω01) = SFF (−ω01)) cor-responds to a noise source whose temperature is muchlarger than !ω01/kB.

The more general case is where our noise source is notin equilibrium; in this case, no general detailed balancerelation holds. However, if we are concerned only witha single particular frequency (given say by the transitionfrequency of our two-level system detector), then it is al-ways possible to define an ‘effective temperature’ for thenoise using Eq. (3.14). In NMR language, this effectivetemperature for the noise will simply be the ‘spin tem-perature’ of our TLS spectrometer once it reaches steadystate after being coupled to the noise source. We willhave more to say about this effective temperature in thesections that follow.

As an aside, we note that a system designed to de-tect the arrival (or emission) of a photon (say) with verygood time resolution must necessarily have very poorphase coherence. This can be achieved with a TLS whosestate is continuously and strongly measured (Schusteret al., 2005). Because of the strong measurement, we willknow immediately when the state of the two-level systemchanges due to absorption or emission of a photon. Onthe other hand, the back-action disturbance of the TLSby the measurement (to be discussed in the next sec-tion) will cause the two level system to have a very shortphase coherence time. As a result, the TLS will have avery broad line width and hence the poor frequency res-olution that must necessarily accompany good temporalresolution.

Finally, we also note that a particular realization ofa TLS quantum noise spectrometer involving a doublequantum dot was discussed by Aguado and Kouwenhoven(2000); here, absorption (but not emission) of energyby the double dot from a noise can lead to a measur-able inelastic current. This system was recently real-ized in experiment (Gustavsson et al., 2007; Onac et al.,2006a). We also note that a more detailed discussionof TLS quantum noise spectrometers can be found inSchoelkopf et al. (2003); this work includes a discussionof various different quantum noise sources which can beimportant in mesoscopic electronic systems, including theback-action quantum noise generated by a single-electrontransistor electrometer.

B. Harmonic oscillator as a spectrum analyzer

Let us now consider what happens when we weaklycouple a quantum harmonic oscillator to our quantumnoise source. Unlike the TLS of the previous subsection,the oscillator has an infinite number of states. However,similar to the TLS, the oscillator still has a well-definedfrequency; as a result, we will see that it too may beused as a spectrum analyzer of quantum noise. Moreover,this example will provide us with a new way to viewquantum noise, and will demonstrate how the concept of

an “effective temperature” of an out-of-equilibrium noisesource (introduced above) can be extremely useful.

Our harmonic oscillator is described by the usualHamiltonian:

H0 =p2

2M+

MΩ2x2

2= !Ω

(c†c +

12

)(3.15)

where a is the lowering operator for the oscillator. Ournoise source acts as a weak, fluctuating force on the os-cillator:

V = AxF = A[xZPF(c + c†)

]F (3.16)

where F is the operator describing the fluctuating noise,and A is again a coupling constant. In complete analogyto the previous subsection, noise in F at the oscillatorfrequency Ω can cause transitions between its eigenstates.We again assume both that A is small, and that ournoise source has a short autocorrelation time, so we mayagain use perturbation theory to derive rates for thesetransitions. There is a rate for increasing the numberof quanta in the oscillator by one, taking a state |n〉 to|n + 1〉:

Γn→n+1 =A2

!2

[(n + 1)x2

ZPF

]SFF [−Ω] ≡ (n + 1)Γ↑

(3.17)As expected, this rate involves the noise at −Ω, as energyis being absorbed from the noise source. The factor inbrackets is just the matrix element for the transition,i.e. |〈n+1|x|n〉|2. Similarly, there is a rate for decreasingthe number of quanta in the oscillator by one:

Γn→n−1 =A2

!2

(nx2

ZPF

)SFF [Ω] ≡ nΓ↓ (3.18)

This rate involves the noise at +Ω, as energy is beingemitted to the noise source.

Given these rates, we may immediately write down asimple master equation which governs the rate of changeof the probability pn(t) that there are n quanta in theoscillator:

d

dtpn = [nΓ↑pn−1 + (n + 1)Γ↓pn+1]

− [nΓ↓ + (n + 1)Γ↑] pn (3.19)

The first two terms describe transitions into the state |n〉from the states |n + 1〉 and |n − 1〉, and hence increasepn. In contrast, the last two terms describe transitionsout of the state |n〉 to the states |n + 1〉 and |n− 1〉, andhence decrease pn.

It is natural to now ask what the stationary state ofthe oscillator is. By solving Eq. (3.19) for d

dtpn = 0, wefind that:

pn = e−n!Ω/(kBTeff )(1− e−!Ω/(kBTeff )

)(3.20)

where

kBTeff [Ω] ≡ !Ω

log[

Γ↓Γ↑

] =!Ω

log[

SF F [Ω]SF F [−Ω]

] (3.21)

12

Eq. (3.20) describes a thermal equilibrium distributionof the oscillator, with an effective oscillator temperatureTeff [Ω] determined by the quantum noise spectrum ofF . This is the same effective temperature that emergedin our discussion of the TLS spectrum analyzer. If ournoise source is in equilibrium, we have seen in the pre-vious subsection that SFF [ω] must obey a condition ofdetailed balance (Eq. (3.14)); in this case, Teff coincideswith the physical temperature of our noise source. Inthe more general case where the noise source is out-of-equilibrium, Teff only serves to characterize the asymme-try of the quantum noise, and will vary with frequency8. Nonetheless, as far as the oscillator is concerned, Teff

acts as a real temperature, and determines the form ofits distribution.

We can learn more about the quantum noise spectrumof F by also looking at the dynamics of the oscillator, asopposed to just its stationary state. In particular, as theaverage energy 〈E〉 of the oscillator is just given by:

〈E(t)〉 =∞∑

n=0

!Ω(

n +12

)pn(t), (3.22)

we can use the master equation Eq. (3.19) to derive anequation for its time dependence. One finds straightfor-wardly:

d

dt〈E〉 = P − γ〈E〉 (3.23)

where

P =!Ω2

(Γ↑ + Γ↓) =A2

4M[SFF [Ω] + SFF [−Ω]](3.24)

γ = Γ↓ − Γ↑ =A2x2

ZPF

!2[SFF [Ω]− SFF [−Ω]](3.25)

The two terms in Eq. (3.23) describe, respectively,heating and damping of the oscillator by the noise source.The heating effect of the noise is completely analogousto what happens classically: a random force causes theoscillator’s momentum to undergo a random walk (i.e. itdiffuses), which in turn causes 〈E〉 to grow linearly intime at rate P . By demanding d〈E〉/dt = 0, we find thatthe combination of damping and heating effects causesthe energy to reach a steady state mean value of

〈E〉 =P

γ. (3.26)

In the quantum case, we see from Eq. (3.24) that itis the symmetric-in-frequency part of the noise spectrumwhich is responsible for this effect, and which thus playsthe role of a classical noise source. This is another reason

8 Note that the effective temperature can become negative if thenoise source prefers emitting energy versus absorbing it; in thepresent case, that would lead to an instability.

why the symmetrized quantum noise is often referred toas the “classical” part of the noise.

Note that in the present description the finite groundstate energy 〈E〉 = !Ω/2 of the harmonic oscillator isdetermined via the balance between the ’heating’ by thezero-point fluctuations of the environment (described bythe symmetrized correlator at T = 0) and the dissipation.It is possible to take an alternative but equally correctviewpoint, where only the deviation 〈δE〉 = 〈E〉 − !Ω/2from the ground state energy is considered. Its evolutionequation

d

dt〈δE〉 = 〈δE〉(Γ↑ − Γ↓) + Γ↑!Ω (3.27)

only contains a decay term at T = 0, leading to 〈δE〉 → 0.We turn now to the damping term in Eq. (3.23). At

first glance, it would appear to be completely quantumin origin, as it involves the asymmetric-in-frequency partof the noise. This latter fact is not surprising: dampinginvolves a net absorption of energy by the noise source,and the asymmetric part of the noise directly charac-terizes the source’s tendency to more frequently absorbrather than emit energy via transitions. In contrast, theclassical picture of damping is that it results from therandom force having a non-zero mean value, i.e.:

〈A · F (t)〉 = −Mγx(t) (3.28)

Both these pictures of damping are of course rigorouslyequivalent. In both the classical and quantum cases, wecan think of damping as originating from the fact thatthe noise source changes in response to the oscillator’smotion, and hence, so does the average value of F . Quan-tum mechanically, we can capture this effect by simplydoing perturbation theory to calculate the change in 〈F 〉brought on by the coupling V = AxF . Working in theinteraction picture, we obtain the standard equations ofquantum linear response:

δ〈A · F (t)〉 = A2

∫dt′χFF (t− t′)〈x(t′)〉 (3.29)

where

χFF (t) =−i

! θ(t)⟨[F (t), F (0)]

⟩(3.30)

Keeping only the part of δ〈F (t)〉 responsible for dissipa-tion (i.e. the part which is in phase with the oscillatorvelocity and thus out-of-phase with its position), and us-ing the fact that the oscillator motion only involves thefrequency Ω, we may write Eq. (3.29) in the familiar formof Eq. (3.28), with:

γ =2A2x2

ZPF

! [−ImχFF [Ω]]

=2A2x2

ZPF

!

[−Im

∫ ∞

−∞dteiΩtχFF (t)

](3.31)

13

Remarkably, a straightforward manipulation ofEq. (3.30) for χFF shows that this expression forγ is exactly equivalent to our previous expression, i.e.:

γ =2A2x2

ZPF

! [−ImχFF [Ω]] =A2x2

ZPF

!2[SFF [Ω]− SFF (−Ω)]

(3.32)Thus, there are two ways of viewing damping that areboth formally and physically equivalent: damping resultsfrom the response of the noise source to the oscillator’smotion, or equivalently, results from the transfer of en-ergy from the oscillator to the noise source via noise-induced transitions.

Returning to Eqs. (3.24) and (3.25), we see that wenow have a new way of looking at quantum noise. Thesymmetric-in-frequency part of the quantum noise spec-trum is analogous to classical noise, and is responsible forheating effects. In contrast, the asymmetric-in-frequencypart of the quantum noise spectrum describes the dissipa-tive response of the noise source to whatever system cou-ples to it. We also see that a harmonic oscillator can serveas a spectrum analyzer of quantum noise–if one can mea-sure both the heating rate P (or equivalently, the meanenergy 〈E〉) and the damping rate γ, one can completelycharacterize the force noise spectrum at both positive andnegative frequencies ±Ω. This idea was recently imple-mented experimentally by Naik et al. (2006), who used ahigh-Q mechanical resonator to probe both the positiveand negative frequency back-action noise produced by asuperconducting single-electron transistor.

Before concluding this section, we mention an impor-tant consequence of the preceding discussion: it imme-diately yields the quantum version of the fluctuation-dissipation theorem (Callen and Welton, 1951). As wehave seen in the previous section, if our noise source is inequilibrium, the positive and negative frequency parts ofthe noise spectrum are strictly related to one another bythe condition of detailed balance (cf. Eq. (3.14)). Thisin turn lets us link the classical, symmetric-in-frequencypart of the noise to the damping, (i.e. the asymmetric-in-frequency part of the noise). Letting β = 1/(kBT ), wehave:

SFF [Ω] ≡ SFF [Ω] + SFF [−Ω]2

(3.33)

=12(1 + e−β!Ω)SFF [Ω]

=12

(1 + e−β!Ω)(1− e−β!Ω)

(1− e−β!Ω)SFF [Ω]

=12

coth(β!Ω/2) (SFF [Ω]− SFF [−Ω])

= coth(β!Ω/2)!ΩM

A2γ[Ω] (3.34)

Thus, in equilibrium, the condition that noise-inducedtransitions obey detailed balance immediately impliesthat noise and damping are related to one another via thetemperature. For T % !Ω, we recover the more familiar

classical version of the fluctuation dissipation theorem:

A2SFF [Ω] = 2kBTMγ (3.35)

The derivation here of the fluctuation dissipation theo-rem is somewhat different than the one usually presentedfor classical systems; there, it is the requirement thatequipartition be obeyed at long times which relates noiseand damping at low frequencies Here, the requirementthat quantum transition rates obey detailed balance givesus a relation between noise and damping which holds atall frequencies. Further insight into the fluctuation dis-sipation theorem is provided in Appendix B.3, where wediscuss it in the simple but instructive context of a trans-mission line terminated by an impedance Z[ω].

C. Practical quantum spectrum analyzers

As we have seen, a ‘quantum spectrum analyzer’ canin principle be constructed from a two level system (ora harmonic oscillator) in which we can separately mea-sure the up and down transition rates between statesdiffering by some precise energy !ω > 0 given by thefrequency of interest. The down transition rate tells usthe noise spectral density at frequency +ω and the uptransition rate tells us the noise spectral density at −ω.While we have already discussed experimental implemen-tation of these ideas using two-level systems and oscilla-tors, similar schemes have been implemented in other sys-tems. A number of recent experiments have made use ofsuperconductor-insulator-superconductor junctions (Bil-langeon et al., 2006; Deblock et al., 2003; Onac et al.,2006b) to measure quantum noise, as the current-voltagecharacteristics of such junctions are very sensitive tothe absorption or emission of energy (so-called photon-assisted transport processes).

In this subsection, we discuss additional methods forthe detection of quantum noise. Recall from Sec. II.Bthat one of the most basic classical noise spectrum ana-lyzers consists of a linear narrow band filter and a squarelaw detector such as a diode. In what follows, we willconsider a quantum treatment of such a device wherewe evade the tricky question of how to model a quantumdiode by simply looking at the energy of the filter circuit.We then turn to various noise detection schemes makinguse of a photomultiplier. We will show that dependingon the detection scheme used, one can measure either thesymmetrized quantum noise spectral density S[ω], or thenon-symmetrized spectral density S[ω].

1. Filter plus diode

Using the simple treatment we gave of a harmonic os-cillator as a quantum spectrum analyzer in Sec. III.B,one can attempt to provide a quantum treatment of theclassical ‘filter plus diode’ spectrum analyzer discussedin Sec. II.B. This approach is due to Lesovik and Loosen

14

(1997) and Gavish et al. (2000). The analysis starts bymodeling the spectrum analyzer’s resonant filter circuitas a harmonic oscillator of frequency Ω weakly coupledto some equilibrium dissipative bath. The oscillator thushas an intrinsic damping rate γ0 , Ω, and is initially ata finite temperature Teq. One then drives this dampedoscillator (i.e. the filter circuit) with the noisy quantumforce F (t) whose spectrum at frequency Ω is to be mea-sured.

In the classical ‘filter plus diode’ spectrum analyzer,the output of the filter circuit was sent to a square lawdetector, whose time-averaged output was then taken asthe measured spectral density. To simplify the analy-sis, we can instead consider how the noise changes theaverage energy of the resonant filter circuit, taking thisquantity as a proxy for the output of the diode. Sureenough, if we subject the filter circuit to purely classicalnoise, it would cause the average energy of the circuit〈E〉 to increase an amount directly proportional to theclassical spectrum SFF [Ω]. We now consider 〈E〉 in thecase of a quantum noise source, and ask how it relates tothe quantum noise spectral density SFF [Ω].

The quantum case is straightforward to analyze usingthe approach of Sec. III.B. Unlike the classical case, thenoise will both lead to additional fluctuations of the filtercircuit and increase its damping rate by an amount γ(c.f. Eq. (3.25)). To make things quantitative, we let neq

denote the average number of quanta in the filter circuitprior to coupling to F (t), i.e.

neq =1

exp(

!ΩkBTeq

)− 1

, (3.36)

and let neff represent the Bose-Einstein factor associatedwith the effective temperature Teff [Ω] of the noise sourceF (t),

neff =1

exp(

!ΩkBTeff [Ω]

)− 1

. (3.37)

One then finds (Gavish et al., 2000; Lesovik and Loosen,1997):

∆〈E〉 = !Ω · γ

γ0 + γ(neff − neq) (3.38)

This equation has an extremely simple interpretation:the first term results from the expected heating effectof the noise, while the second term results from thenoise source having increased the circuit’s damping byan amount γ. Re-expressing this result in terms of thesymmetric and anti-symmetric in frequency parts of thequantum noise spectral density SFF [Ω], we have:

∆〈E〉 =SFF (Ω)−

(neq + 1

2

)(SFF [Ω]− SFF [−Ω])

2m (γ0 + γ)(3.39)

We see that ∆〈E〉 is in general not simply proportionalto the symmetrized noise SFF [Ω]. Thus, the ‘filter plusdiode’ spectrum analyzer does not simply measure thethe symmetrized quantum noise spectral density. Westress that there is nothing particularly quantum aboutthis result. The extra term on the RHS of Eq. (3.39)simply reflects the fact that coupling the noise source tothe filter circuit could change the damping of this circuit;this could easily happen in a completely classical setting.As long as this additional damping effect is minimal, thesecond term in Eq. (3.39) will be minimal, and our spec-trum analyzer will (to a good approximation) measurethe symmetrized noise. Quantitatively, this requires:

neff % neq. (3.40)

We now see where quantum mechanics enters: if the noiseto be measured is close to being zero point noise (i.e.neff → 0), the above condition can never be satisfied, andthus it is impossible to ignore the damping effect of thenoise source on the filter circuit. In the zero point limit,this damping effect (i.e. second term in Eq. (3.39)) willalways be greater than or equal to the expected heatingeffect of the noise (i.e. first term in Eq. (3.39)).

2. Filter plus photomultiplier

We now turn to quantum spectrum analyzers involvinga square law detector we can accurately model– a photo-multiplier. As a first example of such a system, consider aphotomultiplier with a narrow band filter placed in frontof it. The mean photocurrent is then given by

〈I〉 =∫ +∞

−∞dω |f [ω]|2r[ω]SVV[ω], (3.41)

where f is the filter (amplitude) transmission functiondefined previously and r[ω] is the response of the pho-todetector at frequency ω, and SVV represents the elec-tric field spectral density incident upon the photodetec-tor. Naively one thinks of the photomultiplier as a squarelaw detector with the square of the electric field repre-senting the optical power. However, according to theGlauber theory of (ideal) photo-detection (Gardiner andZoller, 2000; Glauber, 2006; Walls and Milburn, 1994),photocurrent is produced if, and only if, a photon isabsorbed by the detector, liberating the initial photo-electron. Glauber describes this in terms of normal or-dering of the photon operators in the electric field auto-correlation function. In our language of noise power atpositive and negative frequencies, this requirement be-comes simply that r[ω] vanishes for ω > 0. Approximat-ing the narrow band filter centered on frequency ±ω0 asin Eq. (2.15), we obtain

〈I〉 = r[−ω0]SVV[−ω0] (3.42)

which shows that this particular realization of a quantumspectrometer only measures electric field spectral density

15

at negative frequencies since the photomultiplier neveremits energy into the noise source. Also one does notsee in the output any ‘vacuum noise’ and so the output(ideally) vanishes as it should at zero temperature. Ofcourse real photomultipliers suffer from imperfect quan-tum efficiencies and have non-zero dark current. Notethat we have assumed here that there are no additionalfluctuations associated with the filter circuit. Our re-sult thus coincides with what we found in the previoussubsection for the ‘filter plus diode’ spectrum analyzer(c.f. Eq. (3.39), in the limit where the filter circuit isinitially at zero temperature (i.e. neq = 0).

3. Double sideband heterodyne power spectrum

At RF and microwave frequencies, practical spectrome-ters often contain heterodyne stages which mix the initialfrequency down to a lower IF frequency (possibly in theclassical regime). Consider a system with a mixer and lo-cal oscillator at frequency ωLO that mixes both the uppersideband input at ωu = ωLO+ωIF and the lower sidebandinput at ωl = ωLO − ωIF down to frequency ωIF. Thiscan be achieved by having a Hamiltonian with a 3-wavemixing term which (in the rotating wave approximation)is given by

V = λ[aIFala†LO + a†IFa†l aLO] + λ[a†IFaua†LO + aIFa†uaLO]

(3.43)The interpretation of this term is that of a Raman pro-cess. Notice that there are two energy conserving pro-cesses that can create an IF photon which could thenactivate the photodetector. First, one can absorb an LOphoton and emit two photons, one at the IF and one atthe lower sideband. The second possibility is to absorban upper sideband photon and create IF and LO photons.Thus we expect from this that the power in the IF chan-nel detected by a photomultiplier would be proportionalto the noise power in the following way

I ∝ S[+ωl] + S[−ωu] (3.44)

since creation of an IF photon involves the signal sourceeither absorbing a lower sideband photon from the mixeror the signal source emitting an upper sideband photoninto the mixer. In the limit of small IF frequency thisexpression would reduce to the symmetrized noise power

I ∝ S[+ωLO] + S[−ωLO] = 2S[ωLO] (3.45)

which is the same as for the ‘classical’ spectrum analyzerwith a square law detector described in Sec. II.B. Forequilibrium noise spectral density from a resistance R0

derived in Appendix C we would then have

SVV[ω] + SVV[−ω] = 2R0!|ω|[2nB(!|ω|) + 1], (3.46)

Assuming our spectrum analyzer has high inputimpedance so that it does not load the noise source, thisvoltage spectrum will determine the output signal of the

analyzer. This symmetrized quantity does not vanish atzero temperature and the output contains the vacuumnoise from the input. This vacuum noise has been seenin experiment. (Schoelkopf et al., 1997)

4. Single sideband heterodyne power spectrum

With proper filtering (and large enough IF frequency)one can eliminate one of the two sidebands from the inputand hence have only one of the two terms in Eq.(3.44).Thus one can detect either the noise at negative frequen-cies, or positive frequencies, or both. For example inthe driven cavity with moveable mirror discussed in Ap-pendix D, the laser beam can be detuned to the red ofthe cavity by precisely the cantilever frequency. In thegood cavity (resolved sideband) limit, then any photonscoming out at the cavity frequency must have come fromdestroying a cantilever phonon and up converting a laserdrive photon to the anti-Stokes sideband which resonateswith the cavity. Cantilever position noise at positive fre-quencies generates Stokes photons but these are so faroff resonance from the cavity that they are severely sup-pressed. Hence the total output power (excluding thelaser frequency) measures only the negative frequencycantilever noise. (Note the output electric field at thecavity frequency also contains incoming vacuum noisereflected by the cavity but this does not contribute tothe photon number detected by a photomultiplier.) Con-versely for blue detuning, any photons must have comefrom creating a cantilever phonon. This measures thepositive frequency cantilever position noise.

IV. QUANTUM MEASUREMENTS

Having introduced both quantum noise and quantumspectrum analyzers, we are now in a position to intro-duce the general topic of quantum measurements. Allpractical measurements are affected by noise. Certainquantum measurements remain limited by quantum noiseeven though they use completely ideal apparatus. As wewill see, the limiting noise here is associated with the factthat canonically conjugate variables are incompatible ob-servables in quantum mechanics.

The simplest, idealized description of a quantum mea-surement, introduced by von Neumann (Bohm, 1989;Haroche and Raimond, 2006; von Neumann, 1932;Wheeler and Zurek, 1984), postulates that the mea-surement process instantaneously collapses the system’squantum state onto one of the eigenstates of the observ-able to be measured. As a consequence, any initial super-position of these eigenstates is destroyed and the valuesof observables conjugate to the measured observable areperturbed. This perturbation is an intrinsic feature ofquantum mechanics and cannot be avoided in any mea-surement scheme, be it of the “projection-type” describedby von Neumann or rather a weak, continuous measure-

16

ment to be analyzed further below.To form a more concrete picture of quantum measure-

ment, we begin by noting that every quantum measure-ment apparatus consists of a macroscopic ‘pointer’ cou-pled to the microscopic system to be measured. (A spe-cific model is discussed in Allahverdyan et al. (2001).)This pointer is sufficiently macroscopic that its positioncan be read out ‘classically’. The interaction between themicroscopic system and the pointer is arranged so thatthe two become strongly correlated. One of the simplestpossible examples of a quantum measurement is that ofthe Stern-Gerlach apparatus which measures the projec-tion of the spin of an S = 1/2 atom along some chosendirection. What is really measured in the experiment isthe final position of the atom on the detector plate. How-ever, the magnetic field gradient in the magnet causesthis position to be perfectly correlated (‘entangled’) withthe spin projection so that the latter can be inferred fromthe former. Suppose for example that the initial state ofthe atom is a product of a spatial wave function ξ0(+r)centered on the entrance to the magnet, and a spin statewhich is the superposition of up and down spins corre-sponding to the eigenstate of σx:

|Ψ0〉 =1√2| ↑〉+ | ↓〉 |ξ0〉. (4.1)

After passing through a magnet with field gradient in thez direction, an atom with spin up is deflected upwardsand an atom with spin down is deflected downwards. Bythe linearity of quantum mechanics, an atom in a spinsuperposition state thus ends up in a superposition ofthe form

|Ψ1〉 =1√2| ↑〉|ξ+〉+ | ↓〉|ξ−〉 , (4.2)

where 〈+r|ξ±〉 = ψ1(+r ± dz) are spatial orbitals peaked inthe plane of the detector. The deflection d is determinedby the device geometry and the magnetic field gradient.The z-direction position distribution of the particle foreach spin component is shown in Fig. 2. If d is suffi-ciently large compared to the wave packet spread then,given the position of the particle, one can unambiguouslydetermine the distribution from which it came and hencethe value of the spin projection of the atom. This is thelimit of a strong ‘projective’ measurement.

In the initial state one has

〈Ψ0|σx|Ψ0〉 = +1, (4.3)

but in the final state one has

〈Ψ1|σx|Ψ1〉 =12〈ξ−|ξ+〉+ 〈ξ+|ξ−〉 (4.4)

For sufficiently large d the states ξ± are orthogonal andthus the act of σz measurement destroys the spin coher-ence

〈Ψ1|σx|Ψ1〉 → 0. (4.5)

+d

zz

FIG. 2 (Color online) Schematic illustration of position dis-tributions of an atom in the detector plane of a Stern-Gerlachapparatus whose field gradient is in the z direction. For smallvalues of the displacement d (described in the text), there issignificant overlap of the distributions and the spin cannot beunambiguously inferred from the position. For large values ofd the spin is perfectly entangled with position and can be in-ferred from the position. This is the limit of strong projectivemeasurement.

This is what we mean by projection or wave function ‘col-lapse’. The result of measurement of the atom positionwill yield a random and unpredictable value of ± 1

2 for thez projection of the spin. This destruction of the coher-ence in the transverse spin components by a strong mea-surement of the longitudinal spin component is the firstof many examples we will see of the Heisenberg uncer-tainty principle in action. Measurement of one variabledestroys information about its conjugate variable. Wewill study several examples in which we understand mi-croscopically how it is that the coupling to the measure-ment apparatus causes the ‘back action’ quantum noisewhich destroys our knowledge of the conjugate variable.

In the special case where the eigenstates of the observ-able we are measuring are also stationary states (i.e. en-ergy eigenstates), measuring the observable a second timewould reproduce exactly the same measurement result,thus providing a way to confirm the accuracy of the mea-surement scheme. These optimal kinds of repeatable mea-surements are called “Quantum Non-Demolition” (QND)measurements (Braginsky and Khalili, 1992, 1996; Bra-ginsky et al., 1980; Peres, 1993). A simple example wouldbe a sequential pair of Stern-Gerlach devices oriented inthe same direction. In the absence of stray magneticperturbations, the second apparatus would always yieldthe same answer as the first. The fact that QND mea-surements are repeatable is of fundamental practical im-portance in overcoming detector inefficiencies (Gambettaet al., 2007). This point is elaborated in the discussionof reaching the quantum limit in practice in Sec. VIII.

A common confusion is to think that a QND measure-ment has no effect on the state of the system being mea-sured. While this is true if the initial state is an eigen-state of the observable, it is not true in general. Consideragain our example of a spin oriented in the x direction.The result of the first σz measurement will be that thestate randomly and completely unpredictably collapsesonto one of the two z projection eigenstates. Howeverall subsequent measurements using the same orientationfor the detectors will always agree with the result of thefirst measurement. Thus QND measurements may affectthe state of the system, but never the value of the ob-servable (once it is determined). Other examples of QND

17

measurements include: (i) measuring the electromagneticfield energy stored inside a cavity by determining the ra-diation pressure exerted on a moving piston (Braginskyand Khalili, 1992), (ii) detecting the presence of a photonin a cavity by its effect on the phase of an atom’s superpo-sition state (Haroche and Raimond, 2006; Nogues et al.,1999), and (iii) the “dispersive” measurement of a qubitstate by its effect on the phase shift of a microwave beam(Blais et al., 2004; Wallraff et al., 2004), which is the firstcanonical example we will describe below.

In contrast to the above, in non-QND measurements,the back-action of the measurement will affect the ob-servable being studied. The canonical example we willconsider below is the position measurement of a harmonicoscillator. Since the position operator does not commutewith the Hamiltonian, the QND criterion is not fulfilled.Other examples of non-QND measurements include: (i)photon counting via photo-detectors that absorb the pho-tons, (ii) continuous measurements where the observabledoes not commute with the Hamiltonian, thus inducinga time-dependence of the measurement result, (iii) mea-surements that can be repeated only after a time longerthan the relaxation (mixing) time of the system.

A. Weak continuous measurements

In discussion “real” quantum measurements, anotherkey notion to introduce is that of weak, continuous mea-surements (Braginsky and Khalili, 1992). Many mea-surements in practice take an extended time-interval tocomplete, which is much longer than the “microscopic”time scales (oscillation periods etc.) of the system. Thereason may be quite simply that the coupling strengthbetween the detector and the system cannot be madearbitrarily large, and one has to wait for the effect ofthe system on the detector to accumulate. For example,in our Stern-Gerlach measurement suppose that we areonly able to achieve small magnetic field gradients andthat consequently, the displacement d cannot be madelarge compared to the wave packet spread (see Fig. 2).In this case the states ξ± would have non-zero overlapand it would not be possible to reliably distinguish them:we thus would only have a “weak” measurement. How-ever, by cascading together a series of such measurementsand taking advantage of the fact that they are QND, wecan eventually achieve an unambiguous strong projec-tive measurement. During this process, the overlap of ξ±would gradually fall to zero corresponding to a smoothcontinuous loss of phase coherence in the transverse spincomponents. Only in this case of weak continuous mea-surements does it make sense to define a measurementrate in terms of a rate of gain of information about thevariable being measured, and the corresponding dephas-ing rate, the rate at which information about the conju-gate variable is being lost. We will see that these rates areintimately related via the Heisenberg uncertainty princi-ple.

While strong projective measurements are often theideal, in some cases one may intentionally desire to havea weak continuous measurement which does not drasti-cally perturb the system. For example in doing continu-ous quantum feedback to control the state of a system, itmay be advantageous to continuously monitor both posi-tion and momentum of a particle, accepting the fact thatsince these are incompatible (i.e. non-commuting) vari-ables, we cannot determine them both precisely. Thereare many practical examples of weak, continuous mea-surement schemes. These include: (i) charge measure-ments, where the current through a device (e.g. quantumpoint contact or single-electron transistor) is modulatedby the presence/absence of a nearby charge, and whereit is necessary to wait for a sufficiently long time to over-come the shot noise and distinguish between the two cur-rent values, (ii) the weak dispersive qubit measurementdiscussed below, (iii) displacement detection of a nano-mechanical beam (e.g. optically or by capacitive couplingto a charge sensor), where one looks at the two quadra-ture amplitudes of the signal produced at the beam’sresonance frequency.

Not surprisingly, quantum noise plays a crucial role indetermining the properties of a weak, continuous quan-tum measurement. For such measurements, noise bothdetermines the back-action effect of the measurement onthe measured system, as well as how quickly informationis acquired in the measurement process. Previously wesaw that a crucial feature of quantum noise is the asym-metry between positive and negative frequencies; we fur-ther saw that this corresponds to the difference betweenabsorption and emission events. For measurements, an-other key aspect of quantum noise will be important:as we will discuss extensively, quantum mechanics placesconstraints on the noise of any system capable of actingas a detector or amplifier. These constraints in turn placerestrictions any weak, continuous measurement, and leaddirectly to quantum limits on how well one can make sucha measurement.

In the rest of this section, we give an introduction tohow one describes a weak, continuous quantum measure-ment, considering the specific examples of using para-metric coupling to a resonant cavity for QND detectionof the state of a qubit and the (necessarily non-QND)detection of the position of a harmonic oscillator. Inthe following section (Sec. V), we derive a very generalquantum mechanical constraint on the noise of any sys-tem capable of acting as a detector, and show how thisconstraint directly leads to the quantum limit on qubitdetection. Finally, in Sec. VI, we will turn to the impor-tant but slightly more involved case of a quantum linearamplifier or position detector. We will show that the ba-sic quantum noise constraint derived Sec. V again leadsto a quantum limit; here, this limit is on how small onecan make the added noise of a linear amplifier.

Before leaving this introductory section, it worthpointing out that the theory of weak continuous mea-surements is sometimes described in terms of some set

18

of auxiliary systems which are sequentially momentarilyweakly coupled to the system being measured. (See Ap-pendix D.) One then envisions a sequence of projectivevon Neumann measurements on the auxiliary variables.The weak entanglement between the system of interestand one of the auxiliary variables leads to a kind of par-tial collapse of the system wave function (more preciselythe density matrix) which is described in mathematicalterms not by projection operators, but rather by POVMs(positive operator valued measures). We will not use thisand the related ‘quantum trajectory’ language here, butdirect the reader to the literature for more informationon this important approach. (Brun, 2002; Haroche andRaimond, 2006; Jordan and Korotkov, 2006; Peres, 1993)

B. Measurement with a parametrically coupled resonantcavity

A very simple yet experimentally practical example ofa quantum detector consists of a resonant optical or RFcavity parametrically coupled to the system being mea-sured. Changes in the variable being measured (e.g. thestate of a qubit or the position of an oscillator) shift thecavity frequency and produce a varying phase shift in thecarrier signal reflected from the cavity. This changingphase shift can be converted (via homodyne interferome-try) into a changing intensity; this can then be detectedusing diodes or photomultipliers.

In this subsection, we will analyze weak, continuousmeasurements made using a parametric cavity detector;this will serve as a good introduction to the more generalapproaches presented in the rest of this review. The cav-ity system is an excellent first case to treat both becauseof its simplicity, and because it is capable of reaching thequantum-limit: it can be used to make a weak, continu-ous measurement as well as is allowed by quantum me-chanics. This is true for both the (QND) measurementof the state of a qubit, and the (non-QND) measure-ment of the position of a harmonic oscillator. Comple-mentary analyses of weak, continuous qubit measurementare given in Makhlin et al. (2000, 2001) (using a single-electron transistor) and in Clerk et al. (2003); Gurvitz(1997); Korotkov (2001b); Korotkov and Averin (2001);Pilgram and Buttiker (2002) (using a quantum point con-tact).

In addition to its pedagogical value, the paramet-ric cavity detector is worth examining because of itswidespread usage in experiment. One important currentrealization is a high-Q microwave cavity used to read outthe state of a superconducting qubit (Blais et al., 2004;Lupascu et al., 2004; Duty et al., 2005; Il’ichev et al.,2003; Izmalkov et al., 2004; Lupascu et al., 2005; Schus-ter et al., 2005; Sillanpaa et al., 2005; Wallraff et al.,2004). Another class of examples are optical cavitieswith a mechanical degree of freedom; here, the cavitycan be used for highly sensitive position measurements.Examples of such systems include those where one of the

cavity mirrors is mounted on a cantilever (Arcizet et al.,2006; Gigan et al., 2006; Kleckner and Bouwmeester,2006). Related systems involve a freely suspended mass(Abramovici et al., 1992; Corbitt et al., 2007), an op-tical cavity with a thin transparent membrane in themiddle (Thompson et al., 2008) and, more generally, anelastically deformable whispering gallery mode resonator(Schliesser et al., 2006). Yet another realization of theparametric cavity detector involves a microwave cavityterminated with a DC SQUID, with part of the SQUIDloop is a doubly-clamped beam (Blencowe and Buks,2007); here, the system acts as a position detector.

The cavity uses interference and the wave nature oflight to convert the input signal to a phase shifted wave.For small phase shifts we have a weak continuous mea-surement. Interestingly, it is the complementary particlenature of light which turns out to be thing which lim-its the measurement. As we will see, it both limits rateat which we can make a measurement (via photon shotnoise in the output beam) and also controls the back ac-tion disturbance of the system being measured (due tophoton shot noise inside the cavity acting on the sys-tem being measured). These two dual aspects are an im-portant part of any weak, continuous quantum measure-ment; hence, understanding the output noise and back-action noise of detectors will be crucial. The questionof reaching the quantum limit in practical measurementswill be discussed in Sec. VIII. For now we simply notethat, if the secondary amplifiers and detectors are notquiet enough for the output noise to be dominated byphoton shot noise and the coupling to the system is notstrong enough to see the back action noise, then the mea-surement cannot be quantum limited.

All of our discussion of measurement imprecision andback action noise in the cavity system will be framed interms of the number phase uncertainty relation for coher-ent states, derived in detail in Appendix G. A coherentphoton state contains a Poisson distribution of the num-ber of photons. Thus if the mean number of photons isN , the fluctuations in the number obey

(∆N)2 = N . (4.6)

Coherent states are over complete and states of differentphase are not orthogonal to each other. As shown inAppendix G this means that there is an uncertainty inany measurement of the phase given by

(∆θ)2 =1

4N. (4.7)

(Equivalently, any homodyne measurement of the phaseof the beam is subject to photon shot noise which leads tothe same phase uncertainty.) Thus coherent states obeythe number-phase uncertainty relation

∆N∆θ =12

(4.8)

analogous to the position momentum uncertainty rela-tion.

19

We now consider the equivalent of this statement interms of noise spectral densities associated with the mea-surement. Consider a continuous photon beam carryingan average photon flux N . As explained in AppendixA, the variance in the number of photons detected growslinearly in time and can be represented in terms of thephoton shot noise spectral density

(∆N)2 = SNN t. (4.9)

Here, SNN represents the spectral density of photon-fluxfluctuations. It is white, and on a physical level, describesphoton shot noise:

SNN = N . (4.10)

In the parametric cavity detector, one needs to read-out the phase of the beam reflected from the cavity; it isthis phase which contains information on the system be-ing measured. As described in Appendix G, homodynemeasurement of this phase is subject to the same pho-ton shot noise fluctuations discussed above. Thus, if thephase of the beam has some nominal small value θ0, theoutput signal from the homodyne detector integrated upto time t will be of the form

I = θ0t +∫ t

0dτ δθ(τ) (4.11)

where δθ is a noise representing the imprecision in ourmeasurement of θ0 due to the photon shot noise in theoutput of the homodyne detector. An unbiased estimateof the phase is

θ =I

t, (4.12)

which obeys

〈θ〉 = θ0 (4.13)

and from the results of Appendix A

(∆θ)2 =Sθθ

t, (4.14)

where Sθθ is the spectral density of the δθ white noise as-sociated with the measurement imprecision. Comparisonwith Eq. (4.7) yields

Sθθ =1

4N. (4.15)

The larger the photon flux in the beam, the larger is thephoton shot noise, but the smaller is the phase noise.We will make repeated use of the fact that the measure-ment imprecision can be represented in terms of a noisespectral density in this manner.

The results above lead us to the fundamentalwave/particle relation for ideal coherent beams

SNNSθθ =14

(4.16)

or in analogy with Eq. (4.8)

√SNNSθθ =

12

(4.17)

Before we study the role that these uncertainty rela-tions play in measurements with high Q cavities, let usconsider the simplest case of reflecting light from a mirrorwithout a cavity. The phase shift of the beam (havingwave vector k) when the mirror moves a distance x is2kx. Thus, the uncertainty in the phase measurementcorresponds to a position imprecision which can again berepresented in terms of a noise spectral density

SIxx =

14k2

Sθθ. (4.18)

Here the superscript I refers to the fact that this is noiserepresenting imprecision in the measurement, not actualfluctuations in the position. We also need to worry aboutback action: each photon hitting the mirror transfers amomentum 2!k to the mirror, so photon shot noise cor-responds to a random back action force noise spectraldensity

SFF = 4!2k2SNN (4.19)

Multiplying these together we have the central result forthe product of the back action force noise and the impre-cision

SFF SIxx = !2SNNSθθ =

!2

4(4.20)

or in analogy with Eq. (4.8)√

SFF SIxx =

!2. (4.21)

Not surprisingly, the situation considered here is as idealas possible. Thus, the RHS above is actually a lowerbound on the product of imprecision and back-actionnoise for any detector; we will prove this rigorously inSec. V.A. Eq. (4.21) thus represents the quantum-limiton the noise of our detector. As we will see shortly, havinga detector with quantum-limited noise is a prerequisitefor reaching the quantum limit on various different mea-surement tasks (e.g. continuous position detection of anoscillator and QND qubit state detection). Note that ingeneral, a given detector will not have quantum limitednoise: we will devote considerable effort in later sectionsto determining the necessary conditions to achieve thelower bound of Eq. (4.21) in a general detector.

We now turn to the story of measurement using a highQ cavity which will be very similar to the above, exceptthat we have to take into account the filtering of thenoise by the cavity response. As discussed in detail inAppendix D, the cavity is simply described as a singlebosonic mode coupled weakly to electromagnetic modesoutside the cavity. The Hamiltonian of the system isgiven by:

H = H0 + !ωc (1 + Az) a†a + Henvt. (4.22)

20

Here, H0 is the unperturbed Hamiltonian of the systemwhose variable z (which is not necessarily a position) isbeing measured, a is the annihilation operator for thecavity mode, and ωc is the cavity resonance frequencyin the absence of the coupling A. We will take both Aand z to be dimensionless. The term Henvt describes theelectromagnetic modes outside the cavity, and their cou-pling to the cavity; it is responsible for both driving anddamping the cavity mode. The damping is parameterizedby rate κ, which tells us how quickly energy leaks out ofthe cavity; we consider the case of a high quality-factorcavity, where Qc ≡ ωc/κ % 1.

Turning to the interaction term in Eq. (4.22), we seethat the parametric coupling strength A determines thechange in frequency of the cavity as the system variablez changes. We are going to consider two cases, one inwhich z represents the continuously varying position of aharmonic oscillator and one in which it represents the dis-crete state of a spin-1/2 qubit. In both cases we assumethat the dynamics of z is slow compared to κ so that thecavity follows the system adiabatically. In this limit thereflected phase shift simply varies slowly in time adia-batically following the instantaneous value of z. We willalso assume that the coupling A is small enough that thephase shifts are always very small and hence the measure-ment is weak. Many photons will have to pass throughthe cavity before much information is gained about thevalue of the phase shift and hence the value of z.

We first consider the case of a ‘one-sided’ cavity whereonly one of the mirrors is semi-transparent, the otherbeing perfectly reflecting (see Fig. 3). In this case, awave incident on the cavity (say, in a one-dimensionalwaveguide) will be perfectly reflected, but with a phaseshift θ determined by the cavity and the value of z. Asshown in Appendix D, the reflection coefficient at thebare cavity frequency ωc is simply given by (Walls andMilburn, 1994)

r = −1 + 2iAQcz

1− 2iAQcz. (4.23)

Note that r has unit magnitude because all photonswhich are incident are reflected if the cavity is lossless.For weak coupling we can write the reflection phase shiftas

r = −eiθ (4.24)

with

θ ≈ 4QcAz = (Aωc)tWDz. (4.25)

We see that the scattering phase shift is simply the fre-quency shift caused by the parametric coupling multi-plied by the Wigner delay time (Wigner, 1955)

tWD = Im∂ ln r

∂ω=

4κ

. (4.26)

Thus the measurement imprecision noise power for a

a(t)bin(t)

bout(t)

FIG. 3 Schematic of a single-sided resonant cavity with inter-nal mode a, driven by input mode bin resulting in the outputfield bout.

given photon flux N incident on the cavity is given by

SIzz =

1(AωctWD)2

Sθθ. (4.27)

The random part of the generalized back action forceconjugate to z is from Eq. (4.22)

Fz ≡ −∂H

∂z= −A!ωc δn (4.28)

where, since z is dimensionless, Fz has units of energy.Here δn = n − n = a†a − 〈a†a〉 represents the photonnumber fluctuations around the mean n inside the cavity.The back action force noise spectral density is thus

SFzFz = (A!ωc)2Snn (4.29)

As shown in Appendix D, the cavity filters the photonshot noise so that at low frequencies ω , κ the numberfluctuation spectral density is simply

Snn = ntWD. (4.30)

The mean photon number in the cavity (also derived inthe Appendix) is

n = NtWD (4.31)

where again N the mean photon flux incident on thecavity. From this it follows that

Snn = Nt2WD = SNN t2WD (4.32)

and

SFzFz = (A!ωctWD)2SNN . (4.33)

Combining this with Eq. (4.27) again yields the sameresult as Eq. (4.21) obtained without the cavity. Theparametric cavity detector (used in this way) is thus aquantum-limited detector: it reaches the quantum-limiton its noise spectral densities.

We will now examine how the quantum limit on thenoise of our detector directly leads to quantum limitson different measurement tasks. In particular, we willconsider the cases of continuous position detection andQND qubit state measurement. For the case of position

21

I(t)

t

ΔI

FIG. 4 (Color online) Distribution of the integrated outputfor the cavity detector, I(t), for the two different qubit states.The separation of the means of the distributions grows linearlyin time, while the width of the distributions only grow as the√

t.

measurements of a mirror, the back action force from thecavity detector is simply the radiation pressure acting onthe mirror (enhanced by the multiple reflections insidethe cavity when tWD is large). The position measurementcase is complicated by the fact that this force will causechanges in the momentum of the mirror. The momen-tum kicks will in turn increase the position fluctuationsover and above the naive measurement imprecision, re-flecting the fact that the measurement is non-QND. Wewill therefore start with the simpler case of QND mea-surement of the state of a qubit.

1. QND measurement of the state of a qubit using a resonantcavity

Here we specialize to the case where the system oper-ator z = σz represents the state of a spin-1/2 quantumbit. Eq. (4.22) becomes

H =12

!ω01σz + +!ωc (1 + Aσz) a†a + Henvt (4.34)

We see that σz commutes with all terms in the Hamilto-nian and is thus a constant of the motion (assuming thatHenvt contains no qubit decay terms so that T1 = ∞) andhence the measurement will be QND. From Eq. (4.25) wesee that the two states of the qubit produce phase shifts±θ0 where

θ0 = AωctWD. (4.35)

As θ0 , 1, it will take many reflected photons beforewe are able to determine the state of the qubit. Thisis a direct consequence of the unavoidable photon shotnoise in the output of the detector, and is a basic featureof weak measurements– information on the input is onlyacquired gradually in time.

Let I(t) be the homodyne signal for the wave reflectedfrom the cavity integrated up to time t. Depending onthe state of the qubit the mean value of I will be

〈I〉 = ±θ0t, (4.36)

and the RMS gaussian fluctuations about the mean willbe

∆I =√

Sθθt. (4.37)

As illustrated in Fig. 4 and discussed extensively inMakhlin et al. (2001), the integrated signal is drawn fromone of two gaussian distributions which are better andbetter resolved with increasing time (as long as the mea-surement is QND). The state of the qubit thus becomesever more reliably determined. The signal energy to noiseenergy ratio becomes

SNR =〈I〉2

(∆I)2=

θ20

Sθθt (4.38)

which can be used to define the measurement rate via

Γmeas ≡SNR

2=

θ20

2Sθθ=

12SI

zz

. (4.39)

There is a certain arbitrariness in the scale factor of 2 inthe definition of the measurement rate, but this particu-lar definition is justified on precise information theoreticgrounds in Appendix E.

While Eq. (4.34) makes it clear that the state of thequbit modulates the cavity frequency, we can easily re-write this equation to show that this same interactionterm is also responsible for the back-action of the mea-surement (i.e. the disturbance of the qubit state by themeasurement process):

H =!2

(ω01 + 2Aωca

†a)σz + !ωca

†a + Henvt (4.40)

We now see that the interaction can also be viewed asproviding a ‘light shift’ (i.e. ac Stark shift) of the qubitsplitting frequency (Blais et al., 2004; Schuster et al.,2005) which contains a constant part 2AnAωc plus a ran-domly fluctuating part

∆ω01 = 2Fz/! (4.41)

which depends on n = a†a, the number of photons in thecavity. During a measurement, n will fluctuate around itsmean and act as a fluctuating back-action ‘force’ on thequbit. In the present QND case, noise in n = a†a cannotcause transitions between the two qubit eigenstates. Thisis the opposite of the situation considered in Sec. III.A,where we wanted to use the qubit as a spectrometer.Despite the lack of any noise-induced transitions, therestill is a back-action here, as noise in n causes the effec-tive splitting frequency of the qubit to fluctuate in time.For weak coupling, the resulting phase diffusion leads tomeasurement-induced dephasing of superpositions in thequbit (Blais et al., 2004; Schuster et al., 2005) accordingto

⟨e−iϕ

⟩=

⟨e−i

R t0 dτ ∆ω01(τ)

⟩. (4.42)

For weak coupling the dephasing rate is slow and thus weare interested in long times t. In this limit the integral

22

is a sum of a large number of statistically independentterms and thus we can take the accumulated phase to begaussian distributed. Using the cumulant expansion wethen obtain

⟨e−iϕ

⟩= exp

(−1

2

⟨[∫ t

0dτ ∆ω01(τ)

]2⟩)

= exp(− 2

!2SFzFz t

). (4.43)

Notice that these integrals are precisely of the form thatwe encountered in the Wiener-Khinchin theorem (see Ap-pendix A). Note also that the noise correlator above isnaturally symmetrized– the quantum asymmetry of thenoise plays no role for this type of coupling. Thus weidentify the dephasing rate

Γϕ =2!2

SFzFz = 2θ20SNN . (4.44)

We may now introduce the fundamental quantum limitfor an ideal QND measurement: at best, one can measureas quickly as one dephases. Using Eqs. (4.39) and (4.44),we see that the cavity reaches this quantum limit:

Γϕ

Γmeas=

4!2

SIzzSFzFz = 4SNNSθθ = 1. (4.45)

As described earlier, this represents the enforcement ofthe Heisenberg uncertainty principle. The faster you gaininformation about one variable, the faster you lose in-formation about the conjugate variable. Note that ingeneral, the ratio Γϕ/Γmeas will be larger than one, asan arbitrary detector will not reach the quantum limiton its noise spectral densities. Such a non-ideal detec-tor produces excess back-action beyond what is requiredquantum mechanically; as we shall repeatedly emphasize,this excess back-action is always associated with wastedinformation.

Before continuing our discussion of the quantum limit,it useful to make an additional remark about dephasing.In addition to the quantum noise point of view presentedabove, there is a second complementary way in which tounderstand the origin of measurement induced dephasing(Stern et al., 1990) which is analogous to our descriptionof loss of transverse spin coherence in the Stern-Gerlachexperiment in Eq. (4.4). The measurement takes the in-cident wave, described by a coherent state |α〉, to a re-flected wave described by a (phase shifted) coherent state|r↑ ·α〉 or |r↓ ·α〉, where r↑/↓ is the qubit-dependent reflec-tion amplitude given in Eq. (4.23). Considering now thefull state of the qubit plus detector, measurement resultsin:

1√2

(| ↑〉+ | ↓〉

)⊗ |α〉 → 1√

2

(e+iω01t/2| ↑〉 ⊗ |r↑ · α〉

+ e−iω01t/2| ↓〉 ⊗ |r↓ · α〉)(4.46)

As |r↑ · α〉 *= |r↓ · α〉, the qubit has become entangledwith the detector: the state above cannot be written as a

product of a qubit state times a detector state. To assessthe coherence of the final qubit state (i.e. the relativephase between ↑ and ↓), one looks at the off-diagonalmatrix element of the qubit’s reduced density matrix:

ρ↓↑ ≡ Tr detector〈↓ |ψ〉〈ψ| ↑〉 (4.47)

=e+iω01t

2〈r↑ · α|r↓ · α〉 (4.48)

=e+iω01t

2exp

[−|α|2

(1− r∗↑r↓

)](4.49)

In Eq. (4.48) we have used the usual expression for theoverlap of two coherent states. We see that the measure-ment reduces the magnitude of ρ↑↓: this is dephasing.The amount of dephasing is directly related to the over-lap between the different detector states that result whenthe qubit is up or down; this overlap can be straightfor-wardly found using Eq. (4.49) and |α|2 = N = Nt, whereN is the mean number of photons that have reflected fromthe cavity after time t. We have

∣∣exp[−|α|2

(1− r∗↑r↓

)]∣∣ = exp[−2Nθ2

0

]≡ exp [−Γϕt]

(4.50)with the dephasing rate Γϕ being given by:

Γϕ = 2θ20N (4.51)

in complete agreement with the previous result inEq.(4.44).

In closing, we note that for strong coupling (outsidethe regime of weak measurements), the light shift perphoton can actually exceed the line width of the qubitand the qubit transition spectrum splits into a series ofresolved lines which have been observed both via ‘quan-tum revivals’ in the time domain (Haroche and Raimond,2006; Raimond et al., 2001) using Rydberg atom cavityQED, and in the frequency domain (Schuster et al., 2007)using superconducting circuit QED.

2. Quantum limit relation for QND qubit state detection

We now return to the quantum limit relation ofEq. (4.45). We have described the weak, continuous QNDmeasurement of a qubit by a cavity with two rates: Γmeas,which tells us how quickly information is acquired, andΓϕ, which tells us how quickly the back-action of the mea-surement decoheres a superposition. As we saw above,these rates are not completely independent: quantummechanics enforces the constraint that the best you canpossibly do is measure as quickly as you dephase (Averin,2003; Clerk et al., 2003; Devoret and Schoelkopf, 2000;Korotkov and Averin, 2001; Makhlin et al., 2001):

Γmeas ≤ Γϕ (4.52)

A quantum limited detector (such as the ideal cavitymeasurement described above) is one which has an equal-ity above; in general, most detectors are very far from

23

this ideal limit, and dephase the qubit much faster thanthey acquire information about its state. We will give arigorous proof of Eq. (4.52) in Sec. V.B; for now, we notethat its heuristic origin rests on the fact that both mea-surement and dephasing rely on the qubit becoming en-tangled with the detector. Consider again Eq. (4.46), de-scribing the evolution of the qubit-detector system whenthe qubit is initially in a superposition of ↑ and ↓. Tosay that we have truly measured the qubit, the two de-tector states |r↑α〉 and |r↓α〉 need to correspond to dif-ferent values of the detector output (i.e. phase shift θ inour example); this necessarily implies they are orthog-onal. This in turn implies that the qubit is completelydephased: ρ↑↓ = 0, just as we saw in Eq. (4.5) in theStern-Gerlach example. Thus, measurement implies de-phasing. The opposite is not true. The two states |r↑α〉and |r↓α〉 could in principle be orthogonal without theircorresponding to different values of the detector output(i.e. θ). For example, the qubit may have become en-tangled with extraneous microscopic degrees of freedomin the detector. Thus, on a heuristic level, the origin ofEq. (4.52) is clear.

Returning to our one-sided cavity system, we see fromEq. (4.45) that the one-sided cavity detector reachesthe quantum limit. It is natural to now ask why thisis the case: is there a general principle in action herewhich allows the one-sided cavity to reach the quantumlimit? The answer is yes: reaching the quantum limitrequires that there is no ‘wasted’ information in the de-tector (Clerk et al., 2003). There should not exist any un-measured quantity in the detector which could have beenprobed to learn more about the state of the qubit. In thesingle-sided cavity detector, information on the state ofthe qubit is only available in (that is, is entirely encodedin) the phase shift of the reflected beam; thus, there is no‘wasted’ information, and the detector does indeed reachthe quantum limit. This principle of ‘no wasted informa-tion’ is discussed extensively in Clerk et al. (2003), andwe will expand on it in what follows.

We now consider a simple variant of the single-sidedcavity detector which fails to reach the quantum limitprecisely due to ‘wasted’ information. Consider a one-dimensional cavity system where both mirrors are slightlytransparent. Now, a wave incident at frequency ωR onone end of the cavity will be partially reflected and par-tially transmitted; if the initial incident wave is describedby a coherent state |α〉, the scattered state can be de-scribed by a tensor product of the reflected wave’s stateand the transmitted wave’s state:

|α〉 → |rσ · α〉|tσ · α〉 (4.53)

where the qubit-dependent reflection and transmissionamplitudes rσ and tσ are given by (Walls and Milburn,

1994):

t↓ =1

1 + 2iAQc(4.54)

r↓ =2iQcA

1 + 2iAQc(4.55)

(4.56)

with t↑ = (t↓)∗ and r↑ = (r↓)∗. Note that the in-cident beam is almost perfectly transmitted: |tσ|2 =1−O(AQc)2.

Similar to the one-sided case, the two-sided cavitycould be used to make a measurement by monitoring thephase of the transmitted wave. Using the expression fortσ above, we find that the qubit-dependent transmissionphase shift is given by:

θ↑/↓ = ±θ0 = ±2AQc (4.57)

where again the two signs correspond to the two differentqubit eigenstates. The phase shift for transmission isonly half as large as in reflection so the Wigner delaytime associated with transmission is

tWD =2κ

. (4.58)

Upon making the substitution of tWD for tWD, the one-sided cavity Eqs. (4.24-4.29) and Eq. (4.24) remain valid.However the internal cavity photon number shot noiseremains fixed so that Eq. (4.30) becomes

Snn = 2ntWD. (4.59)

which means that

Snn = 2N t2WD = 2SNN t2WD (4.60)

and

SFzFz = 2!2A2ω2c t2WDSNN . (4.61)

As a result the back action dephasing doubles relative tothe measurement rate and we have

Γmeas

Γϕ= 2SNNSθθ =

12. (4.62)

Thus the two-sided cavity fails to reach the quantumlimit by a factor of 2.

Using the entanglement picture, we may again alterna-tively calculate the amount of dephasing from the overlapbetween the detector states corresponding to the qubitstates ↑ and ↓ (cf. Eq. (4.48)). We find:

e−Γϕt =∣∣∣〈t↑α|t↓α〉〈r↑α|r↓α〉

∣∣∣ (4.63)

= exp[−|α|2 (1− (t↑)∗t↓ − (r↑)∗r↓)

](4.64)

Note that both the change in the transmission and reflec-tion amplitudes contribute to the dephasing of the qubit.Using the expressions above, we find:

Γϕt = 4(θ0)2|α|2 = 4(θ0)2N = 4(θ0)2Nt = 2Γmeast.(4.65)

24

We thus find that measurement-induced dephasing rateis twice as large as the measurement rate: the two sidedcavity misses the quantum limit by a factor of two inagreement with our quantum noise result that the backaction noise is doubled.

Why does the two-sided cavity fail to reach the quan-tum limit? The answer is clear from Eq. (4.64): eventhough we are not monitoring it, there is information onthe state of the qubit available in the phase of the re-flected wave. Note from Eq. (4.55) that the magnitudeof the reflected wave is weak (∝ A2), but (unlike thetransmitted wave) the difference in the reflection phaseassociated with the two qubit states is large (±π/2). The‘missing information’ in the reflected beam makes a directcontribution to the dephasing rate (i.e. the second term inEq. (4.64)), making it larger than the measurement rateassociated with measurement of the transmission phaseshift. In fact, there is an equal amount of informationin the reflected beam as in the transmitted beam, so thedephasing rate is doubled.

We note in closing that it is often technically easierto work in transmission rather than reflection. One canshow however that one may still reach the quantum limitin transmission by using an asymmetric cavity in whichthe input mirror has much less transmission than theoutput mirror. Most photons are reflected at the input,but those that enter the cavity will almost certainly betransmitted. The price to be made is that the inputcarrier power must be increased.

In summary, we see the departure from the quantumlimit here can be directly traced to “wasted” informa-tion in the detector. This idea holds up for a numberof very different systems: in Clerk et al. (2003), it wasdiscussed in the context of mesoscopic scattering detec-tors (i.e. a generalized quantum point-contact detector),while in Appendix H, we consider the role of informationin the important case of a Mach-Zehnder-interferometerdetector.

3. Measurement of oscillator position using a resonant cavity

In the previous subsections, we studied how a resonantcavity could be used to make a QND measurement of thestate of a qubit. We found that the photon shot noisein the output beam limited the precision with which thecavity phase shift could be determined; it thus set themeasurement rate. We also found that fluctuations inthe cavity photon number produced a random light shiftof the qubit’s energy splitting, causing dephasing. Thisexample of measurement back action causing dephasingis the simplest possible one because the measurement isQND: the back-action does not affect the observable be-ing measured. The measurement is repeatable preciselybecause the back action noise only affects the relativephase of the superposition of the two qubit states, butcauses no transitions between them.

In this subsection we consider the simplest example of

a non-QND measurement, namely the weak continuousmeasurement of the position of a harmonic oscillator. Asa detector, we again use a parametrically-coupled reso-nant cavity, where the position of the oscillator x changesthe frequency of the cavity as per Eq. (4.22) (see, e.g.,Tittonen et al. (1999)). Similar to the qubit case, for asufficiently weak coupling the phase shift of the reflectedbeam from the cavity will depend linearly on the posi-tion x of the oscillator (c.f. Eq. (4.25)); by reading outthis phase, we may thus measure x. The back actionnoise is the same as before, namely photon shot noise inthe cavity. Now however this represents a random forcewhich changes the momentum of the oscillator. Duringthe subsequent time evolution these random force pertur-bations will reappear as random fluctuations in the posi-tion. Thus the measurement is not QND. This will meanthat the minimum uncertainty of even an ideal measure-ment is larger (by exactly a factor of 2) than the ‘true’quantum uncertainty of the position (i.e. the ground stateuncertainty). We will see later that this is an example ofa general principle that a linear ‘phase-preserving’ ampli-fier necessarily adds noise, and that the minimum addednoise exactly doubles the output noise for the case wherethe input is vacuum (i.e. zero-point) noise.

In this subsection we will present a highly simplifiedand heuristic discussion of the standard quantum limitfor position measurement which will convey the essentialideas. A more general and complete discussion of thequantum limit on amplifiers and position detectors willbe presented in Sec. VI.

Before proceeding, it is worthwhile to make some pre-liminary points. First, note that we are speaking hereof a weak continuous measurement of the oscillator po-sition. The measurement is sufficiently weak that theposition undergoes many cycles of oscillation before sig-nificant information is acquired. Thus we are not talkingabout the instantaneous position but rather the overallamplitude and phase, or more precisely the two quadra-ture amplitudes describing the smooth envelope of themotion,

x(t) = X cos(Ωt) + Y sin(Ωt). (4.66)

Comparison with Eq. (2.18) shows that the two quadra-ture amplitudes X and Y are canonically conjugate andhence do not commute with each other

[X, Y ] =i!

MΩ= 2ix2

ZPF. (4.67)

Because the measurement is weak, we are effectively try-ing to measure two incompatible observables simultane-ously. This is also intimately related to the necessity ofoverall measurement uncertainty mentioned above.

It is also worth emphasizing that the quantum limiton continuous position detection is a very different kindof constraint than that on QND qubit state detection.In the latter case, the quantum limit did not limit theaccuracy with which we could measure the state of thequbit. The quantum limit simply set a minimum on the

25

magnitude of the back-action dephasing; as the measure-ment is QND, the size of this back-action had no effect onthe measurement accuracy. We also found that the pre-cise value of the coupling strength to the detector wasnot important: if the coupling is made weaker, it takeslonger to make the measurement, but since it is QND,this does not matter (assuming the qubit has no intrinsicdecay mechanisms). In contrast, for position measure-ments, back-action does affect the quantity being mea-sured, and thus the quantum limit does indeed place alimit on the accuracy of the measurement. We will alsosee that in order to achieve the minimum uncertainty (tobe defined below), one must adjust the coupling strengthto an optimal value. If the coupling is too weak, thephase shift of the cavity is so small that the imprecisionof the measurement is dominated by photon shot noise inthe output. If the coupling is too strong, the back actionnoise perturbs the oscillator so strongly that the positionuncertainty increases beyond the minimum.

We are now ready to start our heuristic analysis of po-sition detection using a cavity detector; further detailsand more rigorous formal analysis are presented in Ap-pendix D.3. Consider first the mechanical oscillator wewish to measure. We take it to be a simple harmonic os-cillator of natural frequency Ω and mechanical dampingrate γ0. For weak damping, and at zero coupling to thedetector, the spectral density of the oscillator’s positionfluctuations is given by Eq. (2.27) with the delta functionreplaced by a Lorentzian9

Sxx[ω] = x2ZPF

nB(!Ω)

γ0

(ω + Ω)2 + (γ0/2)2

+ [nB(!Ω) + 1]γ0

(ω − Ω)2 + (γ0/2)2

. (4.68)

When we now weakly couple the oscillator to the cavity(as per Eq. (4.22), with z = x/xZPF) and drive the cavityon resonance, the phase shift θ of the reflected beam willbe proportional to x (i.e. δθ(t) = [dθ/dx] ·x(t)). As such,the oscillator’s position fluctuations will cause additionalfluctuations of the phase θ, over and above the intrinsicshot-noise induced phase fluctuations Sθθ. We considerthe usual case where the noise spectrometer being usedto measure the noise in θ (i.e. the noise in the homo-dyne current) measures the symmetric-in-frequency noisespectral density; as such, it is the symmetric-in-frequencyposition noise that we will detect. In the classical limit

9 This form is valid only for weak damping because we are as-suming that the oscillator frequency is still sharply defined. Wehave evaluated the Bose-Einstein factor exactly at frequency Ωand we have assumed that the Lorentzian centered at positive(negative) frequency has negligible weight at negative (positive)frequencies.

kBT % !Ω, this is just given by:

Sxx[ω] ≡ 12

(Sxx[ω] + Sxx[−ω])

≈ kBT

2MΩ2

γ0

(|ω|− Ω)2 + (γ0/2)2(4.69)

where as always, we have chosen to define the sym-metrized spectral density as the average of the spectraldensities at positive and negative frequencies. This is incontrast to the convention used in engineering contexts,where one simply takes the sum of the two spectral den-sities. If we ignore back-action effects, we expect to seethis Lorentzian profile riding on top of the backgroundimprecision noise floor; this is illustrated in Fig. 5.

Note that additional stages of amplification would alsoadd noise, and would thus further augment this back-ground noise floor. If we subtract off this noise floor, theFWHM of the curve will give the damping parameter γ0,and the area under the experimental curve

∫ ∞

−∞

dω

2πSxx[ω] =

kBT

MΩ2(4.70)

measures the temperature. What the experimentalistactually plots in making such a curve is the output ofthe entire detector-plus-following-amplifier chain. Impor-tantly, if the temperature is known, then the area of themeasured curve can be used to calibrate the couplingof the detector and the gain of the total overall ampli-fier chain (see, e.g., Flowers-Jacobs et al., 2007; LaHayeet al., 2004). One can thus make a calibrated plot wherethe measured output noise is referred back to the oscilla-tor position. Such a plot also calibrates the backgroundnoise floor in terms of an effective oscillator temperature:this is related to the detector’s “noise temperature” (aquantity to be discussed further in Sec. VI.D).

At zero temperature, Eq. (4.68) yields for the sym-metrized noise spectral density

S0xx[ω] = x2

ZPFγ0/2

(|ω|− Ω)2 + (γ0/2)2. (4.71)

One might expect that one could see this Lorentzian di-rectly in the output noise of the detector (i.e. the θ noise),sitting above the measurement-imprecision noise floor.However, this neglects the effects of measurement backaction. From the classical equation of motion

x = −Ω2x− γ0x +F

M(4.72)

(where F = Fz), we expect the response of the oscillatorto the back action force F at frequency ω produces anadditional displacement

δx[ω] = χxx[ω]F [ω] (4.73)

where χxx[ω] is the mechanical susceptibility

χxx[ω] ≡ 1M

1Ω2 − ω2 − iγ0ω

. (4.74)

26

12

10

8

6

4

2

0

Ouput N

ois

e

!!"#

(arb

. units)

2.01.51.00.50.0Frequency !#$#"

area prop. to T

measurement imprecision

FIG. 5 (Color online) Spectral density of the (symmetrized)output noise Sθθ[ω] of a linear position detector. The oscilla-tor’s noise appears as a Lorentzian sitting above a noise floor(i.e. the measurement imprecision). As discussed in the text,the width of the peak is proportional to the oscillator damp-ing rate γ0, while the area under the peak is proportional totemperature. This latter fact can be used to calibrate theresponse of the detector.

These extra oscillator fluctuations will show up as addi-tional fluctuations in the output of the detector (i.e. thephase shift θ). For simplicity, let us focus on this noise atthe oscillator’s resonance frequency Ω. As a result of thedetector’s back-action, the total measured position noise(i.e. inferred spectral density) at the frequency Ω is givenby:

Sxx,tot[Ω] = S0xx[Ω] +

|χxx[Ω]|2

2[SFF [+Ω] + SFF [−Ω]]

+12

[SI

xx[+Ω] + SIxx[−Ω]

](4.75)

= S0xx[Ω] + Sxx,add[Ω] (4.76)

The first term here is just the intrinsic zero-point noiseof the oscillator:

S0xx[Ω] =

2x2ZPF

γ0= !|χxx[Ω]|. (4.77)

The second term Sxx,add is the total noise added by themeasurement, and includes both the measurement impre-cision SI

xx ≡ SIzzx

2ZPF and the extra fluctuations caused

by the back action. We stress that Sxx,tot corresponds toa position noise spectral density inferred from the outputof the detector: one simply scales the spectral density oftotal output fluctuations Sθθ,tot[Ω] by (dθ/dx)2.

Implicit in Eq. (4.77) is the assumption that the backaction noise and the imprecision noise are uncorrelatedand thus add in quadrature. It is not obvious that this is

correct, since in the case of position measurement usingparametric coupling to a cavity, the back action noiseand output shot noise are both caused by the vacuumnoise in the beam incident on the cavity. It turns outthere are indeed correlations, however the symmetrized(i.e. ‘classical’) correlator SθF does vanish under the rightconditions; we will show this in detail for this partic-ular system in Appendix D and discuss it more gener-ally in Sec. V. Further, Eq. (4.75) assumes that themeasurement does not change the damping rate of theoscillator. One can show that this is indeed the casewhen the cavity is illuminated at its optical resonance,as in this case the back-action force’s quantum noise spec-tral density is symmetric in frequency (see Appendix D).The back-action force thus does not result in any ad-ditional measurement-induced damping of the oscillator(c.f. Eq. (3.32)). Note that this is a special situation:for a generic detector, detector back-action will changethe total damping of the oscillator (and in fact can heator cool the oscillator, depending on the setup); we willdiscuss this further in Sec. VI.D.

Assuming we have a quantum limited detector thatobeys Eq. (4.20) (i.e. SI

xxSFF = !2/4) and that theshot noise is symmetric in frequency, the added positionnoise spectral density at resonance (i.e. second term inEq. (4.76)) becomes:

Sxx,add[Ω] =[|χ[Ω]|2SFF +

!2

41

SFF

]. (4.78)

Recall from Eq. (4.33) that the back action noise is pro-portional to the coupling of the oscillator to the detectorand to the intensity of the drive on the cavity. The addedposition uncertainty noise is plotted in Fig. 6 as a func-tion of SFF . We see that for high drive intensity, the backaction noise dominates the position uncertainty, while forlow drive intensity, the output shot noise (the last termin the equation above) dominates.

The added noise (and hence the total noise Sxx,tot[Ω])is minimized when SFF is tuned to be equal to SFF,opt,with:

SFF,opt =!

2|χxx[Ω]| =!2MΩγ0. (4.79)

We see that the more heavily damped is the oscillator,the less susceptible it is to back action noise and hencethe higher is the optimal coupling. At the optimal point

Sxx,add[Ω] =!

MΩγ0= S0

xx[Ω] (4.80)

Thus, the spectral density of the added position noise isexactly equal to the noise power associated with the os-cillator’s zero-point fluctuations. This represents a mini-mum value for the added noise of any linear position de-tector, and is referred to as the standard quantum limiton position detection [CITE]. Note that this limit onlyinvolves the added noise of the detector, and thus has

27

10

8

6

4

2

a

dd

[ (

ZP

T)2

/ !

0 ]

0.1 1 10

Coupling / opt

Total added position noise Measurement imprecision Contribution from backaction

FIG. 6 (Color online) Noise power of the added position noiseof a linear position detector, evaluated at the oscillator’s reso-nance frequency (Sxx,add[Ω]), as a function of the magnitudeof the back-action noise spectral density SFF . SFF is pro-portional to the oscillator-detector coupling, and in the caseof the cavity detector, is also proportional to the power in-cident on the cavity. The optimal value of SFF is given bySFF,opt = !MΩγ0/2 (c.f. Eq. (4.79)). We have assumed thatthere are no correlations between measurement imprecisionnoise and back-action noise, as is appropriate for the cavitydetector.

nothing to do with the initial temperature of the oscil-lator. We again emphasize that in order to reach thisquantum limit, the detector must be quantum limited:one needs the product SFF SI

xx to be as small as is al-lowed by quantum mechanics (i.e. !2/4). In addition,the measured output noise must be dominated by theoutput noise of the cavity, not by the added noise of fol-lowing amplifier stages. Finally, one must also be able toachieve sufficiently strong coupling to reach the optimumgiven in Eq. (4.79). Note that at this optimal coupling,the measurement imprecision noise and back-action noiseeach make equal contributions to the total measured po-sition uncertainty Sxx,tot. A related, stronger quantumlimit refers to the total inferred position noise from themeasurement, Sxx,tot[ω]. It follows from Eq. (4.80) thatat resonance, the smallest this can be is twice the oscil-lator’s zero point noise:

Sxx,tot[Ω] = 2S0xx[Ω]. (4.81)

Half the noise here is from the oscillator itself, half is fromthe added noise of the detector. Reaching this quantumlimit is even more challenging: one needs both to reachthe quantum limit on the added noise and cool the oscil-lator to its ground state. Note that recent experiments inquantum nanomechanics have come very close to reach-ing the quantum limit on the added noise, but have beenless close in reaching the limit on the total measured

position noise. See Appendix VIII for further discus-sion of practical issues relevant to reaching the quan-tum limit. the minimum (symmetrized) noise power (atthe resonance frequency) is twice the vacuum value– theadded noise from the measurement is precisely equal isthe so called ‘standard quantum limit’ for position mea-surement uncertainty.

Finally, we emphasize that the optimal value of thecoupling derived above was specific to the choice of min-imizing the total position noise power at the resonancefrequency. If a different frequency had been chosen, theoptimal coupling would have been different; one againfinds that the minimum possible added noise correspondsto the ground state noise at that frequency. It is useful toask what the total position noise would be as a functionof frequency, assuming that the coupling has been opti-mized to minimize the noise at the resonance frequency,and that the oscillator is initially in the ground state.From our results above we have

Sxx,tot[ω]

= x2ZPF

γ0/2(|ω|− Ω)2 + (γ0/2)2

+!2

[|χxx[ω]|2

|χxx[Ω]| + |χxx[Ω]|]

≈ x2ZPF

γ0

1 + 3

(γ0/2)2

(|ω|− Ω)2 + (γ0/2)2

(4.82)

which is plotted in Fig. 7. Assuming that the detector isquantum limited, one sees that the Lorentzian peak risesabove the constant background by a factor of three whenthe coupling is optimized to minimize the total noisepower at resonance. This factor of three has a simpleinterpretation. Recall that if we optimize the couplingto reach the quantum limit at resonance, the total addednoise due to the detector (back-action and imprecision)is equal to the oscillator’s zero-point noise. For an opti-mized coupling, back-action and imprecision make equalcontributions to the total added noise; this implies thatback-action noise heats the oscillator from zero tempera-ture so as to increase its position variance by precisely afactor 3/2. It also implies that the background measure-ment imprecision has (at resonance) a value equal to 1/2the oscillator’s zero-point noise. This then yields a peak-to-floor ratio of 3. Note the same maximum ratio is ob-tained when one tries to detect coherent qubit rotationsfrom the noise of a linear detector which is transverselycoupled to the qubit (Korotkov and Averin, 2001); thisis also a non-QND situation.

In Table II, we give a summary of recent experimentswhich approach the quantum limit on weak, continuousposition detector of a mechanical resonator. Note that inmany of these experiments, the effects of detector back-action were not seen. This could either be the result oftoo low of a detector-oscillator coupling, or due to thepresence of excessive thermal noise. As we have shown,the back-action force noise serves to slightly heat the os-cillator. If it is already at an elevated temperature due tothermal noise, this additional heating can be very hardto resolve.

28

2.0

1.5

1.0

0.5

0.01.20.80.40.0

Frequency ω/Ω

equilibriumnoise (T=0)

eq. + backaction noise

total noiseS xx,

tot [ω

] / S0 xx

[Ω]

FIG. 7 (Color online) Spectral density of measured positionfluctuations of a harmonic oscillator, Sxx,tot[ω], as a functionof frequency ω, for a detector which reaches the quantumlimit at the oscillator frequency Ω. We have assumed thatwithout the coupling to the detector, the oscillator would bein its ground state. The y-axis has been normalized by thezero-point position-noise spectral density S0

xx[ω], evaluated atω = Ω. One clearly sees that the total noise at Ω is twice thezero-point value, and that the peak of the Lorentzian risesa factor of three above the background. This backgroundrepresents the measurement-imprecision, and is equal to 1/2of S0

xx(Ω).

In closing, we stress that this subsection has given onlya very rudimentary introduction to the quantum limit onposition detection. A complete discussion which treatsthe important topics of back-action damping, effectivetemperature, noise cross-correlation and power gain isgiven in Sec. VI.D.

V. GENERAL LINEAR RESPONSE THEORY

A. Quantum constraints on noise

In the previous section, we introduced the basic lan-guage and concepts used to describe weak, continuousquantum measurements. We also gave a heuristic discus-sion of how quantum constraints on the noise propertiesof a detector lead to quantum limits on various measure-ment tasks, focusing on the particular case of a resonant-cavity detector. In this section, we will develop this con-nection between quantum limits and noise further and ina more general manner. As before, we will emphasize theidea that reaching the quantum limit requires a detector

having “quantum-ideal” noise properties. The approachhere is different from typical treatments in the quantumoptics literature (Gardiner and Zoller, 2000; Haus, 2000),and uses nothing more than features of quantum linearresponse. Our discussion here will expand upon Clerk(2004); Clerk et al. (2003); somewhat similar approachesto quantum measurement are also discussed in Braginskyand Khalili (1992) and Averin (2003).

In this subsection, we will start by heuristically sketch-ing how constraints on noise (similar to Eq. (4.20) for thecavity detector) can emerge directly from the Heisenberguncertainty principle. We then present a rigorous andgeneral quantum constraint on noise. We introduce boththe notion of a generic linear response detector, and thebasic quantum constraint on detector noise. In the nextsubsection (C), we will discuss how this noise constraintleads to the quantum limit on QND state detection of aqubit. The quantum limit on a linear amplifier (or a po-sition detector) is discussed in detail in the next section.

1. Heuristic weak-measurement noise constraints

As we already stressed at the start of Sec. IV.A, thereis no fundamental quantum limit on the accuracy withwhich a given observable can be measured, at least notwithin the framework of non-relativistic quantum me-chanics. For example, one can, in principle, measure theposition of a particle to arbitrary accuracy in the courseof a projection measurement 10. However, the situationis different when we specialize to continuous, non-QNDmeasurements. Such a measurement can be envisagedas a series of instantaneous measurements, in the limitwhere the spacing between the measurements δt is takento zero. Each measurement in the series has a limitedresolution and perturbs the conjugate variables, therebyaffecting the subsequent dynamics and measurement re-sults. Let us discuss this briefly for the example of aseries of position measurements of a free particle.

After initially measuring the position with an accuracy∆x, the momentum suffers a random perturbation of size∆p ≥ !/(2∆x). Consequently, a second position mea-surement taking place a time δt later will have an addi-tional uncertainty of size δt(∆p/m) ∼ !δt/(m∆x). Thus,when trying to obtain a good estimate of the position byaveraging several such measurements, it is not optimalto make ∆x too small, because otherwise this additionalperturbation, called the “back-action” of the measure-ment device, will become large. The back-action can bedescribed as a random force ∆F = ∆p/δt. A meaningfullimit δt → 0 is obtained by keeping both ∆x2δt ≡ Sxx

10 In relativistic theory, the possibility of pair production wouldnot allow to have a resolution finer than the Compton wave-length (Berestetskii et al., 1982), but this will be irrelevant forour discussion of detectors in the solid state and quantum opticscontexts

29

TABLE II Synopsis of recent experiments approaching the quantum limit on continuous position detection of a mechanicalresonator. The second column corresponds to the best measurement imprecision noise spectral density SI

xx achieved in theexperiment. All spectral densities are at the oscillator’s resonance frequency Ω. As discussed in the text, there is no quantumlimit on how small one can make SI

xx. The third column presents the product of the measured imprecision noise (unlessotherwise noted) and measured back-action noises, divided by !/2; this quantity must be one to achieve the quantum limit onthe added noise.

Experiment Mechanical Imprecision noise Detector Noise Product

frequency [Hz] vs. zero-point noisep

SIxxSFF /(!/2)

Ω/(2π)p

SIxx/S0

xx[Ω]

Cleland et al. (2002) (quantum point contact) 1.5× 106 4.2× 104 back-action not detected

Knobel and Cleland (2003) (single-eletron transistor) 1.2× 108 1.8× 102 back-action not detected

LaHaye et al. (2004) (single-electron transistor) 2.0× 107 5.4 back-action not detected

Naik et al. (2006) (single-electron transistor) 2.2× 107 0.33 8.1× 102

(if SIxx had been limited by SET shot noise) 1.5× 101

Arcizet et al. (2006) (optical cavity) 8.1× 105 19 back-action not detected

Flowers-Jacobs et al. (2007) (atomic point-contact) 4.3× 107 29 1.7× 103

Regal et al. (2008) (microwave cavity) 2.4× 105 30 back-action not detected

Schliesser et al. (2008) (optical cavity) 4.1× 107 9.9 back-action not detected

Poggio et al. (2008) (quantum point contact) 5.2× 103 63 back-action not detected

Etaki et al. (2008) (d.c. SQUID) 2.0× 106 47 back-action not detected

and ∆p2/δt ≡ SFF fixed. In this limit, the deviationsδx(t) describing the finite measurement accuracy andthe fluctuations of the back-action force F can be de-scribed as white noise processes, 〈δx(t)δx(0)〉 = Sxxδ(t)and 〈F (t)F (0)〉 = SFF δ(t). The Heisenberg uncertaintyrelation ∆p∆x ≥ !/2 then implies SxxSFF ≥ !2/4 (Bra-ginsky and Khalili, 1992). Note this is completely analo-gous to the relation Eq. (4.20) we derived for the resonantcavity detector using the fundamental number-phase un-certainty relation. In this section, we will derive rigor-ously more general quantum limit relations on noise spec-tral densities of this form.

2. Generic linear-response detector

To rigorously discuss the quantum limit, we would liketo start with a description of a detector which is as gen-eral as possible. To that end, we will think of a detectoras some physical system (described by some unspecifiedHamiltonian Hdet and some unspecified density matrixρ0) which is time-independent in the absence of couplingto the signal source. The detector has both an inputport, characterized by an operator F , and an outputport, characterized by an operator I (see Fig. 8). Theoutput operator I is simply the quantity which is read-out at the output of the detector (e.g., the current in asingle-electron transistor, or the phase shift in the cavitydetector of the previous section). The input operator F isthe detector quantity which directly couples to the inputsignal (e.g., the qubit), and which causes a back-actiondisturbance of the signal source; in the cavity exampleof the previous section, we had F = n, the cavity pho-ton number. As we are interested in weak couplings, we

input outputoperator

detector

signalsource

"load"

FIG. 8 (Color online) Schematic of a generic linear responsedetector.

will assume a simple bilinear form for the detector-signalinteraction Hamiltonian:

Hint = AxF (5.1)

Here, the operator x (which is not necessarily a positionoperator) carries the input signal. Note that because xbelongs to the signal source, it necessarily commutes withthe detector variables I , F .

We will always assume the coupling strength A to besmall enough that we can accurately describe the outputof the detector using linear response. We thus have:

〈I(t)〉 = 〈I〉0 + A

∫dt′χIF (t− t′)〈x(t′)〉, (5.2)

where 〈I〉0 is the input-independent value of the detectoroutput at zero-coupling, and χIF (t) is the linear-response

30

susceptibility or gain of our detector. Note that in Clerket al. (2003) and Clerk (2004), this gain coefficient isdenoted λ. Using standard time-dependent perturba-tion theory in the coupling Hint, one can easily deriveEq. (5.2), with χIF (t) given by a Kubo-like formula:

χIF (t) = − i

!θ(t)⟨[

I(t), F (0)]⟩

0(5.3)

Here (and in what follows), the operators I and F areHeisenberg operators with respect to the detector Hamil-tonian, and the subscript 0 indicates an expectation valuewith respect to the density matrix of the uncoupled de-tector.

As we have already discussed, there will be unavoid-able noise in both the input and output ports of ourdetector. This noise is subject to quantum mechanicalconstraints, and its presence is what limits our ability tomake a measurement or amplify a signal. We thus need toquantitatively characterize the noise in both these ports.Recall from the discussion in Sec. III.B that it is thesymmetric-in-frequency part of a quantum noise spectraldensity which plays a role akin to classical noise. Wewill thus want to characterize the symmetrized noise cor-relators of our detector (denoted as always with a bar).Redefining these operators so that their average value iszero at zero coupling (i.e. F → F − 〈F 〉0, I → I − 〈I〉0),we have:

SFF [ω] ≡ 12

∫ ∞

−∞dteiωt〈F (t), F (0)〉0 (5.4a)

SII [ω] ≡ 12

∫ ∞

−∞dteiωt〈I(t), I(0)〉0 (5.4b)

SIF [ω] ≡ 12

∫ ∞

−∞dteiωt〈I(t), F (0)〉0 (5.4c)

where , indicates the anti-commutator, SII representsthe intrinsic noise in the output of the detector, and SFF

describes the back-action noise seen by the source of theinput signal. In general, there will be some correlationbetween these two kinds of noise; this is described by thecross-correlator SIF .

Finally, it will also be useful to introduce the reversegain χFI of our detector. This is the response coefficientdescribing an experiment where we couple our input sig-nal to the output port of the detector (i.e. Hint = AxI),and attempt to observe something at the input port(i.e. in 〈F (t)〉). In complete analogy to Eq. (5.2), onewould then have:

〈F (t)〉 = 〈F 〉0 + A

∫dt′χFI(t− t′)〈x(t′)〉 (5.5)

with:

χFI(t) = − i

!θ(t)〈[F (t), I(0)

]〉0 (5.6)

Note that in Clerk et al. (2003) and Clerk (2004), thereverse gain coefficient χFI is denoted λ′.

The reverse gain is indeed something we need to worryabout (and appears in many standard classical electricalamplifiers such as op amps (Boylestad and Nashelsky,2006)). To make a measurement of the output operatorI, we must necessarily couple to it in some manner. IfχFI *= 0, the noise associated with this coupling could inturn lead to additional back-action noise in the operatorF . Even if the reverse gain did nothing but amplify vac-uum noise entering the output port, this would heat upthe system being measured at the input port and henceproduce excess back action. Thus, the ideal situation isto have χFI = 0, implying a high asymmetry between theinput and output of the detector. We note that almostall mesoscopic detectors that have been studied in detail(e.g. single electron transistors and generalized quantumpoint contacts) have been found to have a vanishing re-verse gain: χFI = 0 (Clerk et al., 2003) 11. For thisreason, we will often assume the ideal situation whereχFI = 0 in what follows.

Before proceeding, it is worth emphasizing that thereis a relation between the detector gains χIF and χFI

and the unsymmetrized I-F quantum noise correlator,SIF [ω]. This spectral density, which need not be sym-metric in frequency, is defined as:

SIF [ω] =∫ ∞

−∞dteiωt〈I(t)F (0)〉0 (5.7)

Using the definitions, one can easily show that:

SIF [ω] =12

[SIF [ω] + SIF [−ω]∗] (5.8a)

χIF [ω]− χFI [ω]∗ = − i

! [SIF [ω]− SIF [−ω]∗](5.8b)

Thus, while SIF represents the classical part of the I-Fquantum noise spectral density, the gains χIF , χFI aredetermined by the quantum part of this spectral density.This also demonstrates that though the gains have anexplicit factor of 1/! in their definitions, they have awell defined ! → 0 limit, as the asymmetric-in-frequencypart of SIF [ω] vanishes in this limit.

3. Quantum constraint on noise

Despite having said nothing about the detec-tor’s Hamiltonian or state (except that it is time-independent), we can nonetheless derive a very generalquantum constraint on its noise properties. Note firstthat for purely classical noise spectral densities, one al-ways has the inequality:

SII [ω]SFF [ω]−∣∣SIF [ω]

∣∣2 ≥ 0 (5.9)

11 Note that the fact χFI vanishes for a mesoscopic scattering de-tector is used directly to define the full counting statistics ofcurrent; see Levitov (2003) for a discussion

31

This simply expresses the fact that the correlation be-tween two different noisy quantities cannot be arbitrarilylarge; it follows immediately from the Schwartz inequal-ity. In the quantum case, this simple constraint becomesmodified. Defining for convenience:

χIF [ω] ≡ χIF [ω]− [χFI [ω]]∗ (5.10a)

∆[z] =∣∣1 + z2

∣∣−(1 + |z|2

)

2, (5.10b)

we will show below that the following quantum noise in-equality (involving symmetrized noise correlators) is al-ways valid (see also Eq. (6.36) in (Braginsky and Khalili,1996)):

SII [ω]SFF [ω]−∣∣SIF [ω]

∣∣2 ≥∣∣∣∣!χIF [ω]

2

∣∣∣∣2 (

1 + ∆[

SIF [ω]!χIF [ω]/2

])(5.11)

To interpret this inequality, note that if our detector hasno positive feedback, then Re χIF · χFI ≤ 0, implying| ˜χIF | *= 0. Also note that 1 + ∆[z] ≥ 0. Eq. (5.11)thus tells us that in general, if our detector has gain andno positive feedback, it must have a minimum amount ofback-action and output noise; moreover, these two noisescannot be perfectly anti-correlated. Note that in manycases the last term in Eq. (5.11) (involving ∆[z]) is identi-cally zero; we will comment more on this in what follows.

Though Eq. (5.11) may appear reminiscent to the stan-dard fluctuation dissipation theorem, its origin is quitedifferent: in particular, the quantum noise constraint ap-plies irrespective of whether the detector is in equilib-rium. Eq. (5.11) instead follows directly from Heisen-berg’s uncertainty relation applied to the frequency rep-resentation of the operators I and F . In its most generalform, the Heisenberg uncertainty relation gives a lowerbound for the uncertainties of two observables in terms oftheir commutator and their noise correlator (Gottfried,1966):

(∆A)2(∆B)2 ≥ 14

⟨A, B

⟩2+

14

∣∣∣⟨[

A, B]⟩∣∣∣

2. (5.12)

Here we have assumed 〈A〉 = 〈B〉 = 0. Let us now choosethe Hermitian operators A and B to be given by thecosine-transforms of I and F , respectively, over a finitetime-interval T :

A ≡√

2T

∫ T/2

−T/2dt cos(ωt + δ) I(t) (5.13a)

B ≡√

2T

∫ T/2

−T/2dt cos(ωt) F (t) (5.13b)

Note that we have phase shifted the transform of I rela-tive to that of F by a phase δ. In the limit T → ∞ we

find, at any finite frequency ω *= 0:

(∆A)2 = SII [ω], (∆B)2 = SFF [ω] (5.14a)⟨

A, B⟩

= 2Re eiδSIF [ω] (5.14b)⟨[

A, B]⟩

=∫ +∞

−∞dt cos(ωt + δ)

⟨[I(t), F (0)

]⟩

= i!Re[eiδ

(χIF [ω]− (χFI [ω])∗

)](5.14c)

In the last line, we have simply made use of theKubo formula definitions of the gain and reverse gain(cf. Eqs. (5.3) and (5.6)). As a consequence ofEqs. (5.14a)-(5.14c), the Heisenberg uncertainty relation(5.12) directly yields:

SII [ω]SFF [ω] ≥[Re

(eiδSIF [ω]

)]2+ (5.15)

!2

4[Re eiδ

(χIF [ω]− (χFI [ω])∗

)]2

Maximizing the RHS of this inequality over δ then yieldsthe general quantum noise constraint of Eq. (5.11).

With this derivation, we can now interpret the quan-tum noise constraint Eq. (5.11) as stating that the noiseat a given frequency in two observables, I and F , isbounded by the value of their commutator at that fre-quency. The fact that I and F do not commute is neces-sary for the existence of linear response (gain) from thedetector, but also means that the noise in both I andF cannot be arbitrarily small. A more detailed deriva-tion, yielding additional important insights, is describedin Appendix J.1.

Given the quantum noise constraint of Eq. (5.11), wecan now very naturally define a “quantum-ideal” detector(at a given frequency ω) as one which minimizes the LHSof Eq. (5.11)– a quantum-ideal detector has a minimalamount of noise at frequency ω. We will often be inter-ested in the ideal case where there is no reverse gain (i.e.measuring I does not result in additional back-actionnoise in F ); the condition to have a quantum limiteddetector thus becomes:

SII [ω]SFF [ω]−∣∣SIF [ω]

∣∣2 = (5.16)∣∣∣∣!χIF [ω]

2

∣∣∣∣2 (

1 + ∆[

SIF [ω]!χIF [ω]/2

])

where ∆[z] is given in Eq. (5.10b). Again, as we willdiscuss below, in most cases of interest (e.g. zero fre-quency and/or large amplifier power gain), the last termon the RHS will vanish. In the following sections, wewill demonstrate that the “ideal noise” requirement ofEq. (5.16) is necessary in order to achieve the quantumlimit on QND detection of a qubit, or on the added noiseof a linear amplifier.

Before leaving our general discussion of the quantumnoise constraint, it is worth emphasizing that achievingEq. (5.16) places a very strong constraint on the prop-erties of the detector. In particular, there must exist a

32

tight connection between the input and output ports ofthe detector– in a certain restricted sense, the operators Iand F must be proportional to one another (see Eq. (J13)in Appendix J.1). As is discussed in Appendix J.1, thisproportionality immediately tells us that a quantum-idealdetector cannot be in equilibrium. The proportionalityexhibited by a quantum-ideal detector is parameterizedby a single complex valued number α[ω], whose magni-tude is given by:

|α[ω]|2 = SII [ω]/SFF [ω] (5.17)

While this proportionality requirement may seem purelyformal, it does have a simple heuristic interpretation; asis discussed in Clerk et al. (2003), it may be viewed asa formal expression of the condition of no ‘wasted infor-mation’ discussed in the previous subsection.

4. Role of noise cross-correlations

We now return to the unwieldy second term in theRHS of the quantum noise constraint of Eq. (5.11). Notethat this term is sensitive to the phase of the quantitySIF [ω]/χIF : if this quantity is purely real, then the sec-ond term vanishes, as ∆[z] = 0 if z is real (cf. Eq (5.10b)).The result is a simpler looking noise constraint, as foundelsewhere in the literature (Averin, 2003; Clerk, 2004;Clerk et al., 2003). This simpler noise constraint is al-ways valid in the low-frequency limit ω → 0, as in thislimit χIF , χFI and SIF must all be real.

The situation is somewhat more delicate at finite fre-quencies: for finite ω, the quantity SIF [ω]/χIF [ω] neednot be purely real. In fact, as the astute reader mayhave already realized, if this quantity becomes large andpurely imaginary, then the RHS of Eq. (5.11) will vanish–in this limit, there is no additional quantum constrainton the noise beyond what already exists classically!

Despite this odd behaviour, there is no need to worry.For simplicity, we again focus on the case of a vanishingreverse gain χFI = 0. We show in Appendix J.2 that ifone wishes to have a quantum limited detector (i.e. equal-ity in Eq. (5.11)) and have true amplification, then theratio SIF [ω]/χIF [ω] must also be real. By “true ampli-fication”, we mean that the dimensionless power gain ofthe detector must be much larger than one. The powergain is a measure of how much power is provided by thedetector at its output compared to how much power isdrawn at its input; it will be defined more precisely inSec. VI.D.3. Thus, even at finite frequency, the last termon the RHS of Eq. (5.11) plays no role if we are interestedin a detector which truly amplifies (i.e. has a large powergain). The limit discussed above, where the noise con-straint vanishes because SIF /χIF is imaginary, is of noreal interest, as in this limit, our detector has no powergain. Simply put, if our detector performs no amplifica-tion, then there need not by any quantum constraint onits noise; see Sec. VI for a more complete discussion ofthis point.

Finally, an important corollary to the above point isthat at the quantum limit, the quantity SIF /χIF mustbe real. Thus, at the quantum limit, correlations betweenthe back-action force and the intrinsic output noise fluc-tuations must have the same phase as the gain χIF . Aswe discuss further in Sec. VI.E , this requirement canalso be interpreted in terms of wasted information.

B. Quantum limit on QND detection of a qubit

In Sec. IV.B, we discussed the quantum limit on QNDqubit detection in the specific context of a resonant cavitydetector. We will now show how the full quantum noiseconstraint of Eq. (5.11) directly leads to this quantumlimit for an arbitrary weakly-coupled detector. Similarto Sec. IV.B, we couple the input operator of our genericlinear-response detector to the σz operator of the qubitwe wish to measure (i.e. we take x = σz in Eq. (5.1));we also consider the QND regime, where σz commuteswith the qubit Hamiltonian. As we saw in Sec. IV.B,the quantum limit in this case involves the inequalityΓmeas ≤ Γϕ, where Γmeas is the measurement rate, andΓϕ is the back-action dephasing rate. For the latter quan-tity, we can directly use the results of our calculation forthe cavity system, where we found the dephasing ratewas set by the zero-frequency noise in the cavity pho-ton number (cf. Eq. 4.44). In complete analogy, theback-action dephasing rate here will be determined bythe zero-frequency noise in the input operator F of ourdetector:

Γϕ =2A2

!2SFF [0] (5.18)

The measurement rate (the rate at which informationon the state of qubit is acquired) is also defined in com-plete analogy to what was done for the cavity detector.We imagine we turn the measurement on at t = 0 andstart to integrate up the output I(t) of our detector:

m(t) =∫ t

0dt′I(t′) (5.19)

The probability distribution of the integrated outputm(t) will depend on the state of the qubit; for long times,we may approximate the distribution corresponding toeach qubit state as being gaussian. Noting that we havechosen I so that its expectation vanishes at zero coupling,the average value of 〈m(t)〉 corresponding to each qubitstate is (in the long time limit of interest):

〈m(t)〉↑ = AχIF [0]t (5.20a)〈m(t)〉↓ = −AχIF [0]t (5.20b)

The variance of both distributions is, to leading order,independent of the qubit state:

〈m2(t)〉↑/↓ − 〈m(t)〉2↑/↓ ≡ 〈〈m2(t)〉〉↑/↓ = SII [0]t (5.21)

33

For the last equality above, we have taken the long-timelimit, which results in the variance of m being determinedcompletely by the zero-frequency output noise SII [ω = 0]of the detector. The assumption here is that due tothe weakness of the measurement, the measurement time(i.e. 1/Γmeas) will be much longer than the autocorrela-tion time of the detector’s noise.

We can now define the measurement rate, in completeanalogy to the cavity detector of the previous section(cf. Eq. (4.39)), by how quickly the resolving power ofthe measurement grows 12:

14

[〈m(t)〉↑ − 〈m(t)〉↓]2

〈〈m2(t)〉〉↑ + 〈〈m2(t)〉〉↓≡ Γmeast. (5.22)

This yields:

Γmeas =A2 (χIF [0])2

2SII [0](5.23)

Putting this all together, we find that the “efficiency”ratio η = Γmeas/Γϕ is given by:

η ≡ Γmeas

Γϕ=

!2 (χIF [0])2

4SII [0]SFF [0](5.24)

The quantum-limit bound η ≤ 1 thus follows immedi-ately from the quantum noise constraint of Eq. (5.11).Further, achieving the quantum limit for QND detectionrequires a detector with quantum-ideal noise properties,as defined by Eq. (5.16), as well as a detector with a van-ishing noise cross-correlator: SIF [0] = 0. As discussed inAppendix J.1, achieving this ideal noise condition impliesa tight connection between the input and output ports ofthe detector. This condition also corresponds directly tothe idea that reaching the quantum limit requires thereto be no unused, “wasted information” in the detectoron the state of the qubit (Clerk et al., 2003).

VI. QUANTUM LIMIT ON LINEAR AMPLIFIERS ANDPOSITION DETECTORS

In the previous section, we established the fundamen-tal quantum constraint on the noise of any system ca-pable of acting as a linear detector; we further showedthat this quantum noise constraint directly leads to the“quantum limit” on non-demolition qubit detection us-ing a weakly-coupled detector. In this section, we turnto the more general situation where our detector is aphase-preserving quantum linear amplifier: the input to

12 The strange looking factor of 1/4 here is purely chosen for con-venience; we are defining the measurement rate based on theinformation theoretic definition given in Appendix E. This fac-tor of four is consistent with the definition used in the cavitysystem.

the detector is described some time-dependent operatorx(t) which we wish to have amplified at the output of ourdetector. As we will see, the quantum limit in this caseis a limit on how small one can make the noise added bythe amplifier to the signal. The discussion in this sectionboth furthers and generalizes the heuristic discussion ofposition detection using a cavity detector presented inSec. IV.B.

In this section, we will start by presenting a heuris-tic discussion of quantum constraints on amplification.We will then demonstrate explicitly how the previously-discussed quantum noise constraint leads directly to thequantum limit on the added noise of a phase-preservinglinear amplifier; we will examine both the cases of ageneric linear position detector and a generic voltage am-plifier, following the approach outlined in Clerk (2004).We will also spend time explicitly connecting the linearresponse approach we use here to the bosonic scatteringformulation of the quantum limit favoured by the quan-tum optics community (Caves, 1982; Courty et al., 1999;Grassia, 1998; Haus and Mullen, 1962), paying particularattention to the case of two-port scattering amplifier. Wewill see that there are some important subtleties involvedin converting between the two approaches. In particular,there exists a crucial difference between the case wherethe input signal is tightly coupled to the input of theamplifier (the case usually considered in the quantumoptics community), versus the case where, similar to anideal op-amp, the input signal is only weakly coupled tothe input of the amplifier (the case usually considered inthe solid state community).

A. Preliminaries on amplification

What exactly does one mean by ‘amplification’? Aswe will see (cf. Sec. VI.D.3), a precise definition requiresthat the energy provided at the output of the amplifierbe much larger than the energy drawn at the input ofthe amplifier– the “power gain” of the amplifier mustbe larger than one. For the moment, however, let uswork with the cruder definition that amplification in-volves making some time-dependent signal ‘larger’. Toset the stage, we will first consider an extremely sim-ple classical analogue of a linear amplifier. Imagine the“signal” we wish to amplify is the coordinate x(t) of aharmonic oscillator; we can write this signal as:

x(t) = x(0) cos(ωSt) +p(0)MωS

sin(ωSt) (6.1)

Our signal has two quadrature amplitudes, i.e. the am-plitude of the cosine and sine components of x(t). To“amplify” this signal, we start at t = 0 to parametricallydrive the oscillator by changing its frequency ωS periodi-cally in time: ωS(t) = ω0+δω sin(ωP t), where we assumeδω , ω0. The well-known physical example is a swingwhose motion is being excited by effectively changing thelength of the pendulum at the right frequency and phase.

34

For a “pump frequency” ωP equalling twice the “signalfrequency”, ωP = 2ωS , the resulting dynamics will leadto an amplification of the initial oscillator position, withthe energy provided by the external driving:

x(t) = x(0)eλt cos(ωSt) +p(0)MωS

e−λt sin(ωSt) (6.2)

Thus, one of the quadratures is amplified exponen-tially, at a rate λ = δω/2, while the other one decays.In a quantum-mechanical description, this produces asqueezed state out of an initial coherent state. Such a sys-tem is called a “degenerate parametric amplifier”, and wediscuss its quantum dynamics in more detail in Sec. VI.Gand in Appendix F. We will see that such an amplifier,which only amplifies a single quadrature, is not requiredquantum mechanically to add any noise (Braginsky andKhalili, 1992; Caves, 1982; Caves et al., 1980).

Can we now change this parametric amplificationscheme slightly in order to make both signal quadraturesgrow with time? It turns out this is impossible, as longas we restrict ourselves to a driven system with a singledegree of freedom. The reason in classical mechanics isthat Liouville’s theorem requires phase space volume tobe conserved during motion. In a more formal way, thisis related to the conservation of Poisson brackets, or, inquantum mechanics, to the conservation of commutationrelations. Nevertheless, it is certainly desirable to havean amplifier that acts equally on both quadratures (aso-called “phase-preserving” or “phase-insensitive” am-plifier), since the signal’s phase is often not known be-forehand. The way around the restriction created byLiouville’s theorem is to add more degrees of freedom,such that the phase space volume can expand in bothquadratures (i.e. position and momentum) of the inter-esting signal degree of freedom, while being compressedin other directions. This is achieved most easily by cou-pling the signal oscillator to another oscillator, the “idlermode”. The external driving now modulates the cou-pling between these oscillators, at a frequency that has toequal the sum of the oscillators’ frequencies. The result-ing scheme is called a phase-preserving non-degenerateparametric amplifier (see Appendix F).

Crucially, there is a price to pay for the introduction ofan extra degree of freedom: there will be noise associatedwith the “idler” oscillator, and this noise will contributeto the noise in the output of the amplifier. Classically,one could make the noise associate with the “idler” oscil-lator arbitrarily small by simply cooling it to zero temper-ature. This is not possible quantum-mechanically; thereare always zero-point fluctuations of the idler oscillatorto contend with. It is this noise which sets a funda-mental quantum limit for the operation of the amplifier.We thus have a heuristic accounting for why there is aquantum-limit on the added noise of a phase-preservinglinear amplifier: one needs extra degrees of freedom toamplify both signal quadratures, and such extra degreesof freedom invariably have noise associated with them.

B. Standard Haus-Caves derivation of the quantum limiton a bosonic amplifier

We now make the ideas of the previous subsectionmore precise by quickly sketching the standard deriva-tion of the quantum limit on the noise added by a phase-preserving amplifier. This derivation is originally due toHaus and Mullen (1962), and was both clarified and ex-tended by Caves (1982); the amplifier quantum limit wasalso motivated in a slightly different manner by Heffner(1962)13. While extremely compact, the Haus-Cavesderivation can lead to confusion when improperly ap-plied; we will discuss this in the next subsection, as wellas in Sec. VII, where we apply this argument carefully tothe important case of a two-port quantum voltage am-plifier.

The starting assumption of this derivation is that boththe input and output ports of the amplifier can be de-scribed by sets of bosonic modes. If we focus on a nar-row bandwidth centered on frequency ω, we can describea classical signal E(t) in terms of a complex number adefining the amplitude and phase of the signal (or equiva-lently the two quadrature amplitudes) (Haus, 2000; Hausand Mullen, 1962)

E(t) ∝ i[ae−iωt − a∗e+iωt]. (6.3)

In the quantum case, the two signal quadratures of E(t)(i.e. the real and imaginary parts of a(t)) cannot be mea-sured simultaneously because they are canonically conju-gate; this is in complete analogy to a harmonic oscillator(cf. Eq. (4.66)). As a result a, a∗ must be elevated to thestatus of photon ladder operators: a → a, a∗ → a† .

Consider the simplest case, where there is only a singlemode at both the input and output, with correspondingannhiliation operators a and b 14. It follows that theinput signal into the amplifier is described by the ex-pectation value 〈a〉, while the output signal is describedby 〈b〉. Correspondingly, the symmetrized noise in boththese quantities is described by:

(∆a)2 ≡ 12

⟨a, a†

⟩− |〈a〉|2 (6.4a)

(∆b)2 ≡ 12

⟨b, b†

⟩−

∣∣∣〈b〉∣∣∣2

(6.4b)

To derive a quantum limit on the added noise of theamplifier, one uses two simple facts. First, both the input

13 Note that Caves (1982) provides a thorough discussion of whythe derivation of the amplifier quantum limit given in Heffner(1962) is not rigorously correct

14 To relate this to the linear response detector of Sec. V.A, onecould naively write x, the operator carrying the input signal, as,e.g., x = a + a†, and the output operator I as, e.g., I = b + b†

(we will discuss how to make this correspondence in more detailin Sec. VII)

35

and the output operators must satisfy the usual commu-tation relations:

[a, a†

]= 1 (6.5a)

[b, b†

]= 1 (6.5b)

Second, the linearity of the amplifier and the fact that itis phase preserving (i.e. both signal quadratures are am-plified the same way) implies a simple relation betweenthe output operator b and the input operator a:

b =√

Ga (6.6a)

b† =√

Ga† (6.6b)

where G is the dimensionless “photon number gain” ofthe amplifier. It is immediately clear however this ex-pression cannot possibly be correct as written because itviolates the fundamental bosonic commutation relation[b, b†] = 1. We are therefore forced to write

b =√

Ga + F (6.7a)

b† =√

Ga† + F† (6.7b)

where F is an operator representing additional noiseadded by the amplifier. Based on the discussion of theprevious subsection, we can anticipate what F repre-sents: it is noise associated with the additional degreesof freedom which must invariably be present in a phase-preserving amplifier.

As F represents noise, it has a vanishing expectationvalue; in addition, one also assumes that this noise isuncorrelated with the input signal, implying [F , a] =[F , a†] = 0 and 〈F a〉 = 〈F a†〉 = 0. Insisting that[b, b†] = 1 thus yields:

[F , F†

]= 1−G (6.8)

The question now becomes how small can we make thenoise described by F? Using Eqs. (6.7a) and (6.7b), thenoise at the amplifier output ∆b is given by:

(∆b)2 = G (∆a)2 +12

⟨F , F†

⟩

≥ G (∆a)2 +12

∣∣∣⟨[F , F†]

⟩∣∣∣

≥ G (∆a)2 +|G− 1|

2(6.9)

We have used here a standard inequality to bound theexpectation of F , F†. The first term here is simplythe amplified noise of the input, while the second termrepresents the noise added by the amplifier. Note thatif there is no amplification (i.e. G = 1), there need notbe any added noise. However, in the more relevant caseof large amplification (G % 1), the added noise cannotvanish. It is useful to express the noise at the output asan equivalent noise at (“referred to”) the input by simply

dividing out the photon gain G. Taking the large-G limit,we have:

(∆b)2

G≥ (∆a)2 +

12

(6.10)

Thus, we have a very simple demonstration that an am-plifier with a large photon gain must add at least half aquantum of noise to the input signal. Equivalently, theminimum value of the added noise is simply equal to thezero-point noise associated with the input mode; the totaloutput noise (referred to the input) is at least twice thezero point input noise. Note that both these conclusionsare identical to what we found (by very different means)in our analysis of the resonant cavity position detector inSec. IV.B.3.

As we have already discussed, the added noise oper-ator F is associated with additional degrees of freedom(beyond input and output modes) necessary for phase-preserving amplification. To see this more concretely,note that every linear amplifier is inevitably a non-linearsystem consisting of an energy source and a ‘spigot’ con-trolled by the input signal which redirects the energysource partly to the output channel and partly to someother channel(s). Hence there are inevitably other de-grees of freedom involved in the amplification processbeyond the input and output channels. An explicit ex-ample is the quantum parametric amplifier described inmore detail in Appendix F. Further insights into ampli-fier added noise and its connection to the fluctuation-dissipation theorem can be obtained by considering asimple model where a transmission line is terminated byan effective negative impedance; we discuss this model inAppendix B.4.

To see explicitly the role of the additional degrees offreedom, note first that for G > 1 the RHS of Eq. (6.8) isnegative. Hence the simplest possible form for the addednoise is

F =√

G− 1d† (6.11)

F† =√

G− 1d (6.12)

where d and d† represent a single additional mode ofthe system. This is the minimum number of additionaldegrees of freedom that must inevitably be involved inthe amplification process. Note that for this case, theinequality in Eq. (6.9) is satisfied as an equality, andthe added noise takes on its minimum possible value.If instead we had, say, two additional modes (coupledinequivalently):

F =√

G− 1(cosh θd†1 + sinh θd2) (6.13)

it is straightforward to show that the added noise is in-evitably larger than the minimum. This again can beinterpreted in terms of wasted information, as the extradegrees of freedom are not being monitored as part of themeasurement process and so information is being lost.

36

aout

ain

bin

boutinput line output line

amplifier

FIG. 9 (Color online) Schematic of a two-port bosonic ampli-fier. Both the input and outputs of the amplifier are attachedto transmission lines. The incoming and outgoing wave am-plitudes in the input (output) transmission line are labelledain, aout (bin, bout) respectively. The voltages at the end ofthe two lines (Va, Vb) are linear combinations of incoming andoutgoing wave amplitudes.

TABLE III Two different amplifier modes of operation.

Mode Input Signal Output Signal

s(t) o(t)

Scattering s(t) = ain(t) o(t) = bout(t))

(ain indep. of aout) (bout indep. of bin)

Op-amp s(t) = Va(t) o(t) = Vb(t)

(ain depends on aout) (bout depends on bin)

C. Scattering versus op-amp modes of operation

While extremely elegant and direct, the standardHaus-Caves derivation of the amplifier quantum limit hassome puzzling features. Recall that in our heuristic dis-cussion of position detection (Sec. IV.B.3), we saw thata crucial aspect of the quantum limit was the trade-offbetween back-action noise and measurement imprecisionnoise. We saw that reaching the quantum limit requiredboth a detector with “ideal” noise, as well as an optimiza-tion of the detector-oscillator coupling strength. Some-what disturbingly, none of these ideas appeared explicitlyin the Haus-Caves derivation; this can give the mislead-ing impression that the quantum limit never has anythingto do with back-action. A further confusion comes fromthe fact that many detectors have input and outputs thatcannot be described by a set of bosonic modes. How doesone apply the above arguments to such systems?

The first step in resolving these seeming inconsisten-cies is to realize that there are really two different waysin which one can use a given amplifier or detector. In de-ciding how to couple the input signal (i.e. the signal to beamplified) to the amplifier, and in choosing what quan-tity to measure, the experimentalist essentially enforcesboundary conditions; as we will now show, there are ingeneral two distinct ways in which to do this. For con-creteness, consider the situation depicted in Fig. 9: a two-port voltage amplifier where the input and output portsof the amplifier are attached to one-dimensional trans-mission lines (see App. B for a quick review of quantumtransmission lines). Similar to the previous subsection,

aout

ain

bin

bout

coldload

input line output line

uin uoutvacuum noise

amplifier

circulator

FIG. 10 Illustration of a bosonic two-port amplifier used inthe scattering mode of operation. The “signal” is an incomingwave in the input port of the amplifier, and does not dependon what is coming out of the amplifier. This is achieved byconnecting the input line to a circulator and a “cold load”(i.e. a zero temperature resistor): all that goes back towardsthe source of the input signal is vacuum noise.

we focus on a narrow bandwidth signal centered about afrequency ω. At this frequency, there exists both a right-moving and a left-moving wave in each transmission line.We label the corresponding amplitudes in the input (out-put) line with ain, aout (bin, bout), as per Fig. 9. Quantummechanically, these amplitudes become operators, muchin the same way that we treated the mode amplitude aas an operator in the previous subsection. We will ana-lyze this two-port bosonic amplifier in detail in Sec. VII;here, we will only sketch its operation to introduce thetwo different amplifier operation modes. This will thenallow us to understand the subtleties of the Haus-Cavesquantum limit derivation.

In the first kind of setup, the experimentalist arrangesthings so that ain, the amplitude of the wave incident onthe amplifier’s input port, is precisely equal to the signalto be amplified (i.e. the input signal), irrespective of theamplitude of the wave leaving the input port (i.e. aout).Further, the output signal is taken to be the amplitudeof the outgoing wave exiting the output of the ampli-fier (i.e. bout), again, irrespective of whatever might beentering the output port (see Table III). In this situa-tion, the Haus-Caves description of the quantum limitin the previous subsection is almost directly applicable;we will make this precise in Sec. VII. Back-action isindeed irrelevant, as the prescribed experimental condi-tions mean that it plays no role. We will call this modeof operation the “scattering mode”, as it is most relevantto time-dependent experiments where the experimental-ist launches a signal pulse at the amplifier and looks atwhat exits the output port. One is usually only inter-ested in the scattering mode of operation in cases wherethe source producing the input signal is matched to theinput of the amplifier: only in this case is the input waveain perfectly transmitted into the amplifier. As we willsee in Sec. VII, such a perfect matching requires a rela-tively strong coupling between the signal source and theinput of the amplifier; as such, the amplifier will stronglyenhance the damping of the signal source.

37

The second mode of linear amplifier operation is whatwe call the “op-amp” mode; this is the mode one usu-ally has in mind when thinking of an amplifier whichis weakly coupled to the signal source. The key differ-ence from the “scattering” mode is that here, the inputsignal is not simply the amplitude of a wave incidenton the input port of the amplifier; similarly, the out-put signal is not the amplitude of a wave exiting theoutput port. As such, the Haus-Caves derivation of thequantum limit does not directly apply. For the bosonicamplifier discussed here, the op-amp mode would corre-spond to using the amplifier as a voltage op-amp. Theinput signal would thus be the voltage at the end of theinput transmission line. Recall from Appendix B thateven classically, the voltage at the end of a transmissionline involves the amplitude of both left and right mov-ing waves, i.e. Va(t) ∝ Re [ain(t) + aout(t)]. At first,this might seem quite confusing: if the signal source de-termines Va(t), does this mean it sets the value of bothain(t) and aout(t)? Doesn’t this violate causality? Thesefears are of course unfounded. The signal source enforcesthe value of Va(t) by simply changing ain(t) in responseto the value of aout(t). While there is no violation ofcausality, the fact that the signal source is dynamicallyresponding to what comes out of the amplifier’s inputport implies that back-action is indeed relevant.

The op-amp mode of operation is relevant to the typi-cal situation of weak coupling between the signal sourceand amplifier input. By weak coupling, we mean thatthe amplifier does not appreciably change the dissipationof the signal source. This is analogous to the situationin an ideal voltage op-amp, where the amplifier inputimpedance is much larger than the impedance of the sig-nal source. We stress the op-amp mode and this limit ofweak coupling is the relevant situation in most electricalmeasurements.

Thus, we see that the Haus-Caves formulation of thequantum limit is not directly relevant to amplifiers or de-tectors operated in the usual op-amp mode of operation.We clearly need some other way to describe quantumamplifiers used in this regime. In the remainder of thissection, we will show that the linear-response, quantumnoise approach we have been developing provides a con-venient and general way to discuss the quantum limithere. The linear response approach will allow us to see(similar to Sec. IV.B.3) that reaching the quantum limitdoes indeed require a trade-off between back-action andmeasurement imprecision, and requires use of an ampli-fier with ideal quantum noise properties (cf. Eq. 5.11).This approach also has the added benefit of being directlyapplicable to systems where the input and output of theamplifier are not described by bosonic modes 15. In the

15 Note that the Haus-Caves derivation for the quantum limit of ascattering amplifier has recently been generalized to the case offermionic operators (Gavish et al., 2004).

next section (Sec. VII), we will return to the scatteringdescription of a two-port voltage amplifier, and show ex-plicitly how an amplifier can be quantum-limited whenused in the scattering mode of operation, but miss thequantum limit when used in the op-amp mode of opera-tion.

D. Linear response description of a position detector

In this subsection, we will examine the amplifier quan-tum limit for a two-port linear amplifier in the usual weakcoupling, “op-amp” regime of operation. Our discussionhere will make use of the results we have obtained forthe noise properties of a generic linear response detectorin Sec. V, including the fundamental quantum noise con-straint of Eq. (5.11). The approach we use here has thebenefits of not being restricted to bosonic systems, andof directly involving noise spectral densities, quantitieswhich can be directly measured in experiment. It pro-vides a rough ‘recipe’ of what one needs to do in generalto reach the quantum limit.

For simplicity, we will start with the concrete problemof continuous position detection of a harmonic oscillator;this will generalize the discussion of Sec. IV.B.3, wherewe gave a heuristic discussion of weak, continuous po-sition detection using a resonant cavity detector. Wenow imagine a situation where the generic linear detec-tor introduced in Sec. V.A is weakly coupled at its inputto the position x of a harmonic oscillator (cf. Eq. (5.1))16. We would like to understand the total output noiseof our amplifier in the presence of the oscillator, and,more importantly, how small we can make the ampli-fier’s contribution to this noise. We will discuss the anal-ogous but important quantum limit on the noise temper-ature of a voltage amplifier in Sec. VI.E. Note that thediscussion of a generic position detector presented hereis directly applicable to recent experiments with nano-electromechanical which attempt to reach the quantumlimit (Flowers-Jacobs et al., 2007; Knobel and Cleland,2003; LaHaye et al., 2004; Naik et al., 2006).

1. Detector back-action

We first consider the consequence of noise in the detec-tor input port. As we have already seen in Sec. III.B, thefluctuating back-action force F acting on our oscillatorwill lead to both damping and heating of the oscillator.To model the intrinsic (i.e. detector-independent) heatingand damping of the oscillator, we will also assume thatour oscillator is coupled to an equilibrium heat bath. In

16 For consistency with previous sections, our coupling Hamiltoniandoes not have a minus sign. This is different from the conven-tion of Clerk (2004), where the coupling Hamiltonian is writtenHint = −Ax · F .

38

the weak-coupling limit that we are interested in, one canuse lowest-order perturbation theory in the coupling Ato describe the effects of the back-action force F on theoscillator. While the full quantum theory is somewhatinvolved (see Appendix J.3), it leads to an extremelysimple picture that we can understand even classically.One finds that the oscillator is described by an effectiveLangevin equation17:

Mx(t) = −MΩ2x(t)−Mγ0x(t) + F0(t)

−MA2

∫dt′γ(t− t′)x(t′)−A · F (t) (6.14)

The position x(t) in the above equation is not an op-erator, but is simply a classical variable whose fluctua-tions are driven by the fluctuating forces F (t) and F0(t).Nonetheless, the noise in x calculated from Eq. (6.14)corresponds precisely to Sxx[ω], the symmetrized quan-tum mechanical noise in the operator x (see Appendix J.3for the justification of this statement). The fluctuatingforce exerted by the detector (which represents the heat-ing part of the back-action) is described by A · F (t) inEq. (6.14); it has zero mean, and a spectral density givenby A2SFF [ω] in Eq. (5.4a). The kernel γ(t) describesthe damping effect of the detector. It is given by theasymmetric part of the detector’s quantum noise, as wasderived in Sec. III.B (cf. Eq. ((3.32)). Recall that thedamping effect of the back-action has a simple origin:the average force exerted on the oscillator by the ampli-fier responds to the motion of the oscillator.

Eq. (6.14) also describes the effects of an equilibriumheat bath at temperature T0 which models the intrinsic(i.e. detector-independent) damping and heating of theoscillator. γ0 is the damping arising from this bath, andF0 is the corresponding fluctuating force. The spectraldensity of the F0 noise is determined by γ0 and T0 via thefluctuation-dissipation theorem (cf. Eq. (3.34)). T0 andγ0 have a simple physical significance: they are the tem-perature and damping of the oscillator when the couplingto the detector A is set to zero.

To make further progress, we recall from Sec. III.B thateven though our detector will in general not be in equilib-rium, we may nonetheless assign it an effective tempera-ture Teff [ω] at each frequency (cf. Eq. (3.21)). In general,this effective temperature is not reflective of some phys-ical temperature in the detector; it also does not corre-spond to the “noise temperature” of the detector that willbe discussed in what follows. The effective temperatureof an out-of-equilibrium detector is simply a measure ofthe asymmetry of the detector’s quantum noise. We are

17 Note that we have omitted a back-action term in this equationwhich leads to small renormalizations of the oscillator frequencyand mass. These terms are not important for the following dis-cussion, so we have omitted them for clarity; one can considerM and Ω in this equation to be renormalized quantities. SeeAppendix J.3 for more details.

often interested in the limit where the internal detectortimescales are much faster than the timescales relevantto the oscillator (i.e. Ω−1, γ−1, γ−1

0 ). We may then takethe ω → 0 limit in the expression for Teff , yielding:

2kBTeff ≡SFF (0)Mγ(0)

(6.15)

In this limit, the oscillator position noise calculated fromEq. (6.14) is given by:

Sxx[ω] =1M

2(γ0 + γ)kB

(ω2 − Ω2)2 + ω2(γ + γ0)2γ0T + γTeff

γ0 + γ

(6.16)

This is exactly what would be expected if the oscillatorwere only attached to an equilibrium Ohmic bath with adamping coefficient γΣ = γ0 + γ and temperature T =(γ0T + γTeff)/γΣ.

2. Total output noise

The next step in our analysis is to link fluctuationsin the position of the oscillator (as determined fromEq. (6.14)) to noise in the output of the detector. Asdiscussed in Sec. IV.B.3, the output noise consists of theintrinsic output noise of the detector (i.e. “measurementimprecision noise”) plus the amplified position fluctua-tions in the position of the oscillator. The latter containsboth an intrinsic part and a term due to the response ofthe oscillator to the back action.

To start, imagine that we can treat both the oscillatorposition x(t) and the detector output I(t) as classicallyfluctuating quantities. Using the linearity of the detec-tor’s response, we can then write δItotal, the fluctuatingpart of the detector’s output, as:

δItotal[ω] = δI0[ω] + AχIF [ω] · δx[ω] (6.17)

The first term (δI0) describes the intrinsic (oscillator-independent) fluctuations in the detector output, andhas a spectral density SII [ω]. If we scale this by |χIF |2,we have the measurement imprecision noise discussed inSec. IV.B.3. The second term corresponds to the ampli-fied fluctuations of the oscillator, which are in turn givenby solving Eq. (6.14):

δx[ω] = −[

1/M

(ω2 − Ω2) + iωΩ/Q[ω]

](F0[ω]−A · F [ω])

≡ −χxx[ω](F0[ω]−A · F [ω]) (6.18)

where Q[ω] = Ω/(γ0 + γ[ω]) is the oscillator quality fac-tor. It follows that the spectral density of the total noisein the detector output is given classically by:

SII,tot[ω] = SII [ω]+ |χxx[ω]χIF [ω]|2

(A4SFF [ω] + A2SF0F0 [ω]

)

+ 2A2Re [χxx[ω]χIF [ω]SIF [ω]] (6.19)

39

Ix yFA B

g

γ0 + γ Ag

Aλ Bgy

FD

γout + γld

FIG. 11 (Color online) Schematic of a generic linear-responseposition detector, where a auxiliary oscillator y is driven bythe detector output.

Here, SII ,SFF and SIF are the (classical) detector noisecorrelators calculated in the absence of any coupling tothe oscillator. Note importantly that we have includedthe fact that the two kinds of detector noise (in I and inF ) may be correlated with one another.

To apply the classically-derived Eq. (6.19) to ourquantum detector-plus-oscillator system, we recall fromSec. III.B that symmetrized quantum noise spectral den-sities play the role of classical noise. The LHS ofEq. (6.19) thus becomes SII,tot, the total symmetrizedquantum-mechanical output noise of the detector, whilethe RHS will now contain the symmetrized quantum-mechanical detector noise correlators SFF , SII and SIF ,defined as in Eq. (5.4a). Though this may seem ratherad-hoc, one can easily demonstrate that Eq. (6.19) thusinterpreted would be quantum-mechanically rigorous ifthe detector correlation functions obeyed Wick’s theo-rem. Thus, quantum corrections to Eq. (6.19) will arisesolely from the non-Gaussian nature of the detector noisecorrelators. We expect from the central limit theoremthat such corrections will be small in the relevant limitwhere ω is much smaller that the typical detector fre-quency ∼ kBTeff/!, and neglect these corrections in whatfollows. Note that the validity of Eq. (6.19) for a specificmodel of a tunnel junction position detector has beenexplicitly verified in Clerk and Girvin (2004).

3. Detector power gain

Before proceeding, we need to consider our detectoronce again in isolation, and return to the fundamentalquestion of what we mean by amplification. To be ableto say that our detector truly amplifies the motion of theoscillator, it is not sufficient to simply say the responsefunction χIF must be large (note that χIF is not dimen-sionless!). Instead, true amplification requires that thepower delivered by the detector to a following amplifier bemuch larger than the power drawn by the detector at itsinput– i.e., the detector must have a dimensionless powergain GP [ω] much larger than one. If the power gain wasnot large, we would need to worry about the next stage

in the amplification of our signal, and how much noise isadded in that process. Having a large power gain meansthat by the time our signal reaches the following ampli-fier, it is so large that the added noise of this followingamplifier is unimportant. The power gain is analogous tothe dimensionless photon number gain G that appears inthe standard Haus-Caves description of a bosonic linearamplifier.

To make the above more precise, we start by assum-ing the simple (and usual) case where there is no reversegain: χFI = 0. We will define the power gain GP [ω] ofour generic position detector in a way that is completelyanalogous to the power gain of a voltage amplifier [CITE];alternate and important measures of power gain are dis-cussed in Sec. VIII.B. Imagine we drive the oscillator weare trying to measure with a force 2FD cos ωt; this willcause the output of our detector 〈I(t)〉 to also oscillate atfrequency ω. To optimally detect this signal in the detec-tor output, we further couple the detector output I to asecond oscillator with natural frequency ω, mass M , andposition y: there is a new coupling term in our Hamilto-nian, H ′

int = BI · y, where B is a coupling strength. Theoscillations in 〈I(t)〉 will now act as a driving force on theauxiliary oscillator y (see Fig 11). We can consider theauxiliary oscillator y as a “load” we are trying to drivewith the output of our detector.

To find the power gain, we need to consider both Pout,the power supplied to the output oscillator y from the de-tector, and Pin, the power fed into the input of the ampli-fier. Consider first Pin. This is simply the time-averagedpower dissipation of the input oscillator x caused by theback-action damping γ[ω]. Using a bar to denote a timeaverage, we have:

Pin ≡ Mγ[ω] · x2

= Mγ[ω]ω2|χxx[ω]|2F 2D (6.20)

The oscillator response function χxx[ω] is defined inEq. (6.18); note that it depends on both the back-actiondamping γ[ω] as well as the intrinsic oscillator dampingγ0.

Next, we need to consider the power supplied to the“load” oscillator y at the detector output. This oscilla-tor will have some intrinsic, detector-independent damp-ing γld, as well as a back-action damping γout. In thesame way that the back-action damping γ of the inputoscillator x is determined by the quantum noise in F(cf. Eq. (3.30)-(3.32)), the back-action damping of theload oscillator y is determined by the quantum noise inthe output operator I:

γout[ω] =B2

Mω[−Im χII [ω]]

=B2

M!ω

[SII [ω]− SII [−ω]

2

](6.21)

where χII is the linear-response susceptibility which de-termines how 〈I〉 responds to a perturbation coupling to

40

I:

χII [ω] = − i

!

∫ ∞

0dt

⟨[I(t), I(0)

]⟩eiωt (6.22)

As the oscillator y is being driven on resonance, the rela-tion between y and I is given by y[ω] = χyy[ω]I[ω] withχyy[ω] = −i[ωMγout[ω]]−1. From conservation of energy,we have that the net power flow into the output oscillatorfrom the detector is equal to the power dissipated out ofthe oscillator through the intrinsic damping γld. We thushave:

Pout ≡ Mγld · y2

= Mγldω2|χyy[ω]|2 · |BAχIF χxx[ω]FD|2

=1M

γld

(γld + γout[ω])2· |BAχIF χxx[ω]FD|2

(6.23)

Using the above definitions, we find that the ratio be-tween Pout and Pin is independent of γ0, but depends onγld:

Pout

Pin=

1M2ω2

A2B2|χIF [ω]|2

γout[ω]γ[ω]γld/γout[ω]

(1 + γld/γout[ω])2

(6.24)

We now define the detector power gain GP [ω] as the valueof this ratio maximized over the choice of γld . The max-imum occurs for γld = γout[ω] (i.e. the load oscillator is“matched” to the output of the detector), resulting in:

GP [ω] ≡ max[Pout

Pin

]

=1

4M2ω2

A2B2|χIF |2

γoutγ

=|χIF [ω]|2

4Im χFF [ω] · Im χII [ω](6.25)

In the last line, we have used the relation between thedamping rates γ[ω] and γout[ω] and the linear-responsesusceptibilities χFF [ω] and χII [ω] (c.f. Eqs. (3.31) and(6.21)). We thus find that the power gain is a simple di-mensionless ratio formed by the three different responsecoefficients characterizing the detector, and is indepen-dent of the coupling constants A and B. As we will see insubsection VI.E, it is completely analogous to the powergain of a voltage amplifier, which is also determined bythree parameters: the voltage gain, the input impedanceand the output impedance. Note that there are otherimportant measures of power gain commonly in use inthe engineering community: we will comment on thesein Sec. VIII.B.

4. Simplifications for a quantum-ideal detector

We now consider the important case where our detec-tor has “ideal” quantum noise (i.e. it satisfies the ideal

noise condition of Eq. (5.16)). Fulfilling this conditionimmediately places some powerful constraints on our de-tector.

First, note that we have defined in Eq. (6.15) the effec-tive temperature of our detector based on what happensat the input port; this is the effective temperature seenby the oscillator we are trying to measure. We could alsoconsider the effective temperature of the detector as seenat the output (i.e. by the oscillator y used in definingthe power gain). This “output” effective temperature isdetermined by the quantum noise in the output operatorI:

kBTeff,out[ω] ≡ !ω

log (SII [+ω]/SII [−ω])(6.26)

For a general out-of-equilibrium amplifier, Teff,out doesnot have to be equal to the input effective temperatureTeff defined by Eq. (3.21). However, for a quantum-idealdetector, the effective proportionality between input andoutput operators (cf. Eq. (J13)) immediately yields:

Teff,out[ω] = Teff [ω] (6.27)

Thus, a detector with quantum-ideal noise necessarilyhas the same effective temperature at its input and itsoutput. This is all the more remarkable given that aquantum-ideal detector cannot be in equilibrium, andthus Teff cannot represent a real physical temperature.

Another important simplification for a quantum-idealdetector is the expression for the power gain. Usingthe proportionality between input and output operators(cf. Eq. (J13)), one finds:

GP [ω] =(Im α)2 coth2

(!ω

2kBTeff

)+ (Re α)2

|α|2 (6.28)

where α[ω] is the parameter characterizing a quantumlimited detector in Eq. (5.17); recall that |α[ω]|2 deter-mines the ratio of SII and SFF . It follows immediatelythat for a detector with ‘ideal noise’ to also have a largepower gain (GP % 1), one absolutely needskbTeff % !ω: a large power gain implies a large effectivedetector temperature. In the large GP limit, we have

GP 4[Im α

|α|kBTeff

!ω/2

]2

(6.29)

Thus, the effective temperature of a quantum-ideal detec-tor does more than just characterize the detector back-action– it also determines the power gain.

Finally, an additional consequence of the large GP [ω],large Teff limit is that the gain χIF and noise cross-correlator SIF are in phase: SIF /χIF is purely real, up tocorrections which are small as ω/Teff . This is shown ex-plicitly in Appendix J.2. Thus, we find that a large powergain detector with ideal quantum noise cannot have sig-nificant out-of-phase correlations between its output andinput noises. This last point may be understood in terms

41

of the idea of wasted information: if there were signifi-cant out-of-phase correlations between I and F , it wouldbe possible to improve the performance of the amplifierby using feedback. We will discuss this point more fullyin Sec. VII. Note that as SIF /χIF is real, the last termin the quantum noise constraint of Eq. (5.11) vanishes.

5. Quantum limit on added noise and noise temperature

We now turn to calculating the noise added to oursignal (i.e. 〈x(t)〉) by our generic position detector. Tocharacterize this added noise, it is useful to take the total(symmetrized) noise in the output of the detector, andrefer it back to the input by dividing out the gain of thedetector:

Sxx,tot[ω] ≡ SII,tot[ω]A2|χIF [ω]|2 (6.30)

Sxx,tot[ω] is simply the frequency-dependent spectraldensity of position fluctuations inferred from the outputof the detector. It is this quantity which will directlydetermine the sensitivity of the detector– given a certaindetection bandwidth, what is the smallest variation of xthat can be resolved? The quantity Sxx,tot[ω] will havecontributions both from the intrinsic fluctuations of theinput signal, as well as a contribution due to the detector.We first define Sxx,eq[ω, T ] to be the symmetrized equi-librium position noise of our damped oscillator (whosedamping is γ0 + γ) at temperature T :

Sxx,eq[ω, T ] = ! coth(

!ω

2kBT

)[−Im χxx[ω]] (6.31)

where the oscillator susceptibility χxx[ω] is defined inEq. (6.18). The total inferred position noise may thenbe written:

Sxx,tot[ω] ≡(

γ0

γ0 + γ

)· Sxx,eq[ω, T0]+ Sxx,add[ω] (6.32)

In the usual case where the detector noise can be ap-proximated as being white, this spectral density will con-sist of a Lorentzian sitting atop a constant noise floor(c.f. Fig. 7) The first term in Eq. (6.32) represents posi-tion noise arising from the fluctuating force δF0(t) associ-ated with the intrinsic (detector-independent) dissipationof the oscillator (c.f. Eq. (6.14)). The prefactor of thisterm arises because the strength of the intrinsic Langevinforce acting on the oscillator is proportional to γ0, not toγ0 + γ.

The second term in Eq. (6.32) represents the addedposition noise due to the detector. It has contributionsfrom both from the detector’s intrinsic output noise SII

as well as from the detector’s back-action noise SFF , andmay be written:

Sxx,add[ω] =SII

|χIF |2A2+ A2 |χxx[ω]|2 SFF +

2Re[χ∗IF (χxx[ω])∗ SIF

]

|χIF |2(6.33)

For clarity, we have omitted writing the explicit fre-quency dependence of the gain χIF and noise correlators;they should all be evaluated at the frequency ω. Notethat the first term on the RHS corresponds to the “mea-surement imprecision” noise of our detector, SI

xx(ω).We can now finally address the issue of the quantum

limit in our system. As discussed in Sec. VI.C, the Haus-Caves derivation of the quantum limit (c.f. Sec. VI.B)is not directly applicable to the position detector we aredescribing here; nonetheless, we may use its result toguess what form the quantum limit will take here. TheHaus-Caves argument told us that the added noise of aphase-preserving linear amplifier must be at least as largeas the zero point noise. We thus anticipate that if ourdetector has a large power gain, the spectral density ofthe noise added by the detector (i.e. Sxx,add[ω]) must beat least as big as the zero point noise of our dampedoscillator:

Sxx,add[ω] ≥ limT→0

Sxx,eq[w, T ] = |!Im χxx[ω]| (6.34)

We will now show that the bound above is rigorouslycorrect at each frequency ω. Moreover, the approach wetake will give us a rough idea of what one needs to do toachieve this quantum limit.

The first step is to examine the dependence of theadded noise Sxx,add[ω] (as given by Eq. (6.33)) on thecoupling strength A. If we ignore for a moment thedetector-dependent damping of the oscillator, the sit-uation is the same as the cavity position detector ofSec. IV.B.3: there is an optimal value of the couplingstrength A. For very strong coupling, we will minimizethe effects of the intrinsic output noise of the detector(first term in Eq. (6.33)), but the back-action noise willmake a large contribution (second term in Eq. (6.33)).Conversely, for very weak couplings we minimize theback-action effect, but make the relative importance ofthe intrinsic output noise of the detector quite strong.We would thus expect Sxx,add[ω] to attain a minimumvalue at an optimal choice of coupling A = Aopt whereboth these terms make equal contributions (see Fig. 6).Defining φ[ω] = arg χxx[ω], we thus have the bound:

Sxx,add[ω] ≥ 2|χxx[ω]|[√

SII SFF

|χIF |2+

Re[χ∗IF e−iφ[ω]SIF

]

|χIF |2

]

(6.35)where the minimum value at frequency ω is achievedwhen:

A2opt =

√SII [ω]

|χIF [ω]χxx[ω]|2SFF [ω](6.36)

Using the inequality X2 + Y 2 ≥ 2|XY | we see that thisvalue serves as a lower bound on Sxx,add even in the pres-ence of detector-dependent damping. In the case wherethe detector-dependent damping is negligible, the RHS ofEq. (6.35) is independent of A, and thus Eq. (6.36) canbe satisfied by simply tuning the coupling strength A; in

42

the more general case where there is detector-dependentdamping, the RHS is also a function of A (through the re-sponse function χxx[ω]), and it may no longer be possibleto achieve Eq. (6.36) by simply tuning A18.

While Eq. (6.35) is certainly a bound on the addeddisplacement noise Sxx,add[ω], it does not in itself rep-resent the quantum limit. Reaching the quantum limitrequires more than simply balancing the detector back-action and intrinsic output noises (i.e. the first two termsin Eq. (6.33)); one also needs a detector with “quantum-ideal” noise properties, that is a detector which satis-fies Eq. (5.16) and thus the proportionality conditionof Eq. (J13). Using the quantum noise constraint ofEq. (5.11) to further bound Sxx,add[ω], we obtain:

Sxx,add[ω] ≥ 2∣∣∣∣χxx[ω]χIF

∣∣∣∣×[√(

!|χIF |2

)2 (1 + ∆

[2SIF

!χIF

])+

∣∣SIF

∣∣2

+Re

[χ∗IF e−iφ[ω]SIF

]

|χIF |

](6.37)

where the function ∆[z] is defined in Eq. (5.10b). Theminimum value of Sxx,add[ω] in Eq. (6.37) is now achievedwhen one has both an optimal coupling (i.e. Eq. (6.36))and a quantum limited detector, that is one which satis-fies Eq. (5.11) as an equality. Note again that an arbi-trary detector will not satisfy this ideal noise condition.

Next, we consider the relevant case where our detectoris a good amplifier and has a power gain GP [ω] % 1over the width of the oscillator resonance. As we havediscussed, this implies that the ratio SIF /χIF is purelyreal, up to small !ω/(kBTeff) corrections (see Sec. (V.A.4)and Appendix J.2 for more details). This in turn impliesthat ∆[2SIF /!χIF ] = 0; we thus have:

Sxx,add[ω] ≥

2|χxx[ω]|

√(!2

)2

+(

SIF

χIF

)2

+cos [φ[ω]] SIF

χIF

(6.38)

Finally, as there is no further constraint on SIF /χIF (be-yond the fact that it is real), we can minimize the expres-sion over its value. The minimum Sxx,add[ω] is achievedfor a detector whose cross-correlator satisfies:

SIF [ω]χIF

∣∣∣optimal

= −!2

cot φ[ω], (6.39)

with the minimum value being given by:

Sxx,add[ω]∣∣∣min

= !|Im χxx[ω]| = limT→0

Sxx,eq[ω, T ](6.40)

18 Note that in the heuristic discussion of position detection usinga resonant cavity detector in Sec. IV.B.3, these concerns did notarise as there was no back-action damping.

where Sxx,eq[ω, T ] is the equilibrium contribution toSxx,tot[ω] defined in Eq. (6.31). Thus, in the limit ofa large power gain, we have that at each frequency, theminimum displacement noise added by the detector is pre-cisely equal to the noise arising from a zero temperaturebath. This conclusion is irrespective of the strength ofthe intrinsic (detector-independent) oscillator damping.

We have thus derived the amplifier quantum limit (inthe context of position detection) for a two-port ampli-fier used in the “op-amp” mode of operation. Though wereached a conclusion similar to that given by the Haus-Caves approach, the linear-response, quantum noise ap-proach used is quite different. This approach makes ex-plicitly clear what is needed to reach the quantum limit.We find that to reach the quantum-limit on the addeddisplacement noise Sxx,add[ω] with a large power gain,one needs:

1. A quantum limited detector, that is a detec-tor which satisfies the “ideal noise” condition ofEq. (5.16), and hence the proportionality conditionof Eq. (J13).

2. A coupling A which satisfies Eq. (6.36).

3. A detector cross-correlator SIF which satisfiesEq. (6.39).

Note that condition (1) is identical to what is requiredfor quantum-limited detection of a qubit; it is rather de-manding, and requires that there is no “wasted” informa-tion about the input signal in the detector which is notrevealed in the output (Clerk et al., 2003). Also note thatcot φ changes quickly as a function of frequency acrossthe oscillator resonance, whereas SIF will be roughlyconstant; condition (2) thus implies that it will not bepossible to achieve a minimal Sxx,add[ω] across the en-tire oscillator resonance. A more reasonable goal is tooptimize Sxx,add[ω] at resonance, ω = Ω. As χxx[Ω] isimaginary, Eq. (6.39) tells us that SIF should be zero.Assuming we have a quantum-limited detector with alarge power gain (kBTeff % !Ω), the remaining conditionon the coupling A (Eq. (6.36)) may be written as:

γ[Aopt]γ0 + γ[Aopt]

=∣∣∣∣Im α

α

∣∣∣∣1

2√

GP [Ω]=

!Ω4kBTeff

(6.41)

As γ[A] ∝ A2 is the detector-dependent damping ofthe oscillator, we thus have that to achieve the quantum-limited value of Sxx,add[Ω] with a large power gain,one needs the intrinsic damping of the oscillator to bemuch larger than the detector-dependent damping. Thedetector-dependent damping must be small enough tocompensate the large effective temperature of the detec-tor; if the bath temperature satisfies !Ω/kB , Tbath ,Teff , Eq. (6.41) implies that at the quantum limit, thetemperature of the oscillator will be given by:

Tosc ≡γ · Teff + γ0 · Tbath

γ + γ0→ !Ω

4kB+ Tbath (6.42)

43

input output

Zin Zout

ZS

VS

ZL

IL Vout

λV

λI

V~

I~

FIG. 12 Schematic of a linear voltage amplifier, including areverse gain λI . V and I represent the standard voltage andcurrent noises of the amplifier, as discussed further in the text.The case with reverse gain is discussed in detail in Sec. VII.

Thus, at the quantum limit and for large Teff , the de-tector raises the oscillator’s temperature by !Ω/4kB

19.As expected, this additional heating is only half the zero-point energy; in contrast, the quantum-limited value ofSxx,add[ω] corresponds to the full zero-point result, asit also includes the contribution of the intrinsic outputnoise of the detector.

Finally, we return to Eq. (6.37); this is the constrainton the added noise Sxx,add[ω] before we assumed ourdetector to have a large power gain, and consequentlya large Teff . Note crucially that if we did not requirea large power gain, then there need not be any addednoise. Without the assumption of a large power gain,the ratio SIF /χIF can be made imaginary with a largemagnitude. In this limit, 1 + ∆[2SIF /χIF ] → 0: thequantum constraint on the amplifier noises (e.g. theRHS of Eq. (5.11)) vanishes. One can then easily useEq. (6.37) to show that the added noise Sxx,add[ω] canbe zero. Thus, similar to the results of Heffner (1962),Haus and Mullen (1962) and Caves (1982), in the limitof unit power gain (i.e. small detector effective temper-ature), there is no quantum limit on Sxx,add, as perfectanti-correlations between the two kinds of detector noise(i.e. in I and F ) are possible. Phrased differently, if ourdetector does not amplify, then it need not add any noise.

E. Quantum limit on the noise temperature of a voltageamplifier

In this section, we turn our attention to the quantumlimit on the added noise of a generic linear voltage ampli-fier (Devoret and Schoelkopf, 2000). For such amplifiers,the added noise is usually expressed in terms of the “noisetemperature” of the amplifier; we will define this concept,and demonstrate that, when appropriately defined, thisnoise temperature must be bigger than !ω/(2kB), whereω is the signal frequency. Though the voltage amplifier isclosely analogous to the position detector treated in the

19 If in contrast our oscillator was initially at zero temperature (i.e.Tbath = 0), one finds that the effect of the back-action (at thequantum limit and for GP # 1) is to heat the oscillator to atemperature !Ω/(kB ln 5).

previous section, it is important enough that we will con-sider it again in some detail. In Sec. VII, we will againdiscuss a two-port voltage amplifier using a bosonic scat-tering description similar to that used in formulating theHaus-Caves proof of the amplifier quantum limit; thiswill allow further insights into what is needed to reachthe quantum limit, and how different modes of amplifieroperation are not equivalent. We stress that the generaltreatment presented here can also be applied directly tothe system discussed in Sec. VII.

1. Classical description of a voltage amplifier

Let us begin by recalling the standard schematic de-scription of a voltage amplifier, see Figs. 12. The in-put voltage to be amplified vin(t) is produced by a cir-cuit which has a Thevenin-equivalent impedance Zs, thesource impedance. We stress that we are considering the“op-amp” mode of amplifier operation, and thus the in-put signal does not correspond to the amplitude of awave incident upon the amplifier (see Sec. VI.C). Theamplifier itself has an input impedance Zin and an out-put impedance Zout, as well a voltage gain coefficientλV : assuming no current is drawn at the output (i.e.Zload →∞ in Fig. 12), the output voltage Vout(t) is sim-ply λV times the voltage across the input terminals ofthe amplifier.

The added noise of the amplifier is standardly repre-sented by two noise sources placed at the amplifier input.There is both a voltage noise source V (t) in series withthe input voltage source, and a current noise source I(t)in parallel with input voltage source (Fig. 12). The volt-age noise produces a fluctuating voltage V (t) (spectraldensity SV V [ω]) which simply adds to the signal voltageat the amplifier input, and is amplified at the output; assuch, it is completely analogous to the intrinsic detectoroutput noise SII of our linear response detector. In con-trast, the current noise source of the voltage amplifierrepresents back-action: this fluctuating current (spectraldensity SI I [ω]) flows back across the parallel combinationof the source impedance and amplifier input impedance,producing an additional fluctuating voltage at its input.The current noise is thus analogous to the back-actionnoise SFF of our generic linear response detector.

Putting the above together, the total voltage at theinput terminals of the amplifier is:

vin,tot(t) =Zin

Zin + Zs

[vin(t) + V (t)

]− ZsZin

Zs + ZinI(t)

4 vin(t) + V (t)− ZsI(t) (6.43)

In the second line, we have taken the usual limit ofan ideal voltage amplifier which has an infinite inputimpedance (i.e. the amplifier draws zero current). Thespectral density of the total input voltage fluctuations isthus:

SV V,tot[ω] = Svinvin [ω] + SV V,add[ω]. (6.44)

44

Here Svinvin is the spectral density of the voltage fluc-tuations of the input signal vin(t), and SV V,add is theamplifier’s contribution to the total noise at the input:

SV V,add[ω] = SV V + |Zs|2 SI I − 2Re [Z∗s SV I ](6.45)

For clarity, we have dropped the frequency index for thespectral densities appearing on the RHS of this equation.

It is useful to now consider a narrow bandwidth inputsignal at a frequency ω, and ask the following question: ifthe signal source was simply an equilibrium resistor at atemperature T0, how much hotter would it have to be toproduce a voltage noise equal to SV V,tot[ω]? The result-ing increase in the source temperature is defined as thenoise temperature TN[ω] of the amplifier and is a conve-nient measure of the amplifier’s added noise. It is stan-dard among engineers to define the noise temperature as-suming the initial temperature of the resistor T0 % !ω.One may then use the classical expression for the thermalnoise of a resistor, which yields the definition:

2Re Zs · kBTN[ω] ≡ SV V,tot[ω] (6.46)

Writing Zs = |Zs|eiφ, we have:

2kBTN =1

cos φ

[SV V

|Zs|+ |Zs|SI I − 2Re

(e−iφSV I

)]

(6.47)It is clear from this expression that TN will have a mini-mum as a function of |Zs|. For |Zs| too large, the back-action current noise of the amplifier will dominate TN,while for |Zs| too small, the voltage noise of the ampli-fier (i.e. its intrinsic output noise will dominate). Thesituation is completely analogous to the position detec-tor of the last section; there, we needed to optimize thecoupling strength A to balance back-action and intrinsicoutput noise contributions, and thus minimize the to-tal added noise. Optimizing the source impedance thusyields a completely classical minimum bound on TN:

kBTN ≥√SV V SI I − [Im SV I ]

2 − Re SV I (6.48)

where the minimum is achieved for an optimal sourceimpedance which satisfies:

|Zs[ω]|opt =

√SV V [ω]SI I [ω]

≡ ZN (6.49)

sinφ[ω]∣∣opt

= −Im SV I [ω]√SV V [ω]SI I [ω]

(6.50)

The above equations define the so-called noise impedanceZN. We stress again that the discussion so far in thissubsection has been completely classical.

2. Linear response description

It is easy to connect the classical description of a volt-age amplifier to the quantum mechanical description of a

generic linear response detector; in fact, all that is neededis a “relabeling” of the concepts and quantities we in-troduced in Sec. VI.D when discussing a linear positiondetector. Thus, the quantum voltage amplifier will becharacterized by both an input operator Q and an out-put operator Vout; these play the role, respectively, ofF and I in the position detector. Vout represents theoutput voltage of the amplifier, while Q is the operatorwhich couples to the input signal vin(t) via a couplingHamiltonian:

Hint = vin(t) · Q (6.51)

In more familiar terms, ˆIin−dQ/dt represents the currentflowing into the amplifier 20. The voltage gain of ouramplifier λV will again be given by the Kubo formula ofEq. (5.3), with the substitutions F → Q, I → Vout (wewill assume these substitutions throughout this section).

We can now easily relate the fluctuations of the in-put and output operators to the noise sources used todescribe the classical voltage amplifier. As usual, sym-metrized quantum noise spectral densities S[ω] will playthe role of the classical spectral densities S[ω] appearingin the classical description. First, as the operator Q rep-resents a back-action force, its fluctuations correspond tothe amplifier’s current noise I(t):

SI I [ω] ↔ ω2SQQ[ω]. (6.52)

Similarly, the fluctuations in the operator Vout, when re-ferred back to the amplifier input, will correspond to thevoltage noise V (t) discussed above:

SV V [ω] ↔ SVoutVout [ω]|λV |2

(6.53)

A similar correspondence holds for the cross-correlator ofthese noise sources:

SV I [ω] ↔ +iωSVoutQ[ω]

λV(6.54)

To proceed, we need to identify the input and outputimpedances of the amplifier, and then define its powergain. The first step in this direction is to assume that theoutput of the amplifier (Vout) is connected to an externalcircuit via a term:

H ′int = qout(t) · Vout (6.55)

where iout = dqout/dt is the current in the externalcircuit. We may now identify the input and output

20 Note that one could have instead written the coupling Hamil-

tonian in the more traditional form Hint(t) = φ(t) · ˆIin, whereφ =

Rdt′vin(t′) is the flux associated with the input voltage.

The linear response results we obtain are exactly the same. Weprefer to work with the charge Q in order to be consistent withthe rest of the text.

45

impedances of the amplifier in terms of the damping atthe input and output. Using the Kubo formulae forconductance and resistance yields (cf. Eq. (3.32) andEq. (6.21), with the substitutions F → Q and I → V ):

1/Zin[ω] = iωχQQ[ω] (6.56)

Zout[ω] =χV V [ω]−iω

(6.57)

i.e. 〈Iin〉ω = 1Zin[ω]vin[ω] and 〈V 〉ω = Zout[ω]iout[ω],

where the subscript ω indicates the Fourier transform ofa time-dependent expectation value.

We will consider throughout this section the case of noreverse gain, χQVout = 0. We can define the power gainGP exactly as we did in Sec. VI.D.3 for a linear positiondetector. GP is defined as the ratio of the power deliveredto a load attached to the amplifier output divided bythe power drawn by the amplifier, maximized over theimpedance of the load. One finds:

GP =|λV |2

4Re (Zout)Re (1/Zin)(6.58)

Expressing this in terms of the linear response coefficientsχV V and χQQ, we obtain an expression which is com-pletely analogous to Eq. (6.25) for the power gain for aposition detector:

GP =|λV |2

4Im χQQ · Im χV V(6.59)

Finally, we may again define the effective temperatureTeff [ω] of the amplifier via Eq. (3.21), and define aquantum-limited voltage amplifier as one which satisfiesthe ideal noise condition of Eq. (5.16). For such an am-plifier, the power gain will again be determined by theeffective temperature via Eq. (6.28).

Turning to the noise, we can again calculate the to-tal symmetrized noise at the output port of the amplifierfollowing the same argument used to the get the out-put noise of the position detector (cf. Eq. (6.19)). Aswe did in the classical approach, we will again assumethat the input impedance of the amplifier is much largerthat source impedance: Zin % Zs; we will test this as-sumption for consistency at the end of the calculation.Focusing only on the amplifier contribution to this noise(as opposed to the intrinsic noise of the input signal), andreferring this noise back to the amplifier input, we findthat the symmetrized quantum noise spectral density de-scribing the added noise of the amplifier, SV V,add[ω], sat-isfies the same equation we found for a classical voltageamplifier, Eq.. (6.45), with each classical spectral densityS[ω] being replaced by the corresponding symmetrizedquantum spectral density S[ω] as per Eqs. (6.52) - (6.54).

It follows that the amplifier noise temperature willagain be given by Eq. (6.47), and that the optimalnoise temperature (after optimizing over the sourceimpedance) will be given by Eq. (6.48). Whereas classi-cally nothing more could be said, quantum mechanically,

we now get a further bound from the quantum noise con-straint of Eq. (5.11) and the requirement of a large powergain. The latter requirement tells us that the voltage gainλV [ω] and the cross-correlator SVoutQ must be in phase(cf. Sec. (V.A.4) and Appendix J.2). This in turn meansthat SV I must be purely imaginary. In this case, thequantum noise constraint may be re-written as:

SV V [ω]SI I [ω]−[Im SV I

]2 ≥(

!ω

2

)2

(6.60)

Using these results in Eq. (6.48), we find the ultimatequantum limit on the noise temperature 21:

kBTN[ω] ≥ !ω

2(6.61)

Similar to the case of the position detector, reaching thequantum limit here is not simply a matter of tuning thecoupling (i.e. tuning the source impedance Zs to matchthe noise impedance, cf. Eq. (6.49) - (6.50)); one alsoneeds to have a detector with “ideal” quantum noise, thatis a detector satisfying Eq. (5.16). As we have repeatedlystressed in this review, this is a condition which is notmet by most detectors. It corresponds to the heuristicrequirement that there should not be any “wasted” in-formation in the detector.

Finally, we need to test our initial assumption that|Zs|, |Zin|, taking |Zs| to be equal to its optimal valueZN. Using the proportionality condition of Eq. (J13)and the fact that we are in the large power gain limit(GP [ω] % 1), we find:∣∣∣∣

ZN[ω]Re Zin[ω]

∣∣∣∣ =∣∣∣

α

Im α

∣∣∣!ω

4kBTeff=

12√

GP [ω], 1 (6.62)

It follows that |ZN|,| Zin| in the large power gain, largeeffective temperature regime of interest, thus justifyingthe form of Eq. (6.45). Eq. (6.62) is analogous to thecase of the displacement detector, where we found thatreaching the quantum limit on resonance required thedetector-dependent damping to be much weaker than theintrinsic damping of the oscillator (cf. Eq. (6.41)).

Thus, similar to the situation of the displacement de-tector, the linear response approach allows us both toderive rigorously the quantum limit on the noise tempera-ture TN of an amplifier, and to state conditions that must

21 Note that our definition of the noise temperature TN conformswith that of Devoret and Schoelkopf (2000) and most electricalengineering texts, but is slightly different than that of Caves(1982). Caves assumes the source is initially at zero temperature(i.e. T0 = 0), and consequently uses the full quantum expressionfor its equilibrium noise. In contrast, we have assumed thatkBT0 # !ω. The different definition of the noise temperatureused by Caves leads to the result kBTN ≥ !ω/(ln 3) as opposedto our Eq. (6.61). We stress that the difference between theseresults has nothing to do with physics, but only with how onedefines the noise temperature.

46

be met to reach this limit. To reach the quantum-limitedvalue of TN with a large power gain, one needs both atuned source impedance Zs, and an amplifier which pos-sesses ideal noise properties (cf. Eq. (5.16) and Eq. (J13)).

3. Role of noise cross-correlations

Before leaving the topic of a linear voltage amplifier,we pause to note the role of cross-correlations in currentand voltage noise in reaching the quantum limit. First,note from Eq. (6.50) that in both the classical and quan-tum treatments, the noise impedance ZN of the amplifierwill have a reactive part (i.e. Im ZN *= 0) if there areout-of-phase correlations between the amplifier’s currentand voltage noises (i.e. if Im SV I *= 0). Thus, if suchcorrelations exist, it will not be possible to minimize thenoise temperature (and hence, reach the quantum limit),if one uses a purely real source impedance Zs.

More significantly, note that the final classical expres-sion for the noise temperature TN explicitly involves thereal part of the SV I correlator (cf. Eq. (6.48)). In con-trast, we have shown that in the quantum case, Re SV I

must be zero if one wishes to reach the quantum limitwhile having a large power gain (cf. Appendix J.2); assuch, this quantity does not appear in the final expres-sion for the minimal TN. It also follows that to reach thequantum limit while having a large power gain, an ampli-fier cannot have significant in-phase correlations betweenits current and voltage noise.

This last statement can be given a heuristic explana-tion. If there are out-of-phase correlations between cur-rent and voltage noise, we can easily make use of these byappropriately choosing our source impedance. However,if there are in-phase correlations between current andvoltage noise, we cannot use these simply by tuning thesource impedance. We could however have used them byimplementing feedback in our amplifier. The fact that wehave not done this means that these correlations repre-sent a kind of missing information; as a result, we mustnecessarily miss the quantum limit. In Sec. VII.B, weexplicitly give an example of a voltage amplifier whichmisses the quantum limit due to the presence of in-phasecurrent and voltage fluctuations; we show how this am-plifier can be made to reach the quantum limit by addingfeedback in Sec. I.

F. Near quantum-limited mesoscopic detectors

Having discussed the origin and precise definition ofthe quantum limit on the added noise of a linear, phase-preserving amplifier, we now provide a brief review ofwork examining whether particular detectors are able (inprinciple) to achieve this ideal limit. We will focus onthe “op-amp” mode of operation discussed in Sec. VI.C,where the detector is only weakly coupled to the systemproducing the signal to be amplified. As we have repeat-

edly stressed, reaching the quantum limit in this caserequires the detector to have “quantum ideal noise”, asdefined by Eq. (5.16). Heuristically, this corresponds tothe general requirement of no wasted information: thereshould be no other quantity besides the detector outputthat could be monitored to provide information on the in-put signal (Clerk et al., 2003). We have already given onesimple but relevant example of a detector which reachesthe amplifier quantum limit: the parametric cavity de-tector, discussed extensively in Sec. IV.B. Here, we turnto other more complex detectors.

1. dc SQUID amplifiers

The dc SQUID (superconducting quantum interferencedevice) is a detector based on a superconducting ring hav-ing two Josephson junctions. It can in principle be usedas a near quantum-limited voltage amplifier or flux-to-voltage amplifier. Theoretically, this was investigated us-ing a quantum Langevin approach (Danilov et al., 1983;Koch et al., 1980), as well as more rigorously by usingperturbative techniques (Averin, 2000b) and mappingsto quantum impurity problems (Clerk, 2006). Experi-ments on SQUIDS have also confirmed its potential fornear quantum-limited operation. Muck et al. (2001) wereable to achieve a noise temperature TN approximately1.9 times the quantum limited value at an operating fre-quency of ω = 2π× 519 MHz. Working at lower frequen-cies appropriate to gravitational wave detection applica-tions, Vinante et al. (2001) were able to achieve a TN ap-proximately 200 times the quantum limited value at a fre-quency ω = 2π×1.6 kHz. In practice, it can be difficult toachieve the theoretically-predicted quantum-limited per-formance due to spurious heating caused by the dissipa-tion in the shunt resistances used in the SQUID.

2. Quantum point contact detectors

A quantum point contact (QPC) is a narrow conduct-ing channel formed in a two-dimensional gas. The cur-rent through the constriction is very sensitive to nearbycharges, and thus the QPC acts as a charge-to-currentamplifier. It has been shown theoretically that the QPCcan achieve the amplifier quantum limit , both in theregime where transport is due to tunneling (Gurvitz,1997), as well as in regimes where the transmission is notsmall (Aleiner et al., 1997; Clerk et al., 2003; Korotkovand Averin, 2001; Levinson, 1997; Pilgram and Buttiker,2002). Experimentally, QPCs are in widespread use asdetectors of quantum dot qubits. The back-action de-phasing of QPC detectors was studied in (Buks et al.,1998; Sprinzak et al., 2000); a good agreement was foundwith the theoretical prediction, confirming that the QPChas quantum-limited back-action noise.

47

3. Single-electron transistors and resonant-level detectors

A metallic single-electron transistor (SET) consists ofa small metallic island attached via tunnel junctions tolarger source and drain electrodes. Because of Coulombblockade effects, the conductance of a SET is very sen-sitive to nearby charges, and hence it acts as a sensitivecharge-to-current amplifier. Considerable work has inves-tigated whether metallic SETs can approach the quan-tum limit in various different operating regimes. The-oretically, the performance of a normal-metal SET inthe sequential tunneling regime was studied by Devoretand Schoelkopf (2000); Makhlin et al. (2000); Shnirmanand Schon (1998). In this regime, where transport isvia a sequence of energy-conserving tunnel events, oneis far from optimizing the quantum noise constraint ofEq. (5.16), and hence one cannot reach the quantumlimit (Korotkov, 2001b; Shnirman and Schon, 1998). Ifone instead chooses to work with a normal-metal SETin the regime where transport is due to cotunneling (ahigher-order tunneling process involving a virtual transi-tion), then one can indeed approach the quantum limit(Averin, 2000a; van den Brink, 2002). However, by virtueof being a higher-order process, the related currents andgain factors are small, impinging on the practical util-ity of this regime of operation. It is worth noting thatwhile most theory on SETs assume a dc voltage bias, toenhance bandwidth, experiments are usually conductedusing the rf-SET configuration (Schoelkopf et al., 1998),where the SET changes the damping of a resonant LCcircuit. Korotkov and Paalanen (1999) have shown thatthis mode of operation for a sequential tunneling SETincreases the measurement imprecision noise by approx-imately a factor of 2. The measurement properties ofa normal-metal, sequential-tunneling rf-SET (includingback-action) were studied experimentally in Turek et al.(2005).

Measurement using superconducting SET’s has alsobeen studied. Clerk et al. (2002) have shown that so-called incoherent Cooper-pair tunneling processes in a su-perconducting SET can have a noise temperature whichis approximately a factor of two larger than the quantumlimited value. The measurement properties of supercon-ducting SETs bias at a point of incoherent Cooper-pairtunneling have been probed recently in experiment (Naiket al., 2006; Thalakulam et al., 2004).

The quantum measurement properties of non-interacting resonant level detectors have also been stud-ied theoretically (Averin, 2000b; Clerk and Stone, 2004;Gavish et al., 2006; Mozyrsky et al., 2004). These sys-tems are similar to metallic SET, except that the centralisland only has a single level (as opposed to a continuousdensity of states), and Coulomb-blockade effects are typ-ically neglected. These detectors can reach the quantumlimit in the regime where the voltage and temperatureare smaller than the intrinsic energy broadening of thelevel due to tunneling. They can also reach the quan-tum limit in a large-voltage regime that is analogous to

the cotunneling regime in a metallic SET (Averin, 2000b;Clerk and Stone, 2004).

G. Back-action evasion and noise-free amplification

Having discussed in detail quantum limits on phase-preserving linear amplifiers (i.e. amplifiers which measureboth quadratures of a signal equally well), we now returnto the situation discussed at the very start of Sec. VI.A:imagine we wish only to amplify a single quadrature ofsome time-dependent signal. For this case, there neednot be any added noise from the measurement. Unlikethe case of amplifying both quadratures, Liouville’s the-orem does not require the existence of any additionaldegrees of freedom when amplifying a single quadrature:phase space volume can be conserved during amplifica-tion simply by contracting the unmeasured quadrature(cf. Eq. 6.2). As no extra degrees of freedom are needed,there need not be any extra noise associated with theamplification process.

Alternatively, single-quadrature detection can take aform similar to a QND measurement, where the back-action does not affect the dynamics of the quantity be-ing measured (Bocko and Onofrio, 1996; Braginsky andKhalili, 1992; Braginsky et al., 1980; Caves, 1982; Caveset al., 1980; Thorne et al., 1978). For concreteness, con-sider a high-Q harmonic oscillator with position x(t) andresonant frequency Ω. Its motion may be written interms of quadrature operators defined as in Eq. (4.66):

x(t) = Xδ(t) cos (Ωt + δ) + Yδ(t) sin (Ωt + δ)(6.63)

Here, x(t) is the Heisenberg-picture position operator ofthe oscillator. The quadrature operators can be writtenin terms of the (Schrodinger-picture) oscillator creationand destruction operators as:

Xδ(t) = xZPF

(cei(Ωt+δ) + c†e−i(Ωt+δ)

)(6.64a)

Yδ(t) = −ixZPF

(cei(Ωt+δ) − c†e−i(Ωt+δ)

)(6.64b)

As previously discussed ((c.f. Eq.(4.67)), the two quadra-ture amplitude operators Xδ and Yδ are canonicallyconjugate (c.f. Eq.(4.67)). Making a measurement ofone quadrature amplitude, say Xδ, will thus invariablylead to back-action disturbance of the other, conjugatequadrature Yδ. However, due to the dynamics of a har-monic oscillator, this disturbance will not affect the mea-sured quadrature at later times. One can already see thisfrom the classical equations of motion. Suppose our os-cillator is driven by a time-dependent force F (t) whichonly has appreciable bandwidth near Ω. We may writethis as:

F (t) = FX(t) cos(Ωt + δ) + FY (t) sin(Ωt + δ) (6.65)

48

where FX(t), FY (t) are slowly varying compared to Ω.Using the fact that the oscillator has a high-quality factorQ = Ω/γ, one can easily find the equations of motion:

d

dtXδ(t) = −γ

2Xδ(t)−

FY (t)2mΩ

, (6.66a)

d

dtYδ(t) = −γ

2Yδ(t) +

FX(t)2mΩ

. (6.66b)

Thus, as long as FY (t) and FX(t) are uncorrelated andsufficiently slow, the dynamics of the two quadratures arecompletely independent; in particular, if Yδ is subject toa narrow-bandwidth, noisy force, it is of no consequenceto the evolution of Xδ. An ideal measurement of Xδ willresult in a back-action force having the form in Eq. (6.65)with FY (t) = 0, implying that Xδ(t) will be completelyunaffected by the measurement.

Not surprisingly, if one can measure and amplify Xδ

without any back-action, there need not be any addednoise due to the amplification. In such a setup, theonly added noise is the measurement-imprecision noiseassociated with intrinsic fluctuations of the amplifier out-put. These may be reduced (in principle) to an arbitrar-ily small value by simply increasing the amplifier gain(e.g. by increasing the detector-system coupling): in anideal setup, there is no back-action penalty on the mea-sured quadrature associated with this increase.

The above conclusion can lead to what seems like acontradiction. Imagine we use a back-action evading am-plifier to make a “perfect” measurement of Xδ (i.e. negli-gible added noise). We would then have no uncertainty asto the value of this quadrature. Consequently, would ex-pect the quantum state of our oscillator to be a squeezedstate, where the uncertainty in Xδ is much smaller thanxZPF. However, if there is no back-action acting on Xδ,how is the amplifier able to reduce its uncertainty? Thisseeming paradox can be fully resolved by considering theconditional aspects of an ideal single quadrature mea-surement, where one considers the state of the oscilla-tor given a particular measurement history (Clerk et al.,2008; Ruskov et al., 2005).

It is worth stressing that the possibility of amplifying asingle quadrature without back-action (and hence, with-out added noise) relies crucially on our oscillator resem-bling a perfect harmonic oscillator: the oscillator Q mustbe large, and non-linearities (which could couple the twoquadratures) must be small. In addition, the envelope ofthe non-vanishing back-action force FX(t) must have anarrow bandwidth. One should further note that a veryhigh precision measurement of Xδ will produce a verylarge back action force FX . If the system is not nearlyperfectly harmonic, then the large amplitude impartedto the conjugate quadrature Yδ will inevitably leak backinto Xδ.

Amplifiers or detectors which treat the two signalquadratures differently are known in the quantum op-tics literature as ‘phase sensitive’; we prefer the desig-nation ‘phase-non-preserving’ since they do not preservethe phase of the original signal. Such amplifiers invari-

ably rely on some internal clock (i.e. an oscillator with awell defined phase) which breaks time-translation invari-ance and picks out the phase of the quadrature that willbe amplified (i.e. the choice of δ used to define the twoquadratures in Eq. (6.63)); we will see this explicitly inwhat follows. This leads to an important caveat: evenin a situation where the interesting information is in asingle signal quadrature, to benefit from using a phasenon-preserving amplifier, we must know in advance theprecise phase of this quadrature. If we do not know thisphase, we will have either to revert to a phase-preservingamplification scheme (and thus be susceptible to addednoise) or we would have to develop a sophisticated andhigh speed quantum feedback scheme to dynamicallyadapt the measurement to the correct quadrature in realtime (Armen et al., 2002). In what follows, we will makethe above ideas concrete by considering a few examplesof quantum, phase non-preserving amplifiers.

1. Degenerate parametric amplifier

Perhaps the simplest example of a phase non-preserving amplifier is the degenerate parametric ampli-fier; the classical version of this system was describedat the start of Sec. VI.A (cf. Eq. 6.2). The quantumversion is also extremely simple to treat, and has manysimilarities to the single-mode, phase-preserving ampli-fier discussed in Sec. VI.B. Similar to that system, in thedegenerate parametric amplifier both the amplifier inputand output are described by single bosonic modes, havingannihilation operators a and b respectively. This ampli-fier is operated in the “scattering” mode of operation de-scribed in Sec. VI.C: a describes the input signal, whichis the amplitude of a wave incident on the amplifier, whileb describes the output signal, the amplitude of a waveleaving the amplifier. The dimensionless quadrature op-erators corresponding to the input and output (taking,without loss of generality, the phase δ in Eq. (6.63) to bezero) are given by:

xin =1√2

(a† + a

)

yin =i√2

(a† − a

)

xout =1√2

(b† + b

)

yout =i√2

(b† − b

)(6.67)

These operators are canonically conjugate, satisfying:

[xin, yin] = [xout, yout] = i. (6.68)

With these definitions, the action of the amplifier isdescribed by the input-output relations (compare against

49

Eq. (6.6)):

xout =√

G xin (6.69a)

yout =1√G

yin (6.69b)

These input-output relations are derived in Appendix F.We see that our amplifier clearly treats the two quadra-tures differently, and hence is a phase-sensitive amplifier:one quadrature is amplified and the other is deamplifiedin such a way that the commutation relation can be pre-served without the necessity of any added noise. Notethat the degenerate parametric amplifier relies on hav-ing a “pump” mode which has a large amplitude and analmost-definite phase. This pump mode plays the role ofa clock; in particular, its phase picks out the quadratureof the signal which is amplified (see Appendix F for moredetails).

It is important to stress that while the degenerateparametric amplifier is phase-sensitive and has no addednoise, it is not an example of back-action evasion (seeCaves et al. (1980), footnote on p. 342). This amplifier isoperated in the scattering mode of amplifier operation, amode where (as discussed extensively in Sec. VI.C) back-action is not at all relevant. Recall that in this modeof operation, the amplifier input is perfectly impedancematched to the signal source, and the input signal is sim-ply the amplitude of an incident wave on the amplifier in-put. This mode of operation necessarily requires a strongcoupling between the input signal and the amplifier in-put. If one instead tried to weakly couple the degenerateparametric amplifier to a signal source, and operate itin the “op-amp” mode of operation (c.f. Sec. VI.C), onefinds that there is indeed a back-action disturbance ofthe measured quadrature. We have yet another exam-ple which demonstrates that one must be very carefulto distinguish the “op-amp” and “scattering” mode ofamplifier operation.

2. Double-sideband cavity detector

We now turn to a simple but experimentally-relevantdetector that is truly back-action evading. We will takeas our input signal the position x of a mechanical oscilla-tor. The amplifier setup we consider is almost identicalto the cavity position detector discussed in Sec. IV.B.3:we again have a single-sided resonant cavity whose fre-quency depends linearly on the oscillator’s position, withthe Hamiltonian being given by Eq. (4.22) (with z =x/xZPF). We showed in Sec. IV.B.3 and Appendix D.3that by driving the cavity on resonance, it could be usedto make a quantum limited position measurement: onecan operate it as a phase-preserving amplifier of the me-chanical’s oscillators position, and achieve the minimumpossible amount of added noise. To use the same systemto make a back-action free measurement of one oscillatorquadrature only, one simply uses a different cavity drive.Instead of driving at the cavity resonance frequency ωc,

one drives at the two sidebands associated with the me-chanical motion (i.e. at frequencies ωc ± Ω, where Ω isas always the frequency of the mechanical resonator). Aswe will see, such a drive results in an effective interactionwhich only couples the cavity to one quadrature of theoscillator’s motion. This setup was first proposed as ameans of back-action evasion in Braginsky et al. (1980);further discussion can be found in Braginsky and Khalili(1992); Caves et al. (1980), as well as in Clerk et al.(2008), which gives a fully quantum treatment and con-siders conditional aspects of the measurement. In whatfollows, we sketch the operation of this system followingClerk et al. (2008); details are provided in Appendix D.4.

We will start by requiring that our system be in the“good-cavity” limit, where ωc % Ω % κ (κ is the damp-ing of the cavity mode); we will also require the mechan-ical oscillator to have a high Q-factor. In this regime, thetwo sidebands associated with the mechanical motion atωc±Ω are well-separated from the main cavity resonanceat ωc. Making a single-quadrature measurement requiresthat one drives the cavity at equally at the two sidebandfrequencies. The amplitude of driving field bin enteringthe cavity will be chosen to have the form:

bin(t) = − i√

N

4

(eiδe−i(ωc−Ω)t − e−iδe−i(ωc+Ω)t

)

=

√N

2sin(Ωt + δ)e−iωct. (6.70)

Here, N is the photon number flux associated with thecavity drive (see Appendix D for more details on howto properly include a drive using input-output theory).Such a drive could be produced by taking a signal at thecavity resonance frequency, and amplitude modulating itat the mechanical frequency.

To understand the effect of this drive, note that it sendsthe cavity both photons with frequency (ωc − Ω) andphotons with frequency (ωc + Ω). The first kind of drivephoton can be converted to a cavity photon if a quanta isabsorbed from the mechanical oscillator; the second kindof drive photon can be converted to a cavity photon if aquanta is emitted to the mechanical oscillator. The resultis that we can create a cavity photon by either absorb-ing or emitting a mechanical oscillator quanta. Keepingtrack that there is a well-defined relative phase of ei2δ

between the two kinds of drive photons, we would expectthe double-sideband drive to yield an effective cavity-oscillator interaction of the form:

Veff ∝√

N[a†

(eiδ c + e−iδ c†

)+ h.c.

](6.71a)

∝√

N(a + a†)Xδ (6.71b)

This is exactly what is found in a full calculation (see Ap-pendix D.4). Note that we have written the interactionin an interaction picture in which the fast oscillations ofthe cavity and oscillator operators have been removed.In the second line, we have made use of Eqs. (6.64) to

50

show that the effective interaction only involves the Xδ

oscillator quadrature.We thus see from Eq. (6.71b) that the cavity is only

coupled to the oscillator Xδ quadrature. As shown rig-orously in Appendix subapp:CavityPositionDetector, theresult is that the system only measures and amplifies thisquadrature: the light leaving the cavity has a signatureof Xδ, but not of Yδ. Further, Eq. (6.71b) implies thatthe cavity operator

√N

(a + a†

)will act as a noisy force

on the Yδ quadrature. While this will cause a back-actionheating of Yδ, it will not affect the measured quadratureXδ. We thus have a true back-action evading amplifier:the cavity output light lets one measure Xδ free from anyback-action effect. Note that in deriving Eq. (6.71a), wehave used the fact that the cavity operators have fluctu-ations in a narrow bandwidth ∼ κ , Ω: the back-actionforce noise is slow compared to the oscillator frequency. Ifthis were not the case, we could still have a back-actionheating of the measured Xδ quadrature. Such effects,arising from a non-zero ratio κ/Ω, are treated in Clerket al. (2008).

Finally, as there is no back-action on the measuredXδ quadrature, the only added noise of the amplificationscheme is measurement-imprecision noise (e.g. shot noisein the light leaving the cavity). This added noise canbe made arbitrarily small by increasing the gain of thedetector by, for example, increasing the strength of thecavity drive N . In a real system where κ/ωM is non-zero,the finite-bandwidth of the cavity number fluctuationsleads to a small back-action on the Xδ. As a result, onecannot make the added noise arbitrarily small, as toolarge a cavity drive will heat the measured quadrature.Nonetheless, for a sufficiently small ratio κ/ωM , one canstill beat the standard quantum limit on the added noise(Clerk et al., 2008).

3. Stroboscopic measurements

With sufficiently high bandwidth it should be able todo stroboscopic measurements in sync with the oscillatormotion which could allow one to go below the standardquantum limit in one of the quadratures of motion (Bra-ginsky and Khalili, 1992; Caves et al., 1980). To under-stand this idea, imagine an extreme form of phase sen-sitive detection in which a Heisenberg microscope makesa strong high-resolution measurement which projects theoscillator onto a state of well defined position X0 at timet = 0:

ΨX0(t) =∞∑

n=0

ane−i(n+1/2)Ωt |n〉 , (6.72)

where the coefficients obey an = 〈n|X0〉. Because theposition is well-defined the momentum is extremely un-certain. (Equivalently the momentum kick delivered bythe back action of the microscope makes the oscillatormomentum uncertain.) Thus the wave packet quickly

spreads out and the position uncertainty becomes large.However because of the special feature that the har-monic oscillator levels are evenly spaced, we can see fromEq. (6.72) that the wave packet reassembles itself pre-cisely once each period of oscillation because einΩt = 1for every integer n. (At half periods, the packet re-assembles at position −X0.) Hence stroboscopic mea-surements made once (or twice) per period will be backaction evading and can go below the standard quantumlimit. The only limitations will be the finite anharmonic-ity and damping of the oscillator. Note that the possi-bility of using mesoscopic electron detectors to performstroboscopic measurements has recently received atten-tion (Jordan and Buttiker, 2005; Ruskov et al., 2005).

VII. BOSONIC SCATTERING DESCRIPTION OF ATWO-PORT AMPLIFIER

In this section, we return again to the topic ofSec. VI.E, quantum limits on a quantum voltage ampli-fier. We now discuss the physics in terms of the bosonicvoltage amplifier first introduced in Sec. VI.C. Recallthat in that subsection, we demonstrated that the stan-dard Haus-Caves derivation of the quantum limit was notdirectly relevant to the usual weak-coupling “op-amp”mode of amplifier operation, a mode where the inputsignal is not simply the amplitude of a wave incidenton the amplifier. In this section, we will expand uponthat discussion, giving an explicit discussion of the dif-ferences between the op-amp description of an amplifierpresented in Sec. VI.D, and the scattering description of-ten used in the quantum optics literature (Courty et al.,1999; Grassia, 1998). We will see that what one meansby “back-action” and “added noise” are not the same inthe two descriptions! Further, even though an amplifiermay reach the quantum limit when used in the scatteringmode (i.e. its added noise is as small as allowed by com-mutation relations), it can nonetheless fail to achieve thequantum limit when used in the op-amp mode. Finally,the discussion here will also allow us to highlight impor-tant aspects of the quantum limit not easily discussed inthe more general context of Sec. V.

A. Scattering versus op-amp representations

In the bosonic scattering approach, a generic linearamplifier is modeled as a set of coupled bosonic modes.To make matters concrete, we will consider the specificcase of a voltage amplifier with distinct input and out-put ports, where each port is a semi-infinite transmissionline (see Fig. 9). We gave a heuristic description of thissystem in Sec. VI.C; here, we will investigate its featuresin more detail and with more rigour.

As discussed in Appendix C and Yurke and Denker(1984), a transmission line can be described as a setof non-interacting bosonic modes. Denoting the inputtransmission line with an a and the output transmission

51

line with a b, the current and voltage operators in theselines may be written:

Vq(t) =∫ ∞

0

dω

2π

(Vq[ω]e−iωt + h.c.

)(7.1a)

Iq(t) = σq

∫ ∞

0

dω

2π

(Iq[ω]e−iωt + h.c.

)(7.1b)

with

Vq[ω] =√

!ω

2Zq (qin[ω] + qout[ω]) (7.2a)

Iq[ω] =

√!ω

2Zq(qin[ω]− qout[ω]) (7.2b)

Here, q can be equal to a or b, and we have σa = 1, σb =−1. The operators ain[ω], aout[ω] are canonical bosonicannihilation operators; ain[ω] describes an incoming wavein the input transmission line (i.e. incident on the ampli-fier) having frequency ω, while aout[ω] describes an out-going wave with frequency ω. The operators bin[ω] andbout[ω] describe analogous waves in the output transmis-sion line. We can think of Va as the input voltage toour amplifier, and Vb as the output voltage. Similarly,Ia is the current drawn by the amplifier at the input,and Ib the current drawn at the output of the ampli-fier. Finally, Za (Zb) is the characteristic impedance ofthe input (output) transmission line. Note that we usea slightly different sign convention than in Yurke andDenker (1984).

Amplification of a signal at a particular frequency ωwill in general involve 2N bosonic modes in the amplifier.Four of these modes are simply the frequency-ω modesin the input and output lines (i.e. ain[ω],aout[ω],bin[ω]and bout[ω]). The remaining 2(N − 2) modes describeauxiliary degrees of freedom involved in the amplifica-tion process; these additional modes could correspond tofrequencies different from the signal frequency ω. Theauxiliary modes can also be divided into incoming andoutgoing modes. It is thus convenient to represent themas additional transmission lines attached to the amplifier;these additional lines could be semi-infinite, or could beterminated by active elements.

1. Scattering representation

In general, our generic two-port bosonic amplifier willbe described by a N ×N scattering matrix which deter-mines the relation between the outgoing mode operatorsand incoming mode operators. The form of this matrixis constrained by the requirement that the output modesobey the usual canonical bosonic commutation relations.It is convenient to express the scattering matrix in a form

which only involves the input and output lines explicitly:

(aout[ω]bout[ω]

)=

(s11[ω] s12[ω]s21[ω] s22[ω]

) (ain[ω]bin[ω]

)

+

(Fa[ω]Fb[ω]

)(7.3)

Here Fa[ω] and Fb[ω] are each an unspecified linear com-bination of the auxiliary-line incident mode operators.They thus describe noise in the outgoing modes of theinput and output transmission lines which arises fromthe auxiliary modes involved in the amplification pro-cess. Note the similarity between Eq. (7.3) and Eq. (6.7a)for the simple one-port bosonic amplifier considered inSec. VI.B.

In the quantum optics literature, one typically viewsEq. (7.3) as the defining equation of the amplifier; wewill call this the scattering representation of our ampli-fier. The representation is best suited to the scatteringmode of amplifier operation described in Sec. VI.C. Inthis mode of operation, the experimentalist ensures that〈ain[ω]〉 is precisely equal to the signal to be amplified,irrespective of what is coming out of the amplifier. Sim-ilarly, the output signal from the amplifier is the ampli-tude of the outgoing wave in the output line, 〈bout[ω]〉.If we focus on bout, we have precisely the same situationas described in Sec. 6.10, where we presented the Haus-Caves derivation of the quantum limit (c.f. Eq. (6.7a)). Itthus follows that in the scattering mode of operation, thematrix element s21[ω] represents the gain of our amplifierat frequency ω, |s21[ω]|2 the corresponding “photon num-ber gain”, and Fb the added noise operator of the ampli-fier. The operator Fa represents the back-action noise inthe scattering mode of operation; this back-action has noeffect on the added noise of the amplifier in the scatteringmode.

Similar to Sec. VI.B, one can now apply the standardargument of Haus and Mullen (1962) and Caves (1982)to our amplifier. This argument tells us that since the“out” operators must have the same commutation rela-tions as the “in” operators, the added noise Fb cannot bearbitrarily small in the large gain limit (i.e. |s21| % 1).Note that this version of the quantum limit on the addednoise has nothing to do with back-action. As alreadydiscussed, this is perfectly appropriate for the scatter-ing mode of operation, as in this mode, the experimen-talist ensures that the signal going into the amplifier iscompletely independent of whatever is coming out of theamplifier. This mode of operation could be realized intime-dependent experiments, where a pulse is launchedat the amplifier. This mode is not realized in most weak-coupling amplification experiments, where the signal tobe amplified is not identical to an incident wave ampli-tude.

52

2. Op-amp representation

In the usual op-amp amplifier mode of operation (de-scribed extensively in Sec. V), the input and output sig-nals are not simply incoming/outgoing wave amplitudes;thus, the scattering representation is not an optimal de-scription of our amplifier. The system we are describinghere is a voltage amplifier: thus, in the op-amp mode,the experimentalist would ensure that the voltage at theend of the input line (Va) is equal to the signal to beamplified, and would read out the voltage at the end ofthe output transmission line (Vb) as the output of theamplifier. From Eq. (7.1a), we see that this implies thatthe amplitude of the wave going into the amplifier, ain,will depend on the amplitude of the wave exiting the am-plifier, aout.

Thus, if we want to use our amplifier as a voltage am-plifier, we would like to find a description which is moretailored to our needs than the scattering representationof Eq. (7.3). This can be found by simply re-expressingthe scattering matrix relation of Eq. (7.3) in terms ofvoltages and currents. The result will be what we termthe “op amp” representation of our amplifier, a repre-sentation which is standard in the discussion of classicalamplifiers (see, e.g., Boylestad and Nashelsky (2006)).In this representation, one views Va and Ib as inputsto the amplifier: Va is set by whatever we connect tothe amplifier input, while Ib is set by whatever we con-nect to the amplifier output. In contrast, the outputsof our amplifier are the voltage in the output line, Vb,and the current drawn by the amplifier at the input, Ia.Note that this interpretation of voltages and currents isidentical to how we viewed the voltage amplifier in thelinear-response/quantum noise treatment of Sec. VI.E.

Using Eqs. (7.1a) and (7.1b), and suppressing fre-quency labels for clarity, Eq. (7.3) may be written inthe form:

(Vb

Ia

)=

(λV −Zout1

Zinλ′I

) (Va

Ib

)+

(λV · ˆV

ˆI

)(7.4)

The coefficients in the above matrix are familiar from thediscussion of voltage amplifier in Sec. VI.E. λV [ω] is thevoltage gain of the amplifier, λ′I [ω] is the reverse currentgain of the amplifier, Zout is the output impedance, andZin is the input impedance. The last term on the RHSof Eq. (7.4) describes the two familiar kinds of amplifiernoise. ˆV is the usual voltage noise of the amplifier (re-ferred back to the amplifier input), while ˆI is the usualcurrent noise of the amplifier. Recall that in this stan-dard description of a voltage amplifier (cf. Sec. VI.E),I represents the back-action of the amplifier: the sys-tem producing the input signal responds to these currentfluctuations, resulting in an additional fluctuation in theinput signal going into the amplifier. Similarly, λV · Vrepresents the intrinsic output noise of the amplifier: thiscontribution to the total output noise does not depend onproperties of the input signal. Note that we are using a

sign convention where a positive 〈Ia〉 indicates a currentflowing into the amplifier at its input, while a positive〈Ib〉 indicates a current flowing out of the amplifier at itsoutput. Also note that the operators Va and Ib on theRHS of Eq. (7.4) will have noise; this noise is entirely dueto the systems attached to the input and output of theamplifier, and as such, should not be included in whatwe call the added noise of the amplifier.

Additional important properties of our amplifier fol-low immediately from quantities in the op-amp repre-sentation. As discussed in Sec. VI.D, the most impor-tant measure of gain in our amplifier is the dimensionlesspower gain. This is the ratio between power dissipatedat the output to that dissipated at the input, taking theoutput current IB to be VB/Zout:

GP ≡ (λV )2

4Zin

Zout·(

1 +λV λ′I

2Zin

Zout

)−1

(7.5)

Note that for zero reverse gain, this coincides withEq. (6.58) of Sec. VI.E.

Another important quantity is the loaded inputimpedance: what is the input impedance of the ampli-fier in the presence of a load attached to the output? Inthe presence of reverse current gain λ′I *= 0, the inputimpedance will depend on the output load. Taking theload impedance to be Zload, some simple algebra yields:

1Zin,loaded

=1

Zin+

λ′IλV

Zload + Zout(7.6)

It is of course undesirable to have an input impedancewhich depends on the load. Thus, we see yet again thatit is undesirable to have appreciable reverse gain in ouramplifier (cf. Sec. V.A.2).

3. Converting between representations

Some straightforward algebra now lets us express theop-amp parameters appearing in Eq. (7.4) in terms of thescattering matrix appearing in Eq. (7.3):

λV = 2√

Zb

Za

s21

D(7.7a)

λ′I = 2√

Zb

Za

s12

D(7.7b)

Zout = Zb(1 + s11)(1 + s22)− s12s21

D(7.7c)

1Zin

=1

Za

(1− s11)(1− s22)− s12s21

D(7.7d)

where all quantities are evaluated at the same frequencyω, and D is defined as:

D = (1 + s11)(1− s22) + s12s21 (7.8)

Further, the voltage and current noises in the op-amprepresentation are simple linear combinations of the

53

noises Fa and Fb appearing in the scattering representa-tion:

( ˆVZa · ˆI

)=

√2!ωZa

(− 1

21+s112s21

s22−1D − s12

D

) (Fa

Fb

)

(7.9)

Again, all quantities above are evaluated at frequency ω.

Eq. (7.9) immediately leads to an important conclusionand caveat: what one calls the “back-action” and “addednoise” in the scattering representation (i.e. Fa and Fb )are not the same as the “back-action” and “added noise”defined in the usual op-amp representation. For example,the op-amp back-action ˆI does not in general coincidewith the Fa, the back-action in the scattering picture. Ifwe are indeed interested in using our amplifier as a volt-age amplifier, we are interested in the total added noiseof our amplifier as defined in the op-amp representation.As we saw in Sec. VI.E (cf. Eq. (6.43)), this quantityinvolves both the noises ˆI and ˆV . We thus see explic-itly something already discussed in Sec. VI.C: it is verydangerous to make conclusions about how an amplifierbehaves in the op-amp mode of operation based on itsproperties in the scattering mode of operation. As wewill see, even though an amplifier is “ideal” in the scat-tering mode (i.e. Fa as small as possible), it can nonethe-less fail to reach the quantum limit in the op-amp modeof operation.

In what follows, we will calculate the op-amp noises ˆVand ˆI in a minimal bosonic voltage amplifier, and showexplicitly how this description is connected to the moregeneral linear-response treatment of Sec. VI.E. However,before proceeding, it is worth noting that Eqs. (7.7a)-(7.7d) are themselves completely consistent with linear-response theory. Using linear-response, one would calcu-late the op-amp parameters λV , λ′I , Zin and Zout usingKubo formulas (cf. Eqs. (6.56), (6.57) and the discussionfollowing Eq. (6.51)). These in turn would involve corre-lation functions of Ia and Vb evaluated at zero couplingto the amplifier input and output. Zero coupling meansthat there is no input voltage to the amplifier (i.e. a shortcircuit at the amplifier input, Va = 0) and there is noth-ing at the amplifier output drawing current (i.e. an opencircuit at the amplifier output, Ib = 0). Eq. (7.4) tellsus that in this case, Vb and Ia reduce to (respectively)the noise operators λV

ˆV and ˆI. Using the fact that thecommutators of Fa and Fb are completely determinedby the scattering matrix (cf. Eq. (7.3)), we verify explic-itly in Appendix J.4 that the Kubo formulas yield thesame results for the op-amp gains and impedances asEqs. (7.7a)-(7.7d) above.

B. Minimal two-port scattering amplifier

1. Scattering versus op-amp quantum limit

In this subsection we demonstrate that an amplifierwhich is “ideal” and minimally complex when used in thescattering operation mode fails, when used as a voltageop-amp, to have a quantum limited noise temperature.The system we look at is very similar to the amplifierconsidered by Grassia (1998), though our conclusions aresomewhat different than those found there.

In the scattering representation, one might guess thatan “ideal” amplifier would be one where there are noreflections of signals at the input and output, and noway for incident signals at the output port to reach theinput. In this case, Eq. (7.3) takes the form:

(aout

bout

)=

(0 0√G 0

) (ain

bin

)+

(Fa

Fb

)(7.10)

where we have defined√

G ≡ s21. All quantities aboveshould be evaluated at the same frequency ω; for clarity,we will omit writing the explicit ω dependence of quan-tities throughout this section.

Turning to the op-amp representation, the above equa-tion implies that our amplifier has no reverse gain, andthat the input and output impedances are simply givenby the impedances of the input and output transmissionlines. From Eqs. (7.7), we have:

λV = 2√

Zb

ZaG (7.11a)

λ′I = 0 (7.11b)Zout = Zb (7.11c)

1Zin

=1

Za(7.11d)

We immediately see that our amplifier looks less idealas an op-amp. The input and output impedances arethe same as those of the input and output transmissionline. However, for an ideal op-amp, we would have likedZin →∞ and Zout → 0.

Also of interest are the expressions for the amplifiernoises in the op-amp representation:

( ˆVZa · ˆI

)= −

√2!ωZa

(12 − 1

2√

G

1 0

) (Fa

Fb

)

(7.12)

As s12 = 0, the back-action noise is the same in both theop-amp and scattering representations: it is determinedcompletely by the noise operator Fa. However, the volt-age noise (i.e. the intrinsic output noise) involves both Fa

and Fb. We thus have the unavoidable consequence thatthere will be correlations in ˆI and ˆV . Note that frombasic linear response theory, we know that there mustbe some correlations between ˆI and ˆV if there is to be

54

gain (i.e. λV is given by a Kubo formula involving theseoperators, cf. Eq. (5.3)).

To make further progress, we note again that commu-tators of the noise operators Fa and Fb are completelydetermined by Eq. (7.10) and the requirement that theoutput operators obey canonical commutation relations.We thus have:

[Fa, F†

a

]= 1 (7.13a)

[Fb, F†

b

]= 1− |G| (7.13b)

[Fa, Fb

]= 0 (7.13c)

[Fa, F†

b

]= 0 (7.13d)

We will be interested in the limit of a large powergain, which requires |G|% 1. A minimal solution to theabove equations would be to have the noise operators de-termined by two independent (i.e. mutually commuting)auxiliary input mode operators uin and v†in:

Fa = uin (7.14)

Fb =√|G|− 1v†in (7.15)

Further, to minimize the noise of the amplifier, we takethe operating state of the amplifier to be the vacuum forboth these modes. With these choices, our amplifier is inexactly the minimal form described by Grassia (1998):an input and output line coupled to a negative resis-tance box and an auxiliary “cold load” via a four-portcirculator (see Fig. 13). The negative resistance box isnothing but the single-mode bosonic amplifier discussedin Sec. VI.B; an explicit realization of this element wouldbe the parametric amplifier discussed in Appendix F.The “cold load” is a semi-infinite transmission line whichmodels dissipation due to a resistor at zero-temperature(i.e. its noise is vacuum noise, cf. Appendix C).

Note that within the scattering picture, one would con-clude that our amplifier is ideal: in the large gain limit,the noise added by the amplifier to bout corresponds to asingle quantum at the input:

⟨Fb , F†

b

⟩

|G| =|G|− 1|G|

⟨v†in, vin

⟩→ 1 (7.16)

This however is not the quantity which interests us: aswe want to use this system as a voltage op-amp, we wouldlike to know if the noise temperature defined in the op-amp picture is as small as possible. We are also usuallyinterested in the case of a signal which is weakly cou-pled to our amplifier; here, weak-coupling means thatthe input impedance of the amplifier is much larger thanthe impedance of the signal source (i.e. Zin % Zs). Inthis limit, the amplifier only slightly increases the totaldamping of the signal source.

To address whether we can reach the op-amp quantumlimit in the weak-coupling regime, we can make use of the

aout

ain

bin

bout

ideal1-portamp.

coldload

input line output line

cin cout=G1/2cin+(G-1)1/2v†in

uin uout

vin

vacuum noise

vacuum noisevout

FIG. 13 Schematic of a “minimal” two-port amplifier whichreaches the quantum limit in the scattering mode of operation,but misses the quantum limit when used as a weakly-coupledop-amp. See text for further description

results of the general theory presented in Sec. VI.E. Inparticular, we need to check whether the quantum noiseconstraint of Eq. (6.60) is satisfied, as this is a prereq-uisite for reaching the (weak-coupling) quantum limit.Thus, we need to calculate the symmetrized spectral den-sities of the current and voltage noises, and their cross-correlation. It is easy to confirm from the definitions ofEq. (7.1a) and (7.1b) that these quantities take the form:

SV V [ω] =

⟨ ˆV [ω], ˆV †(ω′)⟩

4πδ(ω − ω′)(7.17a)

SII [ω] =

⟨ ˆI[ω], ˆI†(ω′)⟩

4πδ(ω − ω′)(7.17b)

SV I [ω] =

⟨ ˆV [ω], ˆI†(ω′)⟩

4πδ(ω − ω′)(7.17c)

The expectation values here are over the operating stateof the amplifier; we have chosen this state to be the vac-uum for the auxiliary mode operators uin and vin to min-imize the noise.

Taking |s21|% 1, and using Eqs. (7.14) and (7.15), wehave

SV V [ω] =!ωZa

4(σuu + σvv) =

!ωZa

2(7.18a)

SII [ω] =!ω

Zaσuu =

!ω

Za(7.18b)

SV I [ω] =!ω

2σuu =

!ω

2(7.18c)

where we have defined:

σab ≡⟨ab† + b†a

⟩(7.19)

55

and have used the fact that there cannot be any correla-tions between the operators u and v in the vacuum state(i.e. 〈uv†〉 = 0).

It follows immediately from the above equations thatour minimal amplifier does not optimize the quantumnoise constraint of Eq. (6.60):

SV V [ω]SII [ω]−[Im SV I

]2 = 2×(

!ω

2

)2

. (7.20)

The noise product SV V SII is precisely twice thequantum-limited value. As a result, the general the-ory of Sec. VI.E tells us if one couples an input signalweakly to this amplifier (i.e. Zs , Zin), it is impossibleto reach the quantum limit on the added noise. Thus,while our amplifier is ideal in the scattering mode of op-eration (cf. Eq. (7.16)), it fails to reach the quantumlimit when used in the weak-coupling, op-amp mode ofoperation. Our amplifier’s failure to have “ideal” quan-tum noise also means that if we tried to use it to doQND qubit detection, the resulting back-action dephas-ing would be twice as large as the minimum required byquantum mechanics (cf. Sec. V.B).

One might object to the above conclusions based onthe classical expression for the minimal noise tempera-ture, Eq. (6.48). Unlike the quantum noise constraintof Eq. (6.60), this equation also involves the real partof SV I , and is optimized by our “minimal” amplifier.However, this does not mean that one can achieve anoise temperature of !ω/2 at weak coupling! Recallfrom Sec. VI.E that in the usual process of optimiz-ing the noise temperature, one starts by assuming theweak coupling condition that the source impedance Zs

is much smaller than the amplifier input impedance Zin.One then finds that to minimize the noise temperature,|Zs| should be tuned to match the noise impedance ofthe amplifier ZN ≡

√SV V /SII . However, in our min-

imal bosonic amplifier, it follows from Eqs. (7.18) thatZN = Zin/

√2 ∼ Zin: the noise impedance is on the

order of the input impedance. Thus, it is impossible tomatch the source impedance to the noise impedance whileat the same time satisfying the weak coupling conditionZs , Zin.

Despite its failings, our amplifier can indeed yield aquantum-limited noise temperature in the op-amp modeof operation if we no longer insist on a weak couplingto the input signal. To see this explicitly, imagine weconnected our amplifier to a signal source with sourceimpedance Zs. The total output noise of the amplifier,referred back to the signal source, will now have the form:

ˆVtot = −(

ZsZa

Zs + Za

)ˆI + ˆV (7.21)

Note that this classical-looking equation can be rigor-ously justified within the full quantum theory if one startswith a full description of the amplifier and the signalsource (e.g. a parallel LC oscillator attached in parallel

to the amplifier input). Plugging in the expressions for ˆIand ˆV , we find:

ˆVtot =√

!ω

2

[ (ZsZa

Zs + Za

) (2√Za

uin

)−

√Za(uin − v†in)

]

=√

!ωZa

2

[(Zs − Za

Zs + Za

)uin − v†in

](7.22)

Thus, if one tunes Zs to Za = Zin, the mode uin does notcontribute to the total added noise, and one reaches thequantum limit. Physically speaking, by matching thesignal source to the input line, the “back-action” noisedescribed by Fa = uin does not feed back into the inputof the amplifier. Note that achieving this matching ex-plicitly requires one to be far from weak coupling! HavingZs = Za means that when we attach the amplifier to thesignal source, we will dramatically increase the dampingof the signal source.

2. Why is the op-amp quantum limit not achieved?

Returning to the more interesting case of a weakamplifier-signal coupling, one might still be puzzled asto why our seemingly ideal amplifier misses the quantumlimit. While the mathematics behind Eq. (7.20) is fairlytransparent, it is also possible to understand this resultheuristically. To that end, note again that the amplifiernoise cross-correlation SIV does not vanish in the large-gain limit (cf. Eq. (7.18c)). Correlations between the twoamplifier noises represent a kind of information, as bymaking use of them, we can improve the performance ofthe amplifier. It is easy to take advantage of out-of-phasecorrelations between I and V (i.e. Im SV I) by simplytuning the phase of the source impedance (cf. Eq. (6.47)).However, one cannot take advantage of in-phase noisecorrelations (i.e. Re SV I) as easily. To take advantageof the information here, one needs to modify the ampli-fier itself. By feeding back some of the output voltageto the input, one could effectively cancel out some of theback-action current noise I and thus reduce the over-all magnitude of SII . Hence, the unused information inthe cross-correlator Re SV I represents a kind of wastedinformation: had we made use of these correlations viaa feedback loop, we could have reduced the noise tem-perature and increased the information provided by ouramplifier. The presence of a non-zero Re SV I thus corre-sponds to wasted information, implying that we cannotreach the quantum limit. Recall that within the linear-response approach, we were able to prove rigorously thata large-gain amplifier with ideal quantum noise must haveRe SV I = 0 (cf. the discussion following Eq. (6.29)); thus,a non-vanishing Re SV I rigorously implies that one can-not be at the quantum limit. In Appendix I, we give

56

an explicit demonstration of how feedback may be usedto utilize these cross-correlations to reach the quantumlimit.

Finally, yet another way of seeing that our amplifierdoes not reach the quantum limit (in the weak couplingregime) is to realize that this system does not have awell defined effective temperature. Recall from Sec.VI.Ethat a system with “ideal” quantum noise (i.e. one thatsatisfies Eq. (6.60) as an equality) necessarily has thesame effective temperature at its input and output ports(cf. Eq. (6.27)). Here, that implies the requirement:

|λV |2 · SV V

Zout= ZinSII ≡ 2kBTeff (7.23)

In contrast, our minimal bosonic amplifier has very dif-ferent input and output effective temperatures:

2kBTeff,in = ZinSII =!ω

2(7.24)

2kBTeff,out =|λV |2 · SV V

Zout= 2|G|!ω (7.25)

This huge difference in effective temperatures means thatit is impossible for the system to possess “ideal” quan-tum noise, and thus it cannot reach the weak-couplingquantum limit.

While it implies that one is not at the quantum limit,the fact that Teff,in , Teff,out can nonetheless be viewedas an asset. From a practical point of view, a large Teff,in

can be dangerous. Even though the direct effect of thelarge Teff,in is offset by an appropriately weak couplingto the amplifier (see Eq. (6.42) and following discussion),this large Teff,in can also heat up other degrees of freedomif they couple strongly to the back-action noise of the am-plifier. This can in turn lead to unwanted heating of theinput system. As Teff,in is usually constant over a broadrange of frequencies, this unwanted heating effect can bequite bad. In the minimal amplifier discussed here, thisproblem is circumvented by having a small Teff,in. Theonly price that is payed is that the added noise will be

√2

the quantum limit value. We discuss this issue further inSec. VIII.

VIII. REACHING THE QUANTUM LIMIT IN PRACTICE

A. Importance of QND measurements

The fact that QND measurements are repeatable is offundamental practical importance in overcoming detec-tor inefficiencies (Gambetta et al., 2007). A prototypicalexample is the electron shelving technique (Nagourneyet al., 1986; Sauter et al., 1986) used to measure trappedions. A related technique is used in present implementa-tions of ion-trap based quantum computation. Here the(extremely long-lived) hyperfine state of an ion is readout via state-dependent optical fluorescence. With prop-erly chosen circular polarization of the exciting laser, only

one hyperfine state fluoresces and the transition is cy-cling; that is, after fluorescence the ion almost always re-turns to the same state it was in prior to absorbing the ex-citing photon. Hence the measurement is QND. Typicalexperimental parameters (Wineland et al., 1998) allowthe cycling transition to produce N ∼ 106 fluorescencephotons. Given the photomultiplier quantum efficiencyand typically small solid angle coverage, only a very smallnumber nd will be detected on average. The probabilityof getting zero detections (ignoring dark counts for sim-plicity) and hence misidentifying the hyperfine state isP (0) = e−nd . Even for a very poor overall detection ef-ficiency of only 10−5, we still have nd = 10 and nearlyperfect fidelity F = 1−P (0) ≈ 0.999955. It is importantto note that the total time available for measurementis not limited by the phase coherence time (T2) of thequbit or by the measurement-induced dephasing (Gam-betta et al., 2006; Korotkov, 2001a; Makhlin et al., 2001;Schuster et al., 2005), but rather only by the rate atwhich the qubit makes real transitions between measure-ment (σz) eigenstates. In a perfect QND measurementthere is no measurement-induced state mixing (Makhlinet al., 2001) and the relaxation rate 1/T1 is unaffectedby the measurement process.

B. Power matching versus noise matching

In Sec. VI, we saw that an important part of reach-ing the quantum limit on the added noise of an amplifier(when used in the op-amp mode of operation) is to op-timize the coupling strength to the amplifier. For a po-sition detector, this condition corresponds to tuning thestrength of the back-action damping γ to be much smallerthan the intrinsic oscillator damping (c.f. Eq. (6.41)).For a voltage amplifier, this condition corresponds to tun-ing the impedance of the signal source to be equal to thenoise impedance (c.f. Eq. (6.49)), an impedance which ismuch smaller than the amplifier’s input impedance (c.f.Eq. (6.62)).

In this subsection, we make the simple point that opti-mizing the coupling (i.e. source impedance) to reach thequantum limit is not the same as what one would do tooptimize the power gain. To understand this, we need tointroduce another measure of power gain commonly usedin the engineering community, the available power gainGP,avail. For simplicity, we will discuss this quantity inthe context of a linear voltage amplifier, using the nota-tions of Sec. VI.E; it can be analogously defined for theposition detector of Sec.VI.D. GP,avail tells us how muchpower we are providing to an optimally matched outputload relative to the maximum power we could in principlehave extracted from the source. This is in marked con-trast to the power gain GP , which was calculated usingthe actual power drawn at the amplifier input.

For the available power gain, we first consider Pin,avail.This is the maximum possible power that could be deliv-ered to the input of the amplifier, assuming we optimized

57

both the value of the input impedance Zin and the loadimpedance Zload while keeping Zs fixed. For simplicity,we will take all impedances to be real in our discussion.In general, the power drawn at the input of the amplifieris given by:

Pin =v2in

Zin

ZsZin

(Zs + Zin)2(8.1)

Maximizing this over Zin, we obtain the available inputpower Pin,avail:

Pin,avail =v2in

4Zs(8.2)

The maximum occurs for Zin = Zs.The output power supplied to the load is calculated as

before, keeping Zin and Zs distinct. One has:

Pout =v2load

Zload(8.3)

=v2out

Zload

(Zload

Zout + Zload

)2

(8.4)

=λ2v2

in

Zout

(Zin

Zin + Zs

)2 Zload/Zout

(1 + Zload/Zout)2 (8.5)

As a function of Zload, Pout is maximized when Zload =Zout:

Pout,max =λ2v2

in

4Zout

(Zin

Zin + Zs

)2

(8.6)

The available power gain is now defined as:

GP,avail ≡Pout,max

Pin,avail(8.7)

=λ2Zs

Zout

(Zin

Zin + Zs

)2

(8.8)

= GP4Zs/Zin

(1 + Zs/Zin)2(8.9)

We see that GP,avail is strictly less than or equal to thepower gain GP ; equality is only achieved when Zs = Zin

(i.e. when the source impedance is “power matched” tothe input of the amplifier). The general situation whereGP,avail < GP indicates that we are not drawing as muchpower from the source as we could, and hence the actualpower supplied to the load is not as large as it could be.

Consider now a situation where we have achieved thequantum limit on the added noise. This necessarilymeans that we have “noise matched”, i.e. taken Zs to beequal to the noise impedance ZN. The available powergain in this case is:

GP,avail 4 λ2 ZN

Zout4 2

√GP , GP (8.10)

We have used Eq. (6.62), which tells us that the noiseimpedance is smaller than the input impedance by a large

factor 1/(2√

GP ). Thus, as reaching the quantum limitrequires the use of a source impedance much smaller thanZin, it results in a dramatic drop in the available powergain compared to the case where we “power match” (i.e.take Zs = Zin). In practice, one must decide whether itis more important to minimize the added noise, or max-imize the power provided at the output of the amplifier:one cannot do both at the same time.

IX. CONCLUSIONS

In this review, we have given an introduction to thephysics of quantum noise. We have discussed how onecan measure quantum noise using quantum spectrum an-alyzers, as well as how fundamental constraints on quan-tum noise are directly tied to various quantum limits onmeasurement and amplification. As already mentionedin the introduction, there are many interesting topics re-lated to quantum noise and quantum circuits that wewere not able to cover in this review. In what follows,we give a brief listing of some of these topics, along withrelevant review literature.

Quantum noise has long been of interest in mesoscopicphysics, even before the recent resurgence inspired byconnections to measurement and amplification. In par-ticular, the study of non-equilibrium current fluctuations(i.e. shot noise) of a mesoscopic conductor, and the re-lated full statistics of current fluctuations (“full countingstatistics”), have been intensely studied, as they can re-veal important insights into quantum transport. Bothtopics have been covered in a number of recent reviews(Blanter and Buttiker, 2000; Levitov, 2003; Nazarov,2003). Connections between full counting statistics andquantum measurement have also been explored (Averinand Sukhorukov, 2005; Nazarov and Kindermann, 2003).

There are also several important aspects of weak quan-tum measurement that we have not been able to discussin this review. A relatively recent development has beenthe study of conditional quantum evolution and quantumtrajectories. Such studies attempt to understand the evo-lution of a quantum system in a particular run of an ex-periment: given a particular measurement record, whatis the state of the measured system? Further, can themeasurement record contain evidence of quantum jumpsof the measured system between different states (e.g. amechanical resonator jumping from one Fock state to an-other). Such topics have received substantial attention inthe quantum optics and atomic physics community. Thetopic of quantum jumps and stochastic wavefunction evo-lution is given a tutorial review in Brun, 2002, while anintroduction to conditional quantum evolution and quan-tum feedback is given in Jacobs and Steck, 2006. Thesetopics have also recently been discussed in the contextof condensed matter systems. Studies have examinedcontinuous measurement of a qubit by a quantum pointcontact (Goan and Milburn, 2001; Goan et al., 2001; Ko-rotkov, 1999, 2001b), as well as the possibility of detect-

58

ing quantum jumps in the state of a mechanical resonatorvia QND measurement of its energy (Santamore et al.,2004a,b).

While we have discussed many measurement and am-plification schemes, there are several which we did notaddress. So-called “latching” measurements of the stateof a qubit have recently been used to make measurementsof the state of a superconducting qubit (Siddiqi et al.,2004). For a sufficiently slow input signal, such latch-ing measurements can also be used as an efficient meansof linear amplification, possibly reaching the amplifierquantum limit.

Acknowledgements

This work was supported in part by NSERC, the Cana-dian Institute for Advanced Research (CIFAR), NSAunder ARO contract number W911NF-05-1-0365, theNSF under grants DMR-0653377 and DMR-0603369, theDavid and Lucile Packard Foundation, the W.M. KeckFoundation and the Alfred P. Sloan Foundation. We ac-knowledge the support and hospitality of the Aspen Cen-ter for Physics where part of this work was carried out.F.M. acknowledges support via DIP and through SFB631, SFB/TR 12, the Nanosystems Initiative Munich,and the Emmy-Noether program. S. M. G. acknowledgesthe support of the Centre for Advanced Study at the Nor-wegian Academy of Science and Letters where part of thiswork was carried out.

APPENDIX A: The Wiener-Khinchin Theorem

From the definition of the spectral density in Eqs.(2.2-2.3) we have

SV V [ω] =1T

∫ T

0dt

∫ T

0dt′ eiω(t−t′)〈v(t)v(t′)〉

=1T

∫ T

0dt

∫ +2B(t)

−2B(t)dτ eiωτ 〈v(t + τ/2)v(t− τ/2)〉(A1)

where

B (t) = t if t < T/2= T − t if t > T/2.

If T greatly exceeds the noise autocorrelation time τc

then it is a good approximation to extend the bound B(t)in the second integral to infinity, since the dominant con-tribution is from small τ . Using time translation invari-ance gives

SV V [ω] =1T

∫ T

0dt

∫ +∞

−∞dτ eiωτ 〈v(τ)v(0)〉

=∫ +∞

−∞dτ eiωτ 〈v(τ)v(0)〉 . (A2)

This proves the Wiener-Khinchin theorem stated inEq. (2.4).

A useful application of these ideas is the following.Suppose that we have a noisy signal V (t) = V + η(t)which we begin monitoring at time t = 0. The integratedsignal up to time t given by

I(T ) =∫ T

0dt V (t) (A3)

has mean

〈I(T )〉 = V T. (A4)

Provided that the integration time greatly exceeds theautocorrelation time of the noise, I(T ) is a sum of a largenumber of uncorrelated random variables. The centrallimit theorem tells us in this case that I(t) is gaussiandistributed even if the signal itself is not. Hence theprobability distribution for I is fully specified by its meanand its variance

〈(∆I)2〉 =∫ T

0dtdt′ 〈η(t)η(t′)〉. (A5)

From the definition of spectral density above we have thesimple result that the variance of the integrated signalgrows linearly in time with proportionality constant givenby the noise spectral density at zero frequency

〈(∆I)2〉 = SV V [0]T. (A6)

As a simple application, consider the photon shot noiseof a coherent laser beam. The total number of photonsdetected in time T is

N(T ) =∫ T

0dt N(t). (A7)

The photo-detection signal N(t) is not gaussian, butrather is a point process, that is, a sequence of delta func-tions with random Poisson distributed arrival times andmean photon arrival rate N . Nevertheless at long timesthe mean number of detected photons

〈N(T )〉 = NT (A8)

will be large and the photon number distribution will begaussian with variance

〈(∆N)2〉 = SNN T. (A9)

Since we know that for a Poisson process the variance isequal to the mean

〈(∆N)2〉 = 〈N(T )〉, (A10)

it follows that the shot noise power spectral density is

SNN (0) = N . (A11)

Since the noise is white this result happens to be valid atall frequencies, but the noise is gaussian distributed onlyat low frequencies.

59

APPENDIX B: Modes, Transmission Lines and ClassicalInput/Output Theory

In this appendix we introduce a number of importantclassical concepts about electromagnetic signals whichare essential to understand before moving on to the studyof their quantum analogs. A signal at carrier frequencyω can be described in terms of its amplitude and phaseor equivalently in terms of its two quadrature amplitudes

s(t) = X cos(ωt) + Y sin(ωt). (B1)

We will see in the following that the physical oscillationsof this signal in a transmission line are precisely the sinu-soidal oscillations of a simple harmonic oscillator. Com-parison of Eq. (B1) with Eq. (2.18) shows that we canidentify the quadrature amplitude X with the coordinateof this oscillator and thus the quadrature amplitude Y isproportional to the momentum conjugate to X. Quan-tum mechanically, X and Y become operators X and Ywhich do not commute. Thus their quantum fluctuationsobey the Heisenberg uncertainty relation.

Ordinarily (e.g., in the absence of squeezing), the phasechoice defining the two quadratures is arbitrary and sotheir vacuum (i.e. zero-point) fluctuations are equal

XZPF = YZPF. (B2)

Thus the canonical commutation relation becomes

[X, Y ] = iX2ZPF. (B3)

We will see that the fact that X and Y are canoni-cally conjugate has profound implications both classicallyand quantum mechanically. In particular, the action ofany circuit element (beam splitter, attenuator, amplifier,etc.) must preserve the Poisson bracket (or in the quan-tum case, the commutator) between the signal quadra-tures. This places strong constraints on the propertiesof these circuit elements and in particular, forces everyamplifier to add noise to the signal.

1. Transmission lines and classical input-output theory

We begin by considering a coaxial transmission linemodeled as a perfectly conducting wire with inductanceper unit length of 5 and capacitance to ground per unitlength c as shown in Fig. 14. If the voltage at position xat time t is V (x, t), then the charge density is q(x, t) =cV (x, t). By charge conservation the current I and thecharge density are related by the continuity equation

∂tq + ∂xI = 0. (B4)

The constitutive relation (essentially Newton’s law) givesthe acceleration of the charges

5∂tI = −∂xV. (B5)

We can decouple Eqs. (B4) and (B5) by introducing leftand right propagating modes

V (x, t) = [V→ + V←] (B6)

I(x, t) =1Zc

[V→ − V←] (B7)

where Zc ≡√

5/c is called the characteristic impedanceof the line. In terms of the left and right propagatingmodes, Eqs. (B4) and B5 become

vp∂xV→ + ∂tV→ = 0 (B8)

vp∂xV← − ∂tV← = 0 (B9)

where vp ≡ 1/√

5c is the wave phase velocity. Theseequations have solutions which propagate by uniformtranslation without changing shape since the line is dis-persionless

V→(x, t) = Vout(t−x

vp) (B10)

V←(x, t) = Vin(t +x

vp), (B11)

where Vin and Vout are arbitrary functions of their argu-ments. For an infinite transmission line, Vout and Vin arecompletely independent. However for the case of a semi-infinite line terminated at x = 0 (say) by some system S,these two solutions are not independent, but rather re-lated by the boundary condition imposed by the system.We have

V (x = 0, t) = [Vout(t) + Vin(t)] (B12)

I(x = 0, t) =1Zc

[Vout(t)− Vin(t)], (B13)

from which we may derive

Vout(t) = Vin(t) + ZcI(x = 0, t). (B14)

If the system under study is just an open circuit sothat I(x = 0, t) = 0, then Vout = Vin, meaning that theoutgoing wave is simply the result of the incoming wavereflecting from the open circuit termination. In generalhowever, there is an additional outgoing wave radiatedby the current I that is injected by the system dynamicsinto the line. In the absence of an incoming wave we have

V (x = 0, t) = ZcI(x = 0, t), (B15)

indicating that the transmission line acts as a simple re-sistor which, instead of dissipating energy by Joule heat-ing, carries the energy away from the system as propa-gating waves. The fact that the line can dissipate energydespite containing only purely reactive elements is a con-sequence of its infinite extent. One must be careful withthe order of limits, taking the length to infinity beforeallowing time to go to infinity. In this way the outgoingwaves never reach the far end of the transmission line and

60

Zc , vp

0 dx

V

I

Vin

Vout

V (x,t)

V (x,t)

I(x,t)

V(x,t)

a)

c)

L

a

b)L L

C C C

FIG. 14 a) Coaxial transmission line, indicating voltages andcurrents as defined in the main text. b) Lumped elementrepresentation of a transmission line with capacitance per unitlength c and inductance per unit length ,. c) Discrete LCresonator terminating a transmission line.

reflect back. Since this is a conservative Hamiltonian sys-tem, we will be able to quantize these waves and make aquantum theory of resistors (Caldeira and Leggett, 1983)in Appendix C. The net power flow carried to the rightby the line is

P =1Zc

[V 2out(t)− V 2

in(t)]. (B16)

The fact that the transmission line presents a dissipa-tive impedance to the system means that it causes damp-ing of the system. It also however opens up the possibilityof controlling the system via the input field which par-tially determines the voltage driving the system. Fromthis point of view it is convenient to eliminate the outputfield by writing the voltage as

V (x = 0, t) = 2Vin(t) + ZcI(x = 0, t). (B17)

As we will discuss in more detail below, the first termdrives the system and the second damps it. FromEq. (B14) we see that measurement of the outgoing fieldcan be used to determine the current I(x = 0, t) injectedby the system into the line and hence to infer the systemdynamics that results from the input drive field.

As a simple example, consider the system consisting ofan LC resonator shown in Fig. (14 c). This can be viewedas a simple harmonic oscillator whose coordinate Q is thecharge on the capacitor plate (on the side connected toL0). The current I(x = 0, t) = Q plays the role of thevelocity of the oscillator. The equation of motion for theoscillator is readily obtained from

Q = C0[−V (x = 0+, t)− L0I(x = 0+, t)]. (B18)

Using Eq. (B17) we obtain a harmonic oscillator dampedby the transmission line and driven by the incomingwaves

Q = −Ω20Q− γQ− 2

L0Vin(t), (B19)

where the resonant frequency is Ω20 ≡ 1/

√L0C0. Note

that the term ZcI(x = 0, t) in Eq. (B17) results in thelinear viscous damping rate γ ≡ Zc/L0.

If we solve the equation of motion of the oscillator, wecan predict the outgoing field. In the present instance ofa simple oscillator we have a particular example of thegeneral case where the system responds linearly to theinput field. We can characterize any such system by acomplex, frequency dependent impedance Z[ω] definedby

Z[ω] = −V (x = 0, ω)I(x = 0, ω)

. (B20)

Note the peculiar minus sign which results from our def-inition of positive current flowing to the right (out of thesystem and into the transmission line). Using Eqs. (B12,B13) and Eq. (B20) we have

Vout[ω] = r[ω]Vin[ω], (B21)

where the reflection coefficient r is determined by theimpedance mismatch between the system and the lineand is given by the well known result

r[ω] =Z[ω]− Zc

Z[ω] + Zc. (B22)

If the system is constructed from purely reactive (i.e.lossless) components, then Z[ω] is purely imaginary andthe reflection coefficient obeys |r| = 1 which is consistentwith Eq. (B16) and the energy conservation requirementof no net power flow into the lossless system. For exam-ple, for the series LC oscillator we have been considering,we have

Z[ω] =1

jωC0+ jωL0, (B23)

where, to make contact with the usual electrical engi-neering sign conventions, we have used j = −i. If thedamping γ of the oscillator induced by coupling it to thetransmission line is small, the quality factor of the reso-nance will be high and we need only consider frequenciesnear the resonance frequency Ω0 ≡ 1/

√L0C0 where the

impedance has a zero. In this case we may approximate

Z[ω] ≈ 2jC0Ω2

0

[Ω0 − ω] = 2jL(ω − Ω0) (B24)

which yields for the reflection coefficient

r[ω] =ω − Ω0 + jγ/2ω − Ω0 − jγ/2

(B25)

61

showing that indeed |r| = 1 and that the phase of thereflected signal winds by 2π upon passing through theresonance. 22

Turning to the more general case where the systemalso contains lossy elements, one finds that Z[ω] is nolonger purely imaginary, but has a real part satisfyingRe Z[ω] > 0. This in turn implies via Eq. (B22) that|r| < 1. In the special case of impedance matchingZ[ω] = Zc, all the incident power is dissipated in thesystem and none is reflected. The other two limits ofinterest are open circuit termination with Z = ∞ forwhich r = +1 and and short circuit termination Z = 0for which r = −1.

Finally, if the system also contains an active devicewhich has energy being pumped into it from a separateexternal source, it may under the right conditions be de-scribed by an effective negative resistance Re Z[ω] < 0over a certain frequency range. Eq. (B22) then gives|r| ≥ 1, implying |Vout| > |Vin|. Our system will thus actlike the one-port amplifier discussed in Sec. VI.C: it am-plifies signals incident upon it. We will discuss this ideaof negative resistance further in Sec. B.4; a physical real-ization is provided by the two-port reflection parametricamplifier discussed in Appendix F.

2. Lagrangian, Hamiltonian, and wave modes for atransmission line

Prior to moving on to the case of quantum noise itis useful to review the classical statistical mechanics oftransmission lines. To do this we need to write downthe Lagrangian and then determine the canonical mo-menta and the Hamiltonian. Very conveniently, the sys-tem is simply a large collection of harmonic oscillators(the normal modes) and hence can be readily quantized.This representation of a physical resistor is essentially theone used by Caldeira and Leggett (Caldeira and Leggett,1983) in their seminal studies of the effects of dissipationon tunneling. The only difference between this modeland the vacuum fluctuations in free space is that the rel-ativistic bosons travel in one dimension and do not carrya polarization label. This changes the density of states asa function of frequency, but has no other essential effect.

It is convenient to define a flux variable (Devoret, 1997)

ϕ(x, t) ≡∫ t

−∞dτ V (x, τ), (B26)

where V (x, t) = ∂tϕ(x, t) is the local voltage on the trans-mission line at position x and time t. Each segment ofthe line of length dx has inductance 5 dx and the voltagedrop along it is −dx ∂x∂tϕ(x, t). The flux through this

22 For the case of resonant transmission through a symmetric cav-ity, the phase shift only winds by π.

inductance is thus −dx ∂xϕ(x, t) and the local value ofthe current is given by the constitutive equation

I(x, t) = −15

∂xϕ(x, t). (B27)

The Lagrangian for the system is

L =∫ ∞

0dx

c

2(∂tϕ)2 − 1

25(∂xϕ)2, (B28)

The Euler-Lagrange equation for this Lagrangian is sim-ply the wave equation

v2p∂2

xϕ− ∂2t ϕ = 0. (B29)

The momentum conjugate to ϕ(x) is simply the chargedensity

q(x, t) ≡ δLδ∂tϕ

= c∂tϕ = cV (x, t) (B30)

and so the Hamiltonian is given by

H =∫

dx

12c

q2 +125

(∂xϕ)2

. (B31)

We know from our previous results that the chargedensity consists of left and right moving solutions of ar-bitrary fixed shape. For example we might have for theright moving case

q(t−x/vp) = αk cos[k(x−vpt)]+βk sin[k(x−vpt)]. (B32)

A confusing point is that since q is real valued, we seethat it necessarily contains both eikx and e−ikx termseven if it is only right moving. Note however that fork > 0 and a right mover, the eikx is associated with thepositive frequency term e−iωkt while the e−ikx term isassociated with the negative frequency term e+iωkt whereωk ≡ vp|k|. For left movers the opposite holds. We canappreciate this better if we define

Ak ≡1√L

∫dx e−ikx

1√2c

q(x, t)− i

√k2

25ϕ(x, t)

(B33)where for simplicity we have taken the fields to obey pe-riodic boundary conditions on a length L. Thus we have(in a form which anticipates the full quantum theory)

H =12

∑

k

(A∗kAk + AkA∗k) . (B34)

The classical equation of motion (B29) yields the simpleresult

∂tAk = −iωkAk. (B35)

Thus

q(x, t)

=√

c

2L

∑

k

eikx[Ak(0)e−iωkt + A∗−k(0)e+iωkt

](B36)

=√

c

2L

∑

k

[Ak(0)e+i(kx−ωkt) + A∗k(0)e−i(kx−ωkt)

].

(B37)

62

We see that for k > 0 (k < 0) the wave is right (left)moving, and that for right movers the eikx term is asso-ciated with positive frequency and the e−ikx term is as-sociated with negative frequency. We will return to this

in the quantum case where positive (negative) frequencywill refer to the destruction (creation) of a photon. Notethat the right and left moving voltages are given by

V→ =√

12Lc

∑

k>0

[Ak(0)e+i(kx−ωkt) + A∗k(0)e−i(kx−ωkt)

](B38)

V← =√

12Lc

∑

k<0

[Ak(0)e+i(kx−ωkt) + A∗k(0)e−i(kx−ωkt)

](B39)

The voltage spectral density for the right moving waves is thus

S→V V [ω] =2π

2Lc

∑

k>0

〈AkA∗k〉δ(ω − ωk) + 〈A∗kAk〉δ(ω + ωk) (B40)

The left moving spectral density has the same expression but k < 0.Using Eq. (B16), the above results lead to a net power flow (averaged over one cycle) within a frequency band

defined by a pass filter G[ω] of

P = P→ − P← =vp

2L

∑

k

sgn(k) [G[ωk]〈AkA∗k〉+ G[−ωk]〈A∗kAk〉] . (B41)

3. Classical statistical mechanics of a transmission line

Now that we have the Hamiltonian, we can considerthe classical statistical mechanics of a transmission line inthermal equilibrium at temperature T . Since each modek is a simple harmonic oscillator we have from Eq. (B34)and the equipartition theorem

〈A∗kAk〉 = kBT. (B42)

Using this, we see from Eq. (B40) we see that the rightmoving voltage signal has a simple white noise powerspectrum. Using Eq. (B41) we have for the right movingpower in a bandwidth B (in Hz rather than radians/sec)the very simple result

P→ =vp

2L

∑

k>0

〈G[ωk]A∗kAk + G[−ωk]AkA∗k〉

=kBT

2

∫ +∞

−∞

dω

2πG[ω]

= kBTB. (B43)

where we have used the fact mentioned in connectionwith Eq. (2.15) and the discussion of square law detec-tors that all passive filter functions are symmetric in fre-quency.

One of the basic laws of statistical mechanics is Kirch-hoff’s law stating that the ability of a hot object to emitradiation is proportional to its ability to absorb. Thisfollows from very general thermodynamic arguments con-

cerning the thermal equilibrium of an object with its ra-diation environment and it means that the best possibleemitter is the black body. In electrical circuits this princi-ple is simply a form of the fluctuation dissipation theoremwhich states that the electrical thermal noise producedby a circuit element is proportional to the dissipation itintroduces into the circuit. Consider the example of a ter-minating resistor at the end of a transmission line. If theresistance R is matched to the characteristic impedanceZc of a transmission line, the terminating resistor actsas a black body because it absorbs 100% of the powerincident upon it. If the resistor is held at temperature Tit will bring the transmission line modes into equilibriumat the same temperature (at least for the case where thetransmission line has finite length). The rate at which theequilibrium is established will depend on the impedancemismatch between the resistor and the line, but the finaltemperature will not.

A good way to understand the fluctuation-dissipationtheorem is to represent the resistor R which is terminat-ing the Zc line in terms of a second semi-infinite trans-mission line of impedance R as shown in Fig. (15). Firstconsider the case when the R line is not yet connected tothe Zc line. Then according to Eq. (B22), the open termi-nation at the end of the Zc line has reflectivity |r|2 = 1so that it does not dissipate any energy. Additionallyof course, this termination does not transmit any sig-nals from the R line into the Zc. However when thetwo lines are connected the reflectivity becomes less thanunity meaning that incoming signals on the Zc line see

63

a source of dissipation R which partially absorbs them.The absorbed signals are not turned into heat as in atrue resistor but are partially transmitted into the R linewhich is entirely equivalent. Having opened up this portfor energy to escape from the Zc system, we have alsoallowed noise energy (thermal or quantum) from the Rline to be transmitted into the Zc line. This is completelyequivalent to the effective circuit shown in Fig. (16 a) inwhich a real resistor has in parallel a random currentgenerator representing thermal noise fluctuations of theelectrons in the resistor. This is the essence of the fluc-tuation dissipation theorem.

In order to make a quantitative analysis in terms ofthe power flowing in the two lines, voltage is not the bestvariable to use since we are dealing with more than onevalue of line impedance. Rather we define incoming andoutgoing fields via

Ain =1√Zc

V←c (B44)

Aout =1√Zc

V→c (B45)

Bin =1√R

V→R (B46)

Bout =1√R

V←R (B47)

Normalizing by the square root of the impedance allowsus to write the power flowing to the right in each line inthe simple form

Pc = (Aout)2 − (Ain)2 (B48)PR = (Bin)2 − (Bout)2 (B49)

The out fields are related to the in fields by the s matrix(

Aout

Bout

)= s

(Ain

Bin

)(B50)

Requiring continuity of the voltage and current at theinterface between the two transmission lines, we can solvefor the scattering matrix s:

s =

(+r t

t −r

)(B51)

where

r =R− Zc

R + Zc(B52)

t =2√

RZc

R + Zc. (B53)

Note that |r|2 + |t|2 = 1 as required by energy conserva-tion and that s is unitary with det (s) = −1. By movingthe point at which the phase of the Bin and Bout fieldsare determined one-quarter wavelength to the left, wecan put s into different standard form

s′ =

(+r it

it +r

)(B54)

which has det (s′) = +1.

VR

Vc

VR

Vc

R Zc

FIG. 15 (Color online) Semi-infinite transmission line ofimpedance Zc terminated by a resistor R which is representedas a second semi-infinite transmission line.

As mentioned above, the energy absorbed from the Zc

line by the resistor R is not turned into heat as in atrue resistor but is is simply transmitted into the R line,which is entirely equivalent. Kirchhoff’s law is now easyto understand. The energy absorbed from the Zc line andthe energy transmitted into it by thermal fluctuationsin the R line are both proportional to the absorptioncoefficient

A = 1− |r|2 = |t|2 =4RZc

(R + Zc)2. (B55)

R

IN

SII SVV

R

VN

a) b)

FIG. 16 Equivalent circuits for noisy resistors.

The requirement that the transmission line Zc come toequilibrium with the resistor allows us to readily computethe spectral density of current fluctuations of the randomcurrent source shown in Fig. (16 a). The power dissipatedin Zc by the current source attached to R is

P =∫ +∞

−∞

dω

2πSII [ω]

R2Zc

(R + Zc)2

(B56)

For the special case R = Zc we can equate this to theright moving power P→ in Eq. (B43) because left movingwaves in the Zc line are not reflected and hence cannotcontribute to the right moving power. Requiring P =P→ yields the classical Nyquist result for the currentnoise of a resistor

SII [ω] =2R

kBT (B57)

64

or in the electrical engineering convention

SII [ω] + SII [−ω] =4R

kBT. (B58)

We can derive the equivalent expression for the volt-age noise of a resistor (see Fig. 16 b) by considering thevoltage noise at the open termination of a semi-infinitetransmission line with Zc = R. For an open terminationV→ = V← so that the voltage at the end is given by

V = 2V← = 2V→ (B59)

and thus using Eqs. (B40) and (B42) we find

SV V = 4S→V V = 2RkBT (B60)

which is equivalent to Eq. (B57).

4. Amplification with a transmission line and a negativeresistance

We close our discussion of transmission lines by fur-ther expanding upon the idea mentioned at the end ofApp. B.1 that one can view a one-port amplifier as atransmission line terminated by an effective negative re-sistance. The discussion here will be very general: we willexplore what can be learned about amplification by sim-ply extending the results we have obtained on transmis-sion lines to the case of an effective negative resistance.Our general discussion will not address the important is-sues of how one achieves an effective negative resistanceover some appreciable frequency range: for such ques-tions, one must focus on a specific physical realization,such as the parametric amplifier discussed in Appendix F.

We start by noting that for the case −Zc < R < 0 thepower gain G is given by

G = |r|2 > 1, (B61)

and the s′ matrix introduced in Eq. (B54) becomes

s′ = −( √

G ±√

G− 1±√

G− 1√

G

)(B62)

where the sign choice depends on the branch cut chosenin the analytic continuation of the off-diagonal elements.This transformation is clearly no longer unitary (becausethere is no energy conservation since we are ignoring thework done by the amplifier power supply). Note howeverthat we still have det (s′) = +1. It turns out that thisnaive analytic continuation of the results from positive tonegative resistance is not strictly correct. As we will showin the following, we must be more careful than we havebeen so far in order to insure that the transformationfrom the in fields to the out fields must be canonical.

In order to understand the canonical nature of thetransformation between input and output modes, it isnecessary to delve more deeply into the fact that the

two quadrature amplitudes of a mode are canonicallyconjugate. Following the complex amplitudes definedin Eqs. (B44-B47), let us define a vector of real-valuedquadrature amplitudes for the incoming and outgoingfields

+q in =

X inA

X inB

Y inB

Y inA

, +q out =

XoutA

XoutB

Y outB

Y outA

.

(B63)

The Poisson brackets amongst the different quadratureamplitudes is given by

qini , qin

j ∝ Jij , (B64)

or equivalently the quantum commutators are

[qini , qin

j ] = iX2ZPFJij , (B65)

where

J ≡

0 0 0 +10 0 +1 00 −1 0 0−1 0 0 0

. (B66)

In order for the transformation to be canonical, the samePoisson bracket or commutator relations must hold forthe outgoing field amplitudes

[qouti , qout

j ] = iX2ZPFJij . (B67)

In the case of a non-linear device these relations wouldapply to the small fluctuations in the input and outputfields around the steady state solution. Assuming a lineardevice (or linearization around the steady state solution)we can define a 4 × 4 real-valued scattering matrix s inanalogy to the 2× 2 complex-valued scattering matrix sin Eq. (B51) which relates the output fields to the inputfields

qouti = sijq

inj . (B68)

Eq. (B67) puts a powerful constraint on on the s matrix,namely that it must be symplectic. That is, s and itstranspose must obey

sJ sT = J. (B69)

From this it follows that

det s = ±1. (B70)

This in turn immediately implies Liouville’s theoremthat Hamiltonian evolution preserves phase space volume(since det s is the Jacobian of the transformation whichpropagates the amplitudes forward in time).

Let us further assume that the device is phase preserv-ing, that is that the gain or attenuation is the same for

65

both quadratures. One form for the s matrix consistentwith all of the above requirements is

s =

+ cos θ sin θ 0 0sin θ − cos θ 0 0

0 0 − cos θ sin θ

0 0 sin θ + cos θ

. (B71)

This simply corresponds to a beam splitter and is theequivalent of Eq. (B51) with r = cos θ. As mentioned inconnection with Eq. (B51), the precise form of the scat-tering matrix depends on the choice of planes at whichthe phases of the various input and output waves aremeasured.

Another allowed form of the scattering matrix is:

s′ = −

+ cosh θ + sinh θ 0 0+ sinh θ + cosh θ 0 0

0 0 + cosh θ − sinh θ

0 0 − sinh θ + cosh θ

.

(B72)If one takes cosh θ =

√G, this scattering matrix is

essentially the canonically correct formulation of thenegative-resistance scattering matrix we tried to write inEq. (B62). Note that the off-diagonal terms have changedsign for the Y quadrature relative to the naive expressionin Eq. (B62) (corresponding to the other possible an-alytic continuation choice). This is necessary to satisfythe symplecticity condition and hence make the transfor-mation canonical. The scattering matrix s′ can describeamplification. Unlike the beam splitter scattering ma-trix s above, s′ is not unitary (even though det s′ = 1).Unitarity would correspond to power conservation. Here,power is not conserved, as we are not explicitly trackingthe power source supplying our active system.

The form of the negative-resistance amplifier scatteringmatrix s′ confirms many of the general statements wemade about phase-preserving amplification in Sec. VI.B.First, note that the requirement of finite gain G > 1 andphase preservation makes all the diagonal elements of s′

(i.e. cosh θ ) equal. We see that to amplify the A mode,it is impossible to avoid coupling to the B mode (via thesinh θ term) because of the requirement of symplecticity.We thus see that it is impossible classically or quantummechanically to build a linear phase-preserving amplifierwhose only effect is to amplify the desired signal. Thepresence of the sinh θ term above means that the outputsignal is always contaminated by amplified noise fromat least one other degree of freedom (in this case the Bmode). If the thermal or quantum noise in A and Bare equal in magnitude (and uncorrelated), then in thelimit of large gain where cosh θ ≈ sinh θ, the output noise(referred to the input) will be doubled. This is true forboth classical thermal noise and quantum vacuum noise.

The negative resistance model of an amplifier heregives us another way to think about the noise added byan amplifier: crudely speaking, we can view it as beingdirectly analogous to the fluctuation-dissipation theorem

simply continued to the case of negative dissipation. Justas dissipation can occur only when we open up a newchannel and thus we bring in new fluctuations, so ampli-fication can occur only when there is coupling to an ad-ditional channel. Without this it is impossible to satisfythe requirement that the amplifier perform a canonicaltransformation.

APPENDIX C: Quantum Modes and Noise of aTransmission Line

1. Quantization of a transmission line

Recall from Eq. (B30) and the discussion in AppendixB that the momentum conjugate to the transmission lineflux variable ϕ(x, t) is the local charge density q(x, t).Hence in order to quantize the transmission line modeswe simply promote these two physical quantities to quan-tum operators obeying the commutation relation

[q(x), ϕ(x′)] = −i!δ(x− x′) (C1)

from which it follows that the mode amplitudes definedin Eq. (B33) become quantum operators obeying

[Ak′ , A†k] = !ωkδkk′ (C2)

and we may identify the usual raising and lowering oper-ators by

Ak =√

!ωk bk (C3)

where bk destroys a photon in mode k. The quantumform of the Hamiltonian in Eq. (B34) is thus

H =∑

k

!ωk

[b†k bk +

12

]. (C4)

For the quantum case the thermal equilibrium expressionthen becomes

〈A†kAk〉 = !ωknB(!ωk), (C5)

which reduces to Eq. (B42) in the classical limit !ωk ,kBT .

We have seen previously in Eqs. (B6) that the volt-age fluctuations on a transmission line can be resolvedinto right and left moving waves which are functions of acombined space-time argument

V (x, t) = V→(t− x

vp) + V←(t +

x

vp). (C6)

Thus in an infinite transmission line, specifying V→ ev-erywhere in space at t = 0 determines its value for alltimes. Conversely specifying V→ at x = 0 for all timesfully specifies the field at all spatial points. In prepa-ration for our study of the quantum version of input-output theory in Appendix D, it is convenient to extend

66

Eqs. (B38-B39) to the quantum case:

V→(t) =√

12Lc

∑

k>0

√!ωk

[bk(t)e−iωkt + h.c.

]

=∫ ∞

0

dω

2π

√!ωZc

2

[b→[ω]e−iωt + h.c.

](C7)

In the second line, we have defined:

b→[ω] ≡ 2π

√vp

L

∑

k>0

bkδ(ω − ωk) (C8)

In a similar fashion, we have:

V←(t) =∫ ∞

0

dω

2π

√!ωZc

2

[b←[ω]e−iωt + h.c.

](C9)

b←[ω] ≡ 2π

√vp

L

∑

k<0

bkδ(ω − ωk) (C10)

One can easily verify that among the b→[ω], b←[ω] opera-tors and their conjugates, the only non-zero commutatorsare given by:[b→[ω],

(b→[ω′]

)†]

=[b←[ω],

(b←[ω′]

)†]

= 2πδ(ω − ω′)

(C11)We have taken the continuum limit L →∞ here, allowingus to change sums on k to integrals. We have thus ob-tained the description of a quantum transmission line interms of left and right-moving frequency resolved modes,as used in our discussion of amplifiers in Sec. VII (seeEqs. 7.2). Note that if the right-moving modes are fur-ther taken to be in thermal equilibrium, one finds (again,in the continuum limit):

〈b†→[ω]b→[ω′]〉 = 2πδ(ω − ω′)nB(!ω) (C12a)

〈b→[ω]b†→[ω′]〉 = 2πδ(ω − ω′) [1 + nB(!ω)] .(C12b)

We are typically interested in a relatively narrow bandof frequencies centered on some characteristic drive orresonance frequency Ω0. In this case, it is useful to workin the time-domain, in a frame rotating at Ω0. Fouriertransforming 23 Eqs. (C8) and (C10), one finds:

b→(t) =√

vp

L

∑

k>0

e−i(ωk−Ω0)tbk(0), (C13a)

b←(t) =√

vp

L

∑

k<0

e−i(ωk−Ω0)tbk(0). (C13b)

23 Here and throughout we use a convention which differs from theone commonly used in quantum optics: a[ω] =

R +∞−∞ dt e+iωta(t)

and a†[ω] = [a[−ω]]† =R +∞−∞ dt e+iωta†(t).

These represent temporal right and left moving modes.Note that the normalization factors in Eqs. (C13) hasbeen chosen so that the right moving photon flux at x = 0and time t is given by

〈N〉 = 〈b†→(t)b→(t)〉 (C14)

In the same rotating frame, and within the approxima-tion that all relevant frequencies are near Ω0, Eq. (C7)becomes simply:

V→(t) ≈√

!Ω0Zc

2

[b→(t) + b†→(t)

](C15)

We have already seen that using classical statisticalmechanics, the voltage noise in equilibrium is white.The corresponding analysis of the temporal modes usingEqs. (C13) shows that the quantum commutator obeys

[b→(t), b†→(t′)] = δ(t− t′). (C16)

In deriving this result, we have converted summationsover mode index to integrals over frequency. Further,because (for finite time resolution at least) the integral isdominated by frequencies near +Ω0 we can, within theMarkov (Wigner Weisskopf) approximation, extend thelower limit of frequency integration to minus infinity andthus arrive at a delta function in time. If we further takethe right moving modes to be in thermal equilibrium,then we may similarly approximate:

〈b†→(t′)b→(t)〉 = nB(!Ω0)δ(t− t′) (C17a)

〈b→(t)b†→(t′)〉 = [1 + nB(!Ω0)] δ(t− t′). (C17b)

Equations (C15) to (C17b) indicate that V→(t) can betreated as the quantum operator equivalent of whitenoise; a similar line of reasoning applies mutatis mutan-dis to the left moving modes. We stress that these re-sults rely crucially on our assumption that we are dealingwith a relatively narrow band of frequencies in the vicin-ity of Ω0; the resulting approximations we have madeare known as the Markov approximation. As one can al-ready see from the form of Eqs. (C7,C9), and as will bediscussed further, the actual spectral density of vacuumnoise on a transmission line is not white, but is linear infrequency. The approximation made in Eq. (C16) treatsit as a constant within the narrow band of frequenciesof interest. If the range of frequencies of importance islarge then the Markov approximation is not applicable.

2. Modes and the windowed Fourier transform

While the delta function correlations can make thequantum noise relatively easy to deal with in both thetime and frequency domain, it is sometimes the case thatit is easier to deal with a ‘smoothed’ noise variable. Theintroduction of an ultraviolet cutoff regulates the math-ematical singularities in the noise operators evaluated at

67

equal times and is physically sensible because every realmeasurement apparatus has finite time resolution. A sec-ond motivation is that real spectrum analyzers output atime varying signal which represents the noise power ina certain frequency interval (the ‘resolution bandwidth’)averaged over a certain time interval (the inverse ‘videobandwidth’). The mathematical tool of choice for dealingwith such situations in which time and frequency bothappear is the ‘windowed Fourier transform’. The win-dowed transform uses a kernel which is centered on somefrequency window and some time interval. By summa-tion over all frequency and time windows it is possibleto invert the transformation. The reader is directed to(Mallat, 1999) for the mathematical details.

For our present purposes where we are interested injust a single narrow frequency range centered on Ω0, aconvenient windowed transform kernel for smoothing thequantum noise is simply a box of width ∆t representingthe finite integration time of our detector. In the framerotating at Ω0 we can define

B→j =

1√∆t

∫ tj+1

tj

dτ b→(τ) (C18)

where tj = j(∆t) denotes the time of arrival of the jthtemporal mode at the point x = 0. Recall that b→ hasa photon flux normalization and so B→

j is dimensionless.From Eq. (C16) we see that these smoothed operatorsobey the usual bosonic commutation relations

[B→j , B†→

k ] = δjk. (C19)

The state B†j |0〉 has a single photon occupying basis

mode j, which is centered in frequency space at Ω0 andin time space on the interval j∆t < t < (j +1)∆t. (Thatis, this temporal mode passes the point x = 0 during thejth time interval.) This basis mode is much like a notein a musical score: it has a certain specified pitch and oc-curs at a specified time for a specified duration. Just aswe can play notes of different frequencies simultaneously,we can define other temporal modes on the same time in-terval and they will be mutually orthogonal provided theangular frequency spacing is a multiple of 2π/∆t. Theresult is a set of modes Bm,p labeled by both a frequencyindex m and a time index p. p labels the time interval asbefore, while m labels the angular frequency:

ωm = Ω0 + m2π

∆t(C20)

The result is, as illustrated in Fig. (17), a complete lat-tice of possible modes tiling the frequency-time phasespace, each occupying area 2π corresponding to the time-frequency uncertainty principle.

We can form other modes of arbitrary shapes centeredon frequency Ω0 by means of linear superposition of ourbasis modes (as long as they are smooth on the time scale∆t). Let us define

Ψ =∑

j

ψjB→j . (C21)

area = 2π

t

ω Δt

2π/Δt

B-mp

Bmp

ω=Ω0

m

-m

p

FIG. 17 (Color online) Schematic figure indicating how thevarious modes defined by the windowed Fourier transform tilethe time-frequency plane. Each individual cell corresponds toa different mode, and has an area 2π.

This is also a canonical bosonic mode operator obeying

[Ψ,Ψ†] = 1 (C22)

provided that the coefficients obey the normalization con-dition

∑

j

|ψj |2 = 1. (C23)

We might for example want to describe a mode which iscentered at a slightly higher frequency Ω0 + δΩ (obeying(δΩ)(∆t) << 1) and spread out over a large time intervalT centered at time T0. This could be given for exampleby

ψj = N e−(j−T0/(∆t))2

2T2 e−i(δΩ)(∆t)j (C24)

where N is the appropriate normalization constant.The state having n photons in the mode is simply

1√n!

(Ψ†)n |0〉. (C25)

The concept of ‘wave function of the photon’ is fraughtwith dangers. In the very special case where we re-strict attention solely to the subspace of single photonFock states, we can usefully think of the amplitudes ψjas the ‘wave function of the photon’ (Cohen-Tannoudjiet al., 1989) since it tells us about the spatial mode whichis excited. In the general case however it is essential tokeep in mind that the transmission line is a collection ofcoupled LC oscillators with an infinite number of degreesof freedom. Let us simplify the argument by consideringa single LC oscillator. We can perfectly well write a wavefunction for the system as a function of the coordinate(say the charge q on the capacitor). The ground statewave function χ0(q) is a gaussian function of the coor-dinate. The one photon state created by Ψ† has a wave

68

function χ1(q) ∼ qχ0(q) proportional to the coordinatetimes the same gaussian. In the general case χ is a wavefunctional of the charge distribution q(x) over the entiretransmission line.

Using Eq. (C17a) we have

〈B†→j B→

k 〉 = nB(!Ω0)δjk (C26)

independent of our choice of the coarse-graining time win-dow ∆t. This result allows us to give meaning to thephrase one often hears bandied about in descriptions ofamplifiers that ‘the noise temperature corresponds to amode occupancy of X photons’. This simply means thatthe photon flux per unit bandwidth is X. Equivalentlythe flux in bandwidth B is

N =X

∆t(B∆t) = XB. (C27)

The interpretation of this is that X photons in a tempo-ral mode of duration ∆t pass the origin in time ∆t. Eachmode has bandwidth ∼ 1

∆t and so there are B∆t inde-pendent temporal modes in bandwidth B all occupyingthe same time interval ∆t. The longer is ∆t the longer ittakes a given mode to pass the origin, but the more suchmodes fit into the frequency window.

As an illustration of these ideas, consider the followingelementary question: What is the mode occupancy of alaser beam of power P and hence photon flux N = P

!Ω0?

We cannot answer this without knowing the coherencetime or equivalently the bandwidth. The output of agood laser is like that of a radio frequency oscillator–ithas essentially no amplitude fluctuations. The frequencyis nominally set by the physical properties of the oscilla-tor, but there is nothing to pin the phase which conse-quently undergoes slow diffusion due to unavoidable noiseperturbations. This leads to a finite phase coherence timeτ and corresponding frequency spread 1/τ of the laserspectrum. (A laser beam differs from a thermal sourcethat has been filtered to have the same spectrum in thatit has smaller amplitude fluctuations.) Thus we expectthat the mode occupancy is X = Nτ . A convenient ap-proximate description in terms of temporal modes is totake the window interval to be ∆t = τ . Within the jthinterval we take the phase to be a (random) constant ϕj

so that (up to an unimportant normalization constant)we have the coherent state

∏

j

e√

Xeiϕj B†→j |0〉 (C28)

which obeys

〈B→k 〉 =

√Xeiϕk (C29)

and

〈B†→k B→

k 〉 = X. (C30)

3. Quantum noise from a resistor

Let us consider the quantum equivalent to Eq. (B60),SV V = 2RkBT , for the case of a semi-infinite transmis-sion line with open termination, representing a resistor.From Eq. (B27) we see that the proper boundary con-dition for the ϕ field is ∂xϕ(0, t) = ∂xϕ(L, t) = 0. (Wehave temporarily made the transmission line have a largebut finite length L.) The normal mode expansion thatsatisfies these boundary conditions is

ϕ(x, t) =√

2L

∞∑

n=1

ϕn(t) cos(knx), (C31)

where ϕn is the normal coordinate and kn ≡ πnL . Sub-

stitution of this form into the Lagrangian and carryingout the spatial integration yields a set of independentharmonic oscillators representing the normal modes.

L =∞∑

n=1

c

2ϕ2

n −125

k2nϕ2

n. (C32)

From this we can find the momentum operator pn canon-ically conjugate to the coordinate operator ϕn and quan-tize the system to obtain an expression for the operatorrepresenting the voltage at the end of the transmissionline in terms of the mode creation and destruction oper-ators

V =∞∑

n=1

√!Ωn

Lci(b†n − bn). (C33)

The spectral density of voltage fluctuations is then foundto be

SVV[ω] =2π

L

∞∑

n=1

!Ωn

c

nB(!Ωn)δ(ω + Ωn)

+[nB(!Ωn) + 1]δ(ω − Ωn), (C34)

where nB(!ω) is the Bose occupancy factor for a photonwith energy !ω. Taking the limit L →∞ and convertingthe summation to an integral yields

SVV(ω) = 2Zc!|ω|nB(!|ω|)Θ(−ω)+[nB(!|ω|)+1]Θ(ω)

,

(C35)where Θ is the step function. We see immediately thatat zero temperature there is no noise at negative frequen-cies because energy can not be extracted from zero-pointmotion. However there remains noise at positive frequen-cies indicating that the vacuum is capable of absorbingenergy from another quantum system. The voltage spec-tral density at both zero and non-zero temperature isplotted in Fig. (1).

Eq. (C35) for this ‘two-sided’ spectral density of a re-sistor can be rewritten in a more compact form

SVV[ω] =2Zc!ω

1− e−!ω/kBT, (C36)

69

which reduces to the more familiar expressions in variouslimits. For example, in the classical limit kBT % !ω thespectral density is equal to the Johnson noise result24

SVV[ω] = 2ZckBT, (C37)

in agreement with Eq. (B60). In the quantum limit itreduces to

SVV[ω] = 2Zc!ωΘ(ω). (C38)

Again, the step function tells us that the resistor can onlyabsorb energy, not emit it, at zero temperature.

If we use the engineering convention and add the noiseat positive and negative frequencies we obtain

SVV[ω] + SVV[−ω] = 2Zc!ω coth!ω

2kBT(C39)

for the symmetric part of the noise, which appears in thequantum fluctuation-dissipation theorem (cf. Eq. (3.33)).The antisymmetric part of the noise is simply

SVV[ω]− SVV[−ω] = 2Zc!ω, (C40)

yielding

SVV[ω]− SVV[−ω]SVV[ω] + SVV[−ω]

= tanh!ω

2kBT. (C41)

This quantum treatment can also be applied to anyarbitrary dissipative network (Burkhard et al., 2004; De-voret, 1997). If we have a more complex circuit con-taining capacitors and inductors, then in all of the aboveexpressions, Zc should be replaced by ReZ[ω] where Z[ω]is the complex impedance presented by the circuit.

In the above we have explicitly quantized the stand-ing wave modes of a finite length transmission line. Wecould instead have used the running waves of an infiniteline and recognized that, as the in classical treatment inEq. (B59), the left and right movers are not independent.The open boundary condition at the termination requiresV← = V→ and hence b→ = b←. We then obtain

SV V [ω] = 4S→V V [ω] (C42)

and from the quantum analog of Eq. (B40) we have

SV V [ω] =4!|ω|2cvp

Θ(ω)(nB + 1) + Θ(−ω)nB

= 2Zc!|ω| Θ(ω)(nB + 1) + Θ(−ω)nB(C43)

in agreement with Eq. (C35).

24 Note again that in the engineering convention this would beSVV[ω] = 4ZckBT .

APPENDIX D: Back Action and Input-Output Theory forDriven Damped Cavities

A high Q cavity whose resonance frequency can beparametrically controlled by an external source can actas a very simple quantum amplifier, encoding informa-tion about the external source in the phase and ampli-tude of the output of the driven cavity. For example,in an optical cavity, one of the mirrors could be move-able and the external source could be a force acting onthat mirror. This defines the very active field of optome-chanics, which also deals with microwave cavities cou-pled to nanomechanical systems and other related setups(Arcizet et al., 2006; Brown et al., 2007; Gigan et al.,2006; Harris et al., 2007; Hohberger-Metzger and Kar-rai, 2004; Marquardt et al., 2007, 2006; Meystre et al.,1985; Schliesser et al., 2006; Teufel et al., 2008; Thomp-son et al., 2008; Wilson-Rae et al., 2007). In the case of amicrowave cavity containing a qubit, the state-dependentpolarizability of the qubit acts as a source which shiftsthe frequency of the cavity (Blais et al., 2004; Schusteret al., 2005; Wallraff et al., 2004).

The dephasing of a qubit in a microwave cavity andthe fluctuations in the radiation pressure in an opticalcavity both depend on the quantum noise in the numberof photons inside the cavity. We here use a simple equa-tion of motion method to exactly solve for this quantumnoise in the perturbative limit where the dynamics of thequbit or mirror degree of freedom has only a weak backaction effect on the cavity.

In the following, we first give a basic discussion ofthe cavity field noise spectrum, deferring the detailedmicroscopic derivation to subsequent subsections. Wethen provide a review of the input-output theory fordriven cavities, and employ this theory to analyze theimportant example of a dispersive position measurement,where we demonstrate how the standard quantum limitcan be reached. Finally, we analyze an example wherea modified dispersive scheme is used to detect only onequadrature of a harmonic oscillator’s motion, such thatthis quadrature does not feel any back-action.

1. Photon shot noise inside a cavity and back action

Consider a degree of freedom z coupled parametricallywith strength A to the cavity oscillator

Hint = !ωc(1 + Az) [a†a− 〈a†a〉] (D1)

where following Eq. (4.22), we have taken A to be dimen-sionless, and use z to denote the dimensionless systemvariable that we wish to probe. For example, z couldrepresent the dimensionless position of a mechanical os-cillator

z ≡ x

xZPF. (D2)

We have subtracted the 〈a†a〉 term so that the meanforce on the degree of freedom is zero. To obtain the full

70

Hamiltonian, we would have to add the cavity dampingand driving terms, as well as the Hamiltonian governingthe intrinsic dynamics of the system z. From Eq. (4.29)we know that the back action noise force acting on z isproportional to the quantum fluctuations in the numberof photons n = a†a in the cavity,

Snn(t) = 〈a†(t)a(t)a†(0)a(0)〉 − 〈a†(t)a(t)〉2. (D3)

For the case of continuous wave driving at frequencyωL = ωc + ∆ detuned by ∆ from the resonance, thecavity is in a coherent state |ψ〉 obeying

a(t) = e−iωLt[a + d(t)] (D4)

where the first term is the ‘classical part’ of the mode am-plitude ψ(t) = ae−iωLt determined by the strength of thedrive field, the damping of the cavity and the detuning∆, and d is the quantum part. By definition,

a|ψ〉 = ψ|ψ〉 (D5)

so the coherent state is annihilated by d:

d|ψ〉 = 0. (D6)

That is, in terms of the operator d, the coherent statelooks like the undriven quantum ground state . The dis-placement transformation in Eq. (D4) is canonical since

[a, a†] = 1 ⇒ [d, d†] = 1. (D7)

Substituting the displacement transformation intoEq. (D3) and using Eq. (D6) yields

Snn(t) = n〈d(t)d†(0)〉, (D8)

where n = |a|2 is the mean cavity photon number. If weset the cavity energy damping rate to be κ, such that theamplitude damping rate is κ/2, then the undriven stateobeys

〈d(t)d†(0)〉 = e+i∆te−κ2 |t|. (D9)

This expression will be justified formally in the subse-quent subsection, after introducing input-output theory.We thus arrive at the very simple result

Snn(t) = nei∆t−κ2 |t|. (D10)

The power spectrum of the noise is, via the Wiener-Khinchin theorem (Appendix A), simply the Fouriertransform of the autocorrelation function given inEq. (D10)

Snn[ω] =∫ +∞

−∞dt eiωtSnn(t) = n

κ

(ω + ∆)2 + (κ/2)2.

(D11)As can be seen in Fig. 18a, for positive detuning ∆ =ωL − ωc > 0, i.e. for a drive that is blue-detuned with

Noi

se p

ower

(a)

Frequency

(b)

Noi

se te

mpe

ratu

re

(c)

Frequency

4

2

0-6 0 6 60-6

4

0

-4

3

2

1

5 600 1 2 3 4

-

FIG. 18 (Color online) (a) Noise spectrum of the photon num-ber in a driven cavity as a function of frequency when thecavity drive frequency is detuned from the cavity resonanceby ∆ = +3κ (left peak) and ∆ = −3κ (right peak). (b) Ef-fective temperature Teff of the low frequency noise, ω → 0,as a function of the detuning ∆ of the drive from the cavityresonance. (c) Frequency-dependence of the effective noisetemperature, for different values of the detuning.

respect to the cavity, the noise peaks at negative ω. Thismeans that the noise tends to pump energy into the de-gree of freedom z (i.e. contribute negative damping). Fornegative detuning the noise peaks at positive ω corre-sponding to the cavity absorbing energy from z. Basi-cally, the interaction with z (three wave mixing) tries toRaman scatter the drive photons into the high density ofstates at the cavity frequency. If this is uphill in energy,then z is cooled.

As discussed in Sec. III.B (c.f. Eq. (3.21)), at each fre-quency ω, we can use detailed balance to assign the noisean effective temperature Teff [ω]:

Snn[ω]Snn[−ω]

= e!ω/kBTeff [ω] ⇔

kBTeff [ω] ≡ !ω

log[

Snn[ω]Snn[−ω]

] (D12)

71

or equivalently

Snn[ω]− Snn[−ω]Snn[ω] + Snn[−ω]

= tanh(β!ω/2). (D13)

If z is the coordinate of a harmonic oscillator of frequencyω (or some non-conserved observable of a qubit with levelsplitting ω), then that system will acquire a temperatureTeff [ω] in the absence of coupling to any other environ-ment. In particular, if the characteristic oscillation fre-quency of the system z is much smaller than κ, then wehave the simple result

1kBTeff

= limω→0+

2!ω

Snn[ω]− Snn[−ω]Snn[ω] + Snn[−ω]

= 2d lnSnn[ω]

d!ω

=1!

−4∆∆2 + (κ/2)2

. (D14)

As can be seen in Fig. 18, the asymmetry in the noisechanges sign with detuning, which causes the effectivetemperature to change sign.

First we discuss the case of a positive Teff , where thismechanism can be used to laser cool an oscillating me-chanical cantilever, provided Teff is lower than the in-trinsic equilibrium temperature of the cantilever. (Ar-cizet et al., 2006; Brown et al., 2007; Gigan et al., 2006;Harris et al., 2007; Hohberger-Metzger and Karrai, 2004;Marquardt et al., 2007; Schliesser et al., 2006; Thompsonet al., 2008; Wilson-Rae et al., 2007). A simple classicalargument helps us understand this cooling effect. Sup-pose that the moveable mirror is at the right hand endof a cavity being driven below the resonance frequency.If the mirror moves to the right, the resonance frequencywill fall and the number of photons in the cavity will rise.There will be a time delay however to fill the cavity andso the extra radiation pressure will not be fully effectivein doing work on the mirror. During the return part ofthe oscillation as the mirror moves back to the left, thetime delay in emptying the cavity will cause the mirror tohave to do extra work against the radiation pressure. Atthe end of the cycle it ends up having done net positivework on the light field and hence is cooled. The effect cantherefore be understood as being due to the introductionof some extra optomechanical damping.

The signs reverse (and Teff becomes negative) if thecavity is driven above resonance, and consequently thecantilever motion is heated up. In the absence of in-trinsic mechanical losses, negative values of the effectivetemperature indicate a dynamical instability of the can-tilever (or population inversion in the case of a qubit),where the amplitude of motion grows until it is finallystabilized by nonlinear effects. This can be interpretedas negative damping introduced by the optomechanicalcoupling and can be used to create parametric amplifica-tion of mechanical forces acting on the oscillator.

Finally, we mention that cooling towards the quantumground state of a mechanical oscillator (where phonon

numbers become much less than one), is only possible(Marquardt et al., 2007; Wilson-Rae et al., 2007) in the“far-detuned regime”, where −∆ = ω % κ (in contrastto the ω , κ regime discussed above).

2. Input-output theory for a driven cavity

The results from the previous section can be more for-mally and rigorously derived in a full quantum theoryof a cavity driven by an external coherent source. Thetheory relating the drive, the cavity and the outgoingwaves radiated by the cavity is known as input-outputtheory and the classical description was presented in Ap-pendix B. The present quantum discussion closely followsstandard references on the subject (Walls and Milburn,1994; Yurke, 1984; Yurke and Denker, 1984). The crucialfeature that distinguishes such an approach from manyother treatments of quantum-dissipative systems is thegoal of keeping the bath modes instead of tracing themout. This is obviously necessary for the situations wehave in mind, where the output field emanating from thecavity contains the information acquired during a mea-surement of the system coupled to the cavity. As welearned from the classical treatment, we can eliminatethe outgoing waves in favor of a damping term for thesystem. However we can recover the solution for the out-going modes completely from the solution of the equationof motion of the damped system being driven by the in-coming waves.

In order to drive the cavity we must partially open oneof its ports which exposes the cavity both to the externaldrive and to the vacuum noise outside which permits en-ergy in the cavity to leak out into the surrounding bath.We will formally separate the degrees of freedom into in-ternal cavity modes and external bath modes. Strictlyspeaking, once the port is open, these modes are not dis-tinct and we only have ‘the modes of the universe’ (Gea-Banacloche et al., 1990a,b; Lang et al., 1973). Howeverfor high Q cavities, the distinction is well-defined and wecan model the decay of the cavity in terms of a spon-taneous emission process in which an internal boson isdestroyed and an external bath boson is created. Weassume a single-sided cavity as shown in Fig. 3. For ahigh Q cavity, this physics is accurately captured in thefollowing Hamiltonian

H = Hsys + Hbath + Hint. (D15)

The bath Hamiltonian is

Hbath =∑

q

!ωq b†q bq (D16)

where q labels the quantum numbers of the independentharmonic oscillator bath modes obeying

[bq, b†q′ ] = δq,q′ . (D17)

72

Note that since the bath terminates at the system, thereis no translation invariance, the normal modes are stand-ing not running waves, and the quantum numbers q arenot necessarily wave vectors.

The coupling Hamiltonian is (within the rotating waveapproximation)

Hint = −i!∑

q

[fqa

†bq − f∗q b†qa]. (D18)

For the moment we will leave the system (cavity) Hamil-tonian to be completely general, specifying only that itconsists of a single degree of freedom (i.e. we concentrateon only a single resonance of the cavity with frequencyωc) obeying the usual bosonic commutation relation

[a, a†] = 1. (D19)

(N.B. this does not imply that it is a harmonic oscilla-tor. We will consider both linear and non-linear cavities.)Note that the most general linear coupling to the bathmodes would include terms of the form b†qa

† and bqa butthese are neglected within the rotating wave approxima-tion because in the interaction representation they os-cillate at high frequencies and have little effect on thedynamics.

The Heisenberg equation of motion (EOM) for thebath variables is

˙bq =i

! [H, bq] = −iωqbq + f∗q a (D20)

We see that this is simply the EOM of a harmonic oscil-lator driven by a forcing term due to the motion of thecavity degree of freedom. Since this is a linear system,the EOM can be solved exactly. Let t0 < t be a time inthe distant past before any wave packet launched at thecavity has reached it. The solution of Eq. (D20) is

bq(t) = e−iωq(t−t0)bq(t0) +∫ t

t0

dτ e−iωq(t−τ)f∗q a(τ).

(D21)The first term is simply the free evolution of the bathwhile the second represents the waves radiated by thecavity into the bath.

The EOM for the cavity mode is

˙a =i

! [Hsys, a]−∑

q

fq bq. (D22)

Substituting Eq. (D21) into the last term above yields∑

q

fq bq =∑

q

fqe−iωq(t−t0)bq(t0)

+∑

q

|fq|2∫ t

t0

dτ e−i(ωq−ωc)(t−τ)[e+iωc(τ−t)a(τ)], (D23)

where the last term in square brackets is a slowly varyingfunction of τ . To simplify our result, we note that if

the cavity system were a simple harmonic oscillator offrequency ωc then the decay rate from the n = 1 singlephoton excited state to the n = 0 ground state would begiven by the following Fermi Golden Rule expression

κ(ωc) = 2π∑

q

|fq|2δ(ωc − ωq). (D24)

From this it follows that∫ +∞

−∞

dν

2πκ(ωc + ν)e−iν(t−τ) =

∑

q

|fq|2e−i(ωq−ωc)(t−τ).

(D25)We now make the Markov approximation which assumesthat κ(ν) = κ is a constant over the range of frequenciesrelevant to the cavity so that Eq. (D25) may be repre-sented as

∑

q

|fq|2e−i(ωq−ωc)(t−τ) = κδ(t− τ). (D26)

Using∫ x0

−∞dx δ(x− x0) =

12

(D27)

we obtain for the cavity EOM

˙a =i

! [Hsys, a]− κ

2a−

∑

q

fqe−iωq(t−t0)bq(t0). (D28)

The second term came from the part of the bath motionrepresenting the wave radiated by the cavity and, withinthe Markov approximation, has become a simple lineardamping term for the cavity mode. Note the importantfactor of 2. The amplitude decays at half the rate of theintensity (the energy decay rate κ).

Within the spirit of the Markov approximation it isfurther convenient to treat f ≡

√|fq|2 as a constant and

define the density of states (also taken to be a constant)by

ρ =∑

q

δ(ωc − ωq) (D29)

so that the Golden Rule rate becomes

κ = 2πf2ρ. (D30)

We can now define the so-called ‘input mode’

bin(t) ≡ 1√2πρ

∑

q

e−iωq(t−t0)bq(t0) . (D31)

For the case of a transmission line treated in AppendixC, this coincides with the field b→ moving towards thecavity [see Eq. (C13a)]. We finally have for the cavityEOM

˙a =i

! [Hsys, a]− κ

2a−

√κ bin(t). (D32)

73

Note that when a wave packet is launched from the bathtowards the cavity, causality prevents it from knowingabout the cavity’s presence until it reaches the cavity.Hence the input mode evolves freely as if the cavity werenot present until the time of the collision at which point itbegins to drive the cavity. Since bin(t) evolves under thefree bath Hamiltonian and acts as the driving term in thecavity EOM, we interpret it physically as the input mode.Eq. (D32) is the quantum analog of the classical equa-tion (B19), for our previous example of an LC-oscillatordriven by a transmission line. The latter would also havebeen first order in time if as in Eq. (B35) we had workedwith the complex amplitude A instead of the coordinateQ.

Eq. (D31) for the input mode contains a time labeljust as in the interaction representation. However it isbest interpreted as simply labeling the particular linearcombination of the bath modes which is coupled to thesystem at time t. Some authors even like to think ofthe bath modes as non-propagating while the cavity fliesalong the bath (taken to be 1D) at a velocity v. Thesystem then only interacts briefly with the local modepositioned at x = vt before moving an interacting withthe next local bath mode. We will elaborate on this viewfurther at the end of this subsection.

The expression for the power Pin (energy per time)impinging on the cavity depends on the normalizationchosen in our definition of bin. It can be obtained, forexample, by imagining the bath modes bq to live on a one-dimensional waveguide with propagation velocity v andlength L (using periodic boundary conditions). In thatcase we have to sum over all photons to get the averagepower flowing through a cross-section of the waveguide,Pin =

∑q !ωq(vp/L)

⟨b†q bq

⟩. Inserting the definition for

bin, Eq. (D31), the expression for the input power carriedby a monochromatic beam at frequency ω is

Pin(t) = !ω⟨b†in(t)bin(t)

⟩(D33)

Note that this has the correct dimensions due to ourchoice of normalization for bin (with dimensions

√ω). In

the general case, an integration over frequencies is needed(as will be discussed further below). An analogous for-mula holds for the power radiated by the cavity, to bediscussed now.

The output mode bout(t) is radiated into the bath andevolves freely after the system interacts with bin(t). Ifthe cavity did not respond at all, then the output modewould simply be the input mode reflected off the cav-ity mirror. If the mirror is partially transparent thenthe output mode will also contain waves radiated by thecavity (which is itself being driven by the input modepartially transmitted into the cavity through the mirror)and hence contains information about the internal dy-namics of the cavity. To analyze this output field, lett1 > t be a time in the distant future after the inputfield has interacted with the cavity. Then we can write

an alternative solution to Eq. (D20) in terms of the finalrather than the initial condition of the bath

bq(t) = e−iωq(t−t1)bq(t1)−∫ t1

tdτ e−iωq(t−τ)f∗q a(τ).

(D34)Note the important minus sign in the second term as-sociated with the fact that the time t is now the lowerlimit of integration rather than the upper as it was inEq. (D21).

Defining

bout(t) ≡1√2πρ

∑

q

e−iωq(t−t1)bq(t1), (D35)

we see that this is simply the free evolution of the bathmodes from the distant future (after they have interactedwith the cavity) back to the present, indicating that it isindeed appropriate to interpret this as the outgoing field.Proceeding as before we obtain

˙a =i

! [Hsys, a] +κ

2a−

√κ bout(t). (D36)

Subtracting Eq. (D36) from Eq. (D32) yields

bout(t) = bin(t) +√

κ a(t) (D37)

which is consistent with our interpretation of the out-going field as the reflected incoming field plus the fieldradiated by the cavity out through the partially reflectingmirror.

The above results are valid for any general cavityHamiltonian. The general procedure is to solve Eq. (D32)for a(t) for a given input field, and then solve Eq. (D37)to obtain the output field. For the case of an empty cav-ity we can make further progress because the cavity modeis a harmonic oscillator

Hsys = !ωca†a. (D38)

In this simple case, the cavity EOM becomes

˙a = −iωca−κ

2a−

√κ bin(t). (D39)

Eq. (D39) can be solved by Fourier transformation, yield-ing

a[ω] = −√

κ

i(ωc − ω) + κ/2bin[ω] (D40)

= −√

κχc[ω − ωc]bin[ω] (D41)

and

bout[ω] =ω − ωc − iκ/2ω − ωc + iκ/2

bin[ω] (D42)

which is the result for the reflection coefficient quoted inEq. (4.23). For brevity, here and in the following, we willsometimes use the susceptibility of the cavity, defined as

χc[ω − ωc] ≡1

−i(ω − ωc) + κ/2(D43)

74

For the case of steady driving on resonance where ω = ωc,the above equations yield

bout[ω] =√

κ

2a[ω]. (D44)

In steady state, the incoming power equals the outgoingpower, and both are related to the photon number insidethe single-sided cavity by

P = !ω⟨b†out(t)bout(t)

⟩= !ω

κ

4⟨a†(t)a(t)

⟩(D45)

Note that this does not coincide with the naive expecta-tion, which would be P = !ωκ

⟨a†a

⟩. The reason for this

discrepancy is the the interference between the part of theincoming wave which is promptly reflected from the cav-ity and the field radiated by the cavity. The naive expres-sion becomes correct after the drive has been switched off(where ignoring the effect of the incoming vacuum noise,we would have bout =

√κa). We note in passing that for

a driven two-sided cavity with coupling constants κL andκR (where κ = κL + κR), the incoming power sent intothe left port is related to the photon number by

P = !ωκ2/(4κL)⟨a†a

⟩. (D46)

Here for κL = κR the interference effect completely elim-inates the reflected beam and we have in contrast toEq. (D45)

P = !ωκ

2⟨a†a

⟩. (D47)

Eq. (D39) can also be solved in the time domain toobtain

a(t) = e−(iωc+κ/2)(t−t0)a(t0)

−√

κ

∫ t

t0

dτ e−(iωc+κ/2)(t−τ)bin(τ). (D48)

If we take the input field to be a coherent drive at fre-quency ωL = ωc +∆ so that its amplitude has a classicaland a quantum part

bin(t) = e−iωLt[bin + ξ(t)] (D49)

and if we take the limit t0 →∞ so that the initial tran-sient in the cavity amplitude has damped out, then thesolution of Eq. (D48) has the form postulated in Eq. (D4)with

a = −√

κ

−i∆ + κ/2bin (D50)

and (in the frame rotating at the drive frequency)

d(t) = −√

κ

∫ t

−∞dτ e+(i∆−κ/2)(t−τ)ξ(τ). (D51)

Even in the absence of any classical drive, the inputfield delivers vacuum fluctuation noise to the cavity. No-tice that from Eqs. (D31, D49)

[bin(t), b†in(t′)] = [ξ(t), ξ†(t′)]

=1

2πρ

∑

q

e−i(ωq−ωL)(t−t′)

= δ(t− t′), (D52)

which is similar to Eq. (C16) for a quantum transmissionline. This is the operator equivalent of white noise. UsingEq. (D48) in the limit t0 → −∞ in Eqs. (D4,D51) yields

[a(t), a†(t)] = [d(t), d†(t)]

= κ

∫ t

−∞dτ

∫ t

−∞dτ ′ e−(−i∆+κ/2)(t−τ)

e−(+i∆+κ/2)(t−τ ′)δ(τ − τ ′)= 1 (D53)

as is required for the cavity bosonic quantum degree offreedom. We can interpret this as saying that the cavityzero-point fluctuations arise from the vacuum noise thatenters through the open port. We also now have a simplephysical interpretation of the quantum noise in the num-ber of photons in the driven cavity in Eqs. (D3,D8,D11).It is due to the vacuum noise which enters the cavitythrough the same ports that bring in the classical drive.The interference between the vacuum noise and the clas-sical drive leads to the photon number fluctuations in thecavity.

In thermal equilibrium, ξ also contains thermal radi-ation. If the bath is being probed only over a narrowrange of frequencies centered on ωc (which we have haveassumed in making the Markov approximation) then wehave to a good approximation (consistent with the abovecommutation relation)

〈ξ†(t)ξ(t′)〉 = Nδ(t− t′) (D54)

〈ξ(t)ξ†(t′)〉 = (N + 1)δ(t− t′) (D55)

where N = nB(!ωc) is the thermal equilibrium occupa-tion number of the mode at the frequency of interest. Wecan gain a better understanding of Eq. (D54) by Fouriertransforming it to obtain the spectral density

S[ω] =∫ +∞

−∞dt 〈ξ†(t)ξ(t′)〉eiω(t−t′) = N. (D56)

As mentioned previously, this dimensionless quantity isthe spectral density that would be measured by a photo-multiplier: it represents the number of thermal photonspassing a given point per unit time per unit bandwidth.Equivalently the thermally radiated power in a narrowbandwidth B is

P = !ωNB. (D57)

75

One often hears the confusing statement that the noiseadded by an amplifier is a certain number N of photons(N = 20, say for a good cryogenic HEMT amplifier op-erating at 5 GHz). This means that the excess outputnoise (referred back to the input by dividing by the powergain) produces a flux of N photons per second in a 1 Hzbandwidth, or 106N photons per second in 1 MHz ofbandwidth.

We can gain further insight into input-output theoryby using the following picture. The operator bin(t) repre-sents the classical drive plus vacuum fluctuations whichare just about to arrive at the cavity. We will be ableto show that the output field is simply the input fielda short while later after it has interacted with the cav-ity. Let us consider the time evolution over a short timeperiod ∆t which is very long compared to the inversebandwidth of the vacuum noise (i.e., the frequency scalebeyond which the vacuum noise cannot be treated as con-stant due to some property of the environment) but veryshort compared to the cavity system’s slow dynamics. Inthis circumstance it is useful to introduce the quantumWiener increment related to Eq. (C18)

dW ≡∫ t+∆t

tdτ ξ(τ) (D58)

which obeys

[dW , dW †] = ∆t. (D59)

In the interaction picture (in a displaced frame inwhich the classical drive has been removed) the Hamilto-nian term that couples the cavity to the quantum noiseof the environment is from Eq. (D18)

V = −i!√

κ(a†ξ − aξ†). (D60)

Thus the time evolution operator (in the interaction pic-ture) on the jth short time interval [tj , tj + ∆t] is

Uj = e√

κ(a dcW †−a† dcW ) (D61)

Using this we can readily evolve the incoming temporalmode forward in time by a small step ∆t

dW ′ = U†dW U ≈ dW +√

κ∆t a. (D62)

Recall that in input-output theory we formally definedthe outgoing field as the bath field far in the future prop-agated back (using the free field time evolution) to thepresent, which yielded

bout = bin +√

κa. (D63)

Eq. (D62) is completely equivalent to this. Thus we con-firm our understanding that the incoming field is the bathtemporal mode just before it interacts with the cavity andthe outgoing field is the bath temporal mode just after itinteracts with the cavity.

This leads to the following picture which is especiallyuseful in the quantum trajectory approach to conditionalquantum evolution of a system subject to weak continu-ous measurement (Gardiner et al., 1992; Walls and Mil-burn, 1994). On top of the classical drive bin(t), the bathsupplies to the system a continuous stream of “fresh” har-monic oscillators, each in their ground state (if T = 0).Each oscillator with its quantum fluctuation dW inter-acts briefly for a period ∆t with the system and thenis disconnected to propagate freely thereafter, never in-teracting with the system again. Within this picture itis useful to think of the oscillators arrayed in an infinitestationary line and the cavity flying over them at speedvp and touching each one for a time ∆t.

3. Quantum limited position measurement using a cavitydetector

We will now apply the input-output formalism intro-duced in the previous section to the important example ofa dispersive position measurement, which employs a cav-ity whose resonance frequency shifts in response to themotion of a harmonic oscillator. This physical systemwas considered heuristically in Sec. (IV.B.3). Here wewill present a rigorous derivation using the (linearized)equations of motion for the coupled cavity and oscillatorsystem.

Let the dimensionless position operator

z =1

xZPFx = [c† + c] (D64)

be the coordinate of a harmonic oscillator whose energyis

HM = !ωMc†c (D65)

and whose position uncertainty in the quantum groundstate is xZPF =

√〈0|x2|0〉.

This Hamiltonian could be realized for example bymounting one of the cavity mirrors on a flexible cantilever(see the discussion above).

When the mirror moves, the cavity resonance fre-quency shifts,

ωc = ωc[1 + Az(t)] (D66)

where for a cavity of length L, A = −xZPF/L.Assuming that the mirror moves slowly enough for the

cavity to adiabatically follow its motion (i.e. Ω, κ), theoutgoing light field suffers a phase shift which follows thechanges in the mirror position. This phase shift can bedetected in the appropriate homodyne set up as discussedin Sec. IV.B, and from this phase shift we can determinethe position of the mechanical oscillator. In addition tothe actual zero-point fluctuations of the oscillator, ourmeasurement will suffer from shot noise in the homodynesignal and from additional uncertainty due to the backaction noise of the measurement acting on the oscillator.

76

All of these effects will appear naturally in the derivationbelow.

We begin by considering the optical cavity equationof motion based on Eq. (D32) and the optomechanicalcoupling Hamiltonian in Eq. (D1). These yield

˙a = −iωc(1 + Az)a− κ

2a−

√κbin. (D67)

Let the cavity be driven by a laser at a frequency ωL =ωc +∆ detuned from the cavity by ∆. Moving to a framerotating at ωL we have

˙a = +i(∆−Aωcz)a− κ

2a−

√κbin. (D68)

and we can write the incoming field as a constant pluswhite noise vacuum fluctuations (again, in the rotatingframe)

bin = bin + ξ (D69)

and similarly for the cavity field following Eq. (D4)

a = a + d. (D70)

Substituting these expressions into the equation of mo-tion, we find that the constant classical fields obey

a = −√

κ

κ/2− i∆bin (D71)

and the new quantum equation of motion is, after ne-glecting a small term dz:

˙d = +i∆d− iAωcaz − κ

2d−

√κξ. (D72)

The quantum limit for position measurement will bereached only at zero detuning, so we specialize to thecase ∆ = 0. We also choose the incoming field amplitudeand phase to obey

bin = −i

√N , (D73)

so that

a = +2i

√N

κ, (D74)

where N is the incoming photon number flux. The quan-tum equation of motion for the cavity then becomes

˙d = +gz − κ

2d−

√κξ, (D75)

where the opto-mechanical coupling constant is propor-tional to the laser drive amplitude

g ≡ 2Aωc

√N

κ= Aωc

√n. (D76)

and

n = |a|2 = 4N

κ(D77)

is the mean cavity photon number. Eq. (D75) is easilysolved by Fourier transformation

d[ω] =1

[κ/2− iω]

gz[ω]−

√κξ[ω]

. (D78)

Let us assume that we are in the limit of low mechan-ical frequency relative to the cavity damping, Ω , κ,so that the cavity state adiabatically follows the motionof the mechanical oscillator. Then we obtain to a goodapproximation

d[ω] =2κ

gz[ω]−

√κξ[ω]

(D79)

d†[ω] =2κ

gz[ω]−

√κξ†[ω]

(D80)

The mechanical oscillator equation of motion which isidentical in form to that of the optical cavity

∂tc = −[γ0

2+ iΩ]c−√γ0η(t) +

i

! [Hint, c(t)], (D81)

where Hint is the Hamiltonian in Eq. (D1) and η isthe mechanical vacuum noise from the (zero tempera-ture) bath which is causing the mechanical damping atrate γ0. Using Eq. (D70) and expanding to first orderin small fluctuations yields the equation of motion lin-earized about the steady state solution

∂tc = −[γ0

2+ iΩ]c−√γ0η(t)+2

g√κ

[ξ(t)− ξ†(t)]. (D82)

It is useful to consider an equivalent formulation inwhich we expand the Hamiltonian in Eq. (D1) to secondorder in the quantum fluctuations about the classical so-lution

Hint ≈ !ωcd†d + xF , (D83)

where the force (including the coupling A) is (up to asign)

F = −i!g

xZPF[d− d†]. (D84)

Note that the radiation pressure fluctuations (photonshot noise) inside the cavity provide a forcing term. Thestate of the field inside the cavity in general depends onthe past history of the cantilever position. However forthis special case of driving the cavity on resonance, thedependence of the cavity field on the cantilever history issuch that the latter drops out of the radiation pressure.To see this explicitly, consider the equation of motion forthe force obtained from Eq. (D75)

˙F = −κ

2F + i

!g

xZPF

√κ[ξ − ξ†]. (D85)

77

FIG. 19 (Color online) Phasor diagram for the cavity ampli-tude showing that (for our choice of parameters) the imag-inary quadrature of the vacuum noise ξ interferes with theclassical drive to produce photon number fluctuations whilethe real quadrature produces phase fluctuations which lead tomeasurement imprecision. The quantum fluctuations are il-lustrated in the usual fashion, depicting the Gaussian Wignerdensity of the coherent state in terms of color intensity.

Within our linearization approximation, the position ofthe mechanical oscillator has no effect on the radiationpressure (photon number in the cavity), but of course itdoes affect the phase of the cavity field (and hence theoutgoing field) which is what we measure in the homo-dyne detection.

Thus for this special case z does not appear on theRHS of either Eq. (D85) or Eq. (D82), which means thatthere is no optical renormalization of the cantilever fre-quency (‘optical spring’) or optical damping of the can-tilever. The lack of back-action damping in turn impliesthat the effective temperature Teff of the cavity detectoris infinite (cf. Eq. (3.21)). For this special case of zerodetuning the back action force noise is controlled by asingle quadrature of the incoming vacuum noise (whichinterferes with the classical drive to produce photon num-ber fluctuations). This is illustrated in the cavity ampli-tude phasor diagram of Fig. (19). We see that the vac-uum noise quadrature ξ + ξ† conjugate to F controls thephase noise which determines the measurement impreci-sion (shot noise in the homodyne signal). This will bediscussed further below.

The solution for the cantilever position can again beobtained by Fourier transformation. For frequenciessmall on the scale of κ the solution of Eq. (D85) is

F [ω] =2i!g

xZPF√

κ

ξ[ω]− ξ†[ω]

(D86)

and hence the back action force noise spectral density isat low frequencies

SFF [ω] =4!2g2

x2ZPFκ

(D87)

in agreement with Eq. (4.29).Introducing a quantity proportional to the cantilever

(mechanical) susceptibility (within the rotating wave ap-proximation we are using)

χM[ω − Ω] ≡ 1−i(ω − Ω) + γ0

2

, (D88)

we find from Eq. (D82)

z[ω] = z0[ω]− i

!xZPF χM[ω − Ω]− χM[ω + Ω] F [ω],(D89)

where the equilibrium fluctuations in position are givenby

z0[ω] ≡ −√γ0

χM[ω − Ω]η[ω] + χM[ω + Ω]η†[ω]

.

(D90)We can now obtain the power spectrum Szz describing

the total position fluctuations of the cantilever driven bythe mechanical vacuum noise plus the radiation pressureshot noise. From Eqs. (D89, D90) we find

Sxx[ω]x2

ZPF

= Szz[ω]

= γ0|χ[ω − Ω]|2 (D91)

+x2

ZPF

!2|χM[ω − Ω]− χM[ω + Ω]|2 SFF .

Note that (assuming high mechanical Q, i.e. γ0 , Ω) theequilibrium part has support only at positive frequencieswhile the back action induced position noise is symmetricin frequency reflecting the effective infinite temperatureof the back action noise. Symmetrizing this result withrespect to frequency (and using γ0 , Ω) we have

Sxx[ω] ≈ S0xx[ω]

(1 +

S0xx[Ω]!2

SFF

), (D92)

where S0xx[ω] is the symmetrized spectral density for posi-

tion fluctuations in the ground state given by Eq. (4.71).Now that we have obtained the effect of the back action

noise on the position fluctuations, we must turn our at-tention to the imprecision of the measurement due to shotnoise in the output. The appropriate homodyne quadra-ture variable to monitor to be sensitive to the outputphase shift caused by position fluctuations is

I = bout + b†out, (D93)

which, using the input-output results above, can be writ-ten

I = −(ξ + ξ†) + λx. (D94)

78

We see that the cavity homodyne detector system actsas a position transducer with gain

λ =4g

xZPF√

κ. (D95)

The first term in Eq. (D94) represents the vacuum noisethat mixes with the homodyne local oscillator to producethe shot noise in the output. The resulting measurementimprecision (symmetrized) spectral density referred backto the position of the oscillator is

SIxx =

1λ2

. (D96)

Comparing this to Eq. (D87) we see that we reach thequantum limit relating the imprecision noise to the backaction noise

SIxxSFF =

!2

4(D97)

in agreement with Eq. (4.20).Notice also from Eq. (D94) that the quadrature of the

vacuum noise which leads to the measurement impreci-sion is conjugate to the one which produces the back ac-tion force noise as illustrated previously in Fig. (19). Wesaw in Eq. (2.20) that the two quadratures of motion ofa harmonic oscillator in its ground state have no classical(i.e., symmetrized) correlation. Hence the symmetrizedcross correlator

SIF [ω] = 0 (D98)

vanishes. Because there is no correlation between theoutput imprecision noise and the forces controlling theposition fluctuations, the total output noise referred backto the position of the oscillator is simply

Sxx,tot[ω] = Sxx[ω] + SIxx (D99)

= S0xx[ω]

(1 +

S0xx[Ω]!2

SFF

)+

!2

4SFF.

This expression again clearly illustrates the competitionbetween the back action noise proportional to the drivelaser intensity and the measurement imprecision noisewhich is inversely proportional. We again emphasize thatall of the above relations are particular to the case of zerodetuning of the cavity drive field from the cavity.

The total output noise at some particular frequencywill be a minimum at some optimal drive intensity. Theprecise optimal value depends on the frequency chosen.Typically this is taken to be the mechanical resonancefrequency where we find that the optimal coupling leadsto an optimal back action noise

SFF,opt =!2

2S0xx[Ω]

=!2γ0

4x2ZPF

. (D100)

This makes sense because the higher the damping the lesssusceptible the oscillator is to back action forces. At thisoptimal coupling the total output noise spectral densityat frequency Ω referred to the position is simply twicethe vacuum value

Sxx,tot[Ω] = 2S0xx[Ω], (D101)

in agreement with Eq. (4.81). Evaluation of Eq. (D100)at the optimal coupling yields the graph shown inFig. (7). The background noise floor is due to thefrequency independent imprecision noise with value12 S0

xx[Ω]. The peak value at ω = Ω rises a factor of threeabove this background.

We derived the gain λ in Eq. (D95) by direct solutionof the equations of motion. With the results we have de-rived above, it is straightforward to show that the Kuboformula in Eq. (5.3) yields equivalent results. We havealready seen that the classical (i.e. symmetrized) correla-tions between the output signal I and the force F whichcouples to the position vanishes. However the Kubo for-mula evaluates the quantum (i.e. antisymmetric) corre-lations for the uncoupled system (A = g = 0). Hence wehave

χIF (t) = − i

!θ(t)⟨[−

(ξ(t− δt) + ξ†(t− δt)

),

2i!g

xZPF√

κ

(ξ(0)− ξ†(0)

)]⟩

0

, (D102)

where δt is a small (positive) time representing the delaybetween the time when the vacuum noise impinges onthe cavity and when the resulting outgoing wave reachesthe homodyne detector. (More precisely it also compen-sates for certain small retardation effects neglected in thelimit ω , κ used in several places in the above deriva-tions.) Using the fact that the commutator between thetwo quadratures of the vacuum noise is a delta function,Fourier transformation of the above yields (in the limit

ω δt , 1 the desired result

χIF [ω] = λ. (D103)

Similarly we readily find that the small retardationcauses the reverse gain to vanish. Hence all our resultsare consistent with the requirements needed to reach thestandard quantum limit.

Thus with this study of the specific case of an oscillatorparametrically coupled to a cavity, we have reproduced

79

all of the key results in Sec. (VI.D) derived from com-pletely general considerations of linear response theory.

4. Back-action free single-quadrature detection

We now provide details on the cavity single-quadraturedetection scheme discussed in Sec. VI.G.2. We again con-sider a high-Q cavity whose resonance frequency is mod-ulated by a high-Q mechanical oscillator with co-ordinatex (cf. Eqs. (D1) and (D64)). To use this system for am-plification of a single quadrature, we will consider thetypical case of a fast cavity (ωc % Ω), and take the “goodcavity” limit, where Ω % κ. As explained in the maintext, the crucial ingredient for single-quadrature detec-tion is to take an amplitude-modulated cavity drive de-scribed by the classical input field bin given in Eq. (6.70).As before (cf. Eq. (D4)), we may write the cavity anni-hilation operator a as the sum of a classical piece a(t)and a quantum piece d; only d is influenced by the me-chanical oscillator. a(t) is easily found from the classical(noise-free) equations of motion for the isolated cavity;making use of the conditions ωc % Ω % κ, we have

a(t) 4

√Nκ

2Ωcos (Ωt + δ) e−iωct (D104)

To proceed with our analysis, we work in an interac-tion picture with respect to the uncoupled cavity andoscillator Hamiltonians. Making standard rotating-waveapproximations, the Hamiltonian in the interaction pic-ture takes the simple form corresponding to Eq. (6.71b):

Hint = !A(d + d†

) (eiδ c + e−iδ c†

)

= !A(d + d†

) Xδ

xZPF, (D105)

where

A = A · ωc

√Nκ

4Ω, (D106)

and in the second line, we have made use of the definitionof the quadrature operators Xδ, Yδ given in Eqs. (6.64).The form of Hint was discussed heuristically in the maintext in terms of Raman processes where photons are re-moved from the classical drive bin and either up or downconverted to the cavity frequency via absorption or emis-sion of a mechanical phonon. Alternatively, we can thinkof the drive yielding a time-dependent cavity-oscillatorcoupling which “follows” the Xδ quadrature. Note thatwe made crucial of use of the good cavity limit (κ , Ω)to drop terms in Hint which oscillate at frequencies ±2Ω.These terms represent Raman sidebands which are awayfrom the cavity resonance by a distance ±2Ω. In the goodcavity limit, the density of photon states is negligible sofar off resonance and these processes are suppressed.

Similar to Eqs. (D39) and (D81), the Heisenberg equa-tions of motion (in the rotating frame) follow directly

from Hint and the dissipative terms in the total Hamil-tonian:

∂td = −κ

2d−

√κξ(t)eiωct − iA

(eiδ c + e−iδ c†

)(D107a)

∂tc = −γ0

2c−√γ0η(t)eiΩt − ie−iδ

[A

(d + d†

)− f(t)

]

(D107b)

As before, ξ(t) represents the unavoidable noise in thecavity drive, and η(t), γ0 are the noisy force and damp-ing resulting from an equilibrium bath coupled to themechanical oscillator. Note from Eq. (D107a) that as an-ticipated, the cavity is only driven by one quadrature ofthe oscillator’s motion. We have also included a drivingforce F (t) on the mechanical oscillator which has somenarrow bandwidth centered on the oscillator frequency;this force is parameterized as:

F (t) =2!

xZPFRe

[f(t)e−iΩte−iδ

](D108)

where f(t) is a complex function which is slowly varyingon the scale of an oscillator period.

The equations of motion are easily solved upon Fouriertransformation, resulting in:

Xδ[ω] = −xZPF · χM [ω]

[i (f∗[−ω]− f [ω]) (D109a)

+√

γ0

(eiδ η(ω + Ω) + e−iδ η†(ω − Ω)

)]

Yδ[ω] = ixZPF · χM [ω]

[(−i) (f [ω] + f∗[−ω])(D109b)

+√

γ0

(eiδ η(ω + Ω)− e−iδ η†(ω − Ω)

)

−2iAχc[ω]√

κ(ξ(ω + ωc) + ξ†(ω − ωc)

) ]

where the cavity and mechanical susceptibilities χc, χM

are defined in Eqs. (D43) and (D88).As anticipated, the detected quadrature Xδ is com-

pletely unaffected by the measurement: Eq. (D109a) isidentical to what we would have if there were no couplingbetween the oscillator and the cavity. In contrast, theconjugate quadrature Yδ experiences an extra stochasticforce due to the cavity: this is the measurement back-action.

Turning now to the output field from the cavity bout,we use the input-output relation Eq. (D37) to find in thelab (i.e. non-rotating) frame:

bout[ω] = bout[ω] +[−i(ω − ωc)− κ/2−i(ω − ωc) + κ/2

]ξ[ω]

−iA√

κ

xZPFχc[ω − ωc] · Xδ(ω − ωc)

(D110)

80

The first term on the RHS simply represents the outputfield from the cavity in the absence of the mechanicaloscillator and any fluctuations. It will yield sharp peaksat the two sidebands associated with the drive, ω = ωc±Ω. The second term on the RHS of Eq. (D110) representsthe reflected noise of the incident cavity drive. This noisewill play the role of the “intrinsic ouput noise” of thisamplifier.

Finally, the last term on the RHS of Eq. (D110) is theamplified signal: it is simply the amplified quadratureXδ of the oscillator. This term will result in a peak inthe output spectrum at the resonance frequency of thecavity, ωc. As there is no back-action on the measuredXδ quadrature, the added noise can be made arbitrarilysmall by simply increasing the drive strength N (andhence A).

APPENDIX E: Information Theory and Measurement Rate

Suppose that we are measuring the state of a qubitvia the phase shift ±θ0 from a one-sided cavity. LetI(t) be the homodyne signal integrated up to time t asin Eqs. (4.36-4.37). We would like to understand therelationship between the signal-to-noise ratio defined inEq. (4.38), and the rate at which information about thestate of the qubit is being gained. The probability distri-bution for I conditioned on the state of the qubit σ = ±1is

p(I|σ) =1√

2πSθθtexp

[−(I − σθ0t)2

2Sθθt

]. (E1)

Based on knowledge of this conditional distribution, wenow present two distinct but equivalent approaches togiving an information theoretic basis for the definition ofthe measurement rate.

1. Method I

Suppose we start with an initial qubit density matrix

ρ0 =

(12 00 1

2

). (E2)

After measuring for a time t, the new density matrixconditioned on the results of the measurement is

ρ1 =

(p+ 00 p−

)(E3)

where it will be convenient to parameterize the two prob-abilities by the polarization m ≡ Tr(σzρ1) by

p± =1±m

2. (E4)

The information gained by the measurement is the en-tropy loss25 of the qubit

I = Tr(ρ1 ln ρ1 − ρ0 ln ρ0). (E5)

We are interested in the initial rate of gain of informationat short times θ2

0t , Sθθ where m will be small. In thislimit we have

I ≈ m2

2. (E6)

We must now calculate m conditioned on the measure-ment result I

mI ≡∑

σ

σp(σ|I). (E7)

From Bayes theorem we can express this in terms ofp(I|σ), which is the quantity we know,

p(σ|I) =p(I|σ)p(σ)∑

σ′ p(I|σ′)p(σ′). (E8)

Using Eq. (E1) the polarization is easily evaluated

mI = tanh(

Iθ0

Sθθ

). (E9)

The information gain is thus

II =12

tanh2

(Iθ0

Sθθ

)≈ I2

2

(θ0

Sθθ

)2

(E10)

where the second equality is only valid for small |m|.Ensemble averaging this over all possible measurementresults yields the mean information gain at short times

I ≈ 12

θ20

Sθθt (E11)

which justifies the definition of the measurement rategiven in Eq. (4.39).

2. Method II

An alternative information theoretic derivation is toconsider the qubit plus measurement device to be a sig-naling channel. The two possible inputs to the channelare the two states of the qubit. The output of the chan-nel is the result of the measurement of I. By togglingthe qubit state back and forth, one can send informationthrough the signal channel to another party. The chan-nel is noisy because even for a fixed state of the qubit,

25 It is important to note that we use throughout here the physi-cist’s entropy with the natural logarithm rather than the log base2 which gives the information in units of bits.

81

the measured values of the signal I have intrinsic fluctu-ations. Shannon’s noisy channel coding theorem (Coverand Thomas, 1991) tells us the maximum rate at whichinformation can be reliably sent down the channel by tog-gling the state of the qubit and making measurements ofI. It is natural to take this rate as defining the measure-ment rate for our detector.

The reliable information gain by the receiver on a noisychannel is a quantity known as the ‘mutual information’of the communication channel (Clerk et al., 2003; Coverand Thomas, 1991)

R = −∫ +∞

−∞dI

p(I) ln p(I)−

∑

σ

p(σ) [p(I|σ) ln p(I|σ)]

(E12)The first term is the Shannon entropy in the signal Iwhen we do not know the input signal (the value of thequbit). The second term represents the entropy giventhat we do know the value of the qubit (averaged overthe two possible input values). Thus the first term issignal plus noise, the second is just the noise. Subtractingthe two gives the net information gain. Expanding thisexpression for short times yields

R =18

(〈I(t)〉+ − 〈I(t)〉−)2

Sθθt

=θ20

2Sθθt

= Γmeast (E13)

exactly the same result as Eq. (E11). (Here 〈I(t)〉σ is themean value of I given that the qubit is in state σ.)

APPENDIX F: Quantum Parametric Amplifiers

1. Non-degenerate case

a. Gain and added noise

The so-called non-degenerate parametric amplifier is alinear, phase preserving amplifier which can in principlereach the quantum limit (Gordon et al., 1963; Louisellet al., 1961; Mollow and Glauber, 1967a,b). One possi-ble realization (Yurke et al., 1989) is a cavity with threeinternal resonances that are coupled together by a non-linear element (such as a Josephson junction) whose sym-metry permits three-wave mixing. The three modes arecalled the pump, idler and signal and their energy levelstructure illustrated in Fig. 20 obeys

ωP = ωI + ωS. (F1)

It is important that the system be such that these arethe only three modes to which the cavity couples. Thesystem Hamiltonian is then

Hsys = !(ωPa†PaP + ωIa

†I aI + ωSa†SaS

)

+ i!η(a†Sa†I aP − aSaIa

†P

). (F2)

We have made the rotating wave approximation in thethree-wave mixing term, and without loss of generality,we take the non-linear susceptibility η to be real andpositive. The system is driven at the pump frequency andthe three wave mixing term permits a single pump photonto split into an idler photon and a signal photon. Thisprocess is stimulated by signal photons already presentand leads to gain. A typical mode of operation wouldbe the negative resistance reflection mode (Yurke et al.,1989) in which the input signal is reflected from a non-linear cavity and the reflected beam extracted using acirculator. Since input and output travel on the sameline, the input and output impedances of the amplifierare equal.

The non-linear EOMs become tractable if we assumethe pump has large amplitude and can be treated classi-cally by making the substitution

aP = ψPe−iωPt = ψPe−i(ωI+ωS)t, (F3)

where without loss of generality we take ψP to be realand positive. We note here the important point that ifthis approximation is not valid, then our amplifier wouldin any case not be the linear amplifier which we seek.With this approximation we can hereafter forget aboutthe dynamics of the pump degree of freedom and dealwith the reduced system Hamiltonian

Hsys = !(ωIa

†I aI + ωSa†SaS

)+ i!λ

(a†Sa†Ie

−i(ωI+ωS)t − aSaIe+i(ωI+ωS)t

)(F4)

where λ ≡ ηψP. Transforming to the interaction repre-sentation we are left with the following time-independentquadratic Hamiltonian for the system

Vsys = i!λ(a†Sa†I − aSaI

)(F5)

To get some intuitive understanding of the physics, letus temporarily ignore the damping of the cavity modesdue to their coupling to the external baths. We now havea pair of coupled EOMs for the two modes

˙aS = +λa†I˙a†I = +λaS (F6)

for which the solutions are

aS(t) = cosh(λt)aS(0) + sinh(λt)a†I (0)

a†I (t) = sinh(λt) aS(0) + cosh(λt) a†I (0) (F7)

We see that the amplitude in the signal channel growsexponentially in time and that the effect of the timeevolution is to perform a simple unitary transformationwhich mixes aS with a†I in such a way as to preserve thecommutation relations. We now see the close connec-tion with the form found from very general arguments inEqs. (6.7a-6.12).

82

With this initial result in hand, we are ready to con-sider the presence of finite damping, necessary to feedthe input signal into the amplifier, which will cut off theexponential growth and give a fixed amplitude gain. Fol-lowing our standard Markovian approximation for boththe idler and signal baths, we obtain in analogy with theprevious results the following EOMs in the interactionrepresentation

˙aS = −κS

2aS + λa†I −

√κS bS,in

˙a†I = −κI

2a†I + λ aS −

√κI b†I,in (F8)

where κS and κI are the respective damping rates of thecavity signal mode and the idler mode.

Let us fix our attention on signals inside a frequencywindow δω centered on ωS (hence zero frequency in theinteraction representation). For simplicity, we will firstconsider the case where the signal bandwidth δω is al-most infinitely narrow (i.e. much much smaller than thedamping rate of the cavity modes). It then suffices tofind the steady state solution of these EOMs

aS =2λ

κSa†I −

2√

κSbS,in (F9)

a†I =2λ

κIaS −

2√

κIb†I,in (F10)

from which the output signal is readily computed usingEq. (D37)

bS,out =Q2 + 1Q2 − 1

bS,in +2Q

Q2 − 1b†I,in (F11)

where Q ≡ 2λ√κIκS

is proportional to the pump amplitudeand inversely proportional to the cavity decay rates. Wehave to require Q2 < 1 to make sure that the parametricamplifier does not settle into self-sustained oscillations,i.e. it works below threshold. Under that condition, wecan define the voltage gain

√G0 to be

−√

G0 = (Q2 + 1)/(Q2 − 1), (F12)

such that

bS,out = −√

G0 bS,in −√

G0 − 1 b†I,in (F13)

This is precisely of the Haus-Caves form Eq. (6.12)and shows that the non-degenerate parametric amplifierreaches the quantum limit for minimum added noise. Inthe limit of large gain the output noise (referred to theinput) for a vacuum input signal is precisely doubled.

b. Bandwidth-gain tradeoff

The above results neglected the finite bandwidth δωof the input signal to the amplifier. The gain G0 given

ωP

ωI

ωS

FIG. 20 (Color online) Energy level scheme of the non-degenerate (phase preserving) parametric oscillator.

in Eq. (F12) is only the gain at precisely the mean sig-nal frequency ωS; for a finite bandwidth, we also need tounderstand how the power gain varies as a function offrequency over the entire signal bandwidth. As we willsee, a parametric amplifier suffers from the fact that asone increases the overall magnitude of the gain at thecenter frequency ωS (e.g. by increasing the pump ampli-tude), one simultaneously narrows the frequency rangeover which the gain is appreciable. Heuristically, this isbecause parametric amplification involves increasing thequality factor of the signal mode resonance. This in-crease in quality factor leads to amplification, but it alsoreduces the bandwidth over which aS can respond to theinput signal bS,in.

To deal with a finite signal bandwidth, one simplyFourier transforms Eqs. (F8). The resulting equationsare easily solved and substituted into Eq. (D37), result-ing in a frequency-dependent generalization of the input-output relation given in Eq. (F13):

bS,out[ω] = −g[ω]bS,in[ω]− g′[ω]b†I,in[ω](F14)

Here, g[ω] is the frequency-dependent voltage gain of theamplifier, and g′[ω] satisfies |g′[ω]|2 = |g[ω]|2 − 1. In therelevant limit where G0 = |g[0]|2 % 1 (i.e. large gain atthe signal frequency), one has to an excellent approxima-tion:

g[ω] =

√G0 − i

(κS−κIκI+κS

)(ω/D)

1− i(ω/D), (F15)

with

D =1√G0

κSκI

κS + κI. (F16)

As always, we work in an interaction picture where thesignal frequency has been shifted to zero. D representsthe effective operating bandwidth of the amplifier. Com-ponents of the signal with frequencies (in the rotatingframe) |ω|, D are strongly amplified, while componentswith frequencies |ω| % D are not amplified at all, but

83

can in fact be slightly attenuated. As we already antici-pated, the amplification bandwidth D becomes progres-sively smaller as the pump power and G0 are increased,with the product

√G0D remaining constant. In a para-

metric amplifier increasing the gain via increasing thepump strength comes with a price: the effective operat-ing bandwidth is reduced.

c. Effective temperature

Returning to the behaviour of the paramp at the signalfrequency, we note that Eq. (F13) implies that even forvacuum input to both the signal and idler ports of theparamp, the output will contain a real photon flux. Toquantify this in a simple way, we make use of the tempo-ral modes introduced in our discussion of the windowedFourier transform (c.f. Eq. (C18)). We thus define theinput signal temporal modes as:

BS,in,j =1√∆t

∫ (j+1)∆t

j∆tdτ bS,in(τ); (F17)

the temporal modes BS,out,j and BI,in,j are defined anal-ogously.

With these definitions, we find that the output modewill have a real occupancy even if the input mode isempty:

nS,out = 〈0|B†S,out,jBS,out,j |0〉

= G0〈0|B†S,in,jBS,in,j |0〉+

(G0 − 1)〈0|BI,in,jB†I,in,j |0〉

= G0 − 1. (F18)

The dimensionless mode occupancy nS,out is best thoughtof as a photon flux per unit bandwidth (c.f. Eq.(C26)).This photon flux is equivalent to the photon flux thatwould appear in equilibrium at the very high effectivetemperature (assuming large gain G0)

Teff ≈ !ωSG0. (F19)

Thus, as discussed in Sec. VI.D.4, a high gain amplifiermust have associated with it a large effective temperaturescale. Referring this total output noise back to the input,we have (in the limit G0 % 1):

Teff

G0=

!ωS

2+

!ωS

2=

!ωS

2+ TN. (F20)

This corresponds to the half photon of vacuum noise as-sociated with the signal source, plus the added noise ofa half photon of our phase preserving amplifier (i.e. thenoise temperature TN is equal to its quantum limitedvalue). Here, the added noise is simply the vacuum noiseassociated with the idler port.

The above argument is merely suggestive that the out-put noise looks like an effective temperature. In fact

it is possible to show that the photon number distribu-tion is in the output is precisely that of a Bose-Einsteindistribution at temperature Teff . From Eq. (F2) we seethat the action of the paramp is to destroy a pump pho-ton and create a pair of new photons, one in the signalchannel and one in the idler channel. Using the SU(1,1)symmetry of the quadratic hamiltonian in Eq. (F5) itis possible to show that, for vacuum input, the outputof the paramp is a so-called ‘two-mode squeezed state’of the form (Caves and Schumaker, 1985; Gerry, 1985;Knight and Buzek, 2004)

|Ψout〉 = Z−1/2eαb†Sb†I |0〉 (F21)

where α is a constant related to the gain and, to simplifythe notation, we have dropped the ‘out’ labels on theoperators. The normalization constant Z can be workedout by expanding the exponential and using

(b†S)n|0, 0〉 =√

n!|n, 0〉 (F22)

to obtain

|Ψout〉 = Z−1/2∞∑

n=0

αn|n, n〉 (F23)

and hence

Z =1

1− |α|2 (F24)

so the state is normalizable only for |α|2 < 1.Because this output is obtained by unitary evolution

from the vacuum input state, the output state is a purestate with zero entropy. In light of this, it is interestingto consider the reduced density matrix obtain by tracingover the idler mode. The pure state density matrix is:

ρ = |Ψout〉 〈Ψout|

=∞∑

m,n=0

|n, n〉 αnα∗m

Z〈m, m| (F25)

If we now trace over the idler mode we are left with thereduced matrix for the signal channel

ρS = TrIdler ρ

=∞∑

nS=0

|nS〉|α|2nS

Z〈nS|

=1Z

e−β!ωSa†SaS (F26)

which is a pure thermal equilibrium distribution with ef-fective Boltzmann factor

e−β!ωS = |α|2 < 1. (F27)

The effective temperature can be obtained from the re-quirement that the signal mode occupancy is G0 − 1

1eβ!ωS − 1

= G0 − 1 (F28)

84

which in the limit of large gain reduces to Eq. (F19).This appearance of finite entropy in a subsystem even

when the full system is in a pure state is a purely quan-tum effect. Classically the entropy of a composite systemis at least as large as the entropy of any of its compo-nents. Entanglement among the components allows thislower bound on the entropy to be violated in a quantumsystem.26 In this case the two-mode squeezed state hasstrong entanglement between the signal and idler chan-nels (since their photon numbers are fluctuating identi-cally).

2. Degenerate case

In the degenerate parametric amplifier the idler modeis eliminated, and the non-linearity converts a singlepump photon into two signal photons at frequency ωS =ωP/2. This amplifier is not phase preserving (referred toin the literature by the unfortunate name ‘phase sensi-tive’). Rather it amplifies one quadrature of the signaland attenuates the other. As a result, it is not necessaryto add extra noise to preserve the commutation relations.

The system Hamiltonian is

Hsys = !(ωPa†PaP + ωSa†SaS

)

+ i!η(a†Sa†SaP − aSaSa†P

). (F29)

Treating the pump classically as before, the analog ofEq. (F5) is

Vsys = i!λ

2

(a†Sa†S − aSaS

), (F30)

where λ/2 = ηψP , and the analog of Eq. (F8) is:

˙aS = −κS

2aS + λa†S −

√κS bS,in (F31)

It is useful to define two quadrature modes

xS =1√2

(a†S + aS

)(F32)

yS =i√2

(a†S − aS

)(F33)

which obey [xS, yS] = i. We can define quadrature oper-ators XS,in/out, YS,in/out corresponding to the input andoutput fields in a completely analogous manner.

The steady state solution of Eq. (F31) for the outputfields becomes

XS,out =√

GXS,in (F34)

YS,out =1√G

YS,in, (F35)

26 This paradox has prompted Charles Bennett to remark that aclassical house is at least as dirty as its dirtiest room, but aquantum house can be dirty in every room and still perfectlyclean over all.

where the number gain G is given by

G =[λ + κS/2λ− κS/2

]2

. (F36)

Thus we see that one quadrature is amplified and theother is correspondingly attenuated. The commutationrelation

[XS,out(t), YS,out(t′)] = iδ(t− t′) (F37)

is independent of G without the necessity of adding fur-ther quantum noise.

APPENDIX G: Number Phase Uncertainty

In this appendix, we briefly review the number-phaseuncertainty relation, and from it we derive the relation-ship between the spectral densities describing the photonnumber fluctuations and the phase fluctuations. Con-sider a coherent state labeled by its classical amplitudeα

|α〉 = exp− |α|

2

2

expαa†|0〉. (G1)

This is an eigenstate of the destruction operator

a|α〉 = α|α〉. (G2)

It is convenient to make the unitary displacement trans-formation which maps the coherent state onto a new vac-uum state and the destruction operator onto

a = α + d (G3)

where d annihilates the new vacuum. Then we have

N = 〈N〉 = 〈0|(α∗ + d†)(α + d)|0〉 = |α|2, (G4)

and

(∆N)2 = 〈(N − N)2〉 = |α|2〈0|dd†|0〉 = N . (G5)

Now define the two quadrature amplitudes

X =1√2(a + a†) (G6)

Y =i√2(a† − a). (G7)

Each of these amplitudes can be measured in a homodyneexperiment, for example using the Mach-Zehnder inter-ferometer described in Appendix (H). For convenience,let us take α to be real and positive. Then

〈X〉 =√

2α (G8)

and

〈Y 〉 = 0. (G9)

85

If the phase of this wave undergoes a small modulationdue for example to weak parametric coupling to a qubitthen one can estimate the phase by

〈θ〉 =〈Y 〉〈X〉

. (G10)

The uncertainty will be

(∆θ)2 =〈Y 2〉

(〈X〉)2=

12 〈0|dd†|0〉

2N=

14N

. (G11)

Thus using Eq. (G5) we arrive at the fundamental quan-tum uncertainty relation

∆θ∆N =12. (G12)

Using the input-output theory described in Ap-pendix D we can restate the results above in terms of

noise spectral densities. Let the amplitude of the fieldcoming in to the homodyne detector be

bin = bin + ξ(t) (G13)

where ξ(t) is the vacuum noise obeying

[ξ(t), ξ†(t′)] = δ(t− t′). (G14)

We are using a flux normalization for the field operatorsso

N = 〈b†inbin〉 = |bin|2 (G15)

and

〈N(t)N(0)〉 − N2

= 〈0|(b∗in + ξ†(t))(bin + ξ(t))(b∗in + ξ†(0))(bin + ξ(0))|0〉 −∣∣bin

∣∣4 = Nδ(t). (G16)

From this it follows that the shot noise spectral densityis

SNN = N . (G17)

Similarly the phase can be estimated from the quadra-ture operator

θ =i(b†in − bin)〈b†in + bin〉

= 〈θ〉+ i(ξ† − ξ)

2bin(G18)

which has noise correlator

〈δθ(t)δθ(0)〉 =1

4Nδ(t) (G19)

corresponding to the phase imprecision spectral density

Sθθ =1

4N. (G20)

We thus arrive at the fundamental quantum limit relation

√SθθSNN =

12. (G21)

APPENDIX H: Mach Zehnder Interferometer as aQuantum Limited Detector

In this appendix, we examine the properties of a pro-totypical interferometric detector, the Mach Zehnder in-terferometer (MZI), see Fig. 21, and ask whether this

qubit phaseintensity

FIG. 21 (Color online) Mach-Zehnder interferometer use asa qubit state detector.

system can reach the quantum limit. For simplicity, weconsider using such an interferometer to do QND mea-surement of a qubit by having the qubit influence one armof the interferometer. As follows from the discussion ofSec. IV.B , the conclusions we reach will also apply tothe case where the interferometer is used to do positiondetection of a mechanical oscillator. We show that if oneuses a Fock state input, one reaches the quantum limitas long as the second beam splitter is symmetric. In con-trast, a coherent state input reaches the quantum limitonly in the extreme case of an unbalanced interferometer,

86

where there is a vanishing amplitude in the arm contain-ing the qubit; in the perfectly symmetric case, it missesthe quantum limit by a factor of two. We demonstratethat in both the Fock state and coherent state cases, anydeparture from the quantum limit can be directly tiedto unused information available in the phases of the out-put of the interferometer. The differences between theFock state and coherent state case can be ascribed tothe different types of phase information available at theoutput in each case. Finally, we note that if one wereto couple the qubit (or oscillator) to both arms of theinterferometer, then as long as the second beam-splitteris symmetric, one will reach the quantum limit with acoherent state input; for such a coupling, there is neverany wasted phase information.

Note that the standard treatment of measurement withan interferometer (Caves, 1980, 1981) does not explicitlydistinguish the Fock state and coherent state cases Weshow that there are important differences between thetwo cases. In the case of a Fock state input, one can di-rectly tie the interferometer back-action to the partitionnoise at the first beam splitter. In contrast, in the coher-ent state case, there is no partition noise per se; instead,the intrinsic noise in the input state (i.e. an uncertaintyin number) serves to constrain the measurement.

1. Basic description of the interferometer

The MZI detector consists of two beam splitters, eachof which is described by a 2×2 unitary scattering matrix.The output of the first beam splitter is attached to theinput of the second via two “arms”; the phase shift inone of the arms is determined by the state of the qubitwe wish to measure. Information on this phase shift canthen be determined by looking at the difference betweenthe intensities at MZI’s two output ports.

The scattering matrix S of the MZI determines the re-lation between the annihilation operators describing theMZI input and output ports:

(b1

b2

)= S ·

(a1

a2

)(H1)

where a1 and a2 are annihilation operators for the twoinput ports, and b1 and b2 are annihilation operators forthe two output ports. S is given by:

S = sB · sarms · sA (H2)

where sA, sB are the scattering matrices of the beamsplitters, and sarms is the scattering matrix of the arms.We have:

sA =

( √1− TA i

√TA

i√

TA√

1− TA

)(H3)

sB =

( √1− TB i

√TB

i√

TB√

1− TB

)(H4)

sarms =

(ieiθ 00 1

)(H5)

Here, TA and TB are real numbers between 0 and 1 whichdescribe the transmission probabilities of the two beamsplitters. We have assumed for simplicity that each beamsplitter has time-reversal symmetry and parity symme-try (i.e. the symmetry operation which exchanges “1”and “2” ports). θ = ±θ0 is the phase shift provided bythe qubit; the factor of i in sarms is chosen to yield amaximum sensitivity to the phase θ0 in a measurementof the output intensities.

Letting RA = 1−TA, RB = 1−TB , it will be convenientto parameterize S as:

S = eiη

( √1− T eiβ i

√T e−iφ

i√

T eiφ√

1− T e−iβ

)(H6)

where

e2iη = ieiθ (H7)

T = TARB + TBRA − 2√

TARATBRB sin θ (H8)

T determines the intensity at the two output ports, whileη, β and φ describe the phases of the MZI output. Notethat all of these phases depend on θ, although we do notlist the explicit dependence here.

In what follows, we will use the scattering matrix todetermine the output of the interferometer for differentpossible input states. For simplicity (and as is standardin the quantum optics literature), we will not describethe input to the interferometer using wavepackets, butrather as extended waves. A more appropriate analysisusing wavepackets reaches identical conclusions as theones we present below.

2. Fock state input

We first consider the case where the input of the MZIis prepared in a state having exactly N photons in inputport 1, and zero photons in the input port 2:

|ψin〉 = |N〉a1 |0〉a2 ≡1√N !

(a†1

)N|Ω〉 (H9)

where |Ω〉 is the vacuum. Using Eq. (H1) to convert inputoperators to output operators, and using the binomialtheorem, we have:

87

|ψ±out〉 =

1√N !

([S−1±

]∗11

b†1 +[S−1±

]∗12

b†2

)N|Ω〉

= eiN(η±+φ±)N∑

m=0

eim(β±−φ±)

√√√√(

N

m

)×

√(1− T±)m(iT±)N−m|m〉b1|N −m〉b2 (H10)

Here, ± denotes the two qubit eigenstates, and the twocorresponding phase shifts θ = ±θ0. We see that thestates |ψ±

out〉 yield classical, binomial photon numberstatistics at the output port; this is the direct result ofthe partitioning of photons at the two beam splitters, andthe resulting noise (i.e. uncertainty in the photon num-ber at one of the output ports) is properly termed “shotnoise”.

Note that the phase appearing in the sum over m inEq. (H10), λ± = β± − φ±, is the relative phase differ-ence between the output in port 1 versus port 2. Unlikethe global phase of the wavefunction |ψout〉, this phasedifference is in principle measurable, as the number dif-ference between the two output ports is not fixed. Tolowest order in |θ|, λ± is given by:

λ± = − arctan( √

TARA

(1− 2TA)√

TBRB

)+ (H11)

∓(

(1− 2TB)TARA

(TARB + TBRA) (1− TARB + TBRA)

)θ0

Note that only in the case of a symmetric second beamsplitter (i.e. TB = RB = 1/2) is the phase λ± indepen-dent of the state of the qubit.

a. Measurement rate

The quantity most easily measured at the output ofthe MZI is Ndiff , the difference between the photon num-bers at the two outputs: Ndiff ≡ b†1b1 − b†2b2. Using thedefinition of |ψ±

out〉 above and taking |θ| = θ0 , 1 (i.e. aweak measurement), we have:

〈Ndiff〉± = T±N − (1− T±)N

4 〈Ndiff〉∣∣∣θ=0

± 2NdT

dθθ0

= 〈Ndiff〉∣∣∣θ=0

∓ 4N√

TARATBRBθ0(H12)

The uncertainty (or noise) in the measured quantityNdiff in the state |ψ±

out〉 is just given by the usual binomialexpression; to lowest order in θ it is independent of thequbit state, and is given by:

〈〈N2diff〉〉 ≡ 〈N2

diff〉 − 〈Ndiff〉2

= 4×(T0(1− T0)

)N (H13)

where T0 = T∣∣∣θ=0

.

We can now calculate the measurement rate (see Ap-pendix E) for this setup by asking how long it will taketo resolve the difference between the two possible qubitstates, i.e. to reach the condition:

|〈Ndiff〉+ − 〈Ndiff〉−| ≥ 2√

2 ·√〈〈N2

diff〉〉 (H14)

The numerical prefactor on the rhs has been chosen tomake this definition coincide with our previous definition.In our system, this condition translates into a minimumnumber of photons that are needed in the input state.We find simply:

N ≥ Nmeas =2T0(1− T0)(

dTdθ θ0

)2 (H15)

Note that if one multiplies by eV/h, this expression isidentical to the measurement rate of a quantum point-contact detector (Aleiner et al., 1997; Gurvitz, 1997).Also note that for the Fock state input considered here,an equivalent expression for Nmeas is obtained if one onlymeasures the intensity at one of the output ports. Thisfollows from the fact that the total number of photonsis fixed and hence the fluctuations of the two ports areperfectly anticorrelated. Measuring one determines theother.

b. Dephasing rate

Similar to our analysis in Sec. IV.B, we may calculatethe measurement induced dephasing of the qubit from theoverlap of the two scattering states |ψ±

out〉 (cf. Eq. (4.48));it is this overlap which directly determines how the off-diagonal elements of the qubit’s reduced density matrixdecay. One generally expects an exponential decay:

|〈ψ−|ψ+〉| ∝ exp (−N/Nϕ) (H16)

where Nϕ is the number of input photons needed to de-phase the qubit (i.e. it is analogous to a dephasing timeτϕ).

In our case, we may directly calculate this overlap us-ing Eq. (H10). Letting R = 1− T , we find:

88

|〈ψ−|ψ+〉| =

∣∣∣∣∣∣

(ei(λ+−λ−)

√T−T+ +

√R−R+

)N∣∣∣∣∣∣

(H17)

Expanding λ, T and R in θ, one has:

|〈ψ−|ψ+〉| =

∣∣∣∣∣∣

(1 + 2T

[i

(θ0

dλ

dθ

)−

(θ0

dλ

dθ

)2]−

(θ0

dT

dθ

)21

2T R+ O(θ4

0)

)N∣∣∣∣∣∣

= exp

−N

(

θ0dT

dθ

)21

2T R+ 2T0R0

(θ0

dλ

dθ

)2

+ O(θ40)

(H18)

We thus obtain an exponential decay of the overlap as expected, with a dephasing “rate” 1/Nϕ given by:

1Nϕ

= (θ0)2

(

dT

dθ

)21

2T0R0

+ 2T0R0

(dλ

dθ

)2

= 2T1R1θ20 (H19)

Eq. (H19) for the dephasing rate has the form we wouldexpect from a simple heuristic argument. Each photontraveling down the arm containing the qubit advancesthe qubit’s phase by a deterministic amount 2θ0. How-ever, due to the partition noise at the first beam splitter,there is an uncertainty in the number of photons Nqb

which pass by the qubit, which directly translates intoan uncertainty in the qubit’s phase φqb:

〈〈φ2qb〉〉 = (2θ0)2〈〈N2

qb〉〉 (H20)

= (2θ0)2 (TARAN) (H21)

Treating this noise as Gaussian, this implies that the ex-ponential of the qubit’s phase decays as:

|〈eiφqb〉| = exp(−1

2〈〈φ2

qb〉〉)

= exp(−2θ2

0TARAN),

(H22)a result which is in exact agreement with Eq. (H19).

Turning to the question of the quantum limit (i.e. doesNϕ coincide with Nmeas?), it is more convenient to usethe first expression for the dephasing rate, Eq. (H19). Asthe first term in this equation corresponds exactly to themeasurement rate (cf. Eq. (H15)), and the second term ispositive definite, we find that Nϕ ≤ Nmeas: it takes longerto measure than to dephase the qubit, meaning that theMZI detector with a Fock state input generically fails toreach the quantum limit. As is clear from the calculation,the origin of this failure is the unused or “wasted” infor-mation on the qubit’s state available in the relative phaseλ; this information could be extracted from the output ofthe MZI via a suitable interference experiment. In fact,the second term in Eq. (H19) is precisely the measure-ment rate associated with trying to determine the qubit

state from the relative phase λ. From the number-phaseuncertainty relation, we can find the effective uncertaintyin the phase λ without specifying a particular measure-ment scheme; we have to zeroth order in θ:

〈〈λ2〉〉 =1

4〈〈N2diff〉〉

=1

4T0R0N(H23)

The measurement rate criterion of Eq. (H14) takes theform:

2θ0

(dλ

dθ

)≥ 2

√2√〈〈λ2〉〉 (H24)

Combining equations and solving for the minimal N , wefind:

N ≥ Nmeas,λ =

(2T0R0θ

20

(dλ

dθ

)2)−1

(H25)

Thus, the second term in Eq. (H19) in indeed 1/Nmeas,λ,the measurement rate that would result if we attemptedto measure λ.

While the Fock state input fails to reach the quan-tum limit for a generic choice of parameters, there is animportant set of cases where it does achieve ideality. Itis clear from above that reaching the quantum limit re-quires there to be no information in the phase λ. Turningto Eq. (H11) for this phase, we see that this necessar-ily requires that the second beam splitter be completelysymmetric: T2 = 1/2. Thus, for a symmetric secondbeam splitter and a Fock-state input, one reaches thequantum limit irrespective of the asymmetry in the firstbeam splitter.

89

As a final comment, note that the structure ofEq. (H19) is identical to that for the measurement in-duced dephasing rate of a qubit by a quantum pointcontact detector if one simply multiplies by the inversetimescale eV/h (Gurvitz, 1997).

3. Coherent state input

We now repeat the analysis of the previous section,treating the case where the input of the MZI consists of

a coherent state in the input port ”1”:

|ψin〉 = |α〉a1|0〉a2 = e−|α|2/2 exp

(αa†1

)|Ω〉 (H26)

The magnitude |α|2 corresponds to the average photonnumber in the input: |α|2 = N .

The output state of the MZI is again given by simplyusing the scattering matrix S and the transformation ruleof Eq. (H1) on Eq. (H26). We find:

|ψ±out〉 = e−|α|

2/2 exp(α

([S−1±

]∗11

b†1 +[S−1±

]∗12

b†2

))|Ω〉

= e−|α|2/2 exp

(√Rei(η+β)α · b†1

)exp

(√T ei(η+φ)α · b†2

)|Ω〉

=∣∣∣√

Rei(η+β)α⟩

b1

∣∣∣√

T ei(η+φ)α⟩

b2(H27)

There are some immediate differences worth notingfrom the Fock state case:

• In the coherent state case, the two output portsare not entangled; equivalently, there is no par-tition noise (i.e. shot noise) in the coherent statecase. This follows from the fact that a coherentstate incident onto a beam splitter yields two inde-pendent coherent states (of diminished amplitude)whose number fluctuations are completely uncorre-lated.

• As the total number of photons in the output ofthe interferometer is not fixed, the absolute phaseof the the output in each port can in principle bedetected

a. Measurement rate

We again take the measured quantity at the output ofthe MZI to be Ndiff , the difference in intensities betweenthe two output ports. Unlike the Fock state case, it isimportant the we use both ports; the measurement ratewill be smaller if we only choose to use one.

The average values of the Ndiff in the two qubit states,〈Ndiff〉± is again given by Eq. (H12), with N = |α|2. Thenoise in Ndiff is however different, as the coherent stateshave Poissonian statistics as opposed to binomial. Onefinds to zeroth order in θ0 that the noise in Ndiff is iden-tical to that in the input state |ψin〉, and is completely

independent of the properties of the beam splitter:

〈〈N2diff〉〉 = |α|2 = N (H28)

Thus, unlike the case of the Fock state, the noise in theoutput does not result from scattering in the MZI (i.e.,partitioning of the photons between the two arms by thefirst beam splitter); rather, it simply reflects the intrinsicnoise already present at the input. Note that 〈〈N2

diff〉〉 inthe Fock state case is always less than or equal to thatfound above.

Using the same criteria as in the previous section(cf. Eq. (H14)), we find that the smallest value of Nneeded for a measurement is given by:

N ≥ Nmeas =1

2(

dTdθ θ0

)2 (H29)

We see that for the same dT /dθ, the measurement ratefor the coherent state input is less than or equal to thatin the Fock state case– there is more noise in the coherentstate case, hence the measurement proceeds more slowly.

b. Measurement induced dephasing

We again wish to calculate the wavefunction overlapin Eq. (H16). Using |α|2 = N , we have simply:

|〈ψ−|ψ+〉| =∣∣∣∣exp

(−N

[1−

√R−R+ei(η+−η−)ei(β+−β−) −

√T−T+ei(η+−η−)ei(φ+−φ−)

])∣∣∣∣ (H30)

90

Expanding terms inside the exponential to order θ20, we find for the dephasing “rate” 1/Nϕ:

1Nϕ

= (θ0)2

(

dT

dθ

)21

2T R+ 2T0

(dλT

dθ

)2

+ 2R0

(dλR

dθ

)2

(H31)

where the phases λR and λT are (respectively) the totalphases of the output in port 1 and port 2:

λR = η + β (H32)λT = η + φ (H33)

We can evaluate this expression to find the simple result:

1Nϕ

= 2RAθ20 (H34)

Note that the dephasing rate here is not equal to thatin the Fock state case (cf. Eq. (H19)), and thus can-not be interpreted in terms of partition noise at the firstbeam splitter. This is not surprising– as already dis-cussed, there is no partition noise in the case of a coher-ent state input. Eq. (H34) can instead be simply inter-preted in terms of the number fluctuations of the inputstate. Heuristically, each photon passing the qubit ad-vances its phase by a deterministic amount 2θ0. Similarto the Fock state case, the uncertainty in the number ofphotons passing the qubit, Nqb, directly translates into aphase uncertainty in the qubit (i.e. dephasing). Unlikethe Fock state case, the fluctuations of Nqb are not givenby partition noise, but simply by Poisson statistics:

〈〈N2qb〉〉 = 〈Nqb〉 = RAN (H35)

This leads to an expression for |〈eiφqb〉| which is in exactagreement with Eq. (H34).

Turning now to the question of the quantum limit, itis useful to work with Eq. (H31) for the dephasing rate.Note the close analogy to the corresponding expressionfor Nϕ for the Fock state case, Eq. (H19). The first termdescribes information available in the modulation of T bythe qubit, and corresponds precisely to the measurementrate for the Fock state input (cf. Eq. (H15)). In con-trast, the second and third terms are precisely the mea-surement rates associated with detecting the phase at,respectively, the “1” and “2” output ports. One notesdirectly from Eq. (H31) that in general, Nmeas > Nϕ:the MZI with a coherent state input does not genericallyreach the quantum limit. As usual, this is due to wastedinformation in the detector. Interestingly, even if there isno wasted phase information (i.e λT = λR = 0), one stilldoes not necessarily reach the quantum limit. This is dueto the number uncertainty of the input state: the MZIoutput corresponds to having effectively averaged overinput states with different numbers of photons, a proce-dure which in general results in a loss of information (seeClerk et al. (2003) for a discussion of this point).

For simplicity, consider now the case that was idealfor a Fock state input: an MZI with a symmetric secondbeam splitter, i.e. TB = 1/2. In this case, T0 = 1/2, andthe measurement rate is given by its maximal value:

1Nmeas

= 2TARA(θ0)2 (H36)

This is identical to the expression in the Fock state case,even though the origin is different (i.e. partition noiseversus intrinsic noise in the input state).

For a symmetric second beam splitter, the three termsin the expression for the dephasing rate reduce to:

(θ0)2(

dT

dθ

)21

2T R=

1Nmeas

(H37)

2θ20T0

(dλT

dθ

)2

= R2Aθ2

0 (H38)

2θ20R0

(dλR

dθ

)2

= R2Aθ2

0 (H39)

Combining these to evaluate Eq. (H31), we find again:

1Nϕ

=1

Nmeas+ 2R2

A(θ0)2 = 2RAθ20 (H40)

This yields an alternate interpretation for Eq. (H34) asa sum of measurement rates. It also demonstrates thateven with a symmetric second beam splitter, the MZIdetector with a coherent state input fails to reach thequantum limit: in general, Nmeas > Nϕ. In the com-pletely symmetric case, where TA = 1/2 also, the quan-tum limit is missed by a factor of two. As is clear fromthe calculation, the reason for this failure is unused infor-mation on the state of the qubit available in the phases ofthe two output ports. To approach the quantum limit,one needs to suppress the wasted information in thesephases. From Eq. (H39), we see that this requires one totake the limit RA → 0. In this limit, the amplitude ofthe wave in the arm containing the qubit is vanishinglysmall; as a result, the overall measurement rate is alsosuppressed. Note the sharp difference between the casesof a Fock state input and coherent state input: in the for-mer case, one could reach the quantum limit by simplyhaving TA = 1/2, while in the latter case, one needs toadditionally take the limit of a vanishing measurementrate.

91

4. Symmetric coupling

Our conclusions for the MZI and a coherent state in-put may seem quite depressing: in order to reach thequantum limit, one must have a highly asymmetric firstbeam splitter, with the result that the measurement rateis greatly reduced compared to its optimal value. As afinal concluding postscript on our discussion of the MZI,we point out that there is another, more practical way tohave the MZI with coherent state input reach the quan-tum limit. Instead of using a highly asymmetric firstbeam splitter, one can couple the qubit to both arms ofthe interferometer, such that the scattering matrix forthe arms is now:

sarms =

(ieiθ 00 e−iθ

), (H41)

where again, θ = ±θ0 depending on the state of the qubit.An analysis identical to that presented above now showsthat one reaches the quantum limit for perfectly symmet-ric beam splitters, meaning that one also has an optimalmeasurement rate. For this coupling, there is no longerany wasted phase information in the output of the in-terferometer. While this coupling might be difficult toimagine in the qubit case, it can be realized in the caseof position detection (see Fig. 21).

APPENDIX I: Using feedback to reach the quantum limit

In Sec. (VII.B), we demonstrated that any two portamplifier whose scattering matrix has s11 = s22 = s12 =0 will fail to reach quantum limit when used as a weaklycoupled op-amp; at best, it will miss optimizing the quan-tum noise constraint of Eq. (6.60) by a factor of two.Reaching the quantum limit thus requires at least oneof s11, s22 and s12 to be non-zero. In this subsection,we demonstrate how this may be done. We show thatby introducing a form of negative feedback to the “min-imal” amplifier of the previous subsection, one can takeadvantage of noise correlations to reduce the back-actioncurrent noise SII by a factor of two. As a result, one isable to reach the weak-coupling (i.e. op-amp) quantumlimit. Note that quantum amplifiers with feedback arealso treated in Courty et al. (1999); Grassia (1998).

On a heuristic level, we can understand the need foreither reflections or reverse gain to reach the quantumlimit. A problem with the “minimal” amplifier of thelast subsection was that its input impedance was too lowin comparison to its noise impedance ZN ∼ Za. Fromgeneral expression for the input impedance, Eq. (7.7d),we see that having non-zero reverse gain (i.e. s12 *= 0)and/or non-zero reflections (i.e. s11 *= 0 and/or s22 *= 0)could lead to Zin % Za. This is exactly what occurswhen feedback is used to reach the quantum limit. Keepin mind that having non-vanishing reverse gain is danger-ous: as we discussed earlier, an appreciable non-zero λ′Ican lead to the highly undesirable consequence that the

bz,out

bz,in

az,in

az,out

ideal1-portamp.

coldload

input line output line

cin cout=G1/2cin+(G-1)1/2v†in

vin vout

ax,out bx,in

by,outby,in

ay,inay,out

ax,in bx,outXZ

Ymirror

FIG. 22 Schematic of a modified minimal two-port amplifier,where partially reflecting mirrors have been inserted in theinput and output transmission lines, as well as in the lineleading to the cold load. By tuning the reflection coefficientof the mirror in the cold load arm (mirror Y ), we can in-duce negative feedback which takes advantage of correlationsbetween current and voltage noise. This then allows this sys-tem to reach the quantum limit as a weakly coupled voltageop amp. See text for further description

amplifier’s input impedance depends on the impedanceof the load connected to its output (cf. Eq. (7.6)).

1. Feedback using mirrors

To introduce reverse gain and reflections into the “min-imal” two-port bosonic amplifier of the previous subsec-tion, we will insert mirrors in three of the four arms lead-ing from the circulator: the arm going to the input line,the arm going to the output line, and the arm going tothe auxiliary “cold load” (Fig. 22). Equivalently, onecould imagine that each of these lines is not perfectlyimpedance matched to the circulator. Each mirror willbe described by a 2× 2 unitary scattering matrix:

(aj,out

bj,out

)= Uj ·

(bj,in

aj,in

)(I1)

Uj =

(cos θj − sin θj

sin θj cos θj

)(I2)

Here, the index j can take on three values: j = z for themirror in the input line, j = y for the mirror in the arm

92

going to the cold load, and j = x for the mirror in theoutput line. The mode aj describes the “internal” modewhich exists between the mirror and circulator, while themode bj describes the “external” mode on the other sideof the mirror. We have taken the Uj to be real for con-venience. Note that θj = 0 corresponds to the case of nomirror (i.e. perfect transmission).

It is now a straightforward though tedious exercise toconstruct the scattering matrix for the entire system.From this, one can identify the reduced scattering matrixs appearing in Eq. (7.3), as well as the noise operatorsFj . These may then in turn be used to obtain the op-ampdescription of the amplifier, as well as the commutatorsof the added noise operators. These latter commutatorsdetermine the usual noise spectral densities of the ampli-fier. Details and intermediate steps of these calculationsmay be found in Appendix J.5.

As usual, to see if our amplifier can reach the quantumlimit when used as a (weakly-coupled) op-amp , we needto see if it optimizes the quantum noise constraint ofEq. (6.60). We consider the optimal situation where boththe auxiliary modes of the amplifier (uin and v†in) are inthe vacuum state. The surprising upshot of our analysis(see Appendix J.5) is the following: if we include a smallamount of reflection in the cold load line with the correctphase, then we can reach the quantum limit, irrespectiveof the mirrors in the input and output lines. In particular,if sin θy = −1/

√G, our amplifier optimizes the quantum

noise constraint of Eq. (6.60) in the large gain (i.e. largeG) limit, independently of the values of θx and θy. Notethat tuning θy to reach the quantum limit does not havea catastrophic impact on other features of our amplifier.One can verify that this tuning only causes the voltagegain λV and power gain GP to decrease by a factor oftwo compared to their θy = 0 values (cf. Eqs. (J55) and(J59)). This choice for θy also leads to Zin % Za ∼ ZN

(cf. (J57)), in keeping with our general expectations.Physically, what does this precise tuning of θy corre-

spond to? A strong hint is given by the behaviour of theamplifier’s cross-correlation noise SV I [ω] (cf. Eq. (J60c)).In general, we find that SV I [ω] is real and non-zero.However, the tuning sin θy = −1/

√G is exactly what is

needed to have SV I vanish. Also note from Eq. (J60a)that this special tuning of θy decreases the back-actioncurrent noise precisely by a factor of two compared toits value at θy = 0. A clear physical explanation nowemerges. Our original, reflection-free amplifier had cor-relations between its back-action current noise and out-put voltage noise (cf. Eq. (7.18c)). By introducing nega-tive feedback of the output voltage to the input current(i.e. via a mirror in the cold-load arm), we are able touse these correlations to decrease the overall magnitudeof the current noise (i.e. the voltage fluctuations V par-tially cancel the original current fluctuations I). For anoptimal feedback (i.e. optimal choice of θy), the currentnoise is reduced by a half, and the new current noise isnot correlated with the output voltage noise. Note thatthis is indeed negative (as opposed to positive) feedback–

it results in a reduction of both the gain and the powergain. To make this explicit, in the next section we willmap the amplifier described here onto a standard op-ampwith negative voltage feedback.

2. Explicit examples

To obtain a more complete insight, it is useful to goback and consider what the reduced scattering matrix ofour system looks like when θy has been tuned to reachthe quantum limit. From Eq. (J53), it is easy to see thatat the quantum limit, the matrix s satisfies:

s11 = −s22 (I3a)

s12 =1G

s21 (I3b)

The second equation also carries over to the op-amp pic-ture; at the quantum limit, one has:

λ′I =1G

λV (I4)

One particularly simple limit is the case where thereare no mirrors in the input and output line (θx = θz = 0),only a mirror in the cold-load arm. When this mirror istuned to reach the quantum limit (i.e. sin θy = −1/

√G),

the scattering matrix takes the simple form:

s =

(0 1/

√G√

G 0

)(I5)

In this case, the principal effect of the weak mirror inthe cold-load line is to introduce a small amount of re-verse gain. The amount of this reverse gain is exactlywhat is needed to have the input impedance diverge(cf. Eq. (7.7d)). It is also what is needed to achieve anoptimal, noise-canceling feedback in the amplifier. Tosee this last point explicitly, we can re-write the ampli-fier’s back-action current noise (I) in terms of its originalnoises I0 and V0 (i.e. what the noise operators would havebeen in the absence of the mirror). Taking the relevantlimit of small reflection (i.e. r ≡ sin θy goes to zero as|G| →∞ ), we find that the modification of the currentnoise operator is given by:

I 4 I0 +2√

Gr

1−√

Gr

V0

Za(I6)

As claimed, the presence of a small amount of reflectionr ≡ sin θy in the cold load arm “feeds-back” the originalvoltage noise of the amplifier V0 into the current. Thechoice r = −1/

√G corresponds to a negative feedback,

and optimally makes use of the fact that I0 and V0 arecorrelated to reduce the overall fluctuations in I.

While it is interesting to note that one can reach thequantum limit with no reflections in the input and outputarms, this case is not really of practical interest. The

93

VaVb

R1 R2

+-Gf

FIG. 23 Schematic of a voltage op-amp with negative feed-back.

reverse current gain in this case may be small (i.e. λ′I ∝1/√

G), but it is not small enough: one finds that becauseof the non-zero λ′I , the amplifier’s input impedance isstrongly reduced in the presence of a load (cf. Eq. (7.6)).

There is a second simple limit we can consider which ismore practical. This is the limit where reflections in theinput-line mirror and output-line mirror are both strong.Imagine we take θz = −θx = π/2 − δ/G1/8. If againwe set sin θy = −1/

√G to reach the quantum limit, the

scattering matrix now takes the form (neglecting termswhich are order 1/

√G):

s =

(+1 0

δ2G1/4

2 −1

)(I7)

In this case, we see that at the quantum limit, the reflec-tion coefficients s11 and s22 are exactly what is neededto have the input impedance diverge, while the reversegain coefficient s12 plays no role. For this case of strongreflections in input and output arms, the voltage gain isreduced compared to its zero-reflection value:

λV →√

Zb

Za

(δ

2

)2

G1/4 (I8)

The power gain however is independent of θx, θz, and isstill given by G/2 when θy is tuned to be at the quantumlimit.

3. Op-amp with negative voltage feedback

We now show that a conventional op-amp with feed-back can be mapped onto the amplifier described inthe previous subsection. We will show that tuning thestrength of the feedback in the op-amp corresponds totuning the strength of the mirrors, and that an optimallytuned feedback circuit lets one reach the quantum limit.This is in complete correspondence to the previous sub-section, where an optimal tuning of the mirrors also letsone reach the quantum limit.

More precisely, we consider a scattering description ofa non-inverting op-amp amplifier having negative voltagefeedback. The circuit for this system is shown in Fig. 23.A fraction B of the output voltage of the amplifier is fedback to the negative input terminal of the op-amp. In

practice, B is determined by the two resistors R1 and R2

used to form a voltage divider at the op-amp output. Theop-amp with zero feedback is described by the “ideal”amplifier of Sec. VII.B: at zero feedback, it is describedby Eqs. (7.11a)- (7.11d). For simplicity, we consider therelevant case where:

Zb , R1, R2 , Za (I9)

In this limit, R1 and R2 only play a role through thefeedback fraction B, which is given by:

B =R2

R1 + R2(I10)

Letting Gf denote the voltage gain at zero feedback(B = 0), an analysis of the circuit equations for our op-amp system yields:

λV =Gf

1 + B ·Gf(I11a)

λ′I =B

1 + B ·Gf(I11b)

Zout =Zb

1 + B ·Gf(I11c)

Zin = (1 + B ·Gf )Za (I11d)

GP =G2

f/2B ·Gf + 2Zb/Za

(I11e)

Again, Gf represents the gain of the amplifier in the ab-sence of any feedback, Za is the input impedance at zerofeedback, and Zb is the output impedance at zero feed-back.

Transforming this into the scattering picture yields ascattering matrix s satisfying:

s11 = −s22 = −BGf (Za − (2 + BGf )Zb)BGfZa + (2 + BGf )2Zb

(I12a)

s21 = − 2√

ZaZbGf (1 + BGf )BGfZa + (2 + BGf )2Zb

(I12b)

s12 =B

Gfs21 (I12c)

Note the connection between these equations and the nec-essary form of a quantum limited s-matrix found in theprevious subsection.

Now, given a scattering matrix, one can always find aminimal representation of the noise operators Fa and Fb

which have the necessary commutation relations. Theseare given in general by:

Fa =√

1− |s11|2 − |s12|2 + |l|2 · uin + l · v†in (I13)

Fb =√|s21|2 + |s22|2 − 1 · v†in (I14)

l =s11s∗21 + s12s∗22√|s21|2 + |s22|2 − 1

(I15)

Applying this to the s matrix for our op-amp, and thentaking the auxiliary modes uin and v†in to be in the vac-uum state, we can calculate the minimum allowed SV V

94

and SII for our non-inverting op-amp amplifier. One canthen calculate the product SV V SII and compare againstthe quantum-limited value (SIV is again real). In thecase of zero feedback (i.e. B = 0), one of course findsthat this product is twice as big as the quantum limitedvalue. However, if one takes the large Gf limit whilekeeping B non-zero but finite, one obtains:

SV V SII → (!ω)2(

1− 2B

Gf+ O

(1

Gf

)2)

(I16)

Thus, for a fixed, non-zero feedback ratio B, it is possibleto reach the quantum limit. Note that if B does not tendto zero as Gf tends to infinity, the voltage gain of thisamplifier will be finite. The power gain however will beproportional to Gf and will be large. If one wants alarge voltage gain, one could set B to go to zero withGf i.e. B ∝ 1√

Gf. In this case, one will still reach

the quantum limit in the large Gf limit, and the voltagegain will also be large (i.e. ∝

√Gf ). Note that in all

these limits, the reflection coefficients s11 and s22 tendto −1 and 1 respectively, while the reverse gain tendsto 0. This is in complete analogy to the amplifier withmirrors considered in the previous subsection, in the casewhere we took the reflections to be strong at the inputand at the output (cf. Eq. (I7)). We thus see yet againhow the use of feedback allows the system to reach thequantum limit.

APPENDIX J: Additional Technical Details

This appendix provides further details of calculationspresented in the main text.

1. Proof of Quantum Noise Constraint

Note first that we may write the symmetrized I and Fnoise correlators defined in Eqs. (5.4a) and (5.4b) as sumsover transitions between detector energy eigenstates:

SFF [ω] = π!∑

i,f

〈i|ρ0|i〉 · |〈f |F |i〉|2 [δ(Ef − Ei + !ω) + δ(Ef − Ei − !ω)] (J1)

SII [ω] = π!∑

i,f

〈i|ρ0|i〉 · |〈f |I|i〉|2 [δ(Ef − Ei + !ω) + δ(Ef − Ei − !ω)] (J2)

Here, ρ0 is the stationary density matrix describing thestate of the detector, and |i〉 (|f〉) is a detector energyeigenstate with energy Ei (Ef ). Eq. (J1) expresses thenoise at frequency ω as a sum over transitions. Eachtransition starts with an an initial detector eigenstate|i〉, occupied with a probability 〈i|ρ0|i〉, and ends with afinal detector eigenstate |f〉, where the energy differencebetween the two states is either +!ω or −!ω . Further,each transition is weighted by an appropriate matrix el-ement.

To proceed, we fix the frequency ω > 0, and let theindex ν label each transition |i〉 → |f〉 contributing to thenoise. More specifically, ν indexes each ordered pair ofdetector energy eigenstates states |i〉, |f〉 which satisfyEf − Ei ∈ ±![ω, ω + dω] and 〈i|ρ0|i〉 *= 0. We can nowconsider the matrix elements of I and F which contributeto SII [ω] and SFF [ω] to be complex vectors +v and +w.Letting δ be any real number, let us define:

[+w]ν = 〈f(ν)|F |i(ν)〉 (J3)

[+v]ν =

e−iδ〈f(ν)|I|i(ν)〉 if Ef(ν) − Ei(ν) = +!ω,

eiδ〈f(ν)|I|i(ν)〉 if Ef(ν) − Ei(ν) = −!ω.

(J4)

Introducing an inner product 〈·, ·〉ω via:

〈+a,+b〉ω = π∑

ν

〈i(ν)|ρ0|i(ν)〉 · (aν)∗ bν , (J5)

we see that the noise correlators SII and SFF may bewritten as:

SII [ω]dω = 〈+v,+v〉ω (J6)SFF [ω]dω = 〈+w, +w〉ω (J7)

We may now employ the Cauchy-Schwartz inequality:

〈+v,+v〉ω〈+w, +w〉ω ≥ |〈+v, +w〉ω|2 (J8)

A straightforward manipulation shows that the realpart of 〈+v, +w〉ω is determined by the symmetrized cross-correlator SIF [ω] defined in Eq. (5.4c):

Re 〈+v, +w〉ω = Re[eiδSIF [ω]

]dω (J9)

In contrast, the imaginary part of 〈+v, +w〉ω is independentof SIF ; instead, it is directly related to the gain λ andreverse gain λ′ of the detector:

Im 〈+v, +w〉ω =!2Re

[eiδ

(λ[ω]− [λ′[ω]]∗

)]dω (J10)

95

Substituting Eqs. (J10) and (J9) into Eq. (J8), one im-mediately finds the quantum noise constraint given inEq. (5.15). Maximizing the RHS of this inequality withrespect to the phase δ, one finds that the maximum isachieved for δ = δ0 = − arg(λ) + δ0 with (we assume theideal case where λ′ = 0):

tan 2δ0 = − |SIF | sin 2φ|!λ/2|+ |SIF | cos 2φ

(J11)

where φ = arg(SIF λ∗). At δ = δ0, Eq. (5.15) becomesthe final noise constraint of Eq. (5.11).

The proof given here also allows one to see what mustbe done in order to achieve the “ideal” noise conditionof Eq. (5.16): one must achieve equality in the Cauchy-Schwartz inequality of Eq. (J8). This requires that thevectors +v and +w be proportional to one another; theremust exist a complex factor α (having dimensions [I]/[F ])such that:

+v = α · +w (J12)

Equivalently, we have that

〈f |I|i〉 =

eiδα〈f |F |i〉 if Ef − Ei = +!ω

e−iδα〈f |F |i〉 if Ef − Ei = −!ω.(J13)

for each pair of initial and final states |i〉, |f〉 contribut-ing to SFF [ω] and SII [ω] (cf. Eq. (J1)). Note that thisnot the same as requiring Eq. (J13) to hold for all pos-sible states |i〉 and |f〉. This proportionality conditiontells us that a detector with ideal noise properties has atight connection between its input and output ports; inClerk et al. (2003), this condition was directly tied to theidea that there is no “wasted” information in a quantum-limited detector, an idea that we have discussed in Sec.IV.B and Appendix H.

Finally, for a detector with quantum-ideal noise prop-erties, the magnitude of the constant α can be found fromEq. (5.17). The phase of α can also be determined from:

−Im α

|α| =!|λ|/2√SII SFF

cos δ0 (J14)

For zero frequency or for a large detector effective tem-perature, this simplifies to:

−Im α

|α| =!λ/2√SII SFF

(J15)

Note importantly that to have a non-vanishing gainand power gain, one needs Im α *= 0. This in turn placesa very powerful constraint on a quantum-ideal detectors:all transitions contributing to the noise must be to finalstates |f〉 which are completely unoccupied. To see this,imagine a transition taking an initial state |i〉 = |a〉 toa final state |f〉 = |b〉 makes a contribution to the noise.For a quantum-ideal detector, Eq. (J13) will be satisfied:

〈b|I|a〉 = e±iδα〈b|F |a〉 (J16)

where the plus sign corresponds to Eb > Ea, the mi-nus to Ea > Eb. If now the final state |b〉 was also oc-cupied (i.e. 〈b|ρ0|b〉 *= 0), then the reverse transition|i = b〉 → |f = a〉) would also contribute to the noise.The proportionality condition of Eq. (J13) would nowrequire:

〈a|I|b〉 = e∓iδα〈a|F |b〉 (J17)

As I and F are both Hermitian operators, and as α musthave an imaginary part in order for there to be gain, wehave a contradiction: Eq. (J16) and (J17) cannot bothbe true. It thus follows that the final state of a tran-sition contributing to the noise must be unoccupied inorder for Eq. (J13) to be satisfied and for the detectorto have ideal noise properties. Note that this necessaryasymmetry in the occupation of detector energy eigen-states immediately tells us that a detector or amplifiercannot reach the quantum limit if it is in equilibrium.

2. Simplifications for a Quantum-Limited Detector

In this appendix, we derive the additional constraintson the property of a detector that arise when it satisfiesthe quantum noise constraint of Eq. (5.16).

We begin by noting that a necessary condition for sat-isfying Eq. (5.16) is that the inequality of Eq. (5.15) be-comes an equality for a particular choice of δ. Thus, as-sume Eq. (5.15) holds as an equality for our detector forsome specific value of δ; further, assume the ideal caseof a detector with zero reverse gain, λ′ = 0. Using theproportionality condition between the matrix elements ofI and F (cf. Eq (J13)), the unsymmetrized I-F quantumnoise correlator SIF [ω] (cf. Eq. (5.7)) may be written:

SIF [ω] =

e−iδα∗SFF [ω] if ω > 0,

eiδα∗SFF [ω] if ω < 0(J18)

In the above equation, SFF [ω] is the unsymmetrizedspectral density of the force noise of the detector(cf. Eq. (3.8)); note that SFF [ω] is necessarily real andpositive.

We may now substitute Eq. (J18) into Eqs. (5.8b)-(5.8a); writing SFF [ω] in terms of the detector effectivetemperature Teff (cf. Eq. (3.21)):

!λ[ω]2

= −e−iδ! [−Im χFF [ω]] (J19)[(Im α) coth

(!ω

2kBTeff

)+ i (Re α)

]

SIF [ω] = e−iδ! [−Im χFF [ω]] (J20)[(Re α) coth

(!ω

2kBTeff

)− i (Im α)

]

To proceed, let us write:

e−iδ =λ

|λ|e−iδ (J21)

96

The condition that |λ| is real yields the condition:

tan δ =Re α

Imαtanh

(!ω

2kBTeff

)(J22)

We now consider the relevant limit of a large detectorpower gain GP . GP is determined by Eq. (6.28); the onlyway this can become large is if kBTeff/(!ω) → ∞ whileIm α does not tend to zero. We will thus take the largeTeff limit in the above equations while keeping both α andthe phase of λ fixed. Note that this means the parameterδ must evolve; it tends to zero in the large Teff limit. Inthis limit, we thus find for λ and SIF :

!λ[ω]2

= −2e−iδkBTeffγ[ω] (Im α)

[1 + O

[(!ω

kBTeff

)2]]

(J23)

SIF [ω] = 2e−iδkBTeffγ[ω] (Re α)[1 + O

(!ω

kBTeff

)]

(J24)

Thus, in the large power-gain limit (i.e. large Teff

limit), the gain λ and the noise cross-correlator SIF havethe same phase: SIF /λ is purely real.

3. Derivation of non-equilibrium Langevin equation

In this appendix, we prove that an oscillator weaklycoupled to an arbitrary out-of-equilibrium detector is de-scribed by the Langevin equation given in Eq. (6.14), anequation which associates an effective temperature anddamping kernel to the detector.

We start by defining the oscillator matrix Keldyshgreen function:

G(t) =

(GK(t) GR(t)GA(t) 0

)(J25)

where GR(t− t′) = −iθ(t− t′)〈[x(t), x(t′)]〉, GA(t− t′) =iθ(t′− t)〈[x(t), x(t′)]〉, and GK(t− t′) = −i〈x(t), x(t′)〉.At zero coupling to the detector (A = 0), the oscillatoris only coupled to the equilibrium bath, and thus G0 hasthe standard equilibrium form:

G0[ω] =!m

(−2Im g0[ω] coth

(!ω

2kBTbath

)g0[ω]

g0[ω]∗ 0

)

(J26)where:

g0[ω] =1

ω2 − Ω2 + iωγ0/m(J27)

and where γ0 is the intrinsic damping coefficient, andTbath is the bath temperature.

We next treat the effects of the coupling to the detectorin perturbation theory. Letting Σ denote the correspond-ing self-energy, the Dyson equation for G has the form:

[G[ω]

]−1 =[G0[ω]

]−1 −(

0 ΣA[ω]ΣR[ω] ΣK [ω]

)(J28)

To lowest order in A, Σ[ω] is given by:

Σ[ω] = A2D[ω] (J29)

≡ A2

!

∫dt eiωt (J30)

(0 iθ(−t)〈[F (t), F (0)]〉

−iθ(t)〈[F (t), F (0)]〉 −i〈F (t), F (0)〉

)

Using this lowest-order self energy, Eq. (J28) yields:

GR[ω] =!

m(ω2 − Ω2)−A2Re DR[ω] + iω(γ0 + γ[ω])(J31)

GA[ω] =[GR[ω]

]∗(J32)

GK [ω] = −2iIm GR[ω]×

γ0 coth(

!ω2kBTbath

)+ γ[ω] coth

(!ω

2kBTeff

)

γ0 + γ[ω](J33)

where γ[ω] is given by Eq. (3.25), and Teff [ω] is definedby Eq. (3.21). The main effect of the real part of the re-tarded F Green function DR[ω] in Eq. (J31) is to renor-malize the oscillator frequency Ω and mass m; we simplyincorporate these shifts into the definition of Ω and m inwhat follows.

If Teff [ω] is frequency independent, then Eqs. (J31)- (J33) for G corresponds exactly to an oscillator cou-pled to two equilibrium baths with damping kernels γ0

and γ[ω]. The correspondence to the Langevin equa-tion Eq. (6.14) is then immediate. In the more generalcase where Teff [ω] has a frequency dependence, the cor-relators GR[ω] and GK [ω] are in exact correspondenceto what is found from the Langevin equation Eq. (6.14):GK [ω] corresponds to symmetrized noise calculated fromEq. (6.14), while GR[ω] corresponds to the response co-efficient of the oscillator calculated from Eq. (6.14). Thisagain proves the validity of using the Langevin equationEq. (6.14) to calculate the oscillator noise in the presenceof the detector to lowest order in A.

4. Linear-Response Formulas for a Two-Port BosonicAmplifier

In this appendix, we use the standard linear-responseKubo formulas of Sec. VI.E to derive expressions forthe voltage gain λV , reverse current gain λ′I , inputimpedance Zin and output impedance Zout of a two-port

97

bosonic voltage amplifier (cf. Sec. VII). We recover thesame expressions for these quantities obtained in Sec. VIIfrom the scattering approach. We stress throughout thisappendix the important role played by the causal struc-ture of the scattering matrix describing the amplifier.

In applying the general linear response formulas, wemust bear in mind that these expressions should be ap-plied to the uncoupled detector, i.e. nothing attached tothe detector input or output. In our two-port bosonicvoltage amplifier, this means that we should have ashort circuit at the amplifier input (i.e. no input voltage,Va = 0), and we should have open circuit at the out-put (i.e. Ib = 0, no load at the output drawing current).These two conditions define the uncoupled amplifier. Us-ing the definitions of the voltage and current operators(cf. Eqs. (7.2a) and (7.2b)), they take the form:

ain[ω] = −aout[ω] (J34a)

bin[ω] = bout[ω] (J34b)

The scattering matrix equation Eq. (7.3) then allows usto solve for ain and aout in terms of the added noiseoperators Fa and Fb.

ain[ω] = −1− s22

DFa[ω]− s12

DFb[ω] (J35a)

bin[ω] = −s21

DFa[ω] +

1 + s11

DFb[ω] (J35b)

where D is given in Eq. (7.8), and we have omitted writ-ing the frequency dependence of the scattering matrix.Further, as we have already remarked, the commutatorsof the added noise operators is completely determined bythe scattering matrix and the constraint that output op-erators have canonical commutation relations. The non-vanishing commutators are thus given by:

[Fa[ω], F†

a(ω′)]

= 2πδ(ω − ω′)(1− |s11|2 − |s12|2

)

(J36a)[Fb [ω], F†

b (ω′)]

= 2πδ(ω − ω′)(1− |s21|2 − |s22|2

)

(J36b)[Fa[ω], F†

b (ω′)]

= −2πδ(ω − ω′) (s11s∗21 + s12s

∗22)

(J36c)

The above equations, used in conjunction withEqs. (7.2a) and (7.2b), provide us with all all the infor-mation needed to calculate commutators between currentand voltage operators. It is these commutators which en-ter into the linear-response Kubo formulas. As we willsee, our calculation will crucially rely on the fact thatthe scattering description obeys causality: disturbancesat the input of our system must take some time beforethey propagate to the output. Causality manifests itselfin the energy dependence of the scattering matrix: as afunction of energy, it is an analytic function in the upperhalf complex plane.

a. Input and output impedances

Eq. (6.56) is the linear response Kubo formula forthe input impedance of a voltage amplifier. Recallthat the input operator Q for a voltage amplifier is re-lated to the input current operator Ia via −dQ/dt =Ia (cf. Eq. (6.55)). The Kubo formula for the inputimpedance may thus be re-written in the more familiarform:

Yin,Kubo[ω] ≡ i

ω

(− i

!

∫ ∞

0dt〈[Ia(t), Ia(0)]〉eiωt

)(J37)

where Yin[ω] = 1/Zin[ω],Using the defining equation for Ia (Eq. (7.1b)) and

Eq. (J34a) (which describes an uncoupled amplifier), weobtain:

Yin,Kubo[ω] = (J38)2

Za

∫ ∞

0dteiωt

∫ ∞

0

dω′

2π

ω′

ωΛaa(ω′)

(e−iω′t − eiω′t

)

where we have defined the real function Λaa[ω] for ω > 0via:

[ain[ω], a†in(ω′)

]= 2πδ(ω − ω′)Λaa[ω] (J39)

It will be convenient to also define Λaa[ω] for ω < 0 viaΛaa[ω] = Λaa[−ω]. Eq. (J38) may then be written as:

Yin,Kubo[ω] =2

Za

∫ ∞

0dt

∫ ∞

−∞

dω′

2π

ω′

ωΛaa(ω′)ei(ω−ω′)t

=Λaa[ω]

Za+

i

πωP

∫ ∞

−∞dω′

ω′Λaa(ω′)/Za

ω − ω′

(J40)

Next, by making use of Eq. (J35a) and Eqs. (J36) forthe commutators of the added noise operators, we canexplicitly evaluate the commutator in Eq. (J39) to cal-culate Λaa[ω]. Comparing the result against the resultEq. (7.7d) of the scattering calculation, we find:

Λaa[ω]Za

= Re Yin,scatt[ω] (J41)

where Yin,scatt[ω] is the input admittance of the ampli-fier obtained from the scattering approach. Returning toEq. (J40), we may now use the fact that Yin,scatt[ω] isan analytic function in the upper half plane to simplifythe second term on the RHS, as this term is simply aKramers-Kronig integral:

1πω

P∫ ∞

−∞dω′

ω′Λaa(ω′)/Za

ω − ω′

=1

πωP

∫ ∞

−∞dω′

ω′Re Yin,scatt(ω′)ω − ω′

= Im Yin,scatt[ω] (J42)

98

It thus follows from Eq. (J40) that input impedance cal-culated from the Kubo formula is equal to what we foundpreviously using the scattering approach.

The calculation for the output impedance proceeds inthe same fashion, starting from the Kubo formula givenin Eq. (6.57). As the steps are completely analogous tothe above calculation, we do not present it here. Oneagain recovers Eq. (7.7c), as found previously within thescattering approach.

b. Voltage gain and reverse current gain

Within linear response theory, the voltage gain of theamplifier (λV ) is determined by the commutator betweenthe “input operator” Q and Vb (cf. Eq. (5.3); recall thatQ is defined by dQ/dt = −Ia. Similarly, the reversecurrent gain (λ′I) is determined by the commutator be-tween Ia and Φ, where Φ is defined via dΦ/dt = −Vb

(cf. Eq. (5.6)). Similar to the calculation of the inputimpedance, to properly evaluate the Kubo formulas forthe gains, we must make use of the causal structure ofthe scattering matrix describing our amplifier.

Using the defining equations of the current and volt-age operators (cf. Eqs. (7.1a) and (7.1b)), as well asEqs. (J34a) and (J34b) which describe the uncoupledamplifier, the Kubo formulas for the voltage gain andreverse current gain become:

λV,Kubo[ω] = 4√

Zb

Za(J43)

×∫ ∞

0dt eiωtRe

[∫ ∞

0

dω′

2πΛba(ω′)e−iω′t

]

λ′I,Kubo[ω] = −4√

Zb

Za(J44)

×∫ ∞

0dt eiωtRe

[∫ ∞

0

dω′

2πΛba(ω′)eiω′t

]

where we define the complex function Λba[ω] for ω > 0via:

[bin[ω], a†in(ω′)

]≡ (2πδ(ω − ω′)) Λba[ω] (J45)

We can explicitly evaluate Λba[ω] by using Eqs. (J35a)-(J36) to evaluate the commutator above. Comparing theresult against the scattering approach expressions for thegain and reverse gain (cf. Eqs. (7.7a) and (7.7b)), onefinds:

Λba[ω] = λV,scatt[ω]−[λ′I,scatt[ω]

]∗ (J46)

Note crucially that the two terms above have differentanalytic properties: the first is analytic in the upper halfplane, while the second is analytic in the lower half plane.This follows directly from the fact that the scatteringmatrix is causal.

At this stage, we can proceed much as we did in thecalculation of the input impedance. Defining Λba[ω] for

ω < 0 via Λba[−ω] = Λ∗ba[ω], we can re-write Eqs. (J43)and (J44) in terms of principle part integrals.

λV,Kubo[ω] =Zb

Za

(Λba[ω] +

i

πP

∫ ∞

−∞dω′

Λba(ω′)ω − ω′

)

λ′I,Kubo[ω] = −Zb

Za

(Λba[ω] +

i

πP

∫ ∞

−∞dω′

Λba(ω′)ω + ω′

)

(J47)

Using the analytic properties of the two terms inEq. (J46) for Λba[ω], we can evaluate the principal partintegrals above as Kramers-Kronig relations. One thenfinds that the Kubo formula expressions for the voltageand current gain coincide precisely with those obtainedfrom the scattering approach.

While the above is completely general, it is useful togo through a simpler, more specific case where the role ofcausality is more transparent. Imagine that all the energydependence in the scattering in our amplifier arises fromthe fact that there are small transmission line “stubs” oflength a attached to both the input and output of theamplifier (these stubs are matched to the input and out-put lines). Because of these stubs, a wavepacket incidenton the amplifier will take a time τ = 2a/v to be eitherreflected or transmitted, where v is the characteristic ve-locity of the transmission line. This situation is describedby a scattering matrix which has the form:

s[ω] = e2iωa/v · s (J48)

where s is frequency-independent and real. To furthersimplify things, let us assume that s11 = s22 = s12 = 0.Eqs. (J46) then simplifies to

Λba[ω] = = s21[ω] = s21eiωτ (J49)

where the propagation time τ = 2a/v We then have:

λV [ω0] = 2√

Zb

Zas21

∫ ∞

0dt eiω0tδ(t− τ) (J50)

λI [ω0] = −2√

Zb

Zas21

∫ ∞

0dt eiω0tδ(t + τ) (J51)

If we now do the time integrals and then take the limitτ → 0+, we recover the results of the scattering approach(cf. Eqs. (7.7a) and (7.7b)); in particular, λI = 0. Notethat if we had set τ = 0 from the outset of the calculation,we would have found that both λV and λI are non-zero!

5. Details for Two-port Bosonic Voltage Amplifier withFeedback

In this appendix, we provide more details on the calcu-lations for the bosonic-amplifier-plus-mirrors system dis-cussed in Sec. I. Given that the scattering matrix foreach of the three mirrors is given by Eq. (I2), and thatwe know the reduced scattering matrix for the mirror-free

99

system (cf. Eq. (7.10)), we can find the reduced scatteringmatrix and noise operators for the system with mirrors.One finds that the reduced scattering matrix s is nowgiven by:

s =1M× (J52)

(sin θz +

√G sin θx sin θy − cos θx cos θz sin θy√

G cos θx cos θz sin θx +√

G sin θy sin θz

)

where the denominator M describes multiple reflectionprocesses:

M = 1 +√

G sin θx sin θz sin θy (J53)

Further, the noise operators are given by:

(Fa

Fb

)=

1M

(cos θy cos θz

√G− 1 cos θz sin θx sin θy

−√

G cos θx cos θy sin θz

√G− 1 cos θx

) (uin

v†in

)(J54)

The next step is to convert the above into the op-amprepresentation, and find the gains and impedances of theamplifier, along with the voltage and current noises. Thevoltage gain is given by:

λV =√

ZB

ZA

2√

G

1−√

G sin θy

· 1 + sin θx

cos θx

1− sin θz

cos θz(J55)

while the reverse gain is related to the voltage gain bythe simple relation:

λ′I = − sin θy√G

λV (J56)

The input impedance is determined by the amount ofreflection in the input line and in the line going to thecold load:

Zin = Za1−

√G sin θy

1 +√

G sin θy

· 1 + sin θz

1− sin θz(J57)

Similarly, the output impedance only depends on theamount of reflection in the output line and the in thecold-load line:

Zout = Zb1 +

√G sin θy

1−√

G sin θy

· 1 + sin θx

1− sin θx(J58)

Note that as sin θy tends to −1/√

G , both the inputadmittance and output impedance tend to zero.

Given that we now know the op-amp parameters of ouramplifier, we can use Eq. (7.5) to calculate the amplifier’spower gain GP . Amazingly, we find that the power gainis completely independent of the mirrors in the input andoutput lines:

GP =G

1 + G sin2 θy(J59)

Note that at the special value sin θy = −1√

G (whichallows one to reach the quantum limit), the power gain

is reduced by a factor of two compared to the reflectionfree case (i.e. θy = 0).

Turning to the noise spectral densities, we assume theoptimal situation where both the auxiliary modes uin andv†in are in the vacuum state. We then find that both ˆIand ˆV are independent of the amount of reflection in theoutput line (e.g. θx):

SII =2!ω

Za

[1− sin θz

1 + sin θz

]×

G sin2 θy + cos(2θy)(√

G sin θy − 1)2

(J60a)

SV V = !ωZa

[1 + sin θz

1− sin θz

]×

(3 + cos(2θy)

4− 1

2G

)(J60b)

SV I =√

G(1− 1/G) sin θy + cos2 θy

1−√

G sin θy

(J60c)

As could be expected, introducing reflections in the inputline (i.e. θz *= 0) has the opposite effect on SII versusSV V : if one is enhanced, the other is suppressed.

It thus follows that the product of noise spectraldensities appearing in the quantum noise constraint ofEq. (6.60) is given by (taking the large-G limit):

SII SV V

(!ω)2=

(2− sin2 θy

)· 1 + G sin2 θy(

1−√

G sin θy

)2 (J61)

Note that somewhat amazingly, this product (and hencethe amplifier noise temperature) is completely indepen-dent of the mirrors in the input and output arms (i.e. θz

and θx). This is a result of both SV V and SII havingno dependence on the output mirror (θx), and their hav-ing opposite dependencies on the input mirror (θz). Alsonote that Eq. (J61) does indeed reduce to the result of

100

the last subsection: if θy = 0 (i.e. no reflections in theline going to the cold load), the product SII SV V is equalto precisely twice the quantum limit value of (!ω)2. Forsin(θy) = −1/

√G, the RHS above reduces to one, imply-

ing that we reach the quantum limit for this tuning ofthe mirror in the cold-load arm.

References

Abramovici, A., W. Althouse, R. Drever, Y. Gursel, S. Kawa-mura, F. Raab, D. Shoemaker, L. Sievers, R. Spero,K. Thorne, R. Vogt, R. Weiss, et al., 1992, Science 256,325.

Aguado, R., and L. P. Kouwenhoven, 2000, Phys. Rev. Lett.84, 1986.

Aleiner, I. L., N. S. Wingreen, and Y. Meir, 1997, Phys. Rev.Lett. 79, 3740.

Allahverdyan, A. E., R. Balian, and T. M. Nieuwenhuizen,2001, Phys. Rev. A 64, 032108.

Arcizet, O., P.-F. Cohadon, T. Briant, M. Pinard, andA. Hedimann, 2006, Nature 444, 71.

Armen, M. A., J. K. Au, J. K. Stockton, A. C. Doherty, andH. Mabuchi, 2002, Phys. Rev. Lett. 89, 133602.

Averin, D., 2003, in Quantum Noise in Mesoscopic Systems,edited by Y. Nazarov (Kluwer, Amsterdam), pp. 205–228.

Averin, D. V., 2000a, cond-mat/0010052v1 .Averin, D. V., 2000b, Fortschr. Phys. 48, 1055.Averin, D. V., and E. V. Sukhorukov, 2005, Phys. Rev. Lett.

95, 126803.Berestetskii, V. B., E. M. Lifshitz, and L. P. Pitaevskii,

1982, Quantum Electrodynamics (2nd Edition), volume 4of Course of Theoretical Physics (Butterworth-Heinemann,Oxford).

Billangeon, P. M., F. Pierre, H. Bouchiat, and R. Deblock,2006, Phys. Rev. Lett. 96, 136804.

Blais, A., R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J.Schoelkopf, 2004, Phys. Rev. A 69, 062320.

Blanter, Y. M., and M. Buttiker, 2000, Phys. Rep. 336, 1.Blencowe, M. P., and E. Buks, 2007, Phys. Rev. B 76, 014511.Bocko, M. F., and R. Onofrio, 1996, Rev. Mod. Phys. 68, 755.Bohm, D., 1989, Quantum Theory (Dover, New York).Boylestad, R., and L. Nashelsky, 2006, Electronic Devices and

Circuit Theory (Prentice Hall, New Jersey), 9 edition.Braginsky, V. B., and F. Y. Khalili, 1992, Quantum Measure-

ment (Cambridge University Press, Cambridge).Braginsky, V. B., and F. Y. Khalili, 1996, Rev. Mod. Phys.

68, 1.Braginsky, V. B., Y. I. Vorontsov, and K. P. Thorne, 1980,

Science 209, 547.van den Brink, A. M., 2002, Europhys. Lett. 58, 562.Brown, K. R., J. Britton, R. J. Epstein, J. Chiaverini,

D. Leibfried, and D. J. Wineland, 2007, Phys. Rev. Lett.99, 137205.

Brun, T. A., 2002, Am. J. Phys. 70, 719.Buks, E., R. Schuster, M. Heiblum, D. Mahalu, and V. Uman-

sky, 1998, Nature (London) 391, 871.Burkhard, G., R. H. Koch, and D. P. DiVincenzo, 2004, Phys.

Rev. B 69, 064503.Caldeira, A., and A. Leggett, 1983, Ann. Phys. (NY) 149,

374.Callen, H. B., and T. A. Welton, 1951, Phys. Rev. 83, 34.Caves, C. M., 1980, Phys. Rev. Lett. 45, 75.

Caves, C. M., 1981, Phys. Rev. D 23, 1693.Caves, C. M., 1982, Phys. Rev. D 26, 1817.Caves, C. M., and B. L. Schumaker, 1985, Phys. Rev. A 31,

3068.Caves, C. M., K. S. Thorne, R. W. P. Drever, V. D. Sandberg,

and M. Zimmermann, 1980, Rev. Mod. Phys. 52, 341.Cleland, A. N., J. S. Aldridge, D. C. Driscoll, and A. C. Gos-

sard, 2002, Appl. Phys. Lett. 81, 1699.Clerk, A. A., 2004, Phys. Rev. B 70, 245306.Clerk, A. A., 2006, Phys. Rev. Lett. 96, 056801.Clerk, A. A., and S. M. Girvin, 2004, Phys. Rev. B 70, 121303.Clerk, A. A., S. M. Girvin, A. K. Nguyen, and A. D. Stone,

2002, Phys. Rev. Lett. 89, 176804.Clerk, A. A., S. M. Girvin, and A. D. Stone, 2003, Phys. Rev.

B 67, 165324.Clerk, A. A., F. Marquardt, and K. Jacobs, 2008, New J.

Phys. 10, 095010.Clerk, A. A., and A. D. Stone, 2004, Phys. Rev. B 69, 245303.Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, 1989,

Photons and Atoms - Introduction to Quantum Electrody-namics (Wiley, New York).

Corbitt, T., Y. Chen, E. Innerhofer, H. Muller-Ebhardt,D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf,and N. Mavalvala, 2007, Phys. Rev. Lett 98, 150802.

Courty, J. M., F. Grassia, and S. Reynaud, 1999, Europhys.Lett. 46, 31.

Cover, T., and J. Thomas, 1991, Elements of InformationTheory (Wiley, New York).

Lupascu, A., C. J. M. Verwijs, R. N. Schouten, C. J. P. M.Harmans, and J. E. Mooij, 2004, Phys. Rev. Lett. 93,177006.

Danilov, V. V., K. K. Likharev, and A. B. Zorin, 1983, IEEETrans. Magn. 19, 572.

Deblock, R., E. Onac, L. Gurevich, and L. P. Kouwenhoven,2003, Science 301, 203.

Devoret, M., 1997, in Quantum Fluctuations (Les HouchesSession LXIII) (Elsevier, Amsterdam), pp. 351–86.

Devoret, M. H., and R. J. Schoelkopf, 2000, Nature (London)406, 1039.

Doherty, A. C., S. Habib, K. Jacobs, H. Mabuchi, and S. M.Tan, 2000, Phys. Rev. A 62, 012105.

Duty, T., G. Johansson, K. Bladh, D. Gunnarsson, C. Wilson,and P. Delsing, 2005, Phys. Rev. Lett. 95, 206807.

Etaki, S., M. Poot, I. Mahboob, K. Onomitsu, H. Yamaguchi,and H. S. J. V. der Zant, 2008, Nature Phys. 4, 785.

Flowers-Jacobs, N. E., D. R. Schmidt, and K. W. Lehnert,2007, Phys. Rev. Lett 98, 096804.

Gambetta, J., A. Blais, D. I. Schuster, A. Wallraff, and R. J.Schoelkopf, 2006, Phys. Rev. A 74, 042318.

Gambetta, J., W. A. Braff, A. Wallraff, S. M. Girvin, andR. J. Schoelkopf, 2007, Phys. Rev. A 76, 012325.

Gardiner, C. W., A. S. Parkins, and P. Zoller, 1992, Phys.Rev. A 46, 4363.

Gardiner, C. W., and P. Zoller, 2000, Quantum Noise(Springer, Berlin).

Gavish, U., Y. Levinson, and Y. Imry, 2000, Phys. Rev. B(R) 62, 10637.

Gavish, U., B. Yurke, and Y. Imry, 2004, Phys. Rev. Lett.93, 250601.

Gavish, U., B. Yurke, and Y. Imry, 2006, Phys. Rev. Lett.96, 133602.

Gea-Banacloche, J., N. Lu, L. M. Pedrotti, S. Prasad, M. O.Scully, and K. Wodkiewicz, 1990a, Phys. Rev. A 41, 369.

Gea-Banacloche, J., N. Lu, L. M. Pedrotti, S. Prasad, M. O.

101

Scully, and K. Wodkiewicz, 1990b, Phys. Rev. A 41, 381.Geremia, J., J. K. Stockton, and H. Mabuchi, 2004, Science

304, 270.Gerry, C. S., 1985, Phys. Rev. A 31, 2721.Gigan, S., H. Bohm, M. Paternostro, F. Blaser, J. B.

Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, andA. Zeilinger, 2006, Nature 444, 67.

Glauber, R. J., 2006, Rev. Mod. Phys. 78, 1267.Goan, H.-S., and G. J. Milburn, 2001, Phys. Rev. B 64,

235307.Goan, H. S., G. J. Milburn, H. M. Wiseman, and B. H. Sun,

2001, Phys. Rev. B 63, 125326.Gordon, J. P., W. H. Louisell, and L. R. Walker, 1963, Phys.

Rev. 129, 481.Gottfried, K., 1966, Quantum Mechanics, Volume I: Funda-

mentals (W. A. Benjamin, Inc.).Grassia, F., 1998, Fluctuations quantiques et thermiques dans

les transducteurs electromecaniques, Ph.D. thesis, Univer-site Pierre et Marie Curie.

Gurvitz, S. A., 1997, Phys. Rev. B 56, 15215.Gustavsson, S., M. Studer, R. Leturcq, T. Ihn, K. Ensslin,

D. C. Driscoll, and A. C. Gossard, 2007, Phys. Rev. Lett.99, 206804.

Haroche, S., and J.-M. Raimond, 2006, Exploring the Quan-tum Atoms, Cavities, and Photons (Oxford UniversityPress, Oxford).

Harris, J. G. E., B. M. Zwickl, and A. M. Jayich, 2007, Reviewof Scientific Instruments 78, 013107.

Haus, H. A., 2000, Electromagnetic Noise and Quantum Op-tical Measurements (Springer, New York).

Haus, H. A., and J. A. Mullen, 1962, Phys. Rev. 128, 2407.Heffner, H., 1962, Proc. IRE 50, 1604.Hohberger-Metzger, C., and K. Karrai, 2004, Nature 432,

1002.Il’ichev, E., N. Oukhanski, A. Izmalkov, T. Wagner, M. Gra-

jcar, H.-G. Meyer, A. Y. Smirnov, A. Maassen van denBrink, M. H. S. Amin, and A. M. Zagoskin, 2003, Phys.Rev. Lett. 91, 097906.

Izmalkov, A., M. Grajcar, E. Il’ichev, T. Wagner, H.-G.Meyer, A. Y. Smirnov, M. H. S. Amin, A. M. van den Brink,and A. M. Zagoskin, 2004, Phys. Rev. Lett. 93, 037003.

Jacobs, K., and D. A. Steck, 2006, Contemporary Physics 47,279.

Jordan, A. N., and M. Buttiker, 2005, Phys. Rev. B 71,125333.

Jordan, A. N., and A. N. Korotkov, 2006, Phys. Rev. B 74,085307.

Kleckner, D., and D. Bouwmeester, 2006, Nature 444, 75.Knight, P. L., and V. Buzek, 2004, in Quantum Squeezing,

edited by P. D. Drummond and Z. Ficek (Springer, Berlin),pp. 3–32.

Knobel, R. G., and A. N. Cleland, 2003, Nature (London)424, 291.

Koch, R. H., D. J. van Harlingen, and J. Clarke, 1980, Appl.Phys. Lett. 38, 380.

Korotkov, A. N., 1999, Phys. Rev. B 60, 5737.Korotkov, A. N., 2001a, Phys. Rev. B 63, 085312.Korotkov, A. N., 2001b, Phys. Rev. B 63, 115403.Korotkov, A. N., and D. V. Averin, 2001, Phys. Rev. B 64,

165310.Korotkov, A. N., and M. A. Paalanen, 1999, Appl. Phys. Lett.

74, 4052.LaHaye, M. D., O. Buu, B. Camarota, and K. C. Schwab,

2004, Science 304, 74.

Lang, R., M. O. Scully, and J. Willis E. Lamb, 1973, Phys.Rev. A 7, 1788.

Lesovik, G. B., and R. Loosen, 1997, JETP Lett. 65, 295.Levinson, Y., 1997, Europhys. Lett. 39, 299.Levitov, L. S., 2003, in Quantum Noise in Mesoscopic Sys-

tems, edited by Y. Nazarov (Kluwer, Amsterdam), pp. 373–396.

Louisell, W. H., A. Yariv, and A. E. Siegman, 1961, Phys.Rev. 124, 1646.

Lupascu, A., C. J. P. M. Harmans, and J. E. Mooij, 2005,Phys. Rev. B. 71, 184506.

Makhlin, Y., G. Schon, and A. Shnirman, 2000, Phys. Rev.Lett. 85, 4578.

Makhlin, Y., G. Schon, and A. Shnirman, 2001, Rev. Mod.Phys. 73, 357.

Mallat, S., 1999, A Wavelet Tour of Signal Processing (Aca-demic, San Diego).

Marquardt, F., J. P. Chen, A. A. Clerk, and S. M. Girvin,2007, Phys. Rev. Lett. 99, 093902.

Marquardt, F., J. G. E. Harris, and S. M. Girvin, 2006, Phys.Rev. Lett. 96, 103901.

Meystre, P., E. M. Wright, J. D. McCullen, and E. Vignes,1985, J. Opt. Soc. Am. B 2, 1830.

Mollow, B. R., and R. J. Glauber, 1967a, Phys. Rev. 160,1076.

Mollow, B. R., and R. J. Glauber, 1967b, Phys. Rev. 160,1097.

Mozyrsky, D., I. Martin, and M. B. Hastings, 2004, Phys.Rev. Lett. 92, 018303.

Muck, M., J. B. Kycia, and J. Clarke, 2001, Appl. Phys. Lett.78, 967.

Nagourney, W., J. Sandberg, and H. Dehmelt, 1986, Phys.Rev. Lett. 56, 2797.

Naik, A., O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk,M. P. Blencowe, and K. C. Schwab, 2006, Nature (London)443, 193.

Nazarov, Y. V. (ed.), 2003, Quantum Noise in MesoscopicSystems (Kluwer, Amsterdam).

Nazarov, Y. V., and M. Kindermann, 2003, Eur. Phys. J. B35, 413.

von Neumann, J., 1932, Mathematische Grundlagen derQuantenmechanik (Springer, Berlin).

Nogues, G., A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M.Raimond, and S. Haroche, 1999, Nature (London) 400,239.

Onac, E., F. Balestro, L. H. W. van Beveren, U. Hartmann,Y. V. Nazarov, and L. P. Kouwenhoven, 2006a, Phys. Rev.Lett. 96, 176601.

Onac, E., F. Balestro, B. Trauzettel, C. F. J. Lodewijk, andL. P. Kouwenhoven, 2006b, Phys. Rev. Lett. 96, 026803.

Peres, A., 1993, Quantum Theory: Concepts and Methods(Kluwer, Dorderecht).

Pilgram, S., and M. Buttiker, 2002, Phys. Rev. Lett. 89,200401.

Poggio, M., M. P. Jura, C. L. Degen, M. A. Topinka, H. J.Mamin, D. Goldhaber-Gordon, and D. Rugar, 2008, NaturePhys. 4, 635.

Raimond, J. M., M. Brune, and S. Haroche, 2001, Rev. Mod.Phys. 73, 565.

Regal, C. A., J. D. Teufel, and K. W. Lehnert, 2008, NaturePhys. 4, 555.

Rugar, D., R. Budakian, H. J. Mamin, and B. W. Chui, 2004,Nature (London) 430, 329.

Ruskov, R., and A. N. Korotkov, 2002, Phys. Rev. B 66,

102

041401.Ruskov, R., K. Schwab, and A. N. Korotkov, 2005, Phys. Rev.

B 71, 235407.Santamore, D. H., A. C. Doherty, and M. C. Cross, 2004a,

Phys. Rev. B 70, 144301.Santamore, D. H., H.-S. Goan, G. J. Milburn, and M. L.

Roukes, 2004b, Phys. Rev. A 70, 052105.Sauter, T., W. Neuhauser, R. Blatt, and P. E. Toschek, 1986,

Phys. Rev. Lett. 57, 1696.Schliesser, A., P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J.

Kippenberg, 2006, Phys. Rev. Lett. 97, 243905.Schliesser, A., R. Riviere, G. Anetsberger, O. Arcizet, and

T. J. Kippenberg, 2008, Nature Phys. 4, 415.Schoelkopf, R. J., P. J. Burke, A. A. Kozhevnikov, D. E.

Prober, and M. J. Rooks, 1997, Phys. Rev. Lett. 78, 3370.Schoelkopf, R. J., A. A. Clerk, S. M. Girvin, K. W. Lehnert,

and M. H. Devoret, 2003, in Quantum Noise in MesoscopicSystems, edited by Y. Nazarov (Kluwer, Amsterdam), pp.175–203.

Schoelkopf, R. J., P. Wahlgren, A. A. Kozhevnikov, P. Dels-ing, and D. E. Prober, 1998, Science 280, 1238.

Schuster, D., A. Houck, J. Schreier, A. Wallraff, J. Gambetta,A. Blais, L. Frunzio, B. Johnson, M. Devoret, S. Girvin,and R. Schoelkopf, 2007, Nature 445, 515.

Schuster, D. I., A. Wallraff, A. Blais, L. Frunzio, R.-S. Huang,J. Majer, S. M. Girvin, and R. J. Schoelkopf, 2005, Phys.Rev. Lett. 94, 123602.

Shnirman, A., and G. Schon, 1998, Phys. Rev. B 57, 15400.Siddiqi, I., R. Vijay, F. Pierre, C. M. Wilson, M. Metcalfe,

C. Rigetti, L. Frunzio, and M. H. Devoret, 2004, Phys.Rev. Lett. 93, 207002.

Sillanpaa, M. A., T. Lehtinen, A. Paila, Y. Makhlin,L. Roschier, and P. J. Hakonen, 2005, Phys. Rev. Lett.95, 206806.

Sprinzak, D., E. Buks, M. Heiblum, and H. Shtrikman, 2000,Phys. Rev. Lett. 84, 5820.

Stern, A., Y. Aharonov, and Y. Imry, 1990, Phys. Rev. A 41,3436.

Teufel, J. D., C. A. Regal, and K. W. Lehnert, 2008, New J.Phys. 10, 095002.

Thalakulam, M., Z. Ji, and A. J. Rimberg, 2004, Phys. Rev.Lett. 93, 066804.

Thompson, J. D., B. M. Zwickl, A. M. Jayich, F. Marquardt,S. M. Girvin, and J. G. E. Harris, 2008, Nature (London)452, 06715.

Thorne, K. S., R. W. P. Drever, C. M. Caves, M. Zimmer-mann, and V. D. Sandberg, 1978, Phys. Rev. Lett. 40, 667.

Tittonen, I., G. Breitenbach, T. Kalkbrenner, T. Muller,R. Conradt, S. Schiller, E. Steinsland, N. Blanc, and N. F.de Rooij, 1999, Phys. Rev. A 59, 1038.

Turek, B. A., K. W. Lehnert, A. Clerk, D. Gunnarsson,K. Bladh, P. Delsing, and R. J. Schoelkopf, 2005, Phys.Rev. B 71, 193304.

Vinante, A., R. Mezzena, G. A. Prodi, S. Vitale, M. Cerdonio,P. Falferi, and M. Bonaldi, 2001, Appl. Phys. Lett. 79,2597.

Wallraff, A., D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang,J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf,2004, Nature (London) 431, 162.

Walls, D. F., and G. J. Milburn, 1994, Quantum Optics(Springer, Berlin).

Weiss, U., 1999, Quantum Dissipative Systems (Second Edi-tion) (World Scientific, Singapore).

Wheeler, J. A., and W. H. Zurek, 1984, Quantum Theory and

Measurement (Princeton University Press, Princeton).Wigner, E. P., 1955, Phys. Rev. 98, 145.Wilson-Rae, I., N. Nooshi, W. Zwerger, and T. J. Kippenberg,

2007, Phys. Rev. Lett. 99, 093901.Wineland, D. J., C. Monroe, W. M. Itano, D. Leibfried,

B. King, and D. M. Meekhof, 1998, J. Res. NIST 103,259.

Yurke, B., 1984, Phys. Rev. A 29, 408.Yurke, B., L. R. Corruccini, P. G. Kaminsky, L. W. Rupp,

A. D. Smith, A. H. Silver, R. W. Simon, and E. A. Whit-taker, 1989, Phys. Rev. A 39, 2519.

Yurke, B., and J. S. Denker, 1984, Phys. Rev. A 29, 1419.