REVIEWS OF MODERN PHYSICS, VOLUME 82, APRIL–JUNE 2010

Introduction to quantum noise, measurement, and amplification
A. A. Clerk*
Department of Physics, McGill University, 3600 rue University Montréal, Quebec, Canada
H3A 2T8

M. H. Devoret
Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,
Connecticut 06520-8284, USA

S. M. Girvin
Department of Physics, Yale University, P.O. Box 208120, New Haven,
Connecticut 06520-8120, USA

Florian Marquardt
Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37,
D-80333 München, Germany

R. J. Schoelkopf
Department of Applied Physics, Yale University, P.O. Box 208284, New Haven,
Connecticut 06520-8284, USA

共Published 15 April 2010兲
The topic of quantum noise has become extremely timely due to the rise of quantum information
physics and the resulting interchange of ideas between the condensed matter and atomic, molecular,
optical–quantum optics communities. This review gives a pedagogical introduction to the physics ofquantum noise and its connections to quantum measurement and quantum amplification. After
introducing quantum noise spectra and methods for their detection, the basics of weak continuous
measurements are described. Particular attention is given to the treatment of the standard quantum
limit on linear amplifiers and position detectors within a general linear-response framework. Thisapproach is shown how it relates to the standard Haus-Caves quantum limit for a bosonic amplifier
known in quantum optics and its application to the case of electrical circuits is illustrated, including
mesoscopic detectors and resonant cavity detectors.
DOI: 10.1103/RevModPhys.82.1155

PACS number共s兲: 72.70.⫹m

CONTENTS
I. Introduction
II. Quantum Noise Spectra
A. Introduction to quantum noise
B. Quantum spectrum analyzers
III. Quantum Measurements
A. Weak continuous measurements
B. Measurement with a parametrically coupledresonant cavity
1. QND measurement of the state of a qubit
using a resonant cavity
2. Quantum limit relation for QND qubit state
detection
3. Measurement of oscillator position using aresonant cavity
IV. General Linear-Response Theory
A. Quantum constraints on noise

1. Heuristic weak-measurement noise
constraints
2. Generic linear-response detector
3. Quantum constraint on noise
4. Evading the detector quantum noise
inequality
B. Quantum limit on QND detection of a qubit
V. Quantum Limit on Linear Amplifiers and Position
Detectors
A. Preliminaries on amplification
B. Standard Haus-Caves derivation of the quantum
limit on a bosonic amplifier
C. Nondegenerate parametric amplifier
1. Gain and added noise
2. Bandwidth-gain trade-off
3. Effective temperature
D. Scattering versus op-amp modes of operation
E. Linear-response description of a position detector
1. Detector back-action
2. Total output noise
3. Detector power gain
4. Simplifications for a quantum-ideal detector

1156
1159
1159
1161
1162
1164
1164
1167
1168
1169
1173
1173

*clerk@physics.mcgill.ca

0034-6861/2010/82共2兲/1155共54兲

1155

1173
1174
1175
1176
1177
1178
1178
1179
1181
1181
1182
1182
1183
1185
1185
1186
1186
1188

©2010 The American Physical Society

1156

Clerk et al.: Introduction to quantum noise, measurement, …

5. Quantum limit on added noise and noise
temperature
F. Quantum limit on the noise temperature of a
voltage amplifier
1. Classical description of a voltage amplifier
2. Linear-response description
3. Role of noise cross correlations
G. Near quantum-limited mesoscopic detectors
1. dc superconducting quantum interference
device amplifiers
2. Quantum point contact detectors
3. Single-electron transistors andresonant-level detectors
H. Back-action evasion and noise-free amplification
1. Degenerate parametric amplifier
2. Double-sideband cavity detector
3. Stroboscopic measurements
VI. Bosonic Scattering Description of a Two-Port
AmplifierA. Scattering versus op-amp representations
1. Scattering representation
2. Op-amp representation
3. Converting between representations
B. Minimal two-port scattering amplifier
1. Scattering versus op-amp quantum limit
2. Why is the op-amp quantum limit notachieved?
VII. Reaching the Quantum Limit in PracticeA. Importance of QND measurements
B. Power matching versus noise matching
VIII. Conclusions
Acknowledgments
References

1188
1190
1191
1192
1193
1194
1194
1194
1194
1195
1196
1196
1197
1198
1198
1198
1199
1200
1201
1201
1203
1203
1203
1204
1204
1205
1205

I. INTRODUCTION

Recently several advances have led to a renewed interest in the quantum-mechanical aspects of noise in mesoscopic electrical circuits, detectors, and amplifiers.
One motivation is that such systems can operate simultaneously at high frequencies and at low temperatures,
entering the regime where ប␻ ⬎ kBT. As such, quantum
zero-point fluctuations will play a more dominant role indetermining their behavior than the more familiar thermal fluctuations. A second motivation comes from therelation between quantum noise and quantum measurement. There exists an ever-increasing number of experiments in mesoscopic electronics where one is forced tothink about the quantum mechanics of the detection
process, and about fundamental quantum limits whichconstrain the performance of the detector or amplifier
used.
Given the above, we will focus in this review on discussing what is known as the “standard quantum limit”
共SQL兲 on both displacement detection and amplification. To preclude any possible confusion, it is worthwhile
to state explicitly from the start that there is no limit to
how well one may resolve the position of a particle in an
instantaneous measurement. Indeed, in the typical
Heisenberg microscope setup, one would scatter phoRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

tons off an electron, thereby detecting its position to an
accuracy set by the wavelength of photons used. The
fact that its momentum will suffer a large uncontrolled
perturbation, affecting its future motion, is of no concern here. Only as one tries to further increase the resolution will one finally encounter relativistic effects 共pair
production兲 that set a limit given by the Compton wavelength of the electron. The situation is obviously very
different if one attempts to observe the whole trajectoryof the particle. As this effectively amounts to measurements of both position and momentum, there has to be atrade-off between the accuracies of both, set by the
Heisenberg uncertainty relation. This is enforced inpractice by the uncontrolled perturbation of the momentum during one position measurement adding to thenoise in later measurements, a phenomenon known as“measurement back-action.”
Just such a situation is encountered in “weak measurements” 共Braginsky and Khalili, 1992兲, where one integrates the signal over time, gradually learning more
about the system being measured; this review will focus
on such measurements. There are many good reasons
why one may be interested in doing a weak measurement, rather than an instantaneous, strong, projective
measurement. On a practical level, there may be limitations to the strength of the coupling between the systemand the detector, which have to be compensated by integrating the signal over time. One may also deliberately
opt not to disturb the system too strongly, e.g., to be able
to apply quantum feedback techniques for state control.
Moreover, as one reads out an oscillatory signal overtime, one effectively filters away noise 共e.g., of a technical nature兲 at other frequencies. Finally, consider an example like detection of the collective coordinate of motion of a micromechanical beam. Its zero-point
uncertainty 共ground-state position fluctuation兲 is typically on the order of the diameter of a proton. It is outof the question to reach this accuracy in an instantaneous measurement by scattering photons of such asmall wavelength off the structure, since they would instead resolve the much larger position fluctuations of theindividual atoms comprising the beam 共and induce allkinds of unwanted damage兲, instead of reading out thecenter-of-mass coordinate. The same holds true forother collective degrees of freedom.
The prototypical example we discuss is that of a weakmeasurement detecting the motion of a harmonic oscillator 共such as a mechanical beam兲. The measurement
then actually follows the slow evolution of amplitude
and phase of the oscillations 共or, equivalently, the twoquadrature components兲, and the SQL derives from thefact that these two observables do not commute. It essentially says that the measurement accuracy will be limited to resolving both quadratures down to the scale of
the ground-state position fluctuations, within one mechanical damping time. Note that, in special applications, one might be interested only in one particularquadrature of motion. Then the Heisenberg uncertainty
relation does not enforce any SQL and one may again

Clerk et al.: Introduction to quantum noise, measurement, …

obtain unlimited accuracy, at the expense of renouncing
all knowledge of the other quadrature.
Position detection by weak measurement essentiallyamounts to amplification of the quantum signal up to aclassically accessible level. Therefore, the theory ofquantum limits on displacement detection is intimately
connected to limits on how well an amplifier can work. If
an amplifier does not have any preference for any particular phase of the oscillatory signal, it is called “phasepreserving,” which is the case relevant for amplifying
and thereby detecting both quadratures equally well.1
We derive and discuss the SQL for phase-preserving linear amplifiers 共Haus and Mullen, 1962; Caves, 1982兲.
Quantum mechanics demands that such an amplifieradds noise that corresponds to half a photon added toeach mode of the input signal, in the limit of highphoton-number gain G. In contrast, for small gain, theminimum number of added noise quanta, 共1 − 1 / G兲 / 2,
can become arbitrarily small as the gain is reduced down
to 1 共no amplification兲. One might ask, therefore,
whether it should not be possible to evade the SQL by
being content with small gains. The answer is no, sincehigh gains G Ⰷ 1 are needed to amplify the signal to alevel where it can be read out 共or further amplified兲 using classical devices without their noise having any further appreciable effect, converting 1 input photon intoG Ⰷ 1 output photons. According to Caves, it is necessary to generate an output that “we can lay our grubby,
classical hands on” 共Caves, 1982兲. It is a simple exercise
to show that feeding the input of a first, potentially lowgain amplifier into a second amplifier results in an overall bound on the added noise that is just the one expected for the product of their respective gains.
Therefore, as one approaches the classical level, i.e.,large overall gains, the SQL always applies in its simplified form of half a photon added.
Unlike traditional discussions of the amplifier SQL,
here we devote considerable attention to a generallinear-response approach based on the quantum relation
between susceptibilities and noise. This approach treats
the amplifier or detector as a black box with an input
port coupling to the signal source and an output port toaccess the amplified signal. It is more suited for mesoscopic systems than the quantum optics scattering-type
approach, and it leads us to the quantum noise inequality: a relation between the noise added to the output andthe back-action noise feeding back to the signal source.
In the ideal case 共what we term a “quantum-limited detector”兲, the product of these two contributions reaches
the minimum value allowed by quantum mechanics. We
show that optimizing this inequality on noise is a necessary prerequisite for having a detector achieve the quantum limit in a specific measurement task, such as linear
amplification.
There are several motivations for understanding in
1
In the literature this is often referred to as a “phase insensitive” amplifier. We prefer the term “phase preserving” to avoid
any ambiguity.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1157

principle, and realizing in practice, amplifiers whose
noise reaches this minimum quantum limit. Achieving
the quantum limit on continuous position detection has
been one of the goals of many recent experiments on
quantum electromechanical 共Cleland et al., 2002; Knobel
and Cleland, 2003; LaHaye et al., 2004; Naik et al., 2006;
Flowers-Jacobs et al., 2007; Etaki et al., 2008; Poggio etal., 2008; Regal et al., 2008兲 and optomechanical systems
共Arcizet et al., 2006; Gigan et al., 2006; Schliesser et al.,2008; Thompson et al., 2008; Marquardt and Girvin,
2009兲. As we will show, having a near-quantum-limited
detector would allow one to continuously monitor thequantum zero-point fluctuations of a mechanical resonator. It is also necessary to have a quantum-limited detector is for such tasks as single-spin NMR detection 共Rugar et al., 2004兲, as well as gravitational wave detection
共Abramovici et al., 1992兲. The topic of quantum-limited
detection is also directly relevant to recent activity exploring feedback control of quantum systems 共Wiseman
and Milburn, 1993, 1994; Doherty et al., 2000; Korotkov,
2001b; Ruskov and Korotkov, 2002兲; such schemes require need a close-to-quantum-limited detector.
This review is organized as follows. We start in Sec. II
by providing a review of the basic statistical propertiesof quantum noise, including its detection. In Sec. III weturn to quantum measurements and give a basic introduction to weak continuous measurements. To makethings concrete, we discuss heuristically measurementsof both a qubit and an oscillator using a simple resonantcavity detector, giving an idea of the origin of the quantum limit in each case. Section IV is devoted to a more
rigorous treatment of quantum constraints on noise arising from general quantum linear-response theory. The
heart of the review is contained in Sec. V, where we givea thorough discussion of quantum limits on amplificationand continuous position detection. We also discuss various methods for beating the usual quantum limits on
added noise using back-action evasion techniques. We
are careful to distinguish two very distinct modes of amplifier operation 共the “scattering” versus “op-amp”
modes兲; we expand on this in Sec. VI, where we discussboth modes of operation in a simple two-port bosonic
amplifier. Importantly, we show that an amplifier can bequantum limited in one mode of operation, but fail to bequantum limited in the other mode of operation. Finally,in Sec. VII we highlight a number of practical considerations that one must keep in mind when trying to perform a quantum-limited measurement. Table I provides
a synopsis of the main results discussed in the text as
well as definitions of symbols used.
In addition to the above, we have supplemented themain text with several pedagogical appendixes thatcover basic background topics. Particular attention is
given to the quantum mechanics of transmission linesand driven electromagnetic cavities, topics that are especially relevant given recent experiments making use of
microwave stripline resonators. These appendixes arecontained in a separate online-only supplementary
document 共Clerk et al., 2009兲 共see also http://arxiv.org/
abs/0810.4729兲. In Table II, we list the contents of theseClerk et al.: Introduction to quantum noise, measurement, …

1158

TABLE I. Table of symbols and main results.
Symbolf关␻兴SFF关␻兴
SFF关␻兴

Definition or result
General definitions
Fourier transform of the function or operator f共t兲, defined via f关␻兴 = 兰⬁−⬁dtf共t兲ei␻t„Note that for operators, we use the convention f̂†关␻兴 = 兰⬁−⬁dtf̂†共t兲ei␻t, implying f̂†关␻兴 = 共f̂关−␻兴兲†…
i␻tClassical noise spectral density or power spectrum: SFF关␻兴 = 兰+⬁−⬁dte 具F共t兲F共0兲典i␻tQuantum noise spectral density: SFF关␻兴 = 兰+⬁−⬁dte 具F̂共t兲F̂共0兲典

S̄FF关␻兴
␹AB共t兲

i␻tSymmetrized quantum noise spectral density S̄FF关␻兴 = 21 共SFF关␻兴 + SFF关−␻兴兲 = 21 兰+⬁−⬁dte 具兵F̂共t兲 , F̂共0兲其典General linear-response susceptibility describing the response of A to a perturbation that couples to B; in thequantum case, given by the Kubo formula ␹AB共t兲 = −共i / ប兲␪共t兲具关Â共t兲 , B̂共0兲兴典 关Eq. 共2.14兲兴

A

Coupling constant 共dimensionless兲 between measured system and detector or amplifier,e.g., V̂ = AF共t兲␴ˆ x, V̂ = Ax̂F̂,or V̂ = Aប␻c␴ˆ zâ†â
Mass and angular frequency of a mechanical harmonic oscillator
Zero-point uncertainty of a mechanical oscillator, xZPF = 冑ប / 2M⍀

M,⍀
xZPF
␥0

␻c
␬ , QcTeff关␻兴

Intrinsic damping rate of a mechanical oscillator due to coupling to a bath via V̂ = Ax̂F̂: ␥0 = 共A2 / 2Mប⍀兲共SFF关⍀兴− SFF关−⍀兴兲 关Eq. 共2.12兲兴
Resonant frequency of a cavity
Damping, quality factor of a cavity: Qc = ␻c / ␬Sec. II
Effective temperature at a frequency ␻ for a given quantum noise spectrum, defined via SFF关␻兴 / SFF关−␻兴= exp共ប␻ / kBTeff关␻兴兲 关Eq. 共2.8兲兴
Fluctuation-dissipation theorem relating the symmetrized noise spectrum to the dissipative part for an
equilibrium bath: S̄FF关␻兴 = 21 coth共ប␻ / 2kBT兲共SFF关␻兴 − SFF关−␻兴兲 关Eq. 共2.16兲兴
Sec. IIIṄ␦␪S̄0xx共␻兲Number-phase uncertainty relation for a coherent state: ⌬N⌬␪ 艌 21 关Eqs. 共3.6兲 and 共G12兲兴
Photon-number flux of a coherent beam
Imprecision noise in the measurement of the phase of a coherent beam
Fundamental noise constraint for an ideal coherent beam: SN˙N˙S␪␪ = 41 关Eqs. 共3.8兲 and 共G21兲兴
Symmetrized spectral density of zero-point position fluctuations of a damped harmonic oscillatorS̄xx,tot共␻兲Total output noise spectral density 共symmetrized兲 of a linear position detector, referred back to the oscillatorS̄xx,add共␻兲Added noise spectral density 共symmetrized兲 of a linear position detector, referred back to the oscillator
Sec. IVx̂Input signal

F̂

Fluctuating force from the detector, coupling to x̂ via V̂ = Ax̂F̂
Detector output signal

Î

␣
⌫meas
⌫␸

General quantum constraint on the detector output noise, back-action noise, and gain: S̄II关␻兴S̄FF关␻兴 − 兩S̄IF关␻兴兩2艌 兩ប␹˜ IF关␻兴 / 2兩2共1 + ⌬†S̄IF关␻兴 / ប␭˜ 关␻兴 / 2‡兲 关Eq. 共4.11兲兴, where ␹˜ IF关␻兴 ⬅ ␹IF关␻兴 − †␹FI关␻兴‡* and ⌬关z兴 = 关兩1 + z2兩 − 共1 + 兩z兩2兲兴 / 2„Note that 1 + ⌬关z兴 艌 0 and ⌬ = 0 in most cases of relevance; see discussion around Eq. 共4.17兲…
Complex proportionality constant characterizing a quantum-ideal detector: 兩␣兩2 = S̄II / S̄FF and sin共arg ␣关␻兴兲= ប兩␭关␻兴兩 / 2 / 冑S̄II关␻兴S̄FF关␻兴 关Eqs. 共4.18兲 and 共I17兲兴
Measurement rate 关for a quantum nondemolition 共QND兲 qubit measurement兴 关Eq. 共4.24兲兴
Dephasing rate 共due to measurement back-action兲 关Eqs. 共3.27兲 and 共4.19兲兴
Constraint on weak, continuous QND qubit state detection: ␩ = ⌫meas / ⌫␸ 艋 1 关Eq. 共4.25兲兴
Sec. VG

Photon-number 共power兲 gain, e.g., in Eq. 共5.7兲

Input-output relation for a bosonic scattering amplifier: b̂† = 冑Gâ† + F̂† 关Eq. 共5.7兲兴

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Clerk et al.: Introduction to quantum noise, measurement, …

1159

TABLE I. 共Continued.兲
SymbolDefinition or result

共⌬a兲2

Symmetrized field operator uncertainty for the scattering description of a bosonic amplifier: 共⌬a兲2 ⬅ 21 具兵â , â†其典− 兩具a典兩2
Standard quantum limit for the noise added by a phase-preserving bosonic scattering amplifier in the high-gain
limit, G Ⰷ 1, where 具共⌬a兲2典ZPF = 21 : 共⌬b兲2 / G 艌 共⌬a兲2 + 21 关Eq. 共5.10兲兴
Dimensionless power gain of a linear position detector or voltage amplifier共maximum ratio of the power
delivered by the detector output to a load, vs the power fed into signal source兲:GP关␻兴 = 兩␹IF关␻兴兩2 / 关共4 Im ␹FF关␻兴Im ␹II关␻兴兲 关Eq. 共5.52兲兴
For a quantum-ideal detector, in the high-gain limit: GP ⯝ 关共Im ␣ / 兩␣ 兩 兲共4 kBTeff / ប␻兲兴2 关Eq. 共5.57兲兴

G P关 ␻ 兴S̄xx,eq关␻ , T兴
Aopt

␥关Aopt兴Zin , Zout
Zs
␭V
Ṽ共t兲
Ĩ共t兲
TN
ZN

Intrinsic equilibrium noise S̄xx,eq关␻ , T兴 = ប coth共ប␻ / 2kBT兲共−Im ␹xx关␻兴兲 关Eq. 共5.59兲兴
4
Optimal coupling strength of a linear position detector which minimizes the added noise at frequency ␻: Aopt
关␻兴2= S̄II关␻兴 / 共兩␭关␻兴␹xx关␻兴兩 S̄FF关␻兴兲 关Eq. 共5.64兲兴
Detector-induced damping of a quantum-limited linear position detector at optimal coupling; satisfies ␥关Aopt兴 / ␥0
+ ␥关Aopt兴 = 兩Im ␣ / ␣兩共1 / 冑GP关⍀兴兲 = ប⍀ / 4kBTeff Ⰶ 1 关Eq. 共5.69兲兴
Standard quantum limit for the added noise spectral density of a linear position detector 共valid at each frequency
␻兲: Sxx,add关␻兴 艌 limT→0 Sxx,eq关w , T兴 关Eq. 共5.62兲兴
Effective increase in oscillator temperature due to coupling to the detector back-action, for an ideal detector,
with ប⍀ / kB Ⰶ Tbath Ⰶ Teff: Tosc ⬅ 共␥Teff + ␥0Tbath兲 / 共␥ + ␥0兲 → ប⍀ / 4kB + Tbath 关Eq. 共5.70兲兴
Input and output impedances of a linear voltage amplifier
Impedance of signal source attached to input of a voltage amplifier
Voltage gain of a linear voltage amplifier
Voltage noise of a linear voltage amplifier „Proportional to the intrinsic output noise of the genericlinear-response detector 关Eq. 共5.81兲兴…
Current noise of a linear voltage amplifier „Related to the back-action force noise of the generic linear-response
detector 关Eqs. 共5.80兲兴…
Noise temperature of an amplifier 关defined in Eq. 共5.74兲兴
Noise impedance of a linear voltage amplifier 关Eq. 共5.77兲兴
Standard quantum limit on the noise temperature of a linear voltage amplifier: kBTN关␻兴 艌 ប␻ / 2 关Eq. 共5.89兲兴
Sec. VIV̂a 共V̂b兲

Voltage at the input 共output兲 of the amplifierÎa 共Îb兲

Relation to bosonic mode operators: Eq. 共6.2a兲
Current drawn at the input 共leaving the output兲 of the amplifier

␭I⬘
s关␻兴
ˆ ˆ
Ṽ 共Ĩ兲

Relation to bosonic mode operators: Eq. 共6.2b兲
Reverse current gain of the amplifier
Input-output 2 ⫻ 2 scattering matrix of the amplifier 关Eq. 共6.3兲兴
Relation to op-amp parameters ␭V , ␭I⬘ , Zin , Zout: Eqs. 共6.7兲
Voltage 共current兲 noise operators of the amplifierF̂a关␻兴 , F̂b关␻兴 Input 共output兲 port noise operators in the scattering description 关Eq. 共6.3兲兴
ˆ ˆ
Relation to op-amp noise operators Ṽ , Ĩ: Eq. 共6.9兲

Appendixes. Note that, while some aspects of the topics
discussed in this review have been studied in the quantum optics and quantum dissipative systems communities and are the subject of several comprehensive books
共Braginsky and Khalili, 1992; Weiss, 1999; Gardiner andZoller, 2000; Haus, 2000兲, they are somewhat newer tothe condensed matter physics community; moreover,some of the technical machinery developed in thesefields is not directly applicable to the study of quantum
noise in quantum electronic systems. Finally, note thatwhile this article is a review, there is considerable newRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

material presented, especially in our discussion of quantum amplification 共see Secs. V.D and VI兲.
II. QUANTUM NOISE SPECTRA
A. Introduction to quantum noise

In classical physics, the study of a noisy timedependent quantity invariably involves its spectral density S关␻兴. The spectral density tells us the intensity of thenoise at a given frequency and is directly related to theClerk et al.: Introduction to quantum noise, measurement, …

1160

TABLE II. Contents of online appendix material. Page numbers refer to the supplementary material.
SectionPage

A. Basics of classical and quantum noise
B. Quantum spectrum analyzers: further details
C. Modes, transmission lines and classicalinput-output theory
D. Quantum modes and noise of a transmission line
E. Back-action and input-output theory for driven
damped cavities
F. Information theory and measurement rate
G. Number phase uncertainty
H. Using feedback to reach the quantum limit
I. Additional technical details

1
4
8
15
18
29
30
31
34

2
Gxx共t兲 = xZPF
兵nB共ប⍀兲e+i⍀t + 关nB共ប⍀兲 + 1兴e−i⍀t其, 共2.3兲

autocorrelation function of the noise.2 In a similar fashion, the study of quantum noise involves quantum noise
spectral densities. These are defined in a manner thatmimics the classical case
Sxx关␻兴 =冕+⬁dtei␻t具x̂共t兲x̂共0兲典.

Note, however, that in the quantum case the canonicalcommutation relation between position and momentum
implies there must be some correlations between thetwo, namely, 具x̂共0兲p̂共0兲典 − 具p̂共0兲x̂共0兲典 = iប. These correlations are easily evaluated by writing x̂ and p̂ in terms of
harmonic oscillator ladder operators. We find that inthermal equilibrium 具p̂共0兲x̂共0兲典 = −iប / 2 and 具x̂共0兲p̂共0兲典= +iប / 2. Not only are the position and momentum correlated, but their correlator is imaginary:3 This means
that, despite the fact that the position is Hermitian observable with real eigenvalues, its autocorrelation function is complex and given from Eq. 共2.2兲 by

共2.1兲

2
⬅ ប / 2M⍀ is the RMS zero-point uncertainty
where xZPF
of x in the quantum ground state and nB is the BoseEinstein occupation factor. The complex nature of the
autocorrelation function follows from the fact that theoperator x̂ does not commute with itself at different
times.
Because the correlator is complex it follows that the
spectral density is not symmetric in frequency,−⬁

Here x̂ is a quantum operator 共in the Heisenberg representation兲 whose noise we are interested in, and the angular brackets indicate the quantum statistical average
evaluated using the quantum density matrix. Note that
we use S关␻兴 throughout this review to denote the spectral density of a classical noise, while S关␻兴 will denote aquantum noise spectral density.
As a simple introductory example illustrating important differences from the classical limit, consider the position noise of a simple harmonic oscillator having mass
M and frequency ⍀. The oscillator is maintained in equilibrium with a large heat bath at temperature T via some
infinitesimal coupling, which we ignore in considering
the dynamics. The solutions of the Heisenberg equationsof motion are the same as for the classical case but with
the initial position x and momentum p replaced by thecorresponding quantum operators. It follows that theposition autocorrelation function is
Gxx共t兲 = 具x̂共t兲x̂共0兲典 = 具x̂共0兲x̂共0兲典cos共⍀t兲+ 具p̂共0兲x̂共0兲典

1
sin共⍀t兲.
M⍀

共2.2兲

Classically the second term on the right-hand side
共RHS兲 vanishes because in thermal equilibrium x and p
are uncorrelated random variables. As we will see
shortly for the quantum case, the symmetrized 共sometimes called the “classical”兲 correlator vanishes in thermal equilibrium, just as it does classically: 具x̂p̂ + p̂x̂典 = 0.
2
For readers unfamiliar with the basics of classical noise, acompact review is given in Appendix A 共supplementarymaterial兲.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

2
Sxx关␻兴 = 2␲xZPF
兵nB共ប⍀兲␦共␻ + ⍀兲+ 关nB共ប⍀兲 + 1兴␦共␻ − ⍀兲其.

共2.4兲In contrast, a classical autocorrelation function is always
real, and hence a classical noise spectral density is always symmetric in frequency. Note that in the hightemperature limit kBT Ⰷ ប⍀ we have nB共ប⍀兲 ⬃ nB共ប⍀兲+ 1 ⬃ kBT / ប⍀. Thus, in this limit Sxx关␻兴 becomes symmetric in frequency as expected classically, and coincides
with the classical expression for the position spectral
density 关cf. Eq. 共A12兲兴.
The Bose-Einstein factors suggest a way to understand the frequency asymmetry of Eq. 共2.4兲: the positivefrequency part of the spectral density has to do withstimulated emission of energy into the oscillator and thenegative-frequency part of the spectral density has to dowith emission of energy by the oscillator. That is, thepositive-frequency part of the spectral density is a measure of the ability of the oscillator to absorb energy,
while the negative-frequency part is a measure of theability of the oscillator to emit energy. As we will see,this is generally true, even for nonthermal states. Figure
1 shows this idea for the case of the voltage noise spectral density of a resistor 共see Appendix D.3 for moredetails兲. Note that the result Eq. 共2.4兲 can be extended tothe case of a bath of many harmonic oscillators. As described in Appendix D a resistor can be modeled as an
infinite set of harmonic oscillators and from this model
the Johnson or Nyquist noise of a resistor can be derived.
3

Notice that this occurs because the product of two noncommuting Hermitian operators is not itself a Hermitian operator.

Clerk et al.: Introduction to quantum noise, measurement, …
10

tailed balance relation holds. However, if we are concerned only with a single particular frequency, then it isalways possible to define an “effective temperature” Teff
for the noise using Eq. 共2.7兲, i.e.,8

SVV [ ]/2RkBT

6
4
2
0
-10

-5

<0
emissionby reservoir

0

h /kBT

5

kBTeff关␻兴 ⬅

10>0absorption
by reservoir

FIG. 1. Quantum noise spectral density of voltage fluctuations
across a resistor 共resistance R兲 as a function of frequency atzero temperature 共dashed line兲 and finite temperature 共solidline兲.
B. Quantum spectrum analyzers

The qualitative picture described previously can be
confirmed by considering simple systems which act aseffective spectrum analyzers of quantum noise. The simplest such example is a quantum two-level system 共TLS兲
coupled to a quantum noise source 共Aguado and Kouwenhoven, 2000; Gavish et al., 2000; Schoelkopf et al.,2003兲. With the TLS described as a fictitious spin-1 / 2
particle with spin down 共spin up兲 representing the
ground state 共excited state兲, its Hamiltonian is Ĥ0= 共ប␻01 / 2兲␴ˆ z, where ប␻01 is the energy splitting between
the two states. The TLS is then coupled to an external
noise source via an additional term in the Hamiltonian,V̂ = AF̂␴ˆ x ,共2.5兲

where A is a coupling constant and the operator F̂ represents the external noise source. The coupling Hamiltonian V̂ can lead to the exchange of energy between the
two-level system and noise source and hence transitions
between its two eigenstates. The corresponding Fermi
golden rule transition rates can be compactly expressedin terms of the quantum noise spectral density of F̂,
SFF关␻兴,⌫↑ = 共A /ប 兲SFF关− ␻01兴,

共2.6a兲

⌫↓ = 共A2/ប2兲SFF关+ ␻01兴.

共2.6b兲

21161

2

Here ⌫↑ is the rate at which the qubit is excited from its
ground to excited state and ⌫↓ is the corresponding rate
for the opposite relaxation process. As expected,
positive- 共negative-兲 frequency noise corresponds to absorption 共emission兲 of energy by the noise source. Notethat if the noise source is in thermal equilibrium at temperature T, the transition rates of the TLS must satisfy
the detailed balance relation ⌫↑ / ⌫↓ = e−␤ប␻01, where ␤= 1 / kBT. This in turn implies that in thermal equilibrium
the quantum noise spectral density must satisfy
SFF关+ ␻01兴 = e␤ប␻01SFF关− ␻01兴.

共2.7兲

The more general situation is where the noise source is
not in thermal equilibrium; in this case, no general deRev. Mod. Phys., Vol. 82, No. 2, April–June 2010ប␻.
log†SFF关␻兴/SFF关− ␻兴‡

共2.8兲

Note that for a nonequilibrium system Teff will in general be frequency dependent. In NMR language, Teff will
simply be the “spin temperature” of our TLS spectrometer once it reaches steady state after being coupled to
the noise source.
Another simple quantum noise spectrometer is a harmonic oscillator 共frequency ⍀, mass M, and position x兲
coupled to a noise source 关see, e.g., Schwinger 共1961兲
and Dykman 共1978兲兴. The coupling Hamiltonian is nowV̂ = Ax̂F̂ = A关xZPF共â + â†兲兴F̂,

共2.9兲

where â is the oscillator annihilation operator, F̂ is theoperator describing the fluctuating noise, and A is again
a coupling constant. We see that −AF̂ plays the role of afluctuating force acting on the oscillator. In complete
analogy to the previous section, noise in F̂ at the oscillator frequency ⍀ can cause transitions between the oscillator energy eigenstates. The corresponding Fermi
golden rule transition rates are again simply related to
the noise spectrum SFF关␻兴. Incorporating these rates
into a simple master equation describing the probability
to find the oscillator in a particular energy state, one
finds that the stationary state of the oscillator is a BoseEinstein distribution evaluated at the effective temperature Teff关⍀兴 defined in Eq. 共2.8兲. Further, one can use themaster equation to derive a classical-looking equation
for the average energy 具E典 of the oscillator 共see Appendix B.2兲,d具E典/dt = P − ␥具E典,

共2.10兲

where
P = 共A2/4M兲共SFF关⍀兴 + SFF关− ⍀兴兲 ⬅ A2S̄FF关⍀兴/2M,
共2.11兲
2
␥ = 共A2xZPF/ប2兲共SFF关⍀兴 − SFF关− ⍀兴兲.

共2.12兲

The two terms in Eq. 共2.10兲 describe, heating and damping of the oscillator by the noise source, respectively.
The heating effect of the noise is completely analogous
to what happens classically: a random force causes theoscillator’s momentum to diffuse, which in turn causes
具E典 to grow linearly in time at rate proportional to theforce noise spectral density. In the quantum case, Eq.
共2.11兲 indicates that it is the symmetric-in-frequency partof the noise spectrum, S̄FF关⍀兴, which is responsible forthis effect, and which thus plays the role of a classicalnoise source. This is another reason why S̄FF关␻兴 is often

Clerk et al.: Introduction to quantum noise, measurement, …

1162

referred to as the “classical” part of the noise.4 In contrast, we see that the asymmetric-in-frequency part of
the noise spectrum is responsible for the damping. This
also has a simple heuristic interpretation: damping is
caused by the net tendency of the noise source to absorb, rather than emit, energy from the oscillator.
The damping induced by the noise source may equivalently be attributed to the oscillator’s motion inducing an
average value to 具F典 which is out of phase with x, i.e.,
␦具AF共t兲典 = −M␥ẋ共t兲. Standard quantum linear-response
theory yields

␦具AF̂共t兲典 = A2

冕 ⬘dt ␹FF共t − t⬘兲具x̂共t⬘兲典,

共2.13兲

where we have introduced the susceptibility

␹FF共t兲 = 共− i/ប兲␪共t兲具关F̂共t兲,F̂共0兲兴典.

共2.14兲

Using the fact that the oscillator’s motion involves only
the frequency ⍀, we thus have2
␥ = 共2A2xZPF/ប兲†− Im ␹FF关⍀兴‡.

共2.15兲

A straightforward manipulation of Eq. 共2.14兲 for ␹FFshows that this expression for ␥ is exactly equivalent toour previous expression, Eq. 共2.12兲.
In addition to giving insight on the meaning of thesymmetric and asymmetric parts of a quantum noise
spectral density, the above example also directly yields
the quantum version of the fluctuation-dissipation theorem 共Callen and Welton, 1951兲. As we saw earlier, if ournoise source is in thermal equilibrium, the positive- andnegative-frequency parts of the noise spectrum arestrictly related to one another by the condition of detailed balance 关cf. Eq. 共2.7兲兴. This in turn lets us link theclassical symmetric-in-frequency part of the noise to thedamping 共i.e., the asymmetric-in-frequency part of thenoise兲. Setting ␤ = 1 / kBT and making use of Eq. 共2.7兲, we
haveS̄FF关⍀兴 ⬅

SFF关⍀兴 + SFF关− ⍀兴2= 21 coth共␤ប⍀/2兲共SFF关⍀兴 − SFF关− ⍀兴兲= coth共␤ប⍀/2兲

ប⍀M
␥关⍀兴.
A2

共2.16兲

Thus, in equilibrium, the condition that noise-induced
transitions obey detailed balance immediately implies
that noise and damping are related to one another via
the temperature. For T Ⰷ ប⍀, we recover the more familiar classical version of the fluctuation-dissipation
theorem
Note that with our definition 具F̂2典 = 兰⬁−⬁共d␻ / 2␲兲S̄FF关␻兴. It is
common in engineering contexts to define so-called one-sided
classical spectral densities, which are equal to twice ourdefinition.
4

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

A2S̄FF关⍀兴 = 2kBTM␥ .

共2.17兲

Further insight into the fluctuation-dissipation theorem
is provided in Appendix C.3, where we discuss it in thesimple but instructive context of a transmission line terminated by an impedance Z关␻兴.
We have thus considered two simple examples ofmethods of measure quantum noise spectral densities.
Further details, as well as examples of other quantum
noise spectrum analyzers, are given in Appendix B.

III. QUANTUM MEASUREMENTS

Having introduced both quantum noise and quantum
spectrum analyzers, we are now in a position to introduce the general topic of quantum measurements. All
practical measurements are affected by noise. Certain
quantum measurements remain limited by quantum
noise even though they use completely ideal apparatus.
As we will see, the limiting noise here is associated withthe fact that canonically conjugate variables are incompatible observables in quantum mechanics.
The simplest idealized description of a quantum measurement, introduced by von Neumann 共von Neumann,
1932; Wheeler and Zurek, 1984; Bohm, 1989; Harocheand Raimond, 2006兲, postulates that the measurement
process instantaneously collapses the system’s quantum
state onto one of the eigenstates of the observable to bemeasured. As a consequence, any initial superposition ofthese eigenstates is destroyed and the values of observables conjugate to the measured observable are perturbed. This perturbation is an intrinsic feature of quantum mechanics and cannot be avoided in any
measurement scheme, be it of the projection type described by von Neumann or a rather weak continuous
measurement to be analyzed further below.
To form a more concrete picture of quantum measurement, we begin by noting that every quantum measurement apparatus consists of a macroscopic “pointer”
coupled to the microscopic system to be measured. 关Aspecific model is discussed by Allahverdyan et al.
共2001兲.兴 This pointer is sufficiently macroscopic that its
position can be read out “classically.” The interaction
between the microscopic system and the pointer is arranged so that the two become strongly correlated. Oneof the simplest possible examples of a quantum measurement is that of the Stern-Gerlach apparatus, whichmeasures the projection of the spin of an S = 1 / 2 atomalong some chosen direction. What is really measured in
the experiment is the final position of the atom on thedetector plate. However, the magnetic field gradient in
the magnet causes this position to be perfectly correlated 共“entangled”兲 with the spin projection so that thelatter can be inferred from the former. Suppose, for example that the initial state of the atom is the product ofa spatial wave function ␰0共rជ兲 centered on the entrance tothe magnet and a spin state that is the superposition ofup and down spins corresponding to the eigenstate of ␴ˆ x,

Clerk et al.: Introduction to quantum noise, measurement, …

+d
z

z

FIG. 2. 共Color online兲 Schematic of position distributions of an
atom in the detector plane of a Stern-Gerlach apparatus whose
field gradient is in the z direction. For small values of thedisplacement d 共described in the text兲, there is significant overlap of the distributions and the spin cannot be unambiguously
inferred from the position. For large values of d, the spin isperfectly entangled with position and can be inferred from theposition. This is the limit of strong projective measurement.

兩⌿0典 = 共1/冑2兲兵兩↑典 + 兩↓典其兩␰0典.

共3.1兲

After passing through a magnet with field gradient in thez direction, an atom with spin up is deflected upward
and an atom with spin down is deflected downward. By
the linearity of quantum mechanics, an atom in a spin
superposition state thus ends up in a superposition of
the form兩⌿1典 = 共1/冑2兲兵兩↑典兩␰+典 + 兩↓典兩␰−典其,共3.2兲

where 具rជ 兩 ␰±典 = ␺1共rជ ± dẑ兲 are spatial orbitals peaked in theplane of the detector. The deflection d is determined by
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 sufficiently large compared to the wave packet spread then,given the position of the particle, one can unambiguously determine the distribution from which it came andhence the value of the spin projection of the atom. Thisis the limit of a strong “projective” measurement.
In the initial state one has 具⌿0兩␴ˆ x兩⌿0典 = +1, but in the
final state one has
具⌿1兩␴ˆ x兩⌿1典 = 21 兵具␰−兩␰+典 + 具␰+兩␰−典其.

共3.3兲

For sufficiently large d the states ␰± are orthogonal andthus the act of ␴ˆ z measurement destroys the spin coherence具⌿1兩␴ˆ x兩⌿1典 → 0.

共3.4兲This is what we mean by projection or wave function“collapse.” The result of measurement of the atom position will yield a random and unpredictable value of ± 21
for the z projection of the spin. This destruction of thecoherence in the transverse spin components by a strong
measurement of the longitudinal spin component is thefirst of many examples we will see of the Heisenberg
uncertainty principle in action. Measurement of one
variable destroys information about its conjugate variable. We study several examples in which we understandmicroscopically how it is that the coupling to the measurement apparatus causes the back-action quantum
noise which destroys our knowledge of the conjugate
variable.
In the special case where the eigenstates of the observable we are measuring are also stationary states 共i.e.,energy eigenstates兲, a second measurement of the observable would reproduce exactly the same result, thus
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1163

providing a way to confirm the accuracy of the measurement scheme. These optimal kinds of repeatable measurements are called quantum nondemolition 共QND兲
measurements 共Braginsky et al., 1980; Braginsky andKhalili, 1992, 1996; Peres, 1993兲. A simple example
would be a sequential pair of Stern-Gerlach devices oriented in the same direction. In the absence of stray magnetic perturbations, the second apparatus would always
yield the same answer as the first. In practice, one terms
a measurement QND if the observable being measured
is an eigenstate of the ideal Hamiltonian of the measured system 共i.e., one ignores any couplings between
this system and sources of dissipation兲. This is reasonable if such couplings give rise to dynamics on timescales longer than needed to complete the measurement.
This point is elaborated in Sec. VII, where we discuss
practical considerations related to the quantum limit.
We also discuss in that section the fact that the repeatability of QND measurements is of fundamental practical importance in overcoming detector inefficiencies
共Gambetta et al., 2007兲.
A common confusion is to think that a QND measurement has no effect on the state of the system being measured. While this is true if the initial state is an eigenstate of the observable, it is not true in general. Consider
again our example of a spin oriented in the x direction.
The result of the first ␴ˆ z measurement will be that thestate randomly and completely unpredictably collapses
onto one of the two ␴ˆ z eigenstates: the state is indeed
altered by the measurement. However, all subsequent
measurements using the same orientation for the detectors will always agree with the result of the first measurement. Thus QND measurements may affect thestate of the system, but never the value of the observable 共once it is determined兲. Other examples of QND
measurements include 共i兲 measurement of the electromagnetic field energy stored inside a cavity by determining the radiation pressure exerted on a moving piston
共Braginsky and Khalili, 1992兲, 共ii兲 detection of the presence of a photon in a cavity by its effect on the phase of
an atom’s superposition state 共Nogues et al., 1999;
Haroche and Raimond, 2006兲, and 共iii兲 the “dispersive”
measurement of a qubit state by its effect on the frequency of a driven microwave resonator 共Blais et al.,
2004; Wallraff et al., 2004; Lupaşcu et al., 2007兲, which isthe first canonical example described below.
In contrast to the above, in non-QND measurements
the back-action of the measurement will affect the observable being studied. The canonical example we consider is the position measurement of a harmonic oscillator. Since the position operator does not commute with
the Hamiltonian, the QND criterion is not satisfied.
Other examples of non-QND measurements include 共i兲photon counting via photodetectors that absorb the photons, 共ii兲 continuous measurements where the observable does not commute with the Hamiltonian, thus inducing a time dependence of the measurement result,and 共iii兲 measurements that can be repeated only after atime longer than the energy relaxation time of the system 共e.g., for a qubit, T1兲.

1164

Clerk et al.: Introduction to quantum noise, measurement, …

A. Weak continuous measurements

In discussing “real” quantum measurements, anotherkey notion to introduce is that of weak continuous measurements 共Braginsky and Khalili, 1992兲. Many measurements in practice take an extended time interval to complete, which is much longer than the “microscopic” timescales 共oscillation periods, etc.兲 of the system. The reason may be quite simply that the coupling strength between the detector and the system cannot be made arbitrarily large, and one has to wait for the effect of thesystem on the detector to accumulate. For example, in
our Stern-Gerlach measurement, suppose that we are
only able to achieve small magnetic field gradients andthat, consequently, the displacement d cannot be madelarge compared to the wave packet spread 共see Fig. 2兲.
In this case the states ␰± would have nonzero overlap
and it would not be possible to reliably distinguish them:
we thus would have only a weak measurement. However, by cascading together a series of such measurements and taking advantage of the fact that they are
QND, we can eventually achieve an unambiguous strong
projective measurement: at the end of the cascade, weare certain of which ␴ˆ z eigenstate the spin is in. During
this process, the overlap of ␰± would gradually fall tozero corresponding to a smooth continuous loss of phasecoherence in the transverse spin components. At the endof the process, the QND nature of the measurement ensures that the probability of measuring ␴z = ↑ or ↓ will
accurately reflect the initial wave function. Note that it is
only in this case of weak continuous measurements thatit makes sense to define a measurement rate in terms ofa rate of gain of information about the variable being
measured, and a corresponding dephasing rate, the rate
at which information about the conjugate variable is being lost. We see that these rates are intimately related
via the Heisenberg uncertainty principle.
While strong projective measurements may seem tobe the ideal, there are many cases where one may intentionally desire a weak continuous measurement; as discussed in the Introduction. There are many practical examples of weak continuous measurement schemes.
These include 共i兲 charge measurements, where the current through a device 关e.g., quantum point contact
共QPC兲 or single-electron transistor 共SET兲兴 is modulatedby the presence or absence of a nearby charge, andwhere it is necessary to wait for a sufficiently long timeto overcome the shot noise and distinguish between thetwo current values, 共ii兲 the weak dispersive qubit measurement discussed below, and 共iii兲 displacement detection of a nanomechanical beam 共e.g., optically or by capacitive coupling to a charge sensor兲, where one looks at
the two quadrature amplitudes of the signal produced at
the beam’s resonance frequency.
Not surprisingly, quantum noise plays a crucial role indetermining the properties of a weak continuous quantum measurement. For such measurements, noise both
determines the back-action effect of the measurementon the measured system and how quickly information is
acquired in the measurement process. Previously, we
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

saw that a crucial feature of quantum noise is the asymmetry between positive and negative frequencies; wefurther saw that this corresponds to the difference between absorption and emission events. For measurements, another key aspect of quantum noise will be important: as will be discussed extensively, quantum
mechanics places constraints on the noise of any systemcapable of acting as a detector or amplifier. These constraints in turn place restrictions on any weak continuous measurement, and lead directly to quantum limits
on how well one can make such a measurement.
In the remainder of this section, we give an introduction to how one describes a weak continuous quantum
measurement, considering the specific examples of usingparametric coupling to a resonant cavity for QND detection of the state of a qubit and the 共necessarily nonQND兲 detection of the position of a harmonic oscillator.
In the following section 共Sec. IV兲, we give a derivationof a very general quantum-mechanical constraint on thenoise of any system capable of acting as a detector, andshow how this constraint directly leads to the quantum
limit on qubit detection. Finally, in Sec. V, we turn to theimportant but slightly more involved case of a quantum
linear amplifier or position detector. We show that thebasic quantum noise constraint derived Sec. IV again
leads to a quantum limit; here this limit is on how small
one can make the added noise of a linear amplifier.
Before leaving this section, it is worth pointing outthat the theory of weak continuous measurements issometimes described in terms of some set of auxiliary
systems which are sequentially and momentarily weakly
coupled to the system being measured 共see Appendix
E兲. One then envisions a sequence of projective von
Neumann measurements on the auxiliary variables. The
weak entanglement between the system of interest andone of the auxiliary variables leads to a kind of partial
collapse of the system wave function 共more precisely, thedensity matrix兲, which is described in mathematicalterms not by projection operators, but rather by positive
operator valued measures 共POVMs兲. We will not use
this and the related “quantum trajectory” language here,
but direct the interested reader to the literature formore information on this important approach 共Peres,
1993; Brun, 2002; Haroche and Raimond, 2006; Jordan
and Korotkov, 2006兲.

B. Measurement with a parametrically coupled resonantcavityA simple yet experimentally practical example of aquantum detector consists of a resonant optical or rf
cavity parametrically coupled to the system being measured. Changes in the variable being measured 共e.g., thestate of a qubit or the position of an oscillator兲 shift thecavity frequency and produce a varying phase shift in
the carrier signal reflected from the cavity. This changing
phase shift can be converted 共via homodyne interferometry兲 into a changing intensity; this can then be detectedusing diodes or photomultipliers.

Clerk et al.: Introduction to quantum noise, measurement, …

In this section, we analyze weak continuous measurements made using such a parametric cavity detector; thiswill serve as a good introduction to the more generalapproaches presented later. We show that this detector iscapable of reaching the “quantum limit” meaning that it
can be used to make a weak continuous measurement,
as optimally as is allowed by quantum mechanics. This istrue for both the 共QND兲 measurement of the state of aqubit and the 共non-QND兲 measurement of the positionof a harmonic oscillator. Complementary analyses of
weak continuous qubit measurement are given by Makhlin et al. 共2000, 2001兲 共using a single-electron transistor兲 and by Gurvitz 共1997兲, Korotkov 共2001b兲, Korotkov
and Averin 共2001兲, Pilgram and Büttiker 共2002兲, andClerk et al. 共2003兲 共using a quantum point contact兲. We
focus here on a high-Q cavity detector; weak qubit measurement with a low-Q cavity was studied by Johansson
et al. 共2006兲.
It is worth noting the widespread usage of cavity detectors in experiment. One important current realization
is a microwave cavity used to read out the state of asuperconducting qubit 共Il’ichev et al., 2003; Blais et al.,
2004; Izmalkov et al., 2004; Lupaşcu et al., 2004, 2005;
Wallraff et al., 2004; Duty et al., 2005; Schuster et al.,2005; Sillanpää et al., 2005兲. Another class of examples
are optical cavities used to measure mechanical degreeof freedom. Examples of such systems include thosewhere one of the cavity mirrors is mounted on a cantilever 共Arcizet et al., 2006; Gigan et al., 2006; Kleckner
and Bouwmeester, 2006兲. Related systems involve a
freely suspended mass 共Abramovici et al., 1992; Corbitt
et al., 2007兲, an optical cavity with a thin transparent
membrane in the middle 共Thompson et al., 2008兲, and,more generally, an elastically deformable whispering gallery mode resonator 共Schliesser et al., 2006兲. Systems
where a microwave cavity is coupled to a mechanicalelement are also under active study 共Blencowe andBuks, 2007; Regal et al., 2008; Teufel et al., 2008兲.
We start our discussion with a general observation.
The cavity uses interference and the wave nature of lightto convert the input signal to a phase-shifted wave. For
small phase shifts we have a weak continuous measurement. Interestingly, it is the complementary particle nature of light that turns out to limit the measurement. As
we will see, it both limits the rate at which we can makea measurement 共via photon shot noise in the output
beam兲 and also controls the back-action disturbance ofthe system being measured 共due to photon shot noise
inside the cavity acting on the system being measured兲.
These dual aspects are an important part of any weakcontinuous quantum measurement; hence, an understanding of both the output noise 共i.e., the measurement
imprecision兲 and back-action noise of detectors will be
crucial.
All of our discussion of noise in the cavity system will
be framed in terms of the number-phase uncertainty relation for coherent states. A coherent photon state contains a Poisson distribution of the number of photons,
implying that the fluctuations in photon number obey
共⌬N兲2 = N̄, where N̄ is the mean number of photons. FurRev. Mod. Phys., Vol. 82, No. 2, April–June 20101165

ther, coherent states are overcomplete and states of different phase are not orthogonal to each other; this directly implies 共see Appendix G兲 that there is anuncertainty in any measurement of the phase. For large
N̄, this is given by共⌬␪兲2 = 1/4N̄.

共3.5兲

Thus, large-N̄ coherent states obey the number-phase
uncertainty relation⌬N⌬␪ = 21 ,共3.6兲analogous to the position-momentum uncertainty relation.
Equation 共3.6兲 can also be usefully formulated interms of noise spectral densities associated with themeasurement. Consider a continuous photon beam car¯
rying an average photon flux Ṅ. The variance in thenumber of photons detected grows linearly in time andcan be represented as 共⌬N兲2 = SN˙N˙t, where SN˙N˙ is thewhite-noise spectral density of photon flux fluctuations.
On a physical level, it describes photon shot noise, and is¯given by SN˙N˙ = Ṅ.
Consider now the phase of the beam. Any homodyne
measurement of this phase will be subject to the same
photon shot noise fluctuations discussed above 共see Appendix G for more details兲. Thus, if the phase of thebeam has some nominal small value ␪0, the output signal
from the homodyne detector integrated up to time t will
be of the form I = ␪0t + 兰t0d␶␦␪共␶兲, where ␦␪ is a noise
representing the imprecision in our measurement of ␪0
due to the photon shot noise in the output of the homodyne detector. An unbiased estimate of the phase obtained from I is ␪ = I / t, which obeys 具␪典 = ␪0. Further, one
has 共⌬␪兲2 = S␪␪ / t, where S␪␪ is the spectral density of the␦␪ white noise. Comparison with Eq. 共3.5兲 yields¯
S␪␪ = 1/4Ṅ .

共3.7兲

The above results lead us to the fundamental waveparticle relation for ideal coherent beams,

冑SN˙N˙S␪␪ = 21 .

共3.8兲

Before we study the role that these uncertainty relations play in measurements with high-Q cavities, consider the simplest case of reflection of light from a mirror without a cavity. The phase shift of the beam 共having
wave vector k兲 when the mirror moves a distance x is
2kx. Thus, the uncertainty in the phase measurement
corresponds to a position imprecision which can againIbe represented in terms of a noise spectral density Sxx= S␪␪ / 4k2. Here the superscript I refers to the fact that
this is noise representing imprecision in the measurement, not actual fluctuations in the position. We alsoneed to worry about back-action: each photon hitting
the mirror transfers a momentum 2បk to the mirror, so
photon shot noise corresponds to a random back-action
force noise spectral density SFF = 4ប2k2SN˙N˙. Multiplying

Clerk et al.: Introduction to quantum noise, measurement, …

1166

these together, we have the central result for the productof the back-action force noise and the imprecision,I
SFFSxx= ប2SN˙N˙S␪␪ = ប2/4,

共3.9兲

or in analogy with Eq. 共3.6兲
I
冑SFFSxx= ប/2.

共3.10兲

Not surprisingly, the situation considered here is as ideal
as possible. Thus, the RHS above is actually a lower
bound on the product of imprecision and back-action
noise for any detector capable of significant amplification; we will prove this rigorously in Sec. IV.A Equation
共3.10兲 thus represents the quantum limit on the noise ofour detector. As we will see shortly, having a detector
with quantum-limited noise is a prerequisite for reaching
the quantum limit on various different measurementtasks 共e.g., continuous position detection of an oscillator
and QND qubit state detection兲. Note that in general, agiven detector will have more noise than the quantumlimited value; we devote considerable effort later to determining the conditions needed to achieve the lower
bound of Eq. 共3.10兲.
We now turn to the story of measurement using ahigh-Q cavity; it will be similar to the above discussion,
except that we have to account for the filtering of thenoise by the cavity response. We relegate relevant calculational details to Appendix E. The cavity is simply described as a single bosonic mode coupled weakly to electromagnetic modes outside the cavity. The Hamiltonianof the system is given byĤ = H0 + ប␻c共1 + Aẑ兲â†â + Ĥenvt .

共3.11兲

Here H0 is the unperturbed Hamiltonian of the system
whose variable ẑ 共which is not necessarily a position兲 isbeing measured, â is the annihilation operator for thecavity mode, and ␻c is the cavity resonance frequency inthe absence of the coupling A. We will take both A andẑ to be dimensionless. The term Ĥenvt describes the electromagnetic modes outside the cavity, and their coupling
to the cavity; it is responsible for both driving and damping the cavity mode. The damping is parametrized byrate ␬, which tells us how quickly energy leaks out of thecavity; we consider the case of a high-quality factor cavity, where Qc ⬅ ␻c / ␬ Ⰷ 1.
Turning to the interaction term in Eq. 共3.11兲, we seethat the parametric coupling strength A determines thechange in frequency of the cavity as the system variableẑ changes. We assume for simplicity that the dynamics ofẑ is slow compared to ␬. In this limit the reflected phaseshift simply varies slowly in time adiabatically following
the instantaneous value of ẑ. We also assume that the
coupling A is small enough that the phase shifts are always very small and hence the measurement is weak.
Many photons will have to pass through the cavity before much information is gained about the value of thephase shift and hence the value of ẑ.
We first consider the case of a one-sided cavity where
only one of the mirrors is semitransparent, the otherRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

being perfectly reflecting. In this case, a wave incident
on the cavity 关say, in a one-dimensional 共1D兲 waveguide兴
will be perfectly reflected, but with a phase shift ␪ determined by the cavity and the value of ẑ. The reflection
coefficient at the bare cavity frequency ␻c is simply
given by 共Walls and Milburn, 1994兲
r = − 共1 + 2iAQcẑ兲/共1 − 2iAQcẑ兲.

共3.12兲

Note that r has unit magnitude because all incident photons are reflected if the cavity is lossless. For weak coupling we can write the reflection phase shift as r = −ei␪,where␪ ⬇ 4QcAẑ = 共A␻cẑ兲tWD .

共3.13兲

We see that the scattering phase shift is simply the frequency shift caused by the parametric coupling multiplied by the Wigner delay time 共Wigner, 1955兲
tWD = Im ⳵ ln r/⳵␻ = 4/␬ .

共3.14兲

Thus the measurement-imprecision noise power for a¯
given photon flux Ṅ incident on the cavity is given byI= S␪␪/共A␻ctWD兲2 .
Szz

共3.15兲

The random part of the generalized back-action force
conjugate to ẑ is, from Eq. 共3.11兲,F̂z ⬅ − ⳵Ĥ/⳵ẑ = − Aប␻c␦n̂,

共3.16兲

where, since ẑ is dimensionless, F̂z has units of energy.
Here ␦n̂ = n̂ − n̄ = â†â − 具â†â典 represents the photonnumber fluctuations around the mean n̄ inside the cavity.
The back-action force noise spectral density is thus
SFzFz = 共Aប␻c兲2Snn .

共3.17兲As shown in Appendix E, the cavity filters the photon
shot noise so that at low frequencies ␻ Ⰶ ␬, the number
fluctuation spectral density is simply
Snn = n̄tWD .

共3.18兲

The mean photon number in the cavity is found to be¯¯n̄ = ṄtWD, where again Ṅ is the mean photon flux incident on the cavity. From this it follows thatSFzFz = 共Aប␻ctWD兲2SN˙N˙ .

共3.19兲

Combining this with Eq. 共3.15兲 again yields the same
result as Eq. 共3.10兲 obtained without the cavity. The
parametric cavity detector 共used in this way兲 is thus aquantum-limited detector, meaning that the product of
its noise spectral densities achieves the ideal minimum
value.
We now examine how the quantum limit on the noiseof our detector directly leads to quantum limits on different measurement tasks. In particular, we consider thecases of continuous position detection and QND qubit
state measurement.

Clerk et al.: Introduction to quantum noise, measurement, …

1167

While Eq. 共3.20兲 makes it clear that the state of thequbit modulates the cavity frequency, we can easily rewrite this equation to show that this same interaction
term is also responsible for the back-action of the measurement 共i.e., the disturbance of the qubit state by themeasurement process兲

I(t)
ΔI

t

Ĥ = 共ប/2兲共␻01 + 2A␻câ†â兲␴ˆ z + ប␻câ†â + Ĥenvt . 共3.24兲
FIG. 3. 共Color online兲 Distribution of the integrated output forthe cavity detector I共t兲 for the two different qubit states. The
separation of the means of the distributions grows linearly intime, while the widths of the distributions grow only as 冑t.
1. QND measurement of the state of a qubit using a resonantcavityHere we specialize to the case where the system operator ẑ = ␴ˆ z represents the state of a spin-1 / 2 quantum
bit. Equation 共3.11兲 becomesĤ = 21 ប␻01␴ˆ z + ប␻c共1 + A␴ˆ z兲â†â + Ĥenvt .

共3.20兲

We see that ␴ˆ z commutes with all terms in the Hamiltonian and is thus a constant of the motion 共assumingthat Ĥenvt contains no qubit decay terms so that T1 = ⬁兲and hence the measurement will be QND. From Eq.
共3.13兲 we see that the two states of the qubit produce
phase shifts ±␪0 where␪0 = A␻ctWD .

共3.21兲

As ␪0 Ⰶ 1, we need many reflected photons before weare able to determine the state of the qubit. This is adirect consequence of the unavoidable photon shotnoise in the output of the detector, and is a basic featureof weak measurements—information on the input is acquired only gradually in time.
Let I共t兲 be the homodyne signal for the wave reflectedfrom the cavity integrated up to time t. Depending on
the state of the qubit the mean value of I will be 具I典= ± ␪0t, and the rms Gaussian fluctuations about themean will be ⌬I = 冑S␪␪t. As shown in Fig. 3 and discussed
by in Makhlin et al. 共2001兲, the integrated signal is drawn
from one of two Gaussian distributions, which are betterand better resolved with increasing time 共as long as themeasurement is QND兲. The state of the qubit thus becomes ever more reliably determined. The signal energy
to noise energy ratio becomes
SNR = 具I典2/共⌬I兲2 = ␪20/S␪␪t,

共3.22兲

which can be used to define the measurement rate via
I.
⌫meas ⬅ SNR/2 = ␪20/2S␪␪ = 1/2Szz

共3.23兲

There is a certain arbitrariness in the scale factor of 2
appearing in the definition of the measurement rate; thisparticular choice is motivated by precise informationtheoretic grounds 共as defined, ⌫meas is the rate at which
the “accessible information” grows, Appendix F兲.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

We now see that the interaction can also be viewed asproviding a “light shift” 共i.e., ac Stark shift兲 of the qubit
splitting frequency 共Blais et al., 2004; Schuster et al.,2005兲 which contains a constant part 2An̄A␻c plus a randomly fluctuating part ⌬␻01 = 2F̂z / ប, that depends on n̂= â†â, the number of photons in the cavity. During ameasurement, n̂ will fluctuate around its mean and act asa fluctuating back-action “force” on the qubit. In thepresent QND case, noise in n̂ = â†â cannot cause transitions between the two qubit eigenstates. This is the opposite of the situation considered in Sec. II.B, where we
wanted to use the qubit as a spectrometer. Despite thelack of any noise-induced transitions, there still is aback-action here, as noise in n̂ causes the effective splitting frequency of the qubit to fluctuate in time. For weakcoupling, the resulting phase diffusion leads tomeasurement-induced dephasing of superpositions in
the qubit 共Blais et al., 2004; Schuster et al., 2005兲 according to
具e−i␸典 =冓 冉冕texp − id␶⌬␻01共␶兲

0

冊冔

.

共3.25兲

For weak coupling the dephasing rate is slow and thus
we are interested in long times t. In this limit the integral
is a sum of a large number of statistically independent
terms and thus we can take the accumulated phase to be
Gaussian distributed. Using the cumulant expansion wethen obtain冉 冓冋冕

具e−i␸典 = exp −冉= exp −

1
2

0

册 冔冊
2

t

d␶⌬␻01共␶兲冊

2SF F t .
ប2 z z共3.26兲

Note also that the noise correlator above is naturally
symmetrized—the quantum asymmetry of the noiseplays no role for this type of coupling. Equation 共3.26兲
yields the dephasing rate
⌫␸ = 共2/ប2兲SFzFz = 2␪20SN˙N˙ .

共3.27兲

Using Eqs. 共3.23兲 and 共3.27兲, we find the interesting
conclusion that the dephasing rate and measurement
rates coincide,I⌫␸/⌫meas = 共4/ប2兲Szz
SFzFz = 4SN˙N˙S␪␪ = 1.

共3.28兲

As we will see and prove rigorously, this represents theideal quantum-limited case for QND qubit detection:
the best one can do is measure as quickly as one
dephases. In keeping with our earlier discussions, it represents the enforcement of the Heisenberg uncertainty

Clerk et al.: Introduction to quantum noise, measurement, …

1168

principle. The faster you gain information about one
variable, the faster you lose information about the conjugate variable. Note that, in general, the ratio ⌫␸ / ⌫meas
will be larger than 1, as an arbitrary detector will notreach the quantum limit on its noise spectral densities.
Such a nonideal detector produces excess back-action
beyond what is required quantum mechanically.
In addition to the quantum noise point of view presented above, there is a second complementary way inwhich to understand the origin of measurement-induceddephasing 共Stern et al., 1990兲, which is analogous to ourdescription of loss of transverse spin coherence in the
Stern-Gerlach experiment in Eq. 共3.3兲. The measurement takes the incident wave, described by a coherent
state 兩␣典, to a reflected wave described by a 共phaseshifted兲 coherent state 兩r↑ · ␣典 or 兩r↓ · ␣典, where r↑/↓ is thequbit-dependent reflection amplitude given in Eq. 共3.12兲.
Considering now the full state of the qubit-plus-detector,
the measurement results in a state change:
11

冑2 共兩↑典 + 兩↓典兲 丢 兩␣典 → 冑2 共e+i␻01t/2兩↑典 丢 兩r↑ · ␣典

+ e−i␻01t/2兩↓典 丢 兩r↓ · ␣典兲.

共3.29兲

As 兩r↑ · ␣典 ⫽ 兩r↓ · ␣典, the qubit has become entangled withthe detector: the above state cannot be written as a
product of a qubit state times a detector state. To assess
the coherence of the final qubit state 共i.e., the relative
phase between ↑ and ↓兲, one looks at the off-diagonal
matrix element of the qubit’s reduced density matrix,␳↓↑ ⬅ Trdetector具↓兩␺典具␺兩↑典

共3.30兲

=共e+i␻01t/2兲具r↑ · ␣兩r↓ · ␣典

共3.31兲

=共e+i␻01t/2兲 exp关− 兩␣兩2共1 − r*↑r↓兲兴.

共3.32兲

In Eq. 共3.31兲 we have used the usual expression for theoverlap of two coherent states. We see that the measurement reduces the magnitude of ␳↑↓: this is dephasing.
The amount of dephasing is directly related to the overlap between the different detector states that result
when the qubit is up or down; this overlap can bestraightforwardly found using Eq. 共3.32兲 and 兩␣兩2 = N̄¯
= Ṅt, where N̄ is the mean number of photons that havereflected from the cavity after time t. We have
円exp关− 兩␣兩2共1 − r*↑r↓兲兴円 = exp关− 2N̄␪20兴 ⬅ exp关− ⌫␸t兴,
共3.33兲
with the dephasing rate ⌫␸ given by¯⌫␸ = 2␪20Ṅ

共3.34兲

in complete agreement with the previous result in Eq.
共3.27兲.
2. Quantum limit relation for QND qubit state detection

We now return to the ideal quantum limit relation of
Eq. 共3.28兲. As stated previously, this is a lower bound:Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

quantum mechanics enforces the constraint that in a
QND qubit measurement the best you can possibly do ismeasure as quickly as you dephase 共Devoret and Schoelkopf, 2000; Korotkov and Averin, 2001; Makhlin et al.,
2001; Averin, 2000b, 2003; Clerk et al., 2003兲,⌫meas 艋 ⌫␸ .

共3.35兲

While a detector with quantum-limited noise has anequality above, most detectors will be very far from thisideal limit, and will dephase the qubit faster than theyacquire information about its state. We provide a proofof Eq. 共3.35兲 in Sec. IV.B; for now, we note that its heuristic origin rests on the fact that both measurement anddephasing rely on the qubit becoming entangled withthe detector. Consider again Eq. 共3.29兲, describing theevolution of the qubit-detector system when the qubit isinitially in a superposition of ↑ and ↓. To say that wehave truly measured the qubit, the two detector states兩r↑␣典 and 兩r↓␣典 need to correspond to different values ofthe detector output 共i.e., phase shift ␪ in our example兲;this necessarily implies they are orthogonal. This in turnimplies that the qubit is completely dephased: ␳↑↓ = 0,
just as we saw in Eq. 共3.4兲 in the Stern-Gerlach example.
Thus, measurement implies dephasing. The opposite is
not true. The two states 兩r↑␣典 and 兩r↓␣典 could in principle
be orthogonal without them corresponding to different
values of the detector output 共i.e., ␪兲. For example, thequbit may have become entangled with extraneous microscopic degrees of freedom in the detector. Thus, on aheuristic level, the origin of Eq. 共3.35兲 is clear.
Returning to our one-sided cavity system, we see from
Eq. 共3.28兲 that the one-sided cavity detector reaches thequantum limit. It is natural to now ask why this is thecase: Is there a general principle in action here that allows the one-sided cavity to reach the quantum limit?
The answer is yes: reaching the quantum limit requiresthat there is no “wasted” information in the detector
共Clerk et al., 2003兲. There should not exist any unmeasured quantity in the detector which could have beenprobed to learn more about the state of the qubit. In the
single-sided cavity detector, information on the state of
the qubit is only available in 共that is, is entirely encodedin兲 the phase shift of the reflected beam; thus, there is no“wasted” information, and the detector does indeed
reach the quantum limit.
To make this idea of “no wasted information” more
concrete, we now consider a simple detector that fails toreach the quantum limit precisely due to wasted information. Consider again a 1D cavity system where now
both mirrors are slightly transparent. Now, a wave incident at frequency ␻R on one end of the cavity will be
partially reflected and partially transmitted. If the initial
incident wave is described by a coherent state 兩␣典, thescattered state can be described by a tensor product of
the reflected wave’s state and the transmitted wave’s
state,

Clerk et al.: Introduction to quantum noise, measurement, …

兩␣典 → 兩r␴ · ␣典兩t␴ · ␣典,

共3.36兲

where the qubit-dependent reflection and transmission
amplitudes r␴ and t␴ are given by 共Walls and Milburn,1994兲
t↓ = 1/共1 + 2iAQc兲,

共3.37兲

r↓ = 2iQcA/共1 + 2iAQc兲,共3.38兲

with t↑ = 共t↓兲* and r↑ = 共r↓兲*. Note that the incident beam isalmost perfectly transmitted: 兩t␴兩2 = 1 − O共AQc兲2.
Similar to the one-sided case, the two-sided cavity
could be used to make a measurement by monitoring
the phase of the transmitted wave. Using the expression
for t␴ above, we find that the qubit-dependent transmission phase shift is given by␪˜ ↑/↓ = ± ␪˜ 0 = ± 2AQc ,共3.39兲

where again the two signs correspond to the two different qubit eigenstates. The phase shift for transmission is
only half as large as in reflection so the Wigner delay
time associated with transmission ist̃WD = 2/␬ .

共3.40兲

Upon making the substitution of t̃WD for tWD, the onesided cavity Eqs. 共3.15兲 and 共3.17兲 remain valid. However, the internal cavity photon-number shot noise remains fixed so that Eq. 共3.18兲 becomes
Snn = 2n̄t̃WD .

共3.41兲

which means that¯ 22Snn = 2Ṅt̃WD= 2SN˙N˙t̃WD

共3.42兲and2
SN˙N˙ .
SFzFz = 2ប2A2␻2c t̃WD

共3.43兲

As a result the back-action dephasing doubles relative tothe measurement rate and we have⌫meas/⌫␸ = 2SN˙N˙S␪␪ = 21 .

共3.44兲

Thus the two-sided cavity fails to reach the quantum
limit by a factor of 2.
Using the entanglement picture, we may again alternatively calculate the amount of dephasing from theoverlap between the detector states corresponding to
the qubit states ↑ and ↓ 关cf. Eq. 共3.31兲兴. We finde−⌫␸t = 兩具t↑␣兩t↓␣典具r↑␣兩r↓␣典兩
=exp关− 兩␣兩2共1 − 共t↑兲*t↓ − 共r↑兲*r↓兲兴.

共3.45兲
共3.46兲

Note that both the changes in the transmission and reflection amplitudes contribute to the dephasing of thequbit. Using the above expressions, we find¯⌫␸t = 4共␪˜ 0兲2兩␣兩2 = 4共␪˜ 0兲2N̄ = 4共␪˜ 0兲2Ṅt = 2⌫meast. 共3.47兲
Thus, in agreement with the quantum noise result, thetwo-sided cavity misses the quantum limit by a factor of
2.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 20101169

Why does the two-sided cavity fail to reach the quantum limit? The answer is clear from Eq. 共3.46兲: eventhough we are not monitoring it, there is information on
the state of the qubit available in the phase of the reflected wave. Note from Eq. 共3.38兲 that the magnitude of
the reflected wave is weak 共⬀A2兲, but 共unlike the transmitted wave兲 the difference in the reflection phase associated with the two qubit states is large 共 ± ␲ / 2兲. The“missing information” in the reflected beam makes a direct contribution to the dephasing rate 关i.e., the second
term in Eq. 共3.46兲兴, making it larger than the measurement rate associated with measurement of the transmission phase shift. In fact, there is an equal amount of
information in the reflected beam as in the transmitted
beam, so the dephasing rate is doubled. We thus have aconcrete example of the general principle connecting a
failure to reach the quantum limit to the presence ofwasted information. Note that the application of thisprinciple to generalized quantum point contact detectors
is found in Clerk et al. 共2003兲.
Returning to our cavity detector, we note in closingthat it is often technically easier to work with the transmission of light through a two-sided cavity, rather thanreflection from a one-sided cavity. One can still reachthe quantum limit in the two-sided cavity case if one
uses an asymmetric cavity in which the input mirror has
much less transmission than the output mirror. Most
photons are reflected at the input, but those that enter
the cavity will almost certainly be transmitted. The priceto be paid is that the input carrier power must be increased.
3. Measurement of oscillator position using a resonant cavity

The qubit measurement discussed previously was an
example of a QND measurement: the back-action did
not affect the observable being measured. We now consider the simplest example of a non-QND measurement,
namely, the weak continuous measurement of the position of a harmonic oscillator. The detector will again bea parametrically coupled resonant cavity, where the position of the oscillator x changes the frequency of thecavity as per Eq. 共3.11兲 关see, e.g., Tittonen et al. 共1999兲兴.
Similarly to the qubit case, for a sufficiently weak coupling the phase shift of the reflected beam from the cavity will depend linearly on the position x of the oscillator关cf. Eq. 共3.13兲兴; by reading out this phase, we may thus
measure x. The origin of back-action noise is the same as
before, namely, photon shot noise in the cavity. Now,
however, this represents a random force which changes
the momentum of the oscillator. During the subsequent
time evolution these random force perturbations will reappear as random fluctuations in the position. Thus themeasurement is not QND. This will mean that the minimum uncertainty of even an ideal measurement is larger
共by exactly a factor of 2兲 than the “true” quantum uncertainty of the position 共i.e., the ground-state uncertainty兲. This is known as the standard quantum limit on
weak continuous position detection. It is also an example of a general principle that a linear phase-preserving amplifier necessarily adds noise, and that theminimum added noise exactly doubles the output noisefor the case where the input is vacuum 共i.e., zero-point兲
noise. A more general discussion of the quantum limit
on amplifiers and position detectors will be presented inSec. V.
We start by emphasizing that we are speaking here ofa weak continuous measurement of the oscillator position. The measurement is sufficiently weak that the position undergoes many cycles of oscillation before significant information is acquired. Thus we are not talking
about the instantaneous position but rather the overall
amplitude and phase, or more precisely the two quadrature amplitudes describing the smooth envelope of the
motion,共3.48兲

x̂共t兲 = X̂共t兲cos共⍀t兲 + Ŷ共t兲sin共⍀t兲.

One can easily show that, for an oscillator, the twoquadrature amplitudes X̂ and Ŷ are canonically conjugate and hence do not commute with each other,2
关X̂,Ŷ兴 = iប/M⍀ = 2ixZPF.

共3.49兲

As the measurement is both weak and continuous, it will
yield information on both X̂ and Ŷ. As such, one is effectively trying to simultaneously measure two incompatible observables. This basic fact is intimately related
to the property mentioned above, that even a completely ideal weak continuous position measurement will
have a total uncertainty that is twice the zero-point uncertainty.
We are now ready to start our heuristic analysis of
position detection using a cavity detector; relevant calculational details presented in Appendix E.3. Consider
first the mechanical oscillator we wish to measure. We
take it to be a simple harmonic oscillator of natural frequency ⍀ and mechanical damping rate ␥0. For weakdamping, and at zero coupling to the detector, the spectral density of the oscillator’s position fluctuations isgiven by Eq. 共2.4兲 with the delta function replaced by a
Lorentzian5

再2
Sxx关␻兴 = xZPF
nB共ប⍀兲

␥0
共␻ + ⍀兲2 + 共␥0/2兲2+ 关nB共ប⍀兲 + 1兴冎␥0.
共␻ − ⍀兲2 + 共␥0/2兲2

共3.50兲

When we now weakly couple the oscillator to the cavity 关as per Eq. 共3.11兲, with ẑ = x̂ / xZPF兴 and drive the cavity on resonance, the phase shift ␪ of the reflected beam
will be proportional to x „i.e., ␦␪共t兲 = 关d␪ / dx兴x共t兲…. As
such, the oscillator’s position fluctuations will cause ad5This form is valid only for weak damping because we areassuming that the oscillator frequency is still sharply defined.
We have evaluated the Bose-Einstein factor exactly at frequency ⍀ and have assumed that the Lorentzian centered at
positive 共negative兲 frequency has negligible weight at negative
共positive兲 frequencies.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Output Noise S (ω) (arb. units)

Clerk et al.: Introduction to quantum noise, measurement, …

1170

12
10
8

area prop. to T
measurement
imprecision

6
4
2
0
0.0

0.5
1.0
1.5
Frequency ω / Ω

2.0

FIG. 4. 共Color online兲 Spectral density of the symmetrized
output noise S̄␪␪关␻兴 of a linear position detector. The oscillator’s noise appears as a Lorentzian on top of a noise floor 共i.e.,the measurement imprecision兲. As discussed in the text, thewidth of the peak is proportional to the oscillator damping rate
␥0, while the area under the peak is proportional to temperature. This latter fact can be used to calibrate the response of
the detector.

ditional fluctuations of the phase ␪, over and above theintrinsic shot noise-induced phase fluctuations S␪␪. We
consider the usual case where the noise spectrometer
being used to measure the noise in ␪ 共i.e., the noise in
the homodyne current兲 measures the symmetric-infrequency noise spectral density; as such, it is thesymmetric-in-frequency position noise that we detect. In
the classical limit kBT Ⰷ ប⍀, this is given by1
S̄xx关␻兴 ⬅ 共Sxx关␻兴 + Sxx关− ␻兴兲
2
⬇

␥0
k BT
.
2M⍀2 共兩␻兩 − ⍀兲2 + 共␥0/2兲2

共3.51兲

If we ignore back-action effects, we expect to see thisLorentzian profile riding on top of the background imprecision noise floor; this is shown in Fig. 4.
Note that additional stages of amplification would also
add noise, and would thus further augment this background noise floor. If we subtract off this noise floor, thefull width at half maximum of the curve will give thedamping parameter ␥0, and the area under the experimental curve,

冕

⬁

d␻
k BT
S̄xx关␻兴 =
,
M⍀2
−⬁ 2␲

共3.52兲

measures the temperature. What the experimentalist actually plots in making such a curve is the output of theentire detector-plus-following-amplifier chain. Importantly, if the temperature is known, then the area of themeasured curve can be used to calibrate the coupling of
the detector and the gain of the total overall amplifier
chain 共see, e.g., LaHaye et al., 2004; Flowers-Jacobs etal., 2007兲. One can thus make a calibrated plot where themeasured output noise is referred back to the oscillator
position.

0
2
S̄xx
关␻兴 = xZPF

␥0/2.
共兩␻兩 − ⍀兲2 + 共␥0/2兲2

共3.53兲

One might expect that one could see this Lorentzian
directly in the output noise of the detector 共i.e., the ␪noise兲, above the measurement-imprecision noise floor.
However, this neglects the effects of measurement backaction. From the classical equation of motion we expect
the response of the oscillator to the back-action force
F = Fz / xZPF 关cf. Eq. 共3.16兲兴 at frequency ␻ to produce anadditional displacement ␦x关␻兴 = ␹xx关␻兴F关␻兴, where␹xx关␻兴 is the mechanical susceptibility␹xx关␻兴 ⬅

1
1.
2
M ⍀ − ␻ 2 − i ␥ 0␻

共3.54兲

These extra oscillator fluctuations will show up as additional fluctuations in the output of the detector. For simplicity, we focus on this noise at the oscillator’s resonance frequency ⍀. As a result of the detector’s backaction, the total measured position noise 共i.e., inferred
spectral density兲 at the frequency ⍀ is given by
0S̄xx,tot关⍀兴 = S̄xx关⍀兴 +兩␹xx关⍀兴兩2†SFF关+ ⍀兴 + SFF关− ⍀兴‡
2

1 II关+ ⍀兴 + Sxx
关− ⍀兴‡+ †Sxx2

共3.55兲0关⍀兴 + S̄xx,add关⍀兴.
=S̄xx

共3.56兲

The first term here is just the intrinsic zero-point noiseof the oscillator,02关⍀兴 = 2xZPF/␥0 = ប兩␹xx关⍀兴兩.
S̄xx

共3.57兲

The second term S̄xx,add is the total noise added by themeasurement, and includes both the measurement imI
I 2
⬅ Szz
xZPF and the extra fluctuations caused
precision Sxx
by the back-action. We stress that S̄xx,tot corresponds to a
position noise spectral density inferred from the outputof the detector: one simply scales the spectral density of
total output fluctuations S̄␪␪,tot关⍀兴 by 共d␪ / dx兲 .
Implicit in Eq. 共3.57兲 is the assumption that the backaction noise and the imprecision noise are uncorrelated
and thus add in quadrature. It is not obvious that this iscorrect, since in the cavity detector the back-action noiseand output shot noise are both caused by the vacuum
noise in the beam incident on the cavity. It turns out that
there are indeed correlations, however, the symmetrized2

共i.e., classical兲 correlator S̄␪F does vanish for our choiceof a resonant cavity drive. Further, Eq. 共3.55兲 assumes
that the measurement does not change the damping rateof the oscillator. Again, while this will not be true for an
arbitrary detector, it is the case here for the cavity detector when 共as we have assumed兲 it is driven on resonance. Details justifying both these statements are given
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1171

10
Total added position noise
Measurement imprecision
Contribution from backaction

82Consider now the case where the oscillator is at zerotemperature. Equation 共3.50兲 then yields for the symmetrized noise spectral density

Sxx,add [ (xZPT) / γ0 ]Clerk et al.: Introduction to quantum noise, measurement, …

6
4
2
0.1

1

10

Coupling SFF / SFF,opt
FIG. 5. 共Color online兲 Noise power of the added position noiseof a linear position detector, evaluated at the oscillator’s resonance frequency 共S̄xx,add关⍀兴兲 as a function of the magnitude of
the back-action noise spectral density SFF. SFF is proportional
to the oscillator-detector coupling, and in the case of the cavity
detector is also proportional to the power incident on the cavity. The optimal value of SFF is given by SFF,opt = បM⍀␥0 / 2 关cf.
Eq. 共3.59兲兴. We have assumed that there are no correlations
between measurement-imprecision noise and back-action
noise, as is appropriate for the cavity detector.

in Appendix E; the more general case with nonzero
noise correlations and back-action damping is discussed
in Sec. V.E.
Assuming we have a quantum-limited detector thatI
obeys Eq. 共3.9兲 共i.e., Sxx
SFF = ប2 / 4兲 and that the shotnoise is symmetric in frequency, the added position noise
spectral density at resonance 关i.e., second term in Eq.
共3.56兲兴 becomes冋S̄xx,add关⍀兴 = 兩␹关⍀兴兩2SFF +

册

ប2 1
.
4 SFF

共3.58兲

Recall from Eq. 共3.19兲 that the back-action noise is proportional to the coupling of the oscillator to the detector
and to the intensity of the drive on the cavity. The added
position-uncertainty noise is plotted in Fig. 5 as a function of SFF. We see that for high drive intensity the backaction noise dominates the position uncertainty, while
for low drive intensity the output shot noise 共the lastterm in the equation above兲 dominates.
The added noise 共and hence the total noise S̄xx,tot关⍀兴兲
is minimized when the drive intensity is tuned so thatSFF is equal to SFF,opt, with
SFF,opt = ប/2兩␹xx关⍀兴 = 共ប/2兲M⍀␥0 .

共3.59兲

The more heavily damped is the oscillator, the less susceptible it is to back-action noise and hence the higher isthe optimal coupling. At the optimal coupling strength,
the measurement-imprecision noise and back-action
noise each make equal contributions to the added noise,yielding
0S̄xx,add关⍀兴 = ប/M⍀␥0 = S̄xx关⍀兴.

共3.60兲

Thus, the spectral density of the added position noise isexactly equal to the noise power associated with the oscillator’s zero-point fluctuations. This represents a mini-

Clerk et al.: Introduction to quantum noise, measurement, …

mum value for the added noise of any linear position
detector, and is referred to as the standard quantum
limit on position detection. Note that this limit only involves the added noise of the detector, and thus has
nothing to do with the initial temperature of the oscillator.
We emphasize that to reach the above quantum limit
on weak continuous position detection one needs thedetector itself to be quantum limited, i.e., the productI
SFFSxxmust be as small as is allowed by quantum mechanics, namely, ប2 / 4. Having a quantum-limited detector, however, is not enough: in addition, one must beable to achieve sufficiently strong coupling to reach the
optimum given in Eq. 共3.59兲. Further, the measured output noise must be dominated by the output noise of thecavity, not by the added noise of following amplifier
stages.
A related, stronger quantum limit refers to the total
inferred position noise from the measurement S̄xx,tot关␻兴.
It follows from Eqs. 共3.60兲 and 共3.56兲 that at resonancethe smallest this can be is twice the oscillator’s zeropoint noise0关⍀兴.
S̄xx,tot关⍀兴 = 2S̄xx

共3.61兲

Half the noise here is from the oscillator itself, half isfrom the added noise of the detector. It is even morechallenging to reach this quantum limit: one needs toboth reach the quantum limit on the added noise andcool the oscillator to its ground state.
Finally, we emphasize that the optimal value of thecoupling derived above was specific to the choice ofminimizing the total position noise power at the resonance frequency. If a different frequency had been chosen, the optimal coupling would have been different;
one again finds that the minimum possible added noisecorresponds to the ground-state noise at that frequency.
It is interesting to ask what the total position noise
would be as a function of frequency, assuming that the
coupling has been optimized to minimize the noise at
the resonance frequency, and that the oscillator is initially in the ground state. From our results above wehave
2S̄xx,tot关␻兴 = xZPF

+
⬇

␥0/2
共兩␻兩 − ⍀兲2 + 共␥0/2兲2冋ប 兩␹xx关␻兴兩2+ 兩␹xx关⍀兴兩
2 兩␹xx关⍀兴兩

再

册

冎

2
xZPF
共␥0/2兲2
1+3,
␥0共兩␻兩 − ⍀兲2 + 共␥0/2兲2

共3.62兲

which is plotted in Fig. 6. Assuming that the detector isquantum limited, one sees that the Lorentzian peak rises
above the constant background by a factor of 3 when the
coupling is optimized to minimize the total noise powerat resonance. This represents the best one can do whencontinuously monitoring zero-point position fluctuations. Note that the value of this peak-to-floor ratio is adirect consequence of two simple facts which hold for an
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

2.0

Sxx,tot [ω] / S0xx[Ω]1172

total noise

1.5

eq. + backaction noise

1.0

equilibrium
noise (T=0)

0.5
0.0
0.0

0.4

0.8

Frequency ω/Ω

1.2

FIG. 6. 共Color online兲 Spectral density of measured position
fluctuations of a harmonic oscillator S̄xx,tot关␻兴 as a function of
frequency ␻, for a detector which reaches the quantum limit at
the oscillator frequency ⍀. We have assumed that without the
coupling to the detector the oscillator would be in its groundstate. The y axis has been normalized by the zero-point position noise spectral density S̄0xx关␻兴, evaluated at ␻ = ⍀. One
clearly sees that the total noise at ⍀ is twice the zero-pointvalue, and that the peak of the Lorentzian rises a factor of 3
above the background. This background represents the measurement imprecision and is equal to 1 / 2 of S̄0xx共⍀兲.

optimal coupling at the quantum limit: 共i兲 the total
added noise at resonance 共back-action plus measurement imprecision兲 is equal to the zero-point noise and共ii兲 back-action and measurement imprecision makeequal contributions to the total added noise. Somewhat
surprisingly, the same maximum peak-to-floor ratio is
obtained when one tries to continuously monitor coherent qubit oscillations with a linear detector which istransversely coupled to the qubit 共Korotkov and Averin,
2001兲; this is also a non-QND situation. Finally, if one
only wants to detect the noise peak 共as opposed to making a continuous quantum-limited measurement兲, one
could use two independent detectors coupled to the oscillator and look at the cross correlation between thetwo output noises: in this case, there need not be any
noise floor 共Jordan and Büttiker, 2005a; Doiron et al.,2007兲.
In Table III, we give a summary of recent experiments
which approach the quantum limit on weak continuous
position detection of a mechanical resonator. Note thatin many of these experiments the effects of detector
back-action were not seen. This could either be the result of too low a detector-oscillator coupling or due tothe presence of excessive thermal noise. As shown, theback-action force noise serves to slightly heat the oscillator. If it is already at an elevated temperature due tothermal noise, this additional heating can be hard to resolve.
In closing, we stress that this section has given only arudimentary introduction to the quantum limit on position detection. A complete discussion which treats theClerk et al.: Introduction to quantum noise, measurement, …

1173

TABLE III. 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 S̄Ixx achieved in the experiment. This value is compared against the zero-point position noise spectral density S̄0xx, calculated using the total measured
resonator damping 共which may include a back-action contribution兲. All spectral densities are at the oscillator’s resonance frequency ⍀. As discussed in the text, there is no quantum limit on how small one can make S̄Ixx; for an ideal detector, one needs totune the detector-resonator coupling so that S̄Ixx = S̄0xx / 2 in order to reach the quantum limit on position detection. The thirdcolumn presents the product of the measured imprecision noise 共unless otherwise noted兲 and measured back-action noises, divided
by ប / 2; this quantity must be one to achieve the quantum limit on the added noise.

Experiment
Cleland et al. 共2002兲 共quantum point contact兲
Knobel and Cleland 共2003兲 共single-electron transistor兲
LaHaye et al. 共2004兲 共single-electron transistor兲
Naik et al. 共2006兲 共single-electron transistor兲
共if S̄Ixx had been limited by SET shot noise兲Arcizet et al. 共2006兲 共optical cavity兲
Flowers-Jacobs et al. 共2007兲 共atomic point-contact兲
Regal et al. 共2008兲 共microwave cavity兲
Schliesser et al. 共2008兲 共optical cavity兲
Poggio et al. 共2008兲 共quantum point contact兲
Etaki et al. 共2008兲 共dc SQUID兲
Groblacher et al. 共2009兲 共optical cavity兲
Schliesser et al. 共2009兲 共optical cavity兲Teufel et al. 共2009兲 共microwave cavity兲
aMechanicalfrequency 关Hz兴⍀ / 共2␲兲Imprecisionnoise
zero-point noise冑S̄Ixx / S̄0xx关⍀兴

1.5⫻ 1061.2⫻ 1082.0⫻ 1072.2⫻ 1074.2⫻ 1041.8⫻ 102
5.4
5.3b

8.1⫻ 105
4.3⫻ 1072.4⫻ 105
4.1⫻ 1075.2⫻ 1032.0⫻ 1069.5⫻ 102
6.5⫻ 1071.0⫻ 106

0.87
29
21
0.50
63
47
0.57
5.5
0.63

Detector noiseproducta

冑S̄IxxS̄FF / 共ប / 2兲

8.1⫻ 102
1.5⫻ 101
1.7⫻ 1031.0⫻ 102

A blank value in this column indicates that back-action was not measured in the experiment.
Note that back-action effects dominated the mechanical Q in this measurement, lowering it from 1.2⫻ 105 to ⬃4.2⫻ 102. If onebcompares the imprecision against the zero-point noise of the uncoupled mechanical resonator, one finds

important topics of back-action damping, effective temperature, noise cross correlation, and power gain isgiven in Sec. V.E.
IV. GENERAL LINEAR-RESPONSE THEORY
A. Quantum constraints on noise

In this section, we further develop the connection between quantum limits and noise discussed previously, focusing now on a more general approach. As before, we
emphasize the idea that reaching the quantum limit requires a detector having quantum-ideal noise properties.
The approach here is different from typical treatmentsin the quantum optics literature 共Gardiner and Zoller,
2000; Haus, 2000兲, and uses nothing more than featuresof quantum linear response. Our discussion here will expand upon Clerk et al. 共2003兲 and Clerk 共2004兲; somewhat similar approaches to quantum measurement arealso discussed by Braginsky and Khalili 共1992兲 andAverin 共2003兲.
In this section, we start by heuristically sketching how
constraints on noise 关similar to Eq. 共3.9兲 for the cavity
detector兴 can emerge directly from the Heisenberg uncertainty principle. We then present a rigorous and genRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

冑S̄Ixx / S̄0xx关⍀兴 ⬃ 0.33.

eral quantum constraint on noise. We introduce both thenotion of a generic linear-response detector and the basic quantum constraint on detector noise. Next, we discuss how this noise constraint leads to the quantum limit
on QND state detection of a qubit. The quantum limit
on a linear amplifier 共or a position detector兲 is discussed
in the next section.

1. Heuristic weak-measurement noise constraints

As stressed in the introduction, there is no fundamental quantum limit on the accuracy with which a given
observable can be measured, at least not within theframework of nonrelativistic quantum mechanics. For
example, one can, in principle, measure the position of aparticle to arbitrary accuracy in the course of a projection measurement. However, the situation is different
when we specialize to continuous non-QND measurements. Such a measurement can be envisaged as a series
of instantaneous measurements, in the limit where thespacing between the measurements ␦t is taken to zero.
Each measurement in the series has a limited resolution
and perturbs the conjugate variables, thereby affecting
the subsequent dynamics and measurement results. We

Clerk et al.: Introduction to quantum noise, measurement, …

1174

had F̂ = n̂, the cavity photon number. As we are interested in weak couplings, we assume a simple bilinear
form for the detector-signal interaction Hamiltonian
共4.1兲

Ĥint = Ax̂F̂.
signalsourceinput output
operator"load"

detector
FIG. 7. 共Color online兲 Schematic of a generic linear-response
detector.

discuss this for the example of a series of position measurements of a free particle.
After initially measuring the position with an accuracy⌬x, the momentum suffers a random perturbation ofsize ⌬p 艌 ប / 共2⌬x兲. Consequently, a second position measurement taking place a time ␦t later will have an additional uncertainty of size ␦t共⌬p / m兲 ⬃ ប␦t / m⌬x. Thus,
when one is trying to obtain a good estimate of the position by averaging several such measurements, it is not
optimal to make ⌬x too small, because otherwise thisadditional perturbation, called the back-action of themeasurement device, will become large. The back-action
can be described as a random force ⌬F = ⌬p / ␦t. A meaningful limit ␦t → 0 is obtained by keeping both ⌬x2␦t⬅ S̄xx and ⌬p2 / ␦t ⬅ S̄FF fixed. In this limit, the deviations
␦x共t兲 describing the finite measurement accuracy and thefluctuations of the back-action force F can be described
as white noise processes, 具␦x共t兲␦x共0兲典 = S̄xx␦共t兲 and
具F共t兲F共0兲典 = S̄FF␦共t兲. The Heisenberg uncertainty relation⌬p⌬x 艌 ប / 2 then implies S̄xxS̄FF 艌 ប2 / 4 共Braginsky andKhalili, 1992兲. Note that this is completely analogous to
the relation 共3.9兲 we derived for the resonant cavity detector using the fundamental number-phase uncertainty
relation. In this section, we derive rigorously more general quantum limit relations on noise spectral densitiesof this form.
2. Generic linear-response detector

To rigorously discuss the quantum limit, we start witha description of a detector that is as general as possible.
To that end, we think of a detector as some physical

Here the operator x̂ 共which is not necessarily a position
operator兲 carries the input signal. Note that because x̂
belongs to the signal source, it necessarily commuteswith the detector variables Î , F̂.
We always assume the coupling strength A to be smallenough that we can accurately describe the output of thedetector using linear response.6 We thus have
具Î共t兲典 = 具Î典0 + A

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共4.2兲in the coupling Ĥint, one can easily derive Eq. 共4.2兲, with
␹IF共t兲 given by a Kubo-like formula,

␹IF共t兲 = − 共i/ប兲␪共t兲具关Î共t兲,F̂共0兲兴典0 .

共4.3兲

Here and in what follows the operators Î and F̂ are
Heisenberg operators with respect to the detector
Hamiltonian, and the subscript 0 indicates an expectation value with respect to the density matrix of the uncoupled detector.
As discussed, there will be unavoidable noise in boththe input and output ports of our detector. This noise issubject to quantum-mechanical constraints, and its presence is what limits our ability to make a measurement or
amplify a signal. We thus need to quantitatively characterize the noise in both these ports. Recall from the discussion in Sec. II.B that it is the symmetric-in-frequency
part of a quantum noise spectral density which plays arole akin to classical noise. We thus want to characterize
the symmetrized noise correlators of our detector 共denoted as always with an overbar兲. Redefining these operators so that their average value is zero at zero coupling 共i.e., F̂ → F̂ − 具F̂典0, Î → Î − 具Î典0兲, we have
S̄FF关␻兴 ⬅

S̄II关␻兴 ⬅

ized by an operator F̂, and an output port, characterized

section兲. The input operator F̂ is the detector quantitythat directly couples to the input signal 共e.g., the qubit兲
and that causes a back-action disturbance of the signal
source; in the cavity example of the previous section, we

dt ␹IF共t − t⬘兲具x̂共t⬘兲典,where 具Î典0 is the input-independent value of the detector
output at zero coupling and ␹IF共t兲 is the linear-response
susceptibility or gain of our detector. Note that Clerk et
al. 共2003兲 and Clerk 共2004兲 denoted this gain coefficient␭. Using standard time-dependent perturbation theory

system 共described by some unspecified Hamiltonian Ĥdet
and some unspecified density matrix ␳ˆ 0兲 which is time
independent in the absence of coupling to the signal
source. The detector has both an input port, characterby an operator Î 共see Fig. 7兲. The output operator Î issimply the quantity that is read out at the output of thedetector 共e.g., the current in a single-electron transistor,
or the phase shift in the cavity detector of the previous

冕 ⬘

6

1
2

1
2

冕

⬁

dt ei␻t具兵F̂共t兲,F̂共0兲其典0 ,共4.4a兲

dt ei␻t具兵Î共t兲,Î共0兲其典0 ,共4.4b兲

−⬁

冕

⬁

−⬁

The precise condition for the breakdown of linear response
will depend on specific details of the detector. For example, inthe cavity detector discussed in Sec. III.B, one would need thedimensionless coupling A to satisfy A Ⰶ 1 / Qc具ẑ典 to ensure thatthe nonlinear dependence of the phase shift ␪ on the signal 具ẑ典
is negligible. This translates to the signal modulating the cavity
frequency by an amount much smaller than its linewidth ␬.

Clerk et al.: Introduction to quantum noise, measurement, …

S̄IF关␻兴 ⬅

1
2

冕

⬁

dt ei␻t具兵Î共t兲,F̂共0兲其典0 ,共4.4c兲

−⬁where 兵,其 indicates the anticommutator, S̄II represents
the intrinsic noise in the output of the detector, and S̄FF
describes the back-action noise seen by the source of the
input signal. In general, there will be some correlation
between these two kinds of noise; this is described by
the cross correlator S̄IF.
Finally, we must also allow for the possibility that our
detector could operate in reverse 共i.e., with input andoutput ports playing opposite roles兲. We thus introduce
the reverse gain ␹FI of our detector. This is the response
coefficient describing an experiment where we couple
our input signal to the output port of the detector 共i.e.,Ĥint = Ax̂Î兲, and attempt to observe something at the input port 关i.e., in 具F̂共t兲典兴. In complete analogy to Eq. 共4.2兲,one would then have
具F̂共t兲典 = 具F̂典0 + A冕 ⬘dt ␹FI共t − t⬘兲具x̂共t⬘兲典

共4.5兲

with

␹FI共t兲 = − 共i/ប兲␪共t兲具关F̂共t兲,Î共0兲兴典0 .

共4.6兲

Note that if our detector is in a time-reversal symmetric,
thermal equilibrium state, then Onsager reciprocity re* 共if I and F have thelations would imply either ␹IF = ␹FI* 共if I and Fsame parity under time reversal兲 or ␹IF = −␹FIhave the opposite parity under time reversal兲 关see, e.g.,
Pathria 共1996兲兴. Thus, if the detector is in equilibrium,
the presence of gain necessarily implies the presence ofreverse gain. Nonzero reverse gain is also found in many
standard classical electrical amplifiers such as op-amps
共Boylestad and Nashelsky, 2006兲.
The reverse gain is something that we must worry
about even if we are not interested in operating our detector in reverse. To see why, note that to make a measurement of the output operator Î, we must necessarily
couple to it in some manner. If ␹FI ⫽ 0, the noise associated with this coupling could in turn lead to additional
back-action noise in the operator F̂. Even if the reverse
gain did nothing but amplify vacuum noise entering theoutput port, this would heat up the system being measured at the input port and hence produce excess backaction. Thus, the ideal situation is to have ␹FI = 0, implying a high asymmetry between the input and output ofthe detector, and requiring the detector to be in a state
far from thermodynamic equilibrium. We note that almost all mesoscopic detectors that have been studied indetail 共e.g., single-electron transistors and generalized
quantum point contacts兲 have been found to have a vanishing reverse gain: ␹FI = 0 共Clerk et al., 2003兲. For thisreason, we often focus on the ideal 共but experimentally
relevant兲 situation where ␹FI = 0 in what follows.
Before proceeding, it is worth emphasizing that there
is a relation between the detector gains ␹IF and ␹FI and
Rev. Mod. Phys., Vol. 82, No. 2, April–June 20101175

the unsymmetrized I-F quantum noise correlator, SIF关␻兴.
This spectral density, which need not be symmetric infrequency, is defined as
SIF关␻兴 =冕⬁dt ei␻t具Î共t兲F̂共0兲典0 .

共4.7兲

−⬁

Using the definitions, one can easily show that
S̄IF关␻兴 = 21 †SIF关␻兴 + SIF关− ␻兴*‡,

共4.8a兲

␹IF关␻兴 − ␹FI关␻兴* = − 共i/ប兲†SIF关␻兴 − SIF关− ␻兴*‡.

共4.8b兲

Thus, while S̄IF represents the classical part of the I-F
quantum noise spectral density, the gains ␹IF , ␹FI are determined by the quantum part of this spectral density.
This also demonstrates that though the gains have an
explicit factor of 1 / ប in their definitions, they have awell-defined ប → 0 limit, as the asymmetric-in-frequency
part of SIF关␻兴 vanishes in this limit.
3. Quantum constraint on noise

Despite having said nothing about the detector’s
Hamiltonian or state 共except that it is time independent兲, we can nonetheless derive a general quantum constraint on its noise properties. Note first that for purely
classical noise spectral densities one always has the inequality
S̄II关␻兴S̄FF关␻兴 − 兩S̄IF关␻兴兩2 艌 0.

共4.9兲This simply expresses the fact that the correlation between two different noisy quantities cannot be arbitrarily large; it follows immediately from the Schwartz
inequality. In the quantum case, this simple constraint
becomes modified whenever there is an asymmetry between the detector’s gain and reverse gain. This asymmetry is parametrized by the quantity ␹˜ IF关␻兴,␹˜ IF关␻兴 ⬅ ␹IF关␻兴 − 共␹FI关␻兴兲* .

共4.10兲

We show below that the following quantum noise inequality 共involving symmetrized noise correlators兲 is always valid 关see also Eq. 共6.36兲 in Braginsky and Khalili
共1996兲兴,S̄II关␻兴S̄FF关␻兴 − 兩S̄IF关␻兴兩2
艌

冏

冏冉 冋

ប␹˜ IF关␻兴 2
S̄IF关␻兴
1+⌬
2
˜
ប␹IF关␻兴/2

册冊,

共4.11兲

where⌬关z兴 = 关兩1 + z2兩 − 共1 + 兩z兩2兲兴/2.

共4.12兲

To interpret the quantum noise inequality Eq. 共4.11兲,note that 1 + ⌬关z兴 艌 0. Equation 共4.11兲 thus implies that if
our detector has gain and does not have a perfect sym* 兲, then it
metry between input and output 共i.e., ␹IF ⫽ ␹FI
must in general have a minimum amount of back-action
and output noise; moreover, these two noises cannot be
perfectly anticorrelated. As shown in the following sections, this constraint on the noise of a detector directly

Clerk et al.: Introduction to quantum noise, measurement, …

1176

leads to quantum limits on various different measurement tasks. Note that in the zero-frequency limit ␹˜ IF andS̄IF are both real, implying that the term involving ⌬ in
Eq. 共4.11兲 vanishes. The result is a simpler-looking inequality found elsewhere 共Averin, 2003; Clerk et al.,
2003; Clerk, 2004兲.
While Eq. 共4.11兲 may appear reminiscent of the standard fluctuation-dissipation theorem, its origin is quite
different: in particular, the quantum noise constraint applies irrespective of whether the detector is in equilibrium. Equation 共4.11兲 instead follows directly from
Heisenberg’s uncertainty relation applied to the frequency representation of the operators Î and F̂. In its
most general form, the Heisenberg uncertainty relation
gives a lower bound for the uncertainties of two observables in terms of their commutator and their noise correlator 共Gottfried, 1966兲,
共⌬A兲2共⌬B兲2 艌 41 具兵Â,B̂其典2 + 41 兩具关Â,B̂兴典兩2 .

共4.13兲

Here we have assumed 具Â典 = 具B̂典 = 0. We now choose theHermitian operators Â and B̂ to be given by the cosine
transforms of Î and F̂, respectively, over a finite timeinterval T,
Â ⬅

B̂ ⬅

冑冕
2
T

dt cos共␻t + ␦兲Î共t兲,

共4.14a兲dt cos共␻t兲F̂共t兲.

共4.14b兲

−T/2

冑冕
2
T

T/2

T/2

−T/2

Note that we have phase shifted the transform of Î relative to that of F̂ by a phase ␦. In the limit T → ⬁ we find,at any finite frequency ␻ ⫽ 0
共⌬A兲 = S̄II关␻兴,
2

共⌬B兲 = S̄FF关␻兴,

共4.15a兲2

具兵Â,B̂其典 = 2 Re ei␦S̄IF关␻兴,
具关Â,B̂兴典 =冕

共4.15b兲

+⬁dt cos共␻t + ␦兲具关Î共t兲,F̂共0兲兴典−⬁= iប Re†ei␦„␹IF关␻兴 − 共␹FI关␻兴兲*…‡.

共4.15c兲In the last equality, we have simply made use of the
Kubo formula definitions of the gain and reverse gain
关cf. Eqs. 共4.3兲 and 共4.6兲兴. As a consequence of Eqs.
共4.15a兲, 共4.15b兲, and 共4.15c兲, the Heisenberg uncertainty
relation 共4.13兲 directly yields
S̄II关␻兴S̄FF关␻兴 艌 兵Re共ei␦S̄IF关␻兴兲其2 +− 共␹FI关␻兴兲*…‡2 .

ប2†Re ei␦„␹IF关␻兴
4
共4.16兲Maximizing the RHS of this inequality over ␦ then yields
the general quantum noise constraint of Eq. 共4.11兲.
With this derivation, we can now interpret the quantum noise constraint Eq. 共4.11兲 as stating that the noise
at a given frequency given frequency in two observables
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010Î and F̂ is bounded by the value of their commutator atthat frequency. The fact that Î and F̂ do not commute isnecessary for the existence of linear response 共gain兲
from the detector, but also means that the noise in bothÎ and F̂ cannot be arbitrarily small. A more detailed
derivation, yielding additional important insights, is described in Appendix I.1.
Given the quantum noise constraint of Eq. 共4.11兲, we
can now very naturally define a quantum-ideal detector
共at a given frequency ␻兲 as one which minimizes theleft-hand side 共LHS兲 of Eq. 共4.11兲—a quantum-ideal detector has a minimal amount of noise at frequency ␻. We
are often interested in the ideal case where there is noreverse gain 共i.e., measuring Î does not result in additional back-action noise in F̂兲; the condition to have aquantum-limited detector thus becomesS̄II关␻兴S̄FF关␻兴 − 兩S̄IF关␻兴兩2
=

冏

冏冉 冋

ប␹IF关␻兴 2
S̄IF关␻兴
1+⌬
2
ប␹IF关␻兴/2

册冊,

共4.17兲

where ⌬关z兴 is given in Eq. 共4.12兲. Again, as discussed
below, in most cases of interest 共e.g., zero frequency
and/or large amplifier power gain兲 the last term on the
RHS will vanish. In the following, we demonstrate thatthe ideal noise requirement of Eq. 共4.17兲 is necessary inorder to achieve the quantum limit on QND detection ofa qubit, or on the added noise of a linear amplifier.
Before leaving our general discussion of the quantum
noise constraint, it is worth emphasizing that achieving
Eq. 共4.17兲 places a strong constraint on the properties ofthe detector. In particular, there must exist a tight connection between the input and output ports of thedetector—in a certain restricted sense, the operators Î
and F̂ must be proportional to one another 关see Eq. 共I13兲
in Appendix I.1兴. As is discussed in Appendix I.1, this
proportionality immediately tells us that a quantumideal detector cannot be in equilibrium. The proportionality exhibited by a quantum-ideal detector is parametrized by a single complex-valued number ␣关␻兴, whose
magnitude is given by兩␣关␻兴兩2 = S̄II关␻兴/S̄FF关␻兴.

共4.18兲

While this proportionality requirement may seem purely
formal, it does have a simple heuristic interpretation; as
discussed by Clerk et al. 共2003兲, it may be viewed as aformal expression of the principle that a quantumlimited detector must not contain any wasted information 共cf. Sec. III.B.2兲.
4. Evading the detector quantum noise inequality

We now turn to situations where the RHS of Eq.
共4.11兲 vanishes, implying that there is no additional
quantum constraint on the noise of our detector beyond
what exists classically. In such situations, one could havea detector with perfectly correlated back-action and out-

Clerk et al.: Introduction to quantum noise, measurement, …

put noises 共i.e., S̄FFS̄II = 兩S̄IF兩2兲, or even with a vanishingback-action S̄FF = 0. Perhaps not surprisingly, these situations are not of much utility. As we now show, in cases
where the RHS of Eq. 共4.11兲 vanishes, the detector may
be low noise, but will necessarily be deficient in anotherimportant regard: it will not be good enough that we canignore the extra noise associated with the measurementof the detector output Î. As discussed, the reading out ofÎ will invariably involve coupling the detector output tosome other physical system. In the ideal case, this coupling will not generate any additional back-action on thesystem coupled to the detector’s input port. In addition,
the signal at the detector output should be large enoughthat any noise introduced in measuring Î is negligible;
we already came across this idea in our discussion of theresonant cavity detector 关see discussion following Eq.
共3.60兲兴. This means that we need our detector to truly
amplify the input signal, not simply reproduce it at theoutput with no gain in energy. As now shown, a detector
that evades the quantum constraint of Eq. 共4.11兲 bymaking the RHS of the inequality zero will necessarily
fail in one or both of the above requirements.
The most obvious case where the quantum noise constraint vanishes is for a detector which has equal forward* . As mentioned, this relation
and reverse gains, ␹FI = ␹IFwill necessarily hold if the detector is time-reversal symmetric and in equilibrium, and Î and F̂ have the same
parity under time reversal. In this case, the relatively
large reverse gain implies that in the analysis of a given
measurement task, it is not sufficient to just consider thenoise of the detector: one must necessarily also consider
the noise associated with whatever system is coupled toÎ to read out the detector output, as this noise will be fed
back to the detector input port, causing additional backaction; we give an explicit example of this in the next
subsection, where we discuss QND qubit detection.
* , there is
Even more problematically, when ␹FI = ␹IFnever any amplification by the detector. As discussed inSec. V.E.3, the proper metric of the detector’s ability toamplify is its dimensionless power gain: What is thepower supplied at the output of the detector versus theamount of power drawn at the input from the signal* , one has negative feedback, withsource? When ␹FI = ␹IF
the result that the power gain cannot be larger than 1关see Eq. 共5.53兲兴. There is thus no amplification when
* . Further, if one also insists that the noise con␹FI = ␹IFstraint of Eq. 共4.11兲 is optimized, then one finds thepower gain must be exactly 1; this is explicitly demonstrated in Appendix I.2. The detector thus will simply
act as a transducer, reproducing the input signal at theoutput without any increase in energy. We have here a
specific example of a more general idea that will be discussed in Sec. V: if a detector acts only as a transducer, itneed not add any noise.
At finite frequencies, there is a second way to makethe RHS of the quantum noise constraint of Eq. 共4.11兲
vanish: one needs the quantity S̄IF关␻兴 / ␹˜ IF to be purely
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1177

imaginary, and larger in magnitude than ប / 2. In thiscase, it would again be possible to have the LHS of thenoise constraint of Eq. 共4.11兲 equal to 0. However, one
again finds that in such a case the dimensionless power
gain of the detector is at most equal to 1; it thus does notamplify. This is shown explicitly in Appendix I.2. An
important related statement is that a quantum-limited
detector with a large power gain must have the quantity
S̄IF / ␹IF be real. Thus, at the quantum limit, correlations
between the back-action force and the intrinsic output
noise fluctuations must have the same phase as the gain
␹IF. As discussed in Sec. V.F, this requirement can beinterpreted in terms of the principle of no wasted information introduced in Sec. III.B.2.

B. Quantum limit on QND detection of a qubit

In Sec. III.B, we discussed the quantum limit on QND
qubit detection in the specific context of a resonant cavity detector. We now show how the full quantum noise
constraint of Eq. 共4.11兲 directly leads to this quantum
limit for an arbitrary weakly coupled detector. Similar toSec. III.B, we couple the input operator of our genericlinear-response detector to the ␴ˆ z operator of the qubit
we wish to measure 关i.e., we take x̂ = ␴ˆ z in Eq. 共4.1兲兴; wealso consider the QND regime, where ␴ˆ z commutes with
the qubit Hamiltonian. As we saw in Sec. III.B, thequantum limit in this case involves the inequality ⌫meas艋 ⌫␸, where ⌫meas is the measurement rate and ⌫␸ is the
back-action dephasing rate. For the latter quantity, we
can directly use the results of our calculation for thecavity system, where we found the dephasing rate wasset by the zero-frequency noise in the cavity photon
number 关see Eq. 共3.27兲兴. In complete analogy, the backaction dephasing rate will be determined here by thezero-frequency noise in the input operator F̂ of our detector,⌫␸ = 共2A2/ប2兲S̄FF .

共4.19兲

We omit frequency arguments in this section, as it isalways the zero-frequency susceptibilities and spectral
densities that appear.
The measurement rate 共the rate at which informationon the state of qubit is acquired兲 is also defined in complete 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共t兲 of our detector,

冕 ⬘⬘tm̂共t兲 =

dt Î共t 兲.

共4.20兲

0

The probability distribution of the integrated outputm̂共t兲 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 Î so that its expectation vanishes at zero coupling, the average value of 具m̂共t兲典 corresponding to each
qubit state is 共in the long-time limit of interest兲Clerk et al.: Introduction to quantum noise, measurement, …

1178

具m̂共t兲典↑ = A␹IFt,

具m̂共t兲典↓ = − A␹IFt.

共4.21兲

The variance of both distributions is, to leading order,
independent of the qubit state,
2⬅ 具具m̂2共t兲典典↑/↓ = S̄IIt.
具m̂2共t兲典↑/↓ − 具m̂共t兲典↑/↓共4.22兲

For the last equality above, we have taken the long-time
limit, which results in the variance of m̂ being determined completely by the zero-frequency output noiseS̄II关␻ = 0兴 of the detector. The assumption here is that,
due to the weakness of the measurement, the measurement time 1 / ⌫meas will be much longer than the autocorrelation time of the detector’s noise.
We can now define the measurement rate, in complete
analogy to the cavity detector of the previous section 关cf.
Eq. 共3.23兲兴, by how quickly the resolving power of themeasurement grows,7
1
2
2
2
4 关具m̂共t兲典↑ − 具m̂共t兲典↓兴 /关具具m̂ 共t兲典典↑ + 具具m̂ 共t兲典典↓兴 ⬅ ⌫meast.

共4.23兲
This yields⌫meas = A2共␹IF兲2/2S̄II .

共4.24兲

Putting this all together, we find that the “efficiency”
ratio ␩ = ⌫meas / ⌫␸ is given by␩ ⬅ ⌫meas/⌫␸ = ប2共␹IF兲2/4S̄IIS̄FF .

共4.25兲

In the case where our detector has a vanishing reverse
gain 共i.e., ␹FI = 0兲, the quantum-limit bound ␩ 艋 1 follows
immediately from the quantum noise constraint of Eq.
共4.11兲. We thus see that achieving the quantum limit for
QND qubit detection requires both a detector withquantum-ideal noise properties, as defined by Eq. 共4.17兲,and a detector with a vanishing noise cross correlator:
S̄IF = 0.
If in contrast ␹FI ⫽ 0, it would seem that it is possible
to have ␩ 艌 1. This is of course an invalid inference: as
discussed, ␹FI ⫽ 0 implies that we must necessarily consider the effects of extra noise injected into the detector’s output port when one measures Î, as the reverse
gain will bring this noise back to the qubit, causing extra
dephasing. The result is that one can do no better than␩ = 1. To see this explicitly, consider the extreme case␹IF = ␹FI and S̄II = S̄FF = 0, and suppose we use a second
detector to read out the output Î of the first detector.
This second detector has input and output operators F̂2,Î2; we also take it to have a vanishing reverse gain, sothat we do not have to also worry about how its output
is read out. Coupling of the detectors linearly in thestandard way 共i.e., Hint,2 = ÎF̂2兲, gives the overall gain of

the two detectors in series as ␹I2F2␹IF, while the backaction driving the qubit dephasing is described by thespectral density 共␹FI兲2SF2F2. Using the fact that our second detector must itself satisfy the quantum noise inequality, we have关共␹FI兲2S̄F2F2兴S̄I2I2 艌 共ប2/4兲共␹I2F2␹IF兲2 .

共4.26兲

Thus, the overall chain of detectors satisfies the usual
zero-reverse-gain quantum noise inequality, implyingthat we still have ␩ 艋 1.
V. QUANTUM LIMIT ON LINEAR AMPLIFIERS ANDPOSITION DETECTORSIn the previous section we established the fundamental quantum constraint on the noise of any system capable of acting as a linear detector; we further showedthat this quantum noise constraint directly leads to thequantum limit on nondemolition qubit detection using a
weakly coupled detector. In this section, we turn to themore general situation where our detector is a phasepreserving quantum linear amplifier: the input to the detector is described by some time-dependent operatorx̂共t兲, which we wish to have amplified at the output ofour detector. As we see, the quantum limit in this case isa limit on how small one can make the noise added by
the amplifier to the signal. The discussion in this section
both furthers and generalizes the heuristic discussion of
position detection using a cavity detector presented inSec. III.B.
In this section, we start by presenting a heuristic discussion of quantum constraints on amplification. We
then demonstrate explicitly how the previously discussed quantum noise constraint leads directly to thequantum limit on the added noise of a phase-preserving
linear amplifier; we examine the cases of both a genericlinear position detector and a generic voltage amplifier,following the approach outlined by Clerk 共2004兲. We
also spend time explicitly connecting the linear-response
approach used here to the bosonic scattering formulation of the quantum limit favored by the quantum optics
community 共Haus and Mullen, 1962; Caves, 1982; Grassia, 1998; Courty et al., 1999兲, paying particular attention
to the case of a two-port scattering amplifier. We will seethat there are some important subtleties involved in converting between the two approaches. In particular, there
exists a crucial difference between the case where theinput signal is tightly coupled to the input of the amplifier 共the case usually considered in the quantum optics
community兲, versus the case where, as in an ideal opamp, the input signal is only weakly coupled to the inputof the amplifier 共the case usually considered in the solidstate community兲.
A. Preliminaries on amplification

7

The factor of 1 / 4 here is purely chosen for convenience; weare defining the measurement rate based on the informationtheoretic definition given in Appendix F. This factor of 4 isconsistent with the definition used in the cavity system.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

What exactly does one mean by amplification? As wewill see 共Sec. V.E.3兲, a precise definition requires that theenergy provided at the output of the amplifier be much

Clerk et al.: Introduction to quantum noise, measurement, …

larger than the energy drawn at the input of theamplifier—the power gain of the amplifier must belarger than 1. For the moment, however, we work withthe cruder definition that amplification involves making
some time-dependent signal larger. To set the stage, wefirst consider an extremely simple classical analog of alinear amplifier. Imagine that the “signal” we wish to
amplify is the coordinate x共t兲 of a harmonic oscillator;
we can write this signal as
x共t兲 = x共0兲cos共␻St兲 + 关p共0兲/M␻S兴 sin共␻St兲.

共5.1兲

Our signal has two quadrature amplitudes, i.e., the amplitude of the cosine and sine components of x共t兲. To
amplify this signal, we start at t = 0 to parametricallydrive the oscillator by changing its frequency ␻S periodically in time: ␻S共t兲 = ␻0 + ␦␻ sin共␻Pt兲, where we assume
␦␻ Ⰶ ␻0. The well-known physical example is a swing
whose motion is being excited by effectively changing
the length of the pendulum at the right frequency andphase. For a “pump frequency” ␻P equalling twice the“signal frequency,” ␻P = 2␻S, the resulting dynamics willlead to an amplification of the initial oscillator position,with the energy provided by the external driving,x共t兲 = x共0兲e␭t cos共␻St兲 + 关p共0兲/M␻S兴e−␭t sin共␻St兲.
共5.2兲
Thus, one of the quadratures is amplified exponentially,
at a rate ␭ = ␦␻ / 2, while the other one decays. In aquantum-mechanical description, this produces asqueezed state out of an initial coherent state. Such asystem is called a “degenerate parametric amplifier,”
and we discuss its quantum dynamics in Sec. V.H. We
see that such an amplifier, which amplifies only a single
quadrature, is not required quantum mechanically to
add any noise 共Caves et al., 1980; Caves, 1982; Braginsky
and Khalili, 1992兲.
Can we now change this parametric amplification
scheme slightly in order to make both signal quadratures
grow with time? It turns out that this is impossible aslong as we restrict ourselves to a driven system with asingle degree of freedom. The reason in classical mechanics is that Liouville’s theorem requires phase-spacevolume to be conserved during motion. More formally,
this is related to the conservation of Poisson brackets, orin quantum mechanics to the conservation of commutation relations. Nevertheless, it is certainly desirable tohave an amplifier that acts equally on both quadratures
共a so-called phase-preserving or phase-insensitive amplifier兲, since the signal’s phase is often not known beforehand. The way around the restriction created by Liouville’s theorem is to add more degrees of freedom, suchthat the phase-space volume can expand in both quadratures 共i.e., position and momentum兲 of the interesting
signal degree of freedom, while being compressed inother directions. This is achieved most easily by coupling
the signal oscillator to another oscillator, the “idler
mode.” The external driving now modulates the coupling between these oscillators, at a frequency that has
Rev. Mod. Phys., Vol. 82, No. 2, April–June 20101179

to equal the sum of the oscillators’ frequencies. The resulting scheme is called a phase-preserving nondegenerate parametric amplifier 共see Sec. V.C兲.
Crucially, there is a price to pay for the introduction ofan extra degree of freedom: there will be noise associated with the idler oscillator, and this noise will contribute to the noise in the output of the amplifier. Classically, one could make the noise associated with the idler
oscillator arbitrarily small by simply cooling it to zerotemperature. This is not possible quantum mechanically;
there are always zero-point fluctuations of the idler oscillator to contend with. It is this noise which sets a fundamental quantum limit for the operation of the amplifier. We thus have a heuristic accounting for the
existence of a quantum limit on the added noise of aphase-preserving linear amplifier: one needs extra degrees of freedom to amplify both signal quadratures, andsuch extra degrees of freedom invariably have noise associated with them.
B. Standard Haus-Caves derivation of the quantum limit on abosonic amplifier

We now make the ideas of the previous section more
precise by sketching the standard derivation of thequantum limit on the noise added by a phase-preserving
amplifier. This derivation is originally due to Haus andMullen 共1962兲, and was both clarified and extended by
Caves 共1982兲; the amplifier quantum limit was also motivated in a slightly different manner by Heffner 共1962兲.8
While extremely compact, the Haus-Caves derivation
can lead to confusion when improperly applied; we discuss this in Sec. V.D, as well as in Sec. VI, where weapply this argument carefully to the important case of a
two-port quantum voltage amplifier and discuss the connection to the general linear-response formulation ofSec. IV.
The starting assumption of this derivation is that boththe input and output ports of the amplifier can be described by sets of bosonic modes. If we focus on a narrow bandwidth centered on frequency ␻, we can describe a classical signal E共t兲 in terms of a complex
number a defining the amplitude and phase of the signal
共or equivalently the two quadrature amplitudes兲 共Haus
and Mullen, 1962; Haus, 2000兲
E共t兲 ⬀ i关ae−i␻t − a*e+i␻t兴.

共5.3兲

In the quantum case, the two signal quadratures of E共t兲关i.e., the real and imaginary parts of a共t兲兴 cannot be measured simultaneously because they are canonically conjugate; this is in complete analogy to a harmonic oscillator 关see Eq. 共3.48兲兴. As a result a , a* must be elevated tothe status of photon ladder operators: a → â, a* → a†.
Consider the simplest case where there is only a single
mode at both the input and output, with corresponding
8
Note that Caves 共1982兲 provided a discussion of why thederivation of the amplifier quantum limit given by Heffner
共1962兲 is not rigorously correct.

Clerk et al.: Introduction to quantum noise, measurement, …

1180

operators â and b̂.9 It follows that the input signal into
the amplifier is described by the expectation value 具â典,
while the output signal is described by 具b̂典. Correspondingly, the symmetrized noise in both these quantities is
described by
共⌬a兲2 ⬅ 21 具兵â,â†其典 − 兩具â典兩2 ,共5.4兲

with an analogous definition for 共⌬b兲 .
To derive a quantum limit on the added noise of theamplifier, one uses two simple facts. First, both the input
and the output operators must satisfy the usual commutation relations2

关â,â†兴 = 1,关b̂,b̂†兴 = 1.

共5.5兲

Second, the linearity of the amplifier and the fact that itis phase preserving 共i.e., both signal quadratures are amplified the same way兲 implies a simple relation between
the output operator b̂ and the input operator â,b̂ = 冑Gâ,b̂† = 冑Gâ† ,共5.6兲

where G is the dimensionless photon-number gain of theamplifier. It is clear, however, that this expression cannot
possibly be correct as written because it violates the fundamental bosonic commutation relation 关b̂ , b̂†兴 = 1. We
are therefore forced to writeb̂ = 冑Gâ + F̂,b̂† = 冑Gâ† + F̂† ,共5.7兲

where F̂ is an operator representing additional noise
added by the amplifier. Based on the discussion of theprevious section, we can anticipate what F̂ represents: it
is noise associated with the additional degrees of freedom that must invariably be present in a phasepreserving amplifier.
As F̂ represents noise, it has a vanishing expectation
value; in addition, one also assumes that this noise isuncorrelated with the input signal, implying 关F̂ , â兴= 关F̂ , â†兴 = 0 and 具F̂â典 = 具F̂â†典 = 0. Insisting that 关b̂ , b̂†兴 = 1
thus yields
关F̂,F̂†兴 = 1 − G.

共5.8兲

The question now becomes how small can we makethe noise described by F̂? From Eqs. 共5.7兲, the noise at
the amplifier output ⌬b is given by共⌬b兲2 = G共⌬a兲2 + 21 具兵F̂,F̂†其典 艌 G共⌬a兲2 + 21 兩具关F̂,F̂†兴典兩艌 G共⌬a兲2 + 兩G − 1兩/2.

共5.9兲

We have used here a standard inequality to bound the
expectation of 兵F̂ , F̂†其. The first term here is simply the9

To relate this to the linear-response detector of Sec. IV.A,
one could naively write x̂, the operator carrying the input signal as, x̂ = â + â†, and the output operator Î as, Î = b̂ + b̂† 共wediscuss how to make this correspondence in Sec. VI兲.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

amplified noise of the input, while the second term represents the noise added by the amplifier. Note that if
there is no amplification 共i.e., G = 1兲, there need not be
any added noise. However, in the more relevant case oflarge amplification 共G Ⰷ 1兲, the added noise cannot vanish. It is useful to express the noise at the output as anequivalent noise at the input by simply dividing out thephoton gain G. Taking the large-G limit, we have共⌬b兲2/G 艌 共⌬a兲2 + 21 .

共5.10兲

Thus, we have a simple demonstration that an amplifier
with a large photon gain must add at least half a quantum of noise to the input signal. Equivalently, the minimum value of the added noise is equal to the zero-point
noise associated with the input mode; the total output
noise 共referred to the input兲 is at least twice the zeropoint input noise. Note that both these conclusions are
identical to what we found in our analysis of the resonant cavity position detector in Sec. III.B.3. We discusslater how this conclusion can also be reached using thegeneral linear-response language of Sec. IV 共cf. Secs.
V.E and V.F兲.
As discussed, the added noise operator F is associatedwith additional degrees of freedom 共beyond input andoutput modes兲 necessary for phase-preserving amplification. To see this more concretely, note that every linear
amplifier is inevitably a nonlinear system consisting ofan energy source and a “spigot” controlled by the input
signal, which redirects the energy source partly to the
output channel and partly to some other channel共s兲.
Hence there are inevitably other degrees of freedom involved in the amplification process beyond the input andoutput channels. An explicit example is the quantum
parametric amplifier, discussed in the next subsection.
Further insights into amplifier-added noise and its connection to the fluctuation-dissipation theorem can be obtained by considering a simple model where a transmission line is terminated by an effective negative
impedance; we discuss this model in Appendix C.4.
To see explicitly the role of the additional degrees of
freedom, note first that for G ⬎ 1 the RHS of Eq. 共5.8兲 isnegative. Hence the simplest possible form for theadded noise isF̂ = 冑G − 1d̂†,F̂† = 冑G − 1d̂,

共5.11兲

where d̂ and d̂† represent a single additional mode of thesystem. This is the minimum number of additional degrees of freedom that must inevitably be involved in theamplification process. Note that for this case the inequality in Eq. 共5.9兲 is satisfied as an equality, and theadded noise takes on its minimum possible value. If instead we have, say, two additional modes 共coupled inequivalently兲
F̂ = 冑G − 1共cosh ␪d̂†1 + sinh ␪d̂2兲,共5.12兲

it is straightforward to show that the added noise is inevitably larger than the minimum. This again can be interpreted in terms of wasted information, as the extra
degrees of freedom are not being monitored as part ofClerk et al.: Introduction to quantum noise, measurement, …

this approximation were not valid, then our amplifier
would in any case not be the linear amplifier we seek.
With this approximation we can hereafter ignore the dynamics of the pump degree of freedom and deal with the
reduced system Hamiltonian

ωIωPĤsys = ប共␻IâI†âI + ␻SâS† âS兲 + iប␭共âS† âI†e−i共␻I+␻S兲t

ωS− âSâIe+i共␻I+␻S兲t兲,

FIG. 8. 共Color online兲 Energy level scheme of the nondegenerate 共phase-preserving兲 parametric oscillator.

the measurement process and so information is being
lost.

C. Nondegenerate parametric amplifier
1. Gain and added noise

Before we start our discussion of the connection of
the Haus-Caves formulation of the quantum limit to thegeneral linear-response approach of Sec. IV, it is useful
to consider a specific example. To that end, we analyze
here a nondegenerate parametric amplifier, a linear
phase-preserving amplifier which reaches the quantum
limit on its added noise 共Louisell et al., 1961; Gordon etal., 1963; Mollow and Glauber, 1967a, 1967b兲 and directly realizes the ideas of the previous subsection. One
possible realization 共Yurke et al., 1989兲 is a cavity withthree internal resonances that are coupled together by anonlinear element 共such as a Josephson junction兲 whose
symmetry permits three-wave mixing. The three modes
are called the pump, idler, and signal and their energy
level structure 共shown in Fig. 8兲 obeys ␻P = ␻I + ␻S. The
system Hamiltonian is thenĤsys = ប共␻PâP† âP + ␻IâI†âI + ␻SâS† âS兲 + iប␩共âS† âI†âP
共5.13兲

We have made the rotating wave approximation in thethree-wave mixing term, and without loss of generality
we take the nonlinear susceptibility ␩ to be real andpositive. The system is driven at the pump frequency
and the three-wave mixing term permits a single pump
photon to split into an idler photon and a signal photon.
This process is stimulated by signal photons alreadypresent and leads to gain. A typical mode of operation
would be the negative-resistance reflection mode inwhich the input signal is reflected from a nonlinear cavity and the reflected beam extracted using a circulator
共Yurke et al., 1989; Bergeal et al., 2008兲.
The nonlinear equations of motion 共EOMs兲 become
tractable if we assume the pump has large amplitude andcan be treated classically by making the substitutionâP = ␺Pe−i␻Pt = ␺Pe−i共␻I+␻S兲t ,共5.15兲

where ␭ ⬅ ␩␺P. Transforming to the interaction representation we are left with the following timeindependent quadratic Hamiltonian for the system
V̂sys = iប␭共âS† âI† − âSâI兲.

− âSâIâP† 兲.

1181

共5.16兲

To get some intuitive understanding of the physics, we
temporarily ignore the damping of the cavity modes thatwould result from their coupling to modes outside thecavity. We now have a pair of coupled EOMs for the twomodesâ˙S = + ␭âI† ,â˙I† = + ␭âS ,共5.17兲

for which the solutions areâS共t兲 = cosh共␭t兲âS共0兲 + sinh共␭t兲âI†共0兲,âI†共t兲 = sinh共␭t兲âS共0兲 + cosh共␭t兲âI†共0兲.

共5.18兲

We see that the amplitude in the signal channel grows
exponentially in time and that the effect of the time evolution is to perform a simple unitary transformation
which mixes âS with âI† in such a way as to preserve thecommutation relations. Note the close connection withthe form found from general arguments in Eqs.
共5.7兲–共5.11兲.
We may now tackle the full system which includes thecoupling between the cavity modes and modes external
to the cavity. Such a coupling is of course necessary inorder to feed the input signal into the cavity, as well as
extract the amplified output signal. It will also result in
the damping of the cavity modes, which will cut off the
exponential growth found above and yield a fixed amplitude gain. We present the main results in this section,
relegating details to how one treats the bath modes 关socalled input-output theory 共Walls and Milburn, 1994兲兴 toAppendix E. Working in the standard Markovian limit,
we obtain the following EOMs in the interaction representation:â˙S = − 共␬S/2兲âS + ␭âI† − 冑␬Sb̂S,in ,†.
â˙I† = − 共␬I/2兲âI† + ␭âS − 冑␬Ib̂I,in

共5.19兲

Here ␬S and ␬I are the respective damping rates of thecavity signal mode and the idler mode. The coupling to
extra-cavity modes also lets signals and noise enter thecavity from the baths: this is described by the bosonic

共5.14兲

operators b̂S,in and b̂I,in which drive the signal and idler

where without loss of generality we take ␺P to be realand positive. We note here the important point that if

modes, respectively. b̂S,in describes both the input signal
to be amplified and vacuum noise entering from the bathRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Clerk et al.: Introduction to quantum noise, measurement, …

1182

coupled to the signal mode, whereas b̂I,in simply describes vacuum noise.10
Let us fix our attention on signals inside a frequency
window ␦␻ centered on ␻S 共hence zero frequency in theinteraction representation兲. For simplicity, we first consider the case where the signal bandwidth ␦␻ is almost
infinitely narrow 共i.e., much smaller than the damping
rate of the cavity modes兲. It then suffices to find thesteady state solution of these EOMs,âS = 共2␭/␬S兲âI† − 共2/冑␬S兲b̂S,in ,共5.20兲

†âI† = 共2␭/␬I兲âS − 共2/冑␬I兲b̂I,in.

共5.21兲

The output signal of the nondegenerate paramp is thesignal leaving the cavity signal mode and entering theexternal bath modes; it is described by an operator b̂S,out.
The standard input-output theory treatment of theextra-cavity modes 共Walls and Milburn, 1994兲, presented
in Appendix E, yields the simple relation 关cf. Eq. 共E37兲兴
b̂S,out = b̂S,in + 冑␬SâS .

共5.22兲

The first term corresponds to the reflection of the signal
and noise incident on the cavity from the bath, while thesecond term corresponds to radiation from the cavity
mode into the bath. Using this, we find that the output
signal from the cavity is given by2b̂S,out =

Q +1
2Q †
b̂S,in + 2b̂ ,Q2 − 1Q − 1 I,in

共5.23兲

where Q ⬅ 2␭ / 冑␬I␬S is proportional to the pump amplitude and inversely proportional to the cavity decay
rates. We have to require Q2 ⬍ 1 to make sure that theparametric amplifier does not settle into self-sustained
oscillations, i.e., it works below threshold. Under thatcondition, we can define the photon-number gain G0 via− 冑G0 = 共Q2 + 1兲/共Q2 − 1兲,

共5.24兲

such that†b̂S,out = − 冑G0b̂S,in − 冑G0 − 1b̂I,in.

共5.25兲

In the ideal case, the noise associated with b̂I,in , b̂S,in is
simply vacuum noise. As a result, the input-output relation Eq. 共5.25兲 is precisely of the Haus-Caves form 共5.11兲
for an ideal quantum-limited amplifier. It demonstrates
that the nondegenerate parametric amplifier reaches thequantum limit for minimum added noise. In the limit oflarge gain the output noise 共referred to the input兲 for a
vacuum input signal is precisely doubled.
2. Bandwidth-gain trade-off

The above results neglected the finite bandwidth ␦␻of the input signal to the amplifier. The gain G0 given in
Eq. 共5.24兲 is only the gain at precisely the mean signal
10Note that the b̂ operators are not dimensionless, as b̂†b̂
represents a photon flux 共see Appendix E兲.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

frequency ␻S; for a finite bandwidth, we also need tounderstand how the power gain varies as a function of
frequency over the entire signal bandwidth. As we see, aparametric amplifier suffers from the fact that as one
increases the overall magnitude of the gain at the center
frequency ␻S 共e.g., by increasing the pump amplitude兲,
one simultaneously narrows the frequency range overwhich the gain is appreciable. Heuristically, this is because parametric amplification involves using the pumpenergy to decrease the damping and hence increase thequality factor of the signal mode resonance. This increase in quality factor leads to amplification, but it also
reduces the bandwidth over which âS can respond to the
input signal b̂S,in.
To deal with a finite signal bandwidth, one simply
Fourier transforms Eqs. 共5.19兲. The resulting equations
are easily solved and substituted into Eq. 共5.22兲, resulting in a frequency-dependent generalization of the
input-output relation given in Eq. 共5.25兲,†b̂S,out关␻兴 = − g关␻兴b̂S,in关␻兴 − g⬘关␻兴b̂I,in关␻兴.

共5.26兲

Here g关␻兴 is the frequency-dependent gain of the amplifier, and g⬘关␻兴 satisfies 兩g⬘关␻兴兩2 = 兩g关␻兴兩2 − 1. In the relevant limit where G0 = 兩g关0兴兩2 Ⰷ 1 共i.e., large gain at thesignal frequency兲, one has to a good approximationg关␻兴 =

冑G0 − i关共␬S − ␬I兲/共␬I + ␬S兲兴共␻/D兲
1 − i共␻/D兲,

共5.27兲

withD=1␬ S␬ I冑G 0 ␬ S + ␬ I .

共5.28兲As always, we work in an interaction picture where thesignal frequency has been shifted to zero. D represents
the effective operating bandwidth of the amplifier. Components of the signal with frequencies 共in the rotating
frame兲 兩␻ 兩 Ⰶ D are strongly amplified, while components
with frequencies 兩␻ 兩 Ⰷ D are not amplified at all, but canin fact be slightly attenuated. As already anticipated, theamplification bandwidth D becomes progressively
smaller as the pump power and G0 are increased, withthe product 冑G0D remaining constant. In a parametricamplifier increasing the gain via increasing the pumpstrength comes with a price: the effective operating
bandwidth is reduced.
3. Effective temperature

Recall that in Sec. II.B we introduced the concept ofan effective temperature of a nonequilibrium system,
Eq. 共2.8兲. As we will discuss, this concept plays an important role in quantum-limited amplifiers; the degenerate paramp gives us a first example of this. Returning to
the behavior of the paramp at the signal frequency, we
note that Eq. 共5.25兲 implies that, even for vacuum input
to both the signal and idler ports, the output will containa real photon flux. To quantify this in a simple way, it isClerk et al.: Introduction to quantum noise, measurement, …

useful to introduce temporal modes which describe theinput and output fields during a particular time interval
关j⌬t , 共j + 1兲⌬t兴 共where j is an integer兲,B̂S,in,j =1

冕

共j+1兲⌬t冑⌬t j⌬t

1183

共bS† 兲n兩0,0典 = 冑n!兩n,0典

共5.34兲

to obtain⬁d␶b̂S,in共␶兲,兩⌿out典 = Z−1/2 兺 ␣n兩n,n典

共5.29兲

共5.35兲

n=0

with the temporal modes B̂S,out,j and B̂I,in,j defined analogously. These temporal modes are discussed in Appendix D.2, where we discuss the windowed Fourier transform 关see Eq. 共D18兲兴.
With the above definition, we find that the output
mode will have a real occupancy even if the input modeis empty,†
B̂S,out,j兩0典
n̄S,out = 具0兩B̂S,out,jand hence
Z = 1/共1 − 兩␣兩2兲

so the state is normalizable only for 兩␣兩2 ⬍ 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 interesting toconsider the reduced density matrix obtain by tracing
over the idler mode. The pure-state density matrix is⬁††= G0具0兩B̂S,in,j
B̂S,in,j兩0典 + 共G0 − 1兲具0兩B̂I,in,jB̂I,in,j
兩0典

␳ = 兩⌿out典具⌿out兩 = 兺 兩n,n典

共5.30兲

= G0 − 1.

The dimensionless mode occupancy n̄S,out is best thoughtof as a photon flux per unit bandwidth 关see Eq. 共D26兲兴.
This photon flux is equivalent to the photon flux thatwould appear in equilibrium at the very high effective
temperature 共assuming large gain G0兲

共5.36兲

m,n=0

␣ n␣ ⴱ m
具m,m兩.
Z

共5.37兲

If we now trace over the idler mode we are left with the
reduced density matrix for the signal channel⬁␳˜ S = TrIdler兵␳其 = 兺 兩nS典
nS=0

兩␣兩2nS†
1
具nS兩 ⬅ e−␤ប␻SaSaS ,Z
Z

共5.31兲

共5.38兲

This is an example of a more general principle, discussed
in Sec. V.E.4: a high-gain amplifier must have associatedwith it a large effective temperature scale. Referring thistotal output noise back to the input, we have 共in thelimit G0 Ⰷ 1兲

which is a pure thermal equilibrium distribution witheffective Boltzmann factor

Teff ⬇ ប␻SG0 .

Teff/G0 = ប␻S/2 + ប␻S/2 =ប␻S+ TN .
2† †共5.33兲

where ␣ is a constant related to the gain and, to simplify
the notation, we have dropped the “out” labels on theoperators. The normalization constant Z can be workedout by expanding the exponential and using
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共5.39兲

The effective temperature can be obtained from the requirement that the signal mode occupancy is G0 − 1,
1/共e␤ប␻S − 1兲 = G0 − 1,

共5.32兲

This corresponds to the half photon of vacuum noise
associated with the signal source, plus the added noise ofa half photon of our phase-preserving amplifier 共i.e., thenoise temperature TN is equal to its quantum-limited
value兲. Here the added noise is simply the vacuum noise
associated with the idler port.
The above argument is merely suggestive that the output noise looks like an effective temperature. In fact, it
is possible to show that the photon-number distributionof the output is precisely that of a Bose-Einstein distribution at temperature Teff. From Eq. 共5.13兲 we see thatthe action of the paramp is to destroy a pump photon
and create a pair of new photons, one in the signal channel and one in the idler channel. Using the SU共1,1兲 symmetry of the quadratic hamiltonian in Eq. 共5.16兲 it ispossible to show that, for vacuum input, the output ofthe paramp is a so-called “two-mode squeezed state” ofthe form 共Caves and Schumaker, 1985; Gerry, 1985;
Knight and Buzek, 2004兲兩⌿out典 = Z−1/2e␣bSbI 兩0典,

e−␤ប␻S = 兩␣兩2 ⬍ 1.

共5.40兲

which in the limit of large gain reduces to Eq. 共5.31兲.
This appearance of finite entropy in a subsystem evenwhen the full system is in a pure state is a purely quantum effect. Classically the entropy of a composite system
is at least as large as the entropy of any of its components. Entanglement among the components allows thislower bound on the entropy to be violated in a quantum
system.11 In this case the two-mode squeezed state hasstrong entanglement between the signal and idler channels 共since their photon numbers are fluctuating identically兲.
D. Scattering versus op-amp modes of operation

We now begin to address the question of how thestandard Haus-Caves derivation of the amplifier quantum limit presented in Sec. V.B relates to the generallinear-response approach of Sec. IV. Recall that in Sec.
III.B.3 we already used this latter approach to discuss
position detection with a cavity detector, reaching simi11

This paradox has prompted Charles Bennett to remark thata classical house is at least as dirty as its dirtiest room, but aquantum house can be dirty in every room and still perfectly
clean over all.

Clerk et al.: Introduction to quantum noise, measurement, …

1184

aout

bin

TABLE IV. Two different amplifier modes of operation.

amplifier

ain

bout
input line

output line

FIG. 9. 共Color online兲 Schematic of a two-port bosonic amplifier. Both the inputs and outputs of the amplifier are attached
to transmission lines. The incoming and outgoing wave amplitudes in the input 共output兲 transmission line are labeledâin , âout 共b̂in , b̂out兲, respectively. The voltages at the end of thetwo lines 共V̂a , V̂b兲 are linear combinations of incoming and outgoing wave amplitudes.

lar conclusions 共i.e., at best, the detector adds noise
equal to the zero-point noise兲. In that linear-responsebased discussion, we saw that a crucial aspect of thequantum limit was the trade-off between back-action
noise and measurement-imprecision noise. We saw thatreaching the quantum limit required both a detector
with ideal noise, as well as an optimization of thedetector-oscillator coupling strength. Somewhat disturbingly, none of these ideas appeared explicitly in theHaus-Caves derivation; this can give the misleading impression that the quantum limit never has anything to dowith back-action. A further confusion comes from thefact that many detectors have input and outputs thatcannot be described by a set of bosonic modes. How
does one apply the above arguments to such systems?
The first step in resolving these seeming inconsistencies is to realize that there are really two different ways
in which one can use a given amplifier or detector. In
deciding how to couple the input signal 共i.e., the signal tobe amplified兲 to the amplifier, and in choosing whatquantity to measure, the experimentalist essentially enforces boundary conditions; as now shown, there are in
general two distinct ways in which to do this. For concreteness, consider the situation shown in Fig. 9: a twoport voltage amplifier where the input and output portsof the amplifier are attached to one-dimensional transmission lines 共see Appendix C for a quick review ofquantum transmission lines兲. As in the previous section,
we focus on a narrow bandwidth signal centered about afrequency ␻. At this frequency, there exists both a rightmoving and a left-moving wave in each transmission
line. We label the corresponding amplitudes in the input
共output兲 line with ain , aout 共bin , bout兲, as per Fig. 9. Quantum mechanically these amplitudes become operators,
much in the same way that we treated the mode amplitude a as an operator in the previous section. We analyze this two-port bosonic amplifier in Sec. VI; here we
only sketch its operation to introduce the two different
amplifier operation modes. This will then allow us tounderstand the subtleties of the Haus-Caves quantum
limit derivation.
In the first kind of setup, the experimentalist arranges
things so that ain, the amplitude of the wave incident on
the amplifier’s input port, is precisely equal to the signal
to be amplified 共i.e., the input signal兲, irrespective of theamplitude of the wave leaving the input port 共i.e., aout兲.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Mode
Scattering
Op-amp

Input Signals共t兲

Output Signal
o共t兲

s共t兲 = ain共t兲
共ain indep. of aout兲
s共t兲 = Va共t兲
共ain depends on aout兲

共o共t兲 = bout共t兲兲
共bout indep. of bin兲
o共t兲 = Vb共t兲
共bout depends on bin兲Further, the output signal is taken to be the amplitude of
the outgoing wave exiting the output of the amplifier
共i.e., bout兲, again irrespective of whatever might be entering the output port 共see Table IV兲. In this situation, theHaus-Caves description of the quantum limit in the previous section is almost directly applicable; we make this
precise in Sec. VI. Back-action is indeed irrelevant, as
the prescribed experimental conditions mean that it
plays no role. We call this mode of operation the scattering mode, as it is most relevant to time-dependent
experiments where the experimentalist launches a signal
pulse at the input of the amplifier and looks at what exits
the output port. One is usually only interested in the
scattering mode of operation in cases where the source
producing the input signal is matched to the input of theamplifier: only in this case is the input wave ain perfectlytransmitted into the amplifier. As we see in Sec. VI, sucha perfect matching requires a relatively strong coupling
between the signal source and the input of the amplifier;as such, the amplifier will strongly enhance the dampingof the signal source.
The second mode of linear amplifier operation is what
we call the op-amp mode; this is the mode one usually
has in mind when thinking of an amplifier which is
weakly coupled to the signal source, and will be the next
focus. The key difference from the scattering mode isthat here the input signal is not simply the amplitude ofa wave incident on the input port of the amplifier; similarly, the output signal is not the amplitude of a wave
exiting the output port. As such, the Haus-Caves derivation of the quantum limit does not directly apply. For thebosonic amplifier discussed here the op-amp mode
would correspond to using the amplifier as a voltage opamp. The input signal would thus be the voltage at theend of the input transmission line. Recall that the voltage at the end of a transmission line involves the amplitude of both left- and right-moving waves, i.e., Va共t兲⬀ Re关ain共t兲 + aout共t兲兴. At first this might seem quite confusing: If the signal source determines Va共t兲, does thismean it sets the values of both ain共t兲 and aout共t兲? Does
not this violate causality? The signal source enforces thevalue of Va共t兲 by simply changing ain共t兲 in response to
the value of aout共t兲. While there is no violation of causality, the fact that the signal source is dynamically responding to what comes out of the amplifier’s input port
implies that back-action is indeed relevant.
The op-amp mode of operation is relevant to the typical situation of weak coupling between the signal source
and amplifier input. By weak coupling, we mean here

Clerk et al.: Introduction to quantum noise, measurement, …
cold
load
vacuum noise uin

uout

aout
ain

bin
amplifierinput line

output line

circulator

boutFIG. 10. Illustration of a bosonic two-port amplifier used in
the scattering mode of operation. The signal is an incoming
wave in the input port of the amplifier, and does not depend on
what is coming out of the amplifier. This is achieved by connecting the input line to a circulator and a “cold load” 共i.e., azero-temperature resistor兲: all that goes back toward thesource of the input signal is vacuum noise.

something stronger than just requiring that the amplifier
be linear: we require additionally that the amplifier does
not appreciably change the dissipation of the signal
source. This is analogous to the situation in an ideal voltage op-amp, where the amplifier input impedance ismuch larger than the impedance of the signal source. We
stress that the op-amp mode and this limit of weak coupling is the relevant situation in most electrical measurements.
Thus, we see that the Haus-Caves formulation of thequantum limit is not directly relevant to amplifiers or
detectors operated in the usual op-amp mode of operation. We clearly need some other way to describe quantum amplifiers used in this regime. As we demonstrated
in the remainder of this section, the general linearresponse approach of Sec. IV is exactly what is needed.
To see this, expand the discussion of Sec. IV to include
the concepts of input and output impedance as well as
power gain. The linear-response approach will allow us
to see 共similar to Sec. III.B.3兲 that reaching the quantum
limit in the op-amp mode does indeed require a tradeoff between back-action and measurement imprecision,and requires use of an amplifier with ideal quantum
noise properties 关see Eq. 共4.11兲兴. This approach also has
the added benefit of being directly applicable to systems
where the input and output of the amplifier are not described by bosonic modes.12 In Sec. VI, we return to thescattering description of a two-port voltage amplifier
共Fig. 10兲, and show explicitly how an amplifier can bequantum limited when used in the scattering mode ofoperation, but miss the quantum limit when used in the
op-amp mode of operation.

1185

properties of a generic linear-response detector in Sec.
IV, including the fundamental quantum noise constraintof Eq. 共4.11兲. For simplicity, we start with the problem ofcontinuous position detection of a harmonic oscillator.
Our discussion will thus generalize the discussion of position detection using a cavity detector given in Sec.
III.B.3. We start with a generic detector 共as introducedin Sec. IV.A兲 coupled at its input to the position x̂ of aharmonic oscillator 关see Eq. 共4.1兲兴.13 We want to understand the total output noise of our amplifier in the presence of the oscillator, and more importantly how small
we can make the amplifier’s contribution to this noise.
The resulting lower bound is known as the standard
quantum limit 共SQL兲 on position detection, and is analogous to the quantum limit on the added noise of a voltage amplifier 共discussed in Sec. V.F兲.
1. Detector back-action

We first consider the consequence of noise in the detector input port. As seen in Sec. II.B, the fluctuating
back-action force F̂ acting on our oscillator will lead toboth damping and heating of the oscillator. To model the
intrinsic 共i.e., detector-independent兲 heating and damping of the oscillator, we also assume that our oscillator is
coupled to an equilibrium heat bath. In the weakcoupling limit that we are interested in, one can use
lowest-order perturbation theory in the coupling A todescribe the effects of the back-action force F̂ on theoscillator. A full quantum treatment 共see Appendix I.4兲shows that the oscillator is described by an effective classical Langevin equation,14
Mẍ共t兲 = − M⍀2x共t兲 − M␥0ẋ共t兲 + F0共t兲
− MA2

冕 ⬘

dt ␥共t − t⬘兲ẋ共t⬘兲 − AF共t兲.

共5.41兲

The position x共t兲 in the above equation is not an operator, but is simply a classical variable whose fluctuations
are driven by the fluctuating forces F共t兲 and F0共t兲. Nonetheless, the noise in x calculated from Eq. 共5.41兲 corresponds precisely to S̄xx关␻兴, the symmetrized quantummechanical noise in the operator x̂. The fluctuating force
exerted by the detector 共which represents the heating
part of the back-action兲 is described by AF共t兲 in Eq.
共5.41兲; it has zero mean, and a spectral density given by
A2S̄FF关␻兴 in Eq. 共4.4a兲. The kernel ␥共t兲 describes thedamping effect of the detector. It is given by the asym-

E. Linear-response description of a position detector

In this section, we examine the amplifier quantum
limit for a two-port linear amplifier in the usual weakcoupling, op-amp regime of operation. Our discussion
here will make use of the results obtained for the noise
12
Note that the Haus-Caves derivation for the quantum limitof a scattering amplifier has been generalized to the case offermionic operators 共Gavish et al., 2004兲.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 201013
For consistency with previous sections, our coupling Hamiltonian does not have a minus sign. This is different from theconvention of Clerk 共2004兲, where the coupling Hamiltonian iswritten Hint = −Ax̂ · F̂.
14Note that we have omitted a back-action term in this equation which leads to small renormalizations of the oscillator frequency and mass. These terms are not important for the following discussion, so we have omitted them for clarity; one canconsider M and ⍀ in this equation to be renormalized quantities. See Appendix I.4 for more details.

Clerk et al.: Introduction to quantum noise, measurement, …

1186

metric part of the detector’s quantum noise, as was derived in Sec. II.B 关see Eq. 共2.12兲兴.
Equation 共5.41兲 also describes the effects of an equilibrium heat bath at temperature T0 which models theintrinsic 共i.e., detector-independent兲 damping and heating of the oscillator. The parameter ␥0 is the damping
arising from this bath and F0 is the corresponding fluctuating force. The spectral density of the F0 noise is determined by ␥0 and T0 via the fluctuation-dissipation
theorem 关see Eq. 共2.16兲兴. T0 and ␥0 have a simple physical significance: they are the temperature and dampingof the oscillator when the coupling to the detector A is
set to zero.
To make further progress, we recall from Sec. II.B
that even though our detector will in general not be in
equilibrium, we may nonetheless assign it an effective
temperature Teff关␻兴 at each frequency 关see Eq. 共2.8兲兴.
The effective temperature of an out-of-equilibrium detector is simply a measure of the asymmetry of the detector’s quantum noise. We are often interested in thelimit where the internal detector time scales are much
faster than the time scales relevant to the oscillator 共i.e.,⍀−1 , ␥−1 , ␥−10 兲. We may then take the ␻ → 0 limit in theexpression for Teff, yielding
2kBTeff ⬅ S̄FF共0兲/M␥共0兲.

共5.42兲

In this limit, the oscillator position noise calculated from
Eq. 共5.41兲 is given byS̄xx关␻兴 =2共␥0 + ␥兲kB
1
␥0T + ␥Teff
.
2
2 2
2
2
M 共 ␻ − ⍀ 兲 + ␻ 共 ␥ + ␥ 0兲
␥0 + ␥
共5.43兲

This is exactly what would be expected if the oscillator
were only attached to an equilibrium Ohmic bath with adamping coefficient ␥⌺ = ␥0 + ␥ and temperature T̄= 共␥0T + ␥Teff兲 / ␥⌺.
2. Total output noise

The next step in our analysis is to link fluctuations in
the position of the oscillator 关as determined from Eq.
共5.41兲兴 to noise in the output of the detector. As discussed in Sec. III.B.3, the output noise consists of the
intrinsic output noise of the detector 共i.e.,“measurement-imprecision noise”兲 plus the amplified
position fluctuations in the position of the oscillator. The
latter contains both an intrinsic part and a term due tothe response of the oscillator to the back-action.
To start, imagine that we can treat both the oscillator
position x共t兲 and the detector output I共t兲 as classicallyfluctuating quantities. Using the linearity of the detector’s response, we can then write ␦Itotal, the fluctuating
part of the detector’s output, as␦Itotal关␻兴 = ␦I0关␻兴 + A␹IF关␻兴␦x关␻兴.

共5.44兲

The first term 共␦I0兲 describes the intrinsic 共oscillatorindependent兲 fluctuations in the detector output, andhas a spectral density S̄II关␻兴. If we scale this by 兩␹IF兩2, we
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

have the measurement-imprecision noise discussed inSec. III.B.3. The second term corresponds to the amplified fluctuations of the oscillator, which are in turn given
by solving Eq. 共5.41兲,␦x关␻兴 = −

冋

册

1/M
共F0关␻兴 − AF关␻兴兲
共␻ − ⍀ 兲 + i␻⍀/Q关␻兴22⬅ ␹xx关␻兴共F0关␻兴 − AF关␻兴兲,

共5.45兲

where Q关␻兴 = ⍀ / 共␥0 + ␥关␻兴兲 is the oscillator quality factor. It follows that the spectral density of the total noisein the detector output is given classically by
SII,tot关␻兴 = SII关␻兴 + 兩␹xx关␻兴␹IF关␻兴兩2共A4SFF关␻兴+ A2SF0F0关␻兴兲+ 2A2 Re†␹xx关␻兴␹IF关␻兴SIF关␻兴‡.

共5.46兲

Here SII, SFF, and SIF are the 共classical兲 detector noisecorrelators calculated in the absence of any coupling tothe oscillator. Note importantly that we have included
the fact that the two kinds of detector noise 共in Î and inF̂兲 may be correlated with one another.
To apply the classically derived Eq. 共5.46兲 to our quantum detector-plus-oscillator system, we recall from Sec.
II.B that symmetrized quantum noise spectral densities
play the role of classical noise. The LHS of Eq. 共5.46兲
thus becomes SII,tot, the total symmetrized quantummechanical output noise of the detector, while the RHS
will now contain the symmetrized quantum-mechanicaldetector noise correlators S̄FF, S̄II, and S̄IF, defined as in
Eq. 共4.4a兲. Though this may seem rather ad hoc, one caneasily demonstrate that Eq. 共5.46兲 thus interpreted
would be quantum mechanically rigorous if the detector
correlation functions obeyed Wick’s theorem. Thus,
quantum corrections to Eq. 共5.46兲 will arise solely fromthe non-Gaussian nature of the detector noise correlators. We expect from the central limit, theorem that suchcorrections will be small in the relevant limit, where ␻ ismuch smaller that the typical detector frequency⬃kBTeff / ប, and neglect these corrections in what follows. Note that the validity of Eq. 共5.46兲 for a specificmodel of a tunnel junction position detector has been
explicitly verified by Clerk and Girvin 共2004兲.
3. Detector power gain

Before proceeding, we need to consider our detector
once again in isolation, and return to the fundamental
question of what we mean by amplification. To be able
to say that our detector truly amplifies the motion of theoscillator, it is not sufficient to simply say that the response function ␹IF must be large 共note that ␹IF is notdimensionless兲. Instead, true amplification requires thatthe power delivered by the detector to a following amplifier be much larger than the power drawn by the detector at its input—i.e., the detector must have a dimensionless power gain GP关␻兴 much larger than 1. As
discussed, if the power gain was not large, we would
need to worry about the next stage in the amplification

Clerk et al.: Introduction to quantum noise, measurement, …

Aλ

g
FD

x
A
γ0 + γ␥out关␻兴 = 共B2/M␻兲关− Im ␹II关␻兴兴

Bgy

F

I

Ag

1187

= 共B2/Mប␻兲yBγout + γld冋册

SII关␻兴 − SII关− ␻兴
,2

共5.48兲

where ␹II is the linear-response susceptibility which de-

FIG. 11. 共Color online兲 Schematic of a generic linear-response
position detector, where an auxiliary oscillator y is driven by
the detector output.

termines how 具Î典 responds to a perturbation coupling toÎ␹II关␻兴 = −of our signal, and how much noise is added in that process. Having a large power gain means that by the time
our signal reaches the following amplifier, it is so largethat the added noise of this following amplifier is unimportant. The power gain is analogous to the dimensionless photon-number gain G that appears in the standard
Haus-Caves description of a bosonic linear amplifier关see Eq. 共5.7兲兴.
To make the above more precise, we start with theidea case of no reverse gain ␹FI = 0. We define the power
gain GP关␻兴 of our generic position detector in a way thatis analogous to the power gain of a voltage amplifier.
Imagine we drive the oscillator we are trying to measurewith a force 2FD cos ␻t; this will cause the output of our

i
ប

冕

⬁

dt具关Î共t兲,Î共0兲兴典ei␻t .

共5.49兲

0

As the oscillator y is being driven on resonance, therelation between y and I is given by y关␻兴 = ␹yy关␻兴I关␻兴with ␹yy关␻兴 = −i†␻M␥out关␻兴‡−1. From conservation of energy, we have that the net power flow into the output
oscillator from the detector is equal to the power dissipated out of the oscillator through the intrinsic damping
␥ld. We thus have
Pout ⬅ M␥ldẏ2= M␥ld␻2兩␹yy关␻兴兩2兩BA␹IF␹xx关␻兴FD兩2=

1
␥ld
兩BA␹IF␹xx关␻兴FD兩2 .
M 共␥ld + ␥out关␻兴兲2

共5.50兲

detector 具Î共t兲典 to also oscillate at frequency ␻. To optimally detect this signal in the detector output, we further couple the detector output I to a second oscillator
with natural frequency ␻, mass M, and position y: there

Using the above definitions, we find that the ratio between Pout and Pin is independent of ␥0, but depends on
␥ld,

is a new coupling term in our Hamiltonian Hint
⬘ = BÎ · ŷ,
where B is a coupling strength. The oscillations in 具I共t兲典
will now act as a driving force on the auxiliary oscillator
y 共see Fig 11兲. We can consider the auxiliary oscillator y
as a “load” we are trying to drive with the output of ourdetector.
To find the power gain, we need to consider both Pout,the power supplied to the output oscillator y from thedetector, and Pin, the power fed into the input of theamplifier. Consider first Pin. This is simply the timeaveraged power dissipation of the input oscillator x
caused by the back-action damping ␥关␻兴. Using an overbar to denote a time average, we have

共5.51兲

2
Pin ⬅ M␥关␻兴ẋ2 = M␥关␻兴␻2兩␹xx关␻兴兩2FD.

共5.47兲

Note that the oscillator susceptibility ␹xx关␻兴 depends on
both the back-action damping ␥关␻兴 and the intrinsic oscillator damping ␥0 关see Eq. 共5.45兲兴.
Next, we need to consider the power supplied to theload oscillator y at the detector output. This oscillator
will have some intrinsic detector-independent damping␥ld, as well as a back-action damping ␥out. In the same
way that the back-action damping ␥ of the input oscillator x is determined by the quantum noise in F̂ 关see Eqs.
共2.14兲, 共2.13兲, and 共2.12兲兴, the back-action damping of theload oscillator y is determined by the quantum noise in
the output operator Î,Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1 A2B2兩␹IF关␻兴兩2 ␥ld/␥out关␻兴
Pout
= 2 2
.
Pin M ␻ ␥out关␻兴␥关␻兴 共1 + ␥ld/␥out关␻兴兲2

We now define the detector power gain GP关␻兴 as thevalue of this ratio maximized over the choice of ␥ld. The
maximum occurs for ␥ld = ␥out关␻兴 共i.e., the load oscillator
is “matched” to the output of the detector兲, resulting in冋 册GP关␻兴 ⬅ max

1 A2B2兩␹IF兩2
Pout
=
Pin
4M2␻2 ␥out␥=兩␹IF关␻兴兩2. 共5.52兲
4 Im ␹FF关␻兴Im ␹II关␻兴In the last equality, we have used the relation between
the damping rates ␥关␻兴 and ␥out关␻兴 and the linearresponse susceptibilities ␹FF关␻兴 and ␹II关␻兴 关see Eqs.
共2.15兲 and 共5.48兲兴. We thus find that the power gain is a
simple dimensionless ratio formed by the three different
response coefficients characterizing the detector, and isindependent of the coupling constants A and B. As we
see in Sec. V.F, it is completely analogous to the power
gain of a voltage amplifier, which is also determined bythree parameters: the voltage gain, the input impedance,
and the output impedance. Note that there are otherimportant measures of power gain commonly in use in
the engineering community: we comment on these inSec. VII.B.
Finally, the above results are easily generalized to thecase where the detector’s reverse gain ␹FI is nonvanishing. For simplicity, we present results for the case whereClerk et al.: Introduction to quantum noise, measurement, …

1188

␤ = Re共␹IF␹FI兲 / 兩␹IF兩2 艌 0, implying that there is no positive feedback. Maximizing the ratio of Pout / Pin overchoices of ␥ld now yields
GP,rev = 2GP/共1 + 2␤GP + 冑1 + 4␤GP兲 艋 1/␤ .

共5.53兲

Here GP,rev is the power gain in the presence of reverse
gain, while GP is the zero-reverse gain power gain given
by Eq. 共5.52兲. One can confirm that GP,rev is a monotonic
increasing function of GP, and is bounded by 1 / ␤. As* , there is no additional
noted in Sec. IV.A.4, if ␹FI = ␹IF
quantum noise constraint on our detector beyond what
exists classically 关i.e., the RHS of Eq. 共4.11兲 vanishes兴.
* , the power gain
We now see explicitly that when ␹FI = ␹IFof our detector can be at most 1, as ␤ = 1. Thus, while
there is no minimum back-action noise required byquantum mechanics in this case, there is also no amplification: at best, our detector would act as a transducer.
* and optiNote further that if the detector has ␹FI = ␹IFmizes the inequality of Eq. 共4.11兲, then one can show
GP,rev must be 1 共see Appendix I.2兲: the detector is simply a transducer. This is in keeping with the results obtained using the Haus-Caves approach, which also yields
the conclusion that a noiseless detector is a transducer.
4. Simplifications for a quantum-ideal detector

We now consider the important case where our detector has no reverse gain 共allowing it to have a large powergain兲, and also has ideal quantum noise 关i.e., it satisfies
the ideal noise condition of Eq. 共4.17兲兴. Fulfilling thiscondition immediately places some powerful constraints
on our detector.
First, note that we have defined in Eq. 共5.42兲 the effective temperature of our detector based on what happens at the input port; this is the effective temperature
seen by the oscillator we are trying to measure. We
could also consider the effective temperature of the detector as seen at the output 共i.e., by the oscillator y usedin defining the power gain兲. This output effective temperature is determined by the quantum noise in the output operator Î,
kBTeff,out关␻兴 ⬅ ប␻/log共SII关+ ␻兴/SII关− ␻兴兲.

共5.54兲

For a general out-of-equilibrium amplifier, Teff,out does
not have to be equal to the input effective temperature
Teff defined by Eq. 共2.8兲. However, for a quantum-ideal
detector, the effective proportionality between input andoutput operators 关see Eq. 共I13兲兴 immediately yields
Teff,out关␻兴 = Teff关␻兴.

共5.55兲

Thus, a detector with quantum-ideal noise necessarily
has the same effective temperature at its input and its
output. 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-ideal
detector is the expression for the power gain. Using the
proportionality between input and output operators 关cf.
Eq. 共I13兲兴, one finds
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

G P关 ␻ 兴 =共Im ␣兲2 coth2共ប␻/2kBTeff兲 + 共Re ␣兲2, 共5.56兲兩␣兩2where ␣关␻兴 is the parameter characterizing a quantumlimited detector in Eq. 共4.18兲; recall that 兩␣关␻兴兩2 determines the ratio of SII and SFF. It follows immediatelythat for a detector with ideal noise to also have a large
power gain 共GP Ⰷ 1兲, one absolutely needs kBTeff Ⰷ ប␻: alarge power gain implies a large effective detector temperature. In the large-GP limit, we have
GP ⯝

冋

册

Im ␣ kBTeff 2.
兩␣兩 ប␻/2

共5.57兲

Thus, the effective temperature of a quantum-ideal detector does more than just characterize the detector
back-action—it also determines the power gain.
Finally, an additional consequence of the large-GP关␻兴,large Teff limit is that the gain ␹IF and noise cross correlator S̄IF are in phase: S̄IF / ␹IF is purely real, up tocorrections that are as small as ␻ / Teff. This is shown
explicitly in Appendix I.3. Thus, we find that a large
power gain detector with ideal quantum noise cannot
have significant out-of-phase correlations between its
output and input noises. This last point may be understood in terms of the idea of wasted information: if there
were significant out-of-phase correlations between Î andF̂, it would be possible to improve the performance of
the amplifier by using feedback. We discuss this point in
Sec. VI. Note that because S̄IF / ␹IF is real the last term in
the quantum noise constraint of Eq. 共4.11兲 vanishes.
5. Quantum limit on added noise and noise
temperature

We now turn to calculating the noise added to oursignal 关i.e., 具x̂共t兲典兴 by our generic position detector. To
characterize this added noise, it is useful to take the total
共symmetrized兲 noise in the output of the detector andrefer it back to the input by dividing out the gain of thedetector,S̄xx,tot关␻兴 ⬅ S̄II,tot关␻兴/A2兩␹IF关␻兴兩2 .

共5.58兲

S̄xx,tot关␻兴 is simply the frequency-dependent spectral
density of position fluctuations inferred from the outputof the detector. It is this quantity that will directly determine the sensitivity of the detector–given a certain detection bandwidth, what is the smallest variation of xthat can be resolved? The quantity S̄xx,tot关␻兴 will havecontributions from the intrinsic fluctuations of the input
signal as well as a contribution due to the detector. We
first define S̄xx,eq关␻ , T兴 to be the symmetrized equilibrium position noise of our damped oscillator 共whose
damping is ␥0 + ␥兲 at temperature T,

Clerk et al.: Introduction to quantum noise, measurement, …

S̄xx,eq关␻,T兴 = ប coth共ប␻/2kBT兲共− Im ␹xx关␻兴兲,

共5.59兲

where the oscillator susceptibility ␹xx关␻兴 is defined in
Eq. 共5.45兲. The total inferred position noise may then be
written as
S̄xx,tot关␻兴 ⬅ 关␥0/共␥0 + ␥兲兴 · S̄xx,eq关␻,T0兴 + S̄xx,add关␻兴.
共5.60兲
In the usual case where the detector noise can be approximated as being white, this spectral density will consist of a Lorentzian sitting atop a constant noise floor
共see Fig. 6兲. The first term in Eq. 共5.60兲 represents position noise arising from the fluctuating force ␦F0共t兲 associated with the intrinsic 共detector-independent兲 dissipation of the oscillator 关see Eq. 共5.41兲兴. The prefactor ofthis term arises because the strength of the intrinsic
Langevin force acting on the oscillator is proportional to␥0, not to ␥0 + ␥.
The second term in Eq. 共5.60兲 represents the added
position noise due to the detector. It has contributions

1189

tion is the same as the cavity position detector of Sec.
III.B.3: there is an optimal value of the couplingstrength A which corresponds to a trade-off between imprecision noise and back-action 关i.e., the first and second
terms in Eq. 共5.61兲兴. We thus expect S̄xx,add关␻兴 to attain aminimum value at an optimal choice of coupling A= Aopt where both these terms make equal contributions
共see Fig. 5兲. Defining ␾关␻兴 = arg ␹xx关␻兴, we thus have theboundS̄xx,add关␻兴 艌 2兩␹xx关␻兴兩+冋冑S̄IIS̄FF
兩␹IF兩2* e−i␾关␻兴S̄ 兴
Re关␹IF
IF兩␹IF兩2

册,

共5.63兲

where the minimum value at frequency ␻ is achieved
when

冑

2
Aopt
= S̄II关␻兴/兩␹IF关␻兴␹xx关␻兴兩2S̄FF关␻兴.

共5.64兲

both from the detector’s intrinsic output noise S̄II from

Using the inequality X2 + Y2 艌 2 兩 XY兩 we see that thisthe detector’s back-action noise S̄FF, and may be written
asvalue serves as a lower bound on S̄xx,add even in thepresence of detector-dependent damping. In the casewhere the detector-dependent damping is negligible, theRHS of Eq. 共5.63兲 is independent of A, and thus Eq.
共5.64兲 can be satisfied by simply tuning the couplingstrength A; in the more general case where there is
detector-dependent damping, the RHS is also a functionof A 共through the response function ␹xx关␻兴兲, and it may
no longer be possible to achieve Eq. 共5.64兲 by simply
tuning A.15
While Eq. 共5.63兲 is certainly a bound on the added

S̄xx,add关␻兴 =S̄II+ A2兩␹xx兩2S̄FF兩␹IF兩2A2+* 共␹ 兲*S̄ 兴
2 Re关␹IF
xx
IF

兩␹IF兩2.

共5.61兲

For clarity, we have omitted writing the explicit frequency dependence of the gain ␹IF, susceptibility ␹xx,and noise correlators; they should all be evaluated at thefrequency ␻. Note that the first term on the RHS corresponds to the measurement-imprecision noise of our deI
共␻兲.
tector, S̄xx
We can now finally address the quantum limit on theadded noise in this setup. As discussed in Sec. V.D, theHaus-Caves derivation of the quantum limit 共cf. Sec.
V.B兲 is not directly applicable to the position detector
we are describing here; nonetheless, we may use its result to guess what form the quantum limit will take here.
The Haus-Caves argument told us that the added noiseof a phase-preserving linear amplifier must be at least as
large as the zero-point noise. We thus anticipate that, if
our detector has a large power gain, the spectral densityof the noise added by the detector 共i.e., S̄xx,add关␻兴兲 mustbe at least as large as the zero-point noise of ourdamped oscillator,S̄xx,add关␻兴 艌 lim S̄xx,eq关w,T兴 = 兩ប Im ␹xx关␻兴兩.
T→0

共5.62兲

We now show that the bound above is rigorously correctat each frequency ␻.
The first step is to examine the dependence of theadded noise S̄xx,add关␻兴 关as given by Eq. 共5.61兲兴 on the
coupling strength A. If we ignore for a moment thedetector-dependent damping of the oscillator, the situaRev. Mod. Phys., Vol. 82, No. 2, April–June 2010displacement noise S̄xx,add关␻兴, it does not in itself represent the quantum limit. Reaching the quantum limit requires more than simply balancing the detector backaction and intrinsic output noises 关i.e., the first two terms
in Eq. 共5.61兲兴; one also needs a detector with quantumideal noise properties, that is a detector which satisfies Eq.
(4.17). Using the quantum noise constraint of Eq. 共4.11兲
to further bound S̄xx,add关␻兴, we obtainS̄xx,add关␻兴 艌 2

冏

⫻

+

␹xx关␻兴
␹IF冏

冋冑冉 冊 冉 冋 册冊
册ប兩␹IF兩 2
2S̄IF
1+⌬
2
ប␹IF

* e−i␾关␻兴S̄ 兴
Re关␹IF
IF

兩␹IF兩

,

+ 兩S̄IF兩2
共5.65兲

where the function ⌬关z兴 is defined in Eq. 共4.12兲. The
minimum value of S̄xx,add关␻兴 in Eq. 共5.65兲 is now
achieved when one has both an optimal coupling 关i.e.,
15
Note that, in the heuristic discussion of position detection
using a resonant cavity detector in Sec. III.B.3, these concerns
did not arise as there was no back-action damping.

Clerk et al.: Introduction to quantum noise, measurement, …

1190

Eq. 共5.64兲兴 and a quantum-limited detector, that is one
which satisfies Eq. 共4.11兲 as an equality.
Next, we consider the relevant case where our detector is a good amplifier and has a power gain GP关␻兴 Ⰷ 1over the width of the oscillator resonance. As discussed,
this implies that the ratio S̄IF / ␹IF is purely real, up tosmall ប␻ / kBTeff corrections 共see Sec. IV.A.4 and Appendix I.3 for more details兲. This in turn implies thatacross the entire oscillator resonance. A more reasonable goal is to optimize S̄xx,add关␻兴 at resonance, ␻ = ⍀. As

␹xx关⍀兴 is imaginary, Eq. 共5.67兲 tells us that S̄IF should be
zero. Assuming we have a quantum-limited detector
with a large power gain 共kBTeff Ⰷ ប⍀兲, the remaining
condition on the coupling A 关Eq. 共5.64兲兴 may be written
as

冏 冏

Im ␣
1
␥关Aopt兴ប⍀.
==冑␥0 + ␥关Aopt兴
␣ 2 GP关⍀兴 4kBTeff⌬关2S̄IF / ប␹IF兴 = 0; we thus haveS̄xx,add关␻兴 艌 2兩␹xx关␻兴兩+冋冑冉 冊 冉 冊
册
ប 2
S̄IF
+
2
␹IF

cos†␾关␻兴‡S̄IF
.
␹IF

2

As ␥关A兴 ⬀ A2 is the detector-dependent damping of
the oscillator, we thus have that to achieve the quantum共5.66兲

Finally, as there is no further constraint on S̄IF / ␹IF 共beyond the fact that it is real兲, we can minimize the expression over its value. The minimum S̄xx,add关␻兴 is achieved
for a detector whose cross correlator satisfies
兩S̄IF关␻兴/␹IF兩optimal = − 共ប/2兲 cot ␾关␻兴,limited value of S̄xx,add关⍀兴 with a large power gain, one
needs the intrinsic damping of the oscillator to be muchlarger than the detector-dependent damping. The
detector-dependent damping must be small enough tocompensate the large effective temperature of the detector; if the bath temperature satisfies ប⍀ / kB Ⰶ TbathⰆ Teff, Eq. 共5.69兲 implies that at the quantum limit thetemperature of the oscillator will be given by

共5.67兲

Tosc ⬅ 共␥ · Teff + ␥0 · Tbath兲/共␥ + ␥0兲 → ប⍀/4kB + Tbath .
共5.70兲

with the minimum value given by
S̄xx,add兩关␻兴兩min = ប兩Im ␹xx关␻兴兩 = lim S̄xx,eq关␻,T兴, 共5.68兲
T→0

where S̄xx,eq关␻ , T兴 is the equilibrium contribution toS̄xx,tot关␻兴 defined in Eq. 共5.59兲. Thus, in the limit of alarge power gain, we have that at each frequency theminimum displacement noise added by the detector isprecisely equal to the noise arising from a zerotemperature bath. This conclusion is irrespective of thestrength of the intrinsic 共detector-independent兲 oscillator
damping.
We have thus derived the amplifier quantum limit 共inthe context of position detection兲 for a two-port amplifier used in the op-amp mode of operation. Though wereached a conclusion similar to that given by the HausCaves approach, the linear-response, quantum noise approach used is quite different. This approach makes explicitly clear what is needed to reach the quantum limit.
We find that to reach the quantum limit on the added
displacement noise S̄xx,add关␻兴 with a large power gain,
one needs 共1兲 a quantum-limited detector, that is, a detector which satisfies the ideal noise condition of Eq.
共4.17兲, and hence the proportionality condition of Eq.
共I13兲; 共2兲 a coupling A which satisfies Eq. 共5.64兲; and 共3兲
a detector cross correlator S̄IF that satisfies Eq. 共5.67兲.
Recall that condition 共1兲 is identical to what is required for quantum-limited detection of a qubit; it israther demanding, and requires that there is no wasted
information about the input signal in the detector which
is not revealed in the output 共Clerk et al., 2003兲. Also
note that cot ␾ changes quickly as a function of frequency across the oscillator resonance, whereas S̄IF will
be roughly constant; condition 共2兲 thus implies that it
will not be possible to achieve a minimal S̄xx,add关␻兴
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共5.69兲

Thus, at the quantum limit and for large Teff, the detector raises the oscillator’s temperature by ប⍀ / 4kB.16 As
expected, this additional heating is only half the zeropoint energy; in contrast, the quantum-limited value ofS̄xx,add关␻兴 corresponds to the full zero-point result, as it
also includes the contribution of the intrinsic output
noise of the detector.
Finally, we return to Eq. 共5.65兲; this is the constraint
on the added noise S̄xx,add关␻兴 before we assumed our
detector to have a large power gain, and consequently alarge Teff. Note crucially that if we did not require alarge power gain, then there need not be any added
noise. Without the assumption of a large power gain, theratio S̄IF / ␹IF can be made imaginary with a large magnitude. In this limit, 1 + ⌬关2S̄IF / ␹IF兴 → 0: the quantum constraint on the amplifier noises 关e.g., the RHS of Eq.
共4.11兲兴 vanishes. One can then easily use Eq. 共5.65兲 toshow that the added noise S̄xx,add关␻兴 can be zero. Thisconfirms a general conclusion that we have seen several
times now 共see Secs. IV.A.4 and V.B兲: if a detector does
not amplify 共i.e., the power gain is unity兲, it need not
produce any added noise.
F. Quantum limit on the noise temperature of a voltage
amplifier

We now turn our attention to the quantum limit on
the added noise of a generic linear voltage amplifier
16

If in contrast our oscillator was initially at zero temperature
共i.e., Tbath = 0兲, one finds that the effect of the back-action 共at
the quantum limit and for GP Ⰷ 1兲 is to heat the oscillator to atemperature ប⍀ / kB ln 5.

Clerk et al.: Introduction to quantum noise, measurement, …
~
V
vin
ZS

Zin
input

λV

Zout
~
I

λI

output

IL

Vout

ZL

FIG. 12. Schematic of a linear voltage amplifier, including areverse gain ␭I. Ṽ and Ĩ represent the standard voltage andcurrent noises of the amplifier, as discussed in the text. The
case with reverse gain is discussed in Sec. VI.

used in the op-amp mode of operation 关see, e.g., Devoret and Schoelkopf 共2000兲兴. For such amplifiers, theadded noise is usually expressed in terms of the “noise
temperature” of the amplifier; we define this concept
and demonstrate that, when appropriately defined, thisnoise temperature must be bigger than ប␻ / 共2kB兲, where␻ is the signal frequency. Though the voltage amplifier isclosely analogous to the position detector treated previously, its importance makes it worthy of a separate discussion. As in the previous section, our discussion here
will use the general linear-response approach. In contrast, in Sec. VI, we present the bosonic scattering description of a two-port voltage amplifier, a description
similar to that used in formulating the Haus-Caves proofof the amplifier quantum limit. We will then be in a good
position to contrast the linear-response and scattering
approaches and will see there that the scattering and the
op-amp modes of operation discussed here are notequivalent. We stress that the general treatment presented here can also be applied directly to the system
discussed in Sec. VI.
1. Classical description of a voltage amplifier

We begin by recalling the standard schematic description of a voltage amplifier 共see Fig. 12兲. The input voltage to be amplified vin共t兲 is produced by a circuit which
has a Thevenin-equivalent impedance Zs, the source impedance. We stress that we are considering the op-amp
mode of amplifier operation, and thus the input signal
does not correspond to the amplitude of a wave incident
upon the amplifier 共see Sec. V.D兲. The amplifier itself
has an input impedance Zin and an output impedanceZout, as well as a voltage gain coefficient ␭V: assuming nocurrent is drawn at the output 共i.e., Zload → ⬁ in Fig. 12兲,
the output voltage Vout共t兲 is simply ␭V times the voltage
across the input terminals of the amplifier.
The added noise of the amplifier is usually represented by two noise sources placed at the amplifier input. There is both a voltage noise source Ṽ共t兲 in series
with the input voltage source and a current noise sourceĨ共t兲 in parallel with the input voltage source 共Fig. 12兲.
The voltage noise produces a fluctuating voltage Ṽ共t兲
共spectral density SV˜V˜关␻兴兲, which simply adds to the signal voltage at the amplifier input, and is amplified at theoutput; as such, it is completely analogous to the intrinsic detector output noise SII of our linear-response detector. In contrast, the current noise source of the voltRev. Mod. Phys., Vol. 82, No. 2, April–June 20101191

age amplifier represents back-action: this fluctuating
current 共spectral density SI˜I˜关␻兴兲 flows back across theparallel combination of the source impedance and amplifier input impedance, producing an additional fluctuating voltage at its input. The current noise is thus analogous to the back-action noise SFF of our generic linearresponse detector.
Putting the above together, the total voltage at the
input terminals of the amplifier isvin,tot共t兲 =

Zin
ZsZin
关vin共t兲 + Ṽ共t兲兴 −Ĩ共t兲Zin + Zs
Zs + Zin⯝ vin共t兲 + Ṽ共t兲 − ZsĨ共t兲.

共5.71兲

In the second line, we have taken the usual limit of an
ideal voltage amplifier which has an infinite input impedance 共i.e., the amplifier draws zero current兲. The
spectral density of the total input voltage fluctuations is
thus
SVV,tot关␻兴 = Svinvin关␻兴 + SVV,add关␻兴.

共5.72兲

Here Svinvin is the spectral density of the voltage fluctuations of the input signal vin共t兲 and SVV,add is the amplifier’s contribution to the total noise at the input
SVV,add关␻兴 = SV˜V˜ + 兩Zs兩2SI˜I˜ − 2 Re关Zs*SV˜I˜兴.

共5.73兲

For clarity, we have dropped the frequency index for the
spectral densities appearing on the RHS of this equation.
It is useful now to consider a narrow bandwidth input
signal at a frequency ␻, and ask the following question:
if the signal source was simply an equilibrium resistor ata temperature T0, how much hotter would it have to be
to produce a voltage noise equal to SVV,tot关␻兴? The resulting increase in the source temperature is defined as
the noise temperature TN关␻兴 of the amplifier and is aconvenient measure of the amplifier’s added noise. It is
standard among engineers to define the noise temperature with the assumption that the initial temperature of
the resistor T0 Ⰷ ប␻. One may then use the classical expression for the thermal noise of a resistor, which yields
the definition
2 Re ZskBTN关␻兴 ⬅ SVV,tot关␻兴.
Writing Zs = 兩Zs兩ei␾, we have
2kBTN =

冋

共5.74兲册SV˜V˜1+ 兩Zs兩SI˜I˜ − 2 Re共e−i␾SV˜I˜兲 .
cos ␾ 兩Zs兩
共5.75兲

It is clear from this expression that TN will have a minimum as a function of 兩Zs兩. For 兩Zs兩 too large, the backaction current noise of the amplifier will dominate TN,
while for 兩Zs兩 too small, the voltage noise of the amplifier
共i.e., its intrinsic output noise兲 will dominate. The situation is completely analogous to that of the position detector of the last section; there we needed to optimize
the coupling strength A to balance back-action and intrinsic output noise contributions and thus minimize theClerk et al.: Introduction to quantum noise, measurement, …

1192

total added noise. Optimization of the source impedancethus yields a completely classical minimum bound on
T N,
kBTN 艌 冑SV˜V˜SI˜I˜ − 关Im SV˜I˜兴2 − Re SV˜I˜ ,共5.76兲

where the minimum is achieved for an optimal source
impedance that satisfies兩Zs关␻兴兩opt = 冑SV˜V˜关␻兴/SI˜I˜关␻兴 ⬅ ZN ,sin ␾兩关␻兴兩opt = − Im SV˜I˜关␻兴/冑SV˜V˜关␻兴SI˜I˜关␻兴.

共5.77兲
共5.78兲

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

It is easy to connect the classical description of a voltage amplifier to the quantum-mechanical description ofa generic linear-response detector; in fact, all that isneeded is a “relabeling” of the concepts and quantities
introduced in Sec. V.E when discussing a linear position
detector. Thus, the quantum voltage amplifier will becharacterized by both an input operator Q̂ and an output operator V̂out; these play the roles of F̂ and Î in theposition detector, respectively. V̂out represents the output voltage of the amplifier, while Q̂ is the operator
which couples to the input signal vin共t兲 via a coupling
Hamiltonian
共5.79兲

ˆIn more familiar terms, Ĩin − dQ̂ / dt represents the current flowing into the amplifier.17 The voltage gain of ouramplifier ␭V will again be given by the Kubo formula of
Eq. 共4.3兲, with the substitutions F̂ → Q̂ , Î → V̂out 共we assume these substitutions throughout this section兲.
We can now easily relate the fluctuations of the input
and output operators to the noise sources used to describe the classical voltage amplifier. As usual, symmetrized quantum noise spectral densities S̄关␻兴 will play therole of the classical spectral densities S关␻兴 appearing in
the classical description. First, as the operator Q̂ represents a back-action force, its fluctuations correspond to
the amplifier’s current noise Ĩ共t兲,
17Note that one could have instead written the couplingˆ
Hamiltonian in the more traditional form Ĥint共t兲 = ␾共t兲Ĩin,
where ␾ = 兰dt⬘vin共t⬘兲 is the flux associated with the input voltage. The linear-response results we obtain are exactly thesame. We prefer to work with the charge Q̂ in order to beconsistent with the rest of the text.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共5.80兲

Similarly, the fluctuations in the operator V̂out, when referred back to the amplifier input, will correspond to thevoltage noise Ṽ共t兲 discussed above,
SV˜V˜关␻兴 ↔ S̄VoutVout关␻兴/兩␭V兩2 .

共5.81兲

A similar correspondence holds for the cross correlatorof these noise sources,
SV˜I˜关␻兴 ↔ + i␻S̄VoutQ关␻兴/␭V .

共5.82兲

To proceed, we need to identify the input and output
impedances of the amplifier, and then define its power
gain. The first step in this direction is to assume that theoutput of the amplifier 共V̂out兲 is connected to an external
circuit via a term

⬘ = qout共t兲V̂out ,Ĥint

2. Linear-response descriptionĤint = vin共t兲Q̂.

SI˜I˜关␻兴 ↔ ␻2S̄QQ关␻兴.

共5.83兲

where ĩout = dqout / dt is the current in the external circuit.
We may now identify the input and output impedancesof the amplifier in terms of the damping at the input andoutput. Use of the Kubo formulas for conductance andresistance yields 关cf. Eq. 共2.12兲 and Eq. 共5.48兲, with thesubstitutions F̂ → Q̂ and Î → V̂兴
1/Zin关␻兴 = i␻␹QQ关␻兴,

共5.84兲

Zout关␻兴 = ␹VV关␻兴/共− i␻兲,

共5.85兲

i.e., 具Ĩin典␻ = 共1 / Zin关␻兴兲vin关␻兴 and 具V典␻ = Zout关␻兴ĩout关␻兴,where the subscript ␻ indicates the Fourier transform ofa time-dependent expectation value.
We consider throughout this section the case of noreverse gain, ␹QVout = 0. We can define the power gain GP
exactly as in Sec. V.E.3 for a linear position detector. GP
is defined as the ratio of the power delivered to a load
attached to the amplifier output divided by the powerdrawn by the amplifier, maximized over the impedanceof the load. One finds
GP = 兩␭V兩2/4 Re共Zout兲Re共1/Zin兲.

共5.86兲

Expressing this in terms of the linear-response coefficients ␭VV and ␹QQ, we obtain an expression that is completely analogous to Eq. 共5.52兲 for the power gain for aposition detector
GP = 兩␭V兩2/4 Im ␹QQ Im ␹VV .

共5.87兲

Finally, we define again the effective temperature Teff关␻兴of the amplifier via Eq. 共2.8兲, and define a quantumlimited voltage amplifier as one that satisfies the idealnoise condition of Eq. 共4.17兲. For such an amplifier, thepower gain will be determined by the effective temperature via Eq. 共5.56兲.
Turning to the noise, we calculate the total symmetrized noise at the output port of the amplifier following
the same argument used to get the output noise of theposition detector 关cf. Eq. 共5.46兲兴. As we did in the clas-

Clerk et al.: Introduction to quantum noise, measurement, …

sical approach, we assume that the input impedance of
the amplifier is much larger than the source impedance:
Zin Ⰷ Zs; we test this assumption for consistency at theend of the calculation. Focusing only on the amplifiercontribution to this noise 共as opposed to the intrinsic
noise of the input signal兲, and referring this noise backto the amplifier input, we find that the symmetrized
quantum noise spectral density describing the added

冏

1193

冏冏 冏

1
Z N关 ␻ 兴
␣
ប␻
=
=
Ⰶ 1.
Re Zin关␻兴Im ␣ 4kBTeff 2冑GP关␻兴
共5.90兲

S̄关␻兴 as per Eqs. 共5.80兲–共5.82兲.
It follows that the amplifier noise temperature willagain be given by Eq. 共5.75兲, and that the optimal noise
temperature 共after optimizing over the source impedance兲 will be given by Eq. 共5.76兲. Whereas classicallynothing more could be said, quantum mechanically we
now get a further bound from the quantum noise constraint of Eq. 共4.11兲 and the requirement of a large
power gain. The latter requirement tells us that the volt-

It follows that 兩ZN兩 Ⰶ 兩Zin兩 in the large-power-gain, largeeffective-temperature regime of interest, thus justifying
the form of Eq. 共5.73兲. Equation 共5.90兲 is analogous tothe case of the displacement detector, where we foundthat reaching the quantum limit on resonance required
the detector-dependent damping to be much weaker
than the intrinsic damping of the oscillator 关cf. Eq.
共5.69兲兴.
Thus, similarly to the situation of the displacement
detector, the linear-response approach allows us both toderive rigorously the quantum limit on the noise temperature TN of an amplifier and to state conditions thatmust be met to reach this limit. To reach the quantumlimited value of TN with a large power gain, one needs
both a tuned source impedance Zs and an amplifier thatpossesses ideal noise properties 关cf. Eqs. 共4.17兲 and Eq.
共I13兲兴.

age gain ␭V关␻兴 and the cross correlator S̄Vout Q must bein phase 共cf. Sec. IV.A.4 and Appendix I.3兲. This in turn

3. Role of noise cross correlations

noise of the amplifier S̄VV,add关␻兴 satisfies the same equation we found for a classical voltage amplifier, Eq. 共5.73兲,with each classical spectral density S关␻兴 replaced by thecorresponding symmetrized quantum spectral density

means that S̄V˜I˜ must be purely imaginary. In this case,the quantum noise constraint may be rewritten asS̄V˜V˜关␻兴S̄I˜I˜关␻兴 − 关Im S̄V˜I˜兴2 艌 共ប␻/2兲2 .

共5.88兲

Using these results in Eq. 共5.76兲, we find the ultimate
quantum limit on the noise temperature,18
kBTN关␻兴 艌 ប␻/2

共5.89兲

As for the position detector, reaching the quantum limit
here is not simply a matter of tuning the coupling 关i.e.,tuning the source impedance Zs to match the noise impedance, cf. Eq. 共5.77兲 and 共5.78兲兴; one also needs tohave an amplifier with ideal quantum noise, that is, anamplifier satisfying Eq. 共4.17兲.
Finally, we need to test our initial assumption that
兩Zs兩 Ⰶ 兩Zin兩, taking 兩Zs兩 to be equal to its optimal value
ZN. Using the proportionality condition of Eq. 共I13兲 andthe fact that we are in the large-power-gain limit
共GP关␻兴 Ⰷ 1兲, we find18
Note that our definition of the noise temperature TN conforms with that of Devoret and Schoelkopf 共2000兲 and mostelectrical engineering texts, but is slightly different from that ofCaves 共1982兲. Caves assumed the source is initially at zero temperature 共i.e., T0 = 0兲, and consequently used the full quantum
expression for its equilibrium noise. In contrast, we have assumed that kBT0 Ⰷ ប␻. The different definition of the noise
temperature used by Caves leads to the result kBTN
艌 ប␻ / 共ln 3兲 as opposed to our Eq. 共5.89兲. We stress that thedifference between these results has nothing to do with physics, but only with how one defines the noise temperature.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Before leaving the topic of a linear voltage amplifier,we pause to note the role of cross correlations in current
and voltage noise in reaching the quantum limit. First,
note from Eq. 共5.78兲 that in both the classical and quantum treatments the noise impedance ZN of the amplifier
will have a reactive part 共i.e., Im ZN ⫽ 0兲 if there areout-of-phase correlations between the amplifier’s current
and voltage noises 共i.e., if Im SVI ⫽ 0兲. Thus, if such correlations exist, it will not be possible to minimize thenoise temperature 共and hence reach the quantum limit兲,
if one uses a purely real source impedance Zs.
More significantly, note that the final classical expression for the noise temperature TN explicitly involves thereal part of the SVI correlator 关cf. Eq. 共5.76兲兴. In contrast,
we have shown that in the quantum case Re S̄VI must be
zero if one wishes to reach the quantum limit while having a large power gain 共see Appendix I.3兲; as such, thisquantity does not appear in the final expression for theminimal TN. It also follows that to reach the quantum
limit while having a large power gain, an amplifier cannot have significant in-phase correlations between its
current and voltage noise.
This last statement can be given a heuristic explanation. If there are out-of-phase correlations between current and voltage noise, we can easily make use of theseby appropriately choosing our source impedance. However, if there are in-phase correlations between current
and voltage noise, we cannot use these simply by tuning
the source impedance. We could, however, have usedthem by implementing feedback in our amplifier. The
fact that we have not done this means that these correlations represent a kind of missing information; as a result, we must necessarily miss the quantum limit. In Sec.
VI.B, we explicitly give an example of a voltage amplifier which misses the quantum limit due to the presence

1194

Clerk et al.: Introduction to quantum noise, measurement, …

of in-phase current and voltage fluctuations; we show
how this amplifier can be made to reach the quantum
limit by adding feedback in Appendix H.
G. Near quantum-limited mesoscopic detectors

Having discussed the origin and precise definition of
the quantum limit on the added noise of a linear, phasepreserving amplifier, we now provide a review of work
examining whether particular detectors are able in principle to achieve this ideal limit. We focus on the op-amp
mode of operation discussed in Sec. V.D, where the detector is only weakly coupled to the system producing
the signal to be amplified. As repeatedly stressed, reaching the quantum limit in this case requires the detector
to have quantum-ideal noise, as defined by Eq. 共4.17兲.
Heuristically, this corresponds to the general requirement of no wasted information: there should be no otherquantity besides the detector output that could be monitored to provide information on the input signal 共Clerk
et al., 2003兲. We have already given one simple but relevant example of a detector which reaches the amplifier
quantum limit: the parametric cavity detector, discussed
in Sec. III.B. Here we turn to other more complex detectors.
1. dc superconducting quantum interference device amplifiers

The dc superconducting quantum interference device
共SQUID兲 is a detector based on a superconducting ring
having two Josephson junctions. It can in principle beused as a near quantum-limited voltage amplifier orflux-to-voltage amplifier. Theoretically, this was investigated using a quantum Langevin approach 共Koch et al.,
1981; Danilov et al., 1983兲, as well as more rigorously byusing perturbative techniques 共Averin, 2000b兲 and mappings to quantum impurity problems 共Clerk, 2006兲. Experiments on SQUIDS have also confirmed their potential for near-quantum-limited operation. Mück et al.
共2001兲 were able to achieve a noise temperature TN approximately 1.9 times the quantum-limited value at an
operating frequency of ␻ = 2␲ ⫻ 519 MHz. Working at
lower frequencies appropriate to gravitational wave detection applications, Vinante et al. 共2001兲 were able toachieve a TN approximately 200 times the quantumlimited value at a frequency ␻ = 2␲ ⫻ 1.6 kHz; more recently, the same group achieved a TN approximately tentimes the quantum limit at a frequency ␻ = 2␲⫻ 1.6 kHz 共Falferi et al., 2008兲. In practice, it can be difficult to achieve the theoretically predicted quantumlimited performance due to spurious heating caused by
the dissipation in the shunt resistances used in the
SQUID. This effect can be significantly ameliorated,
however, by adding cooling fins to the shunts 共Wellstood
et al., 1994兲.
2. Quantum point contact detectors

A quantum point contact 共QPC兲 is a narrow conducting channel formed in a two-dimensional gas. The current through the constriction is sensitive to nearby
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

charges, and thus the QPC acts as a charge-to-current
amplifier. It has been shown theoretically that the QPC
can achieve the amplifier quantum limit, both in the regime where transport is due to tunneling 共Gurvitz, 1997兲,and in regimes where the transmission is not small
共Aleiner et al., 1997; Levinson, 1997; Korotkov andAverin, 2001; Pilgram and Büttiker, 2002; Clerk et al.,2003兲. Experimentally, QPCs are in widespread use asdetectors of quantum dot qubits. The back-action
dephasing of QPC detectors was studied by Buks et al.
共1998兲 and Sprinzak et al. 共2000兲; good agreement was
found with the theoretical prediction, confirming that
the QPC has quantum-limited back-action noise.
3. Single-electron transistors and resonant-level detectors

A metallic single-electron transistor 共SET兲 consists ofa small metallic island attached via tunnel junctions tolarger source and drain electrodes. Because of
Coulomb-blockade effects, the conductance of a SET issensitive to nearby charges, and hence it acts as a sensitive charge-to-current amplifier. Considerable work has
investigated whether metallic SETs can approach thequantum limit in various different operating regimes.
Theoretically, the performance of a normal-metal SET
in the sequential tunneling regime was studied by Shnirman and Schön 共1998兲, Devoret and Schoelkopf 共2000兲,
Makhlin et al. 共2000兲, Aassime et al. 共2001兲, and Johansson et al. 共2002, 2003兲. In this regime, where transport is
via a sequence of energy-conserving tunnel events, one
is far from optimizing the quantum noise constraint of
Eq. 共4.17兲, and hence one cannot reach the quantum
limit 共Shnirman and Schön, 1998; Korotkov, 2001b兲. If
one instead chooses to work with a normal-metal SET inthe cotunneling regime 共a higher-order tunneling process involving a virtual transition兲, then one can indeed
approach the quantum limit 共Averin, 2000a; van den
Brink, 2002兲. However, by virtue of being a higher-orderprocess, the related currents and gain factors are small,
impinging on the practical utility of this regime of operation. It is worth noting that while most theory on SETs
assume a dc voltage bias, to enhance bandwidth, experiments are usually conducted using the rf-SET configuration 共Schoelkopf et al., 1998兲, where the SET changes
the damping of a resonant LC circuit. Korotkov andPaalanen 共1999兲 showed that this mode of operation fora sequential tunneling SET increases the measurementimprecision noise by approximately a factor of 2. The
measurement properties of a normal-metal, sequentialtunneling rf-SET 共including back-action兲 were studied
experimentally by Turek et al. 共2005兲.
Measurement using superconducting SETs has also
been studied. Clerk et al. 共2002兲 showed that so-called
incoherent Cooper-pair tunneling processes in a superconducting SET can have a noise temperature which isapproximately a factor of 2 larger than the quantumlimited value. The measurement properties of superconducting SETs biased at a point of incoherent Cooperpair tunneling have been probed recently in experiment
共Thalakulam et al., 2004; Naik et al., 2006兲.

Clerk et al.: Introduction to quantum noise, measurement, …

The quantum measurement properties of phasecoherent noninteracting resonant level detectors havealso been studied theoretically 共Averin, 2000b; Clerk
and Stone, 2004; Mozyrsky et al., 2004; Gavish et al.,2006兲. These systems are similar to metallic SET, exceptthat the central island only has a single level 共as opposed
to a continuous density of states兲, and Coulombblockade effects are typically neglected. These detectors
can reach the quantum limit in the regime where thevoltage and temperature are smaller than the intrinsic
energy broadening of the level due to tunneling. They
can also reach the quantum limit in a large-voltage regime that is analogous to the cotunneling regime in a
metallic SET 共Averin, 2000b; Clerk and Stone, 2004兲.
The influence of dephasing processes on such a detector
was studied by Clerk and Stone 共2004兲.
H. Back-action evasion and noise-free amplification

Having discussed in detail quantum limits on phasepreserving linear amplifiers 共i.e., amplifiers which measure both quadratures of a signal equally well兲, we now
return to the situation discussed in Sec. V.A: imagine wewish only to amplify a single quadrature of some timedependent signal. For this case, there need not be any
added noise from the measurement. Unlike the case of
amplifying both quadratures, Liouville’s theorem does
not require the existence of any additional degrees of
freedom when amplifying a single quadrature: phasespace volume can be conserved during amplification
simply by contracting the unmeasured quadrature 关see
Eq. 共5.2兲兴. 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 backaction does not affect the dynamics of the quantity being
measured 共Thorne et al., 1978; Braginsky et al., 1980;
Caves et al., 1980; Caves, 1982; Braginsky and Khalili,
1992; Bocko and Onofrio, 1996兲. For concreteness, consider a high-Q harmonic oscillator with position x共t兲 andresonant frequency ⍀. Its motion may be written interms of quadrature operators defined as in Eq. 共3.48兲,x̂共t兲 = X̂␦共t兲cos共⍀t + ␦兲 + Ŷ␦共t兲sin共⍀t + ␦兲.

共5.91兲

Here x̂共t兲 is the Heisenberg-picture position operator of
the oscillator. The quadrature operators can be written
in terms of the 共Schrödinger-picture兲 oscillator creationand destruction operators as
X̂␦共t兲 = xZPF共ĉei共⍀t+␦兲 + ĉ†e−i共⍀t+␦兲兲,

共5.92a兲Ŷ␦共t兲 = − ixZPF共ĉei共⍀t+␦兲 − ĉ†e−i共⍀t+␦兲兲.

共5.92b兲

As previously discussed, the two quadrature amplitude operators X̂␦ and Ŷ␦ are canonically conjugate 关cf.
Eq. 共3.49兲兴. Making a measurement of one quadratureamplitude, say X̂␦, will thus invariably lead to backaction disturbance of the other conjugate quadrature Ŷ␦.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1195

However, due to the dynamics of a harmonic oscillator,
this disturbance will not affect the measured quadrature
at later times. One can already see this from the classicalequations of motion. Suppose our oscillator is driven bya time-dependent force F共t兲 which has appreciable bandwidth only near ⍀. We may write this as
F共t兲 = FX共t兲cos共⍀t + ␦兲 + FY共t兲sin共⍀t + ␦兲,

共5.93兲

where FX共t兲, FY共t兲 are slowly varying compared to ⍀.
Using the fact that the oscillator has a high-quality factor Q = ⍀ / ␥, one can easily find the equations of motion,dX␦共t兲/dt = − 共␥/2兲X␦共t兲 − FY共t兲/2m⍀,

共5.94a兲

dY␦共t兲/dt = − 共␥/2兲Y␦共t兲 + FX共t兲/2m⍀.

共5.94b兲Thus, as long as FY共t兲 and FX共t兲 are uncorrelated andsufficiently slow, the dynamics of the two quadratures
are completely independent; in particular, if Y␦ is subject
to a narrow-bandwidth, noisy force, it is of no consequence to the evolution of X␦. An ideal measurement ofX␦ will result in a back-action force having the form of
Eq. 共5.93兲 with FY共t兲 = 0, implying that X␦共t兲 will be completely unaffected by the measurement.
Not surprisingly, if one can measure and amplify X␦
without any back-action, there need not be any added
noise due to the amplification. In such a setup, the only
added noise is the measurement-imprecision noise associated with intrinsic fluctuations of the amplifier output.
These may be reduced in principle to an arbitrarily smallvalue by simply increasing the amplifier gain 共e.g., byincreasing the detector-system coupling兲: in an ideal
setup, there is no back-action penalty on the measured
quadrature associated with this increase.
The above conclusion can lead to what seems like acontradiction. Imagine we use a back-action evading
amplifier to make a “perfect” measurement of X␦ 共i.e.,
negligible added noise兲. We would then have no uncertainty as to the value of this quadrature. Consequently,
we would expect the quantum state of our oscillator to
be a squeezed state, where the uncertainty in X␦ is muchsmaller than xZPF. However, if there is no back-action
acting on X␦, how is the amplifier able to reduce its uncertainty? This seeming paradox can be fully resolved byconsidering the conditional aspects of an ideal singlequadrature measurement, where one considers the stateof the oscillator given a particular measurement history
共Ruskov et al., 2005; Clerk et al., 2008兲.
It is worth stressing that the possibility of amplifying asingle quadrature without back-action 共and hence without added noise兲 relies crucially on our oscillator resembling a perfect harmonic oscillator: the oscillator Q mustbe large, and nonlinearities 共which could couple the twoquadratures兲 must be small. In addition, the envelope ofthe nonvanishing back-action force FX共t兲 must have anarrow bandwidth. One should further note that a very
high-precision measurement of X␦ will produce a verylarge back-action force FX. If the system is not nearlyClerk et al.: Introduction to quantum noise, measurement, …

1196

perfectly harmonic, then the large amplitude imparted
to the conjugate quadrature Y␦ will inevitably leak backinto X␦.
Amplifiers or detectors that treat the two signal
quadratures differently are known in the quantum optics
literature as phase sensitive; we prefer the designation
phase nonpreserving since they do not preserve thephase of the original signal. Such amplifiers invariably
rely on some internal clock 共i.e., an oscillator with awell-defined phase兲 which breaks time-translation invariance and picks out the phase of the quadrature thatwill be amplified 关i.e., the choice of ␦ used to define thetwo quadratures in Eq. 共5.91兲兴; we see this explicitly inwhat follows. This leads to an important caveat: even ina situation where the interesting information is in asingle signal quadrature, to benefit from using a phasenonpreserving amplifier, we must know in advance theprecise phase of this quadrature. If we do not know thisphase, we either have to revert to a phase-preserving
amplification scheme 共and thus be susceptible to added
noise兲 or we would have to develop a sophisticated andhigh speed quantum feedback scheme to dynamically
adapt the measurement to the correct quadrature in realtime 共Armen et al., 2002兲. In what follows, we make theabove ideas concrete by considering a few examples ofquantum phase-nonpreserving amplifiers.19
1. Degenerate parametric amplifier

Perhaps the simplest example of a phase nonpreserving amplifier is the degenerate parametric amplifier; theclassical version of this system was described in Sec. V.A
关see Eq. 共5.2兲兴. The setup is similar to the nondegenerateparametric amplifier discussed in Sec. V.C, except thatthe idler mode is eliminated, and the nonlinearity converts a single pump photon into two signal photons at
frequency ␻S = ␻P / 2. As we now show, the resulting dynamics causes one signal quadrature to be amplified
while the other is attenuated, in such a way that it is notnecessary to add extra noise to preserve the canonicalcommutation relations.
The system Hamiltonian isĤsys = ប共␻PâP† âP + ␻SâS† âS兲 + iប␩共âS† âS† âP − âSâSâP† 兲.
共5.95兲
If the pump is treated classically as before, the analog of
Eq. 共5.16兲 isV̂sys = iប共␭/2兲共âS† âS† − âSâS兲,

共5.96兲

where ␭ / 2 = ␩␺P, and the analog of Eq. 共5.19兲 isâ˙S = − 共␬S/2兲âS + ␭âS† − 冑␬Sb̂S,in .

共5.97兲

19
One could in principle generalize the linear-response approach of Sec. V to deal with phase-nonpreserving detectors.
However, as such detectors are not time-translational invariant, such a description becomes rather cumbersome and is notparticularly helpful. We prefer instead to present concrete
examples.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

The dimensionless quadrature operators corresponding to the signal mode arex̂S = 共1/冑2兲共âS† + âS兲,ŷS = 共i/冑2兲共âS† − âS兲,

共5.98兲

which obey 关x̂S , ŷS兴 = i. We can define quadrature operators X̂S,in/out, ŶS,in/out corresponding to the input and output fields in a completely analogous manner.
The steady state solution of Eq. 共5.97兲 for the output
fields becomes
X̂S,out = 冑GX̂S,in,

ŶS,out = ŶS,in/冑G,

共5.99兲

where the number gain G is given by
G = 关共␭ + ␬S/2兲/共␭ − ␬S/2兲兴2 .

共5.100兲We thus see clearly that the amplifier treats the twoquadratures differently. One quadrature is amplified andthe other attenuated, with the result that the commutation relation can be preserved without the necessity of
extra degrees of freedom and added noise. Note that thelarge-amplitude pump mode has played the role of aclock in the degenerate paramp: it is the phase of thepump that picks out which quadrature of the signal willbe amplified.
Before ending our discussion here, it is important tostress that while the degenerate parametric amplifier isphase sensitive and has no added noise, it is not an example of back-action evasion 关see Caves et al. 共1980兲,
footnote on p. 342兴. This amplifier is operated in the
scattering mode of amplifier operation, a mode where
共as discussed in Sec. V.D兲 back-action is not relevant at
all. Recall that in this mode of operation the amplifier
input is perfectly impedance matched to the signal
source, and the input signal is simply the amplitude of anincident wave on the amplifier input. This mode of operation necessarily requires a strong coupling between
the input signal and the amplifier input 共i.e., 具b̂S,in典兲. If
one instead tried to weakly couple the degenerate parametric amplifier to a signal source, and operate it in the
op-amp mode of operation 共cf. Sec. V.D兲, one finds that
there is indeed a back-action disturbance of the measured quadrature. We have yet another example which
demonstrates that one must be careful to distinguish the
op-amp and scattering modes of amplifier operation.
2. Double-sideband cavity detector

We now turn to a simple but experimentally relevant
detector that is truly back-action evading. We take as
our input signal the position x̂ of a mechanical oscillator.
The amplifier setup we consider is almost identical tothe cavity position detector discussed in Sec. III.B.3: weagain have a single-sided resonant cavity whose frequency depends linearly on the oscillator’s position, with
the Hamiltonian given by Eq. 共3.11兲 共with ẑ = x̂ / xZPF兲.
We showed in Sec. III.B.3 and Appendix E.3 that, by
driving the cavity on resonance, it could be used to makea quantum-limited position measurement: one can operate it as a phase-preserving amplifier of the mechanicaloscillator’s position and achieve the minimum possible

Clerk et al.: Introduction to quantum noise, measurement, …

amount of added noise. To use the same system to makea back-action free measurement of one oscillator
quadrature only, one simply uses a different cavity drive.
Instead of driving at the cavity resonance frequency ␻c,one drives at the two sidebands associated with the mechanical motion 共i.e., at frequencies ␻c ± ⍀, where ⍀ is as
always the frequency of the mechanical resonator兲. As
we see, such a drive results in an effective interactionthat couples the cavity to only one quadrature of theoscillator’s motion. This setup was first proposed as ameans of back-action evasion by Braginsky et al. 共1980兲;
further discussion can be found in Caves et al. 共1980兲 andBraginsky and Khalili 共1992兲, as well as in Clerk et al.
共2008兲, which gives a fully quantum treatment and considers conditional aspects of the measurement. In what
follows, we sketch the operation of this system following
Clerk et al. 共2008兲; details are provided in Appendix E.4.
We start by requiring that our system be in the “goodcavity” limit, where ␻c Ⰷ ⍀ Ⰷ ␬ 共␬ is the damping of thecavity mode兲; we also require the mechanical oscillator
to have a high-Q factor. In this regime, the two sidebands associated with the mechanical motion at ␻c ± ⍀are well separated from the main cavity resonance at ␻c.
Making a single-quadrature measurement requires that
one drives the cavity equally at the two sideband frequencies. The amplitude of the driving field b̄in entering
the cavity will be chosen to have the formb̄in共t兲 = −=冑

i Ṅ i␦ −i共␻ −⍀兲t −i␦ −i共␻ +⍀兲t−e e c 兲
共e e c4

冑Ṅ2sin共⍀t + ␦兲e−i␻ct .

共5.101兲

Here Ṅ is the photon-number flux associated with thecavity drive 共see Appendix E for more details on how toproperly include a drive using input-output theory兲.
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
sends the cavity both photons with frequency ␻c − ⍀ andphotons with frequency ␻c + ⍀. The first kind of drive
photon can be converted to a cavity photon if a quantum
is absorbed from the mechanical oscillator; the second
kind of drive photon can be converted to a cavity photon if a quantum is emitted to the mechanical oscillator.
The result is that we can create a cavity photon by either
absorbing or emitting a mechanical oscillator quantum.
If there is a well-defined relative phase of ei2␦ between
the two kinds of drive photons, we would expect the
double-sideband drive to yield an effective cavityoscillator interaction of the form冑

Veff ⬀ Ṅ关â†共ei␦ĉ + e−i␦ĉ†兲 + H.c.兴
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共5.102a兲冑⬀ Ṅ共â + â†兲X̂␦ .

1197

共5.102b兲

This is exactly what is found in a full calculation 共see
Appendix E.4兲. Note that we have written the interaction in an interaction picture in which the fast oscillations of the cavity and oscillator operators have been
removed. In the second line, we have made use of Eqs.
共5.92兲 to show that the effective interaction involves only
the X̂␦ oscillator quadrature.
We thus see from Eq. 共5.102b兲 that the cavity is
coupled only to the oscillator X␦ quadrature. As shown
rigorously in Appendix E.3, the result is that the system
only measures and amplifies this quadrature: the lightleaving the cavity has a signature of X̂␦ but not of Ŷ␦.
Further, Eq. 共5.102b兲 implies that the cavity operator

冑Ṅ共â + â†兲 will act as a noisy force on the Y␦ quadrature.

While this will cause a back-action heating of Y␦, it will
not affect the measured quadrature X␦. We thus have atrue back-action-evading amplifier: the cavity output
light lets one measure X␦ free from any back-action effect. Note that, in deriving Eq. 共5.102a兲, we have used
the fact that the cavity operators have fluctuations in anarrow bandwidth ⬃␬ Ⰶ ⍀: the back-action force noise isslow compared to the oscillator frequency. If this were
not the case, we could still have a back-action heating of
the measured X␦ quadrature. Such effects, arising from anonzero ratio ␬ / ⍀, are treated in Clerk et al. 共2008兲.
Finally, as there is no back-action on the measured X␦
quadrature, the only added noise of the amplification
scheme is measurement-imprecision noise 共e.g., shotnoise in the light leaving the cavity兲. This added noise
can be made arbitrarily small by increasing the gain of
the detector by, for example, increasing the strength ofthe cavity drive Ṅ. In a real system where ␬ / ␻M is nonzero, the finite bandwidth of the cavity number fluctuations leads to a small back-action on the X␦. As a result,
one cannot make the added noise arbitrarily small, as
too large a cavity drive will heat the measured quadrature. Nonetheless, for a sufficiently small ratio ␬ / ␻M,
one can still beat the standard quantum limit on theadded noise 共Clerk et al., 2008兲.

3. Stroboscopic measurements

With sufficiently high bandwidth it should be possible
to do stroboscopic measurements synchronized with theoscillator motion, which could allow one to go below thestandard quantum limit in one quadrature of motion
共Caves et al., 1980; Braginsky and Khalili, 1992兲. To understand this idea, imagine an extreme form of phasesensitive detection in which a Heisenberg microscope
makes a strong high-resolution measurement which
projects the oscillator onto a state of well-defined position X0 at time t = 0,Clerk et al.: Introduction to quantum noise, measurement, …

1198⬁⌿X0共t兲 = 兺 ane−i共n+1/2兲⍀t兩n典,

共5.103兲

n=0where the coefficients obey an = 具n 兩 X0典. Because the position is well defined the momentum is extremely uncertain. 共Equivalently, the momentum kick delivered by the
back-action of the microscope makes the oscillator momentum uncertain.兲 Thus the wave packet quickly
spreads out and the position uncertainty becomes large.
However, because of the special feature that the harmonic oscillator levels are evenly spaced, we can see
from Eq. 共5.103兲 that the wave packet re-assembles itself
precisely once each period of oscillation because ein⍀t= 1 for every integer n. 共At half periods, the packet reassembles at position −X0.兲 Hence stroboscopic measurements made once 共or twice兲 per period will be backaction evading and can go below the standard quantum
limit. The only limitations will be the finite anharmonicity and damping of the oscillator. Note that the possibility of using mesoscopic electron detectors to perform
stroboscopic measurements has recently received attention 共Jordan and Büttiker, 2005b; Ruskov et al., 2005兲.
VI. BOSONIC SCATTERING DESCRIPTION OF A
TWO-PORT AMPLIFIER

In this section, we return again to the topic of Sec.
V.F, quantum limits on a quantum voltage amplifier. We
now discuss the physics in terms of the bosonic voltage
amplifier first introduced in Sec. V.D. Recall that in that
section we demonstrated that the standard Haus-Caves
derivation of the quantum limit was not directly relevantto the usual weak-coupling op-amp mode of amplifier
operation, a mode where the input signal is not simply
the amplitude of a wave incident on the amplifier. In thissection, we expand upon that discussion, giving an explicit discussion of the differences between the op-amp
description of an amplifier presented in Sec. V.E, andthe scattering description often used in the quantum optics literature 共Grassia, 1998; Courty et al., 1999兲. We seethat what one means by back-action and added noise are
not the same in the two descriptions! Further, eventhough an amplifier may reach the quantum limit whenused in the scattering mode 共i.e., its added noise is as
small as allowed by commutation relations兲, it can nonetheless fail to achieve the quantum limit when used in
the op-amp mode. Finally, the discussion here will also
allow us to highlight important aspects of the quantum
limit not easily discussed in the more general context ofSec. IV.

mission line can be described as a set of noninteracting
bosonic modes 共see Appendix D for a quick review兲.
Denoting the input transmission line with an a and theoutput transmission line with a b, the current and voltage operators in these lines may be written asV̂q共t兲 =冕d␻
共V̂q关␻兴e−i␻t + H.c.兲,2
␲
0

共6.1a兲

冕

共6.1b兲

Îq共t兲 = ␴q
with

⬁

⬁

d␻
共Îq关␻兴e−i␻t + H.c.兲,
0 2␲

冑
冑

V̂q关␻兴 =ប␻
Zq共q̂in关␻兴 + q̂out关␻兴兲,2

共6.2a兲

Îq关␻兴 =ប␻共q̂in关␻兴 − q̂out关␻兴兲.
2Zq

共6.2b兲

Here q can be equal to a or b, and we have ␴a = 1,
␴b = −1. The operators âin关␻兴, âout关␻兴 are bosonic annihilation operators; âin关␻兴 describes an incoming wave in
the input transmission line 共i.e., incident on the amplifier兲 having frequency ␻, while âout关␻兴 describes an outgoing wave with frequency ␻. The operators b̂in关␻兴 andb̂out关␻兴 describe analogous waves in the output transmission line. We can think of V̂a as the input voltage to ouramplifier and V̂b as the output voltage. Similarly, Îa is thecurrent drawn by the amplifier at the input and Îb is thecurrent drawn at the output of the amplifier. Finally, Za
共Zb兲 is the characteristic impedance of the input 共output兲
transmission line.
As we have seen, amplification invariably requires additional degrees of freedom. Thus, to amplify a signal ata particular frequency ␻, there will be 2N bosonicmodes involved, where the integer N is necessarily
larger than 2. Four of these modes are simply thefrequency-␻ modes in the input and output lines 共i.e.,âin关␻兴, âout关␻兴, b̂in关␻兴, and b̂out关␻兴兲. The remaining 2共N− 2兲 modes describe auxiliary degrees of freedom involved in the amplification process; these additionalmodes could correspond to frequencies different fromthe signal frequency ␻. The auxiliary modes can also be
divided into incoming and outgoing modes. It is thus
convenient to represent them as additional transmission
lines attached to the amplifier; these additional lines
could be semi-infinite or could be terminated by active
elements.

A. Scattering versus op-amp representations

1. Scattering representation

In the bosonic scattering approach, a generic linear
amplifier is modeled as a set of coupled bosonic modes.
To make matters concrete, we consider the specific caseof a voltage amplifier with distinct input and output
ports, where each port is a semi-infinite transmission line
共see Fig. 9兲. We start by recalling that a quantum trans-

In general, our generic two-port bosonic amplifier will
be described by an N ⫻ N scattering matrix, which determines the relation between the outgoing and incomingmode operators. The form of this matrix is constrainedby the requirement that the output modes obey theusual canonical bosonic commutation relations. It is con-

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Clerk et al.: Introduction to quantum noise, measurement, …

venient to express the scattering matrix in a form which
only involves the input and output lines explicitly,

冉 冊冉
âout关␻兴b̂out关␻兴=s11关␻兴 s12关␻兴s21关␻兴 s22关␻兴冊冉 冊 冉
âin关␻兴b̂in关␻兴+F̂a关␻兴
F̂b关␻兴

冊

.

共6.3兲

1199

2. Op-amp representation

In the usual op-amp amplifier mode of operation 共described in Sec. IV兲, the input and output signals are notsimply incoming and outgoing wave amplitudes; thus,
the scattering representation is not an optimal description of our amplifier. The system we are describing here
is a voltage amplifier: thus, in the op-amp mode, theexperimentalist would ensure that the voltage at the endof the input line 共V̂a兲 is equal to the signal to be amplified, and would read out the voltage at the end of theHere F̂a关␻兴 and F̂b关␻兴 are each an unspecified linear
combination of the incoming auxiliary modes introduced
above. They thus describe noise in the outgoing modesof the input and output transmission lines which arises
from the auxiliary modes involved in the amplification
process. Note the similarity between Eqs. 共6.3兲 and 共5.7兲
for the simple one-port bosonic amplifier considered inSec. V.B.
In the quantum optics literature, one typically views
Eq. 共6.3兲 as the defining equation of the amplifier; wecall this the scattering representation of our amplifier.
The representation is best suited to the scattering modeof amplifier operation described in Sec. V.D. In thismode of operation, the experimentalist ensures that具âin关␻兴典 is precisely equal to the signal to be amplified,irrespective of what is coming out of the amplifier. Similarly, the output signal from the amplifier is the ampli-

output transmission line 共V̂b兲 as the output of the amplifier. From Eq. 共6.1a兲, we see that this implies that theamplitude of the wave going into the amplifier, ain, will
depend on the amplitude of the wave exiting the amplifier, aout.
Thus, if we want to use our amplifier as a voltage
amplifier, we want to find a description that is more tailored to our needs than the scattering representation of
Eq. 共6.3兲. This can be found by simply re-expressing thescattering matrix relation of Eq. 共6.3兲 in terms of voltages and currents. The result will be what we term theop amp representation of our amplifier, a representation
which is standard in the discussion of classical amplifiers关see, e.g., Boylestad and Nashelsky 共2006兲兴. In this rep-

tude of the outgoing wave in the output line, 具b̂out关␻兴典. If

input, while Îb is set by whatever we connect to the amplifier output. In contrast, the outputs of our amplifier

we focus on b̂out, we have precisely the same situation as
described in Sec. 5.10, where we presented the HausCaves derivation of the quantum limit 关see Eq. 共5.7兲兴. It
follows that in the scattering mode of operation the matrix element s21关␻兴 represents the gain of our amplifier at
frequency ␻, 兩s21关␻兴兩2 the corresponding “photonnumber gain,” and F̂b the added noise operator of theamplifier. The operator F̂a represents the back-action
noise in the scattering mode of operation; this backaction has no effect on the added noise of the amplifierin the scattering mode.
Similar to Sec. V.B, one can now apply the standard
argument of Haus and Mullen 共1962兲 and Caves 共1982兲
to our amplifier. This argument tells us that since the“out” operators must have the same commutation relations as the “in” operators, the added noise F̂b cannot
be arbitrarily small in the large-gain limit 共i.e., 兩s21兩 Ⰷ 1兲.
Note that this version of the quantum limit on the added
noise has nothing to do with back-action. As discussed,
this is perfectly appropriate for the scattering mode ofoperation, as in this mode the experimentalist ensuresthat the signal going into the amplifier is completely independent of whatever is coming out of the amplifier.
This mode of operation could be realized in timedependent experiments, where a pulse is launched at theamplifier. This mode is not realized in most weakcoupling amplification experiments, where the signal tobe amplified is not identical to an incident wave amplitude.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

resentation, one views V̂a and Îb as inputs to the amplifier: V̂a is set by whatever we connect to the amplifier

are the voltage in the output line, V̂b, and the current
drawn by the amplifier at the input, Îa. Note that thisinterpretation of voltages and currents is identical to theway we viewed the voltage amplifier in the linearresponse and quantum noise treatment of Sec. V.F.
Using Eqs. 共6.1a兲 and 共6.1b兲 and suppressing frequency labels for clarity, Eq. 共6.3兲 may be written explicitly in terms of the voltages and current in the input
共V̂a , Îa兲 and output 共V̂b , Îb兲 transmission lines,

冉冊冢
V̂b
Îa

␭V − Zout
= 1
␭I⬘
Zin

冣冉 冊 冢 冣
V̂a
Îb

+

ˆ
␭V · Ṽ
ˆ
Ĩ

.

共6.4兲

The coefficients in the above matrix are completely determined by the scattering matrix of Eq. 共6.3兲 关see Eqs.
共6.7兲 below兴; moreover, they are familiar from the discussion of a voltage amplifier in Sec. V.F. ␭V关␻兴 is thevoltage gain of the amplifier, ␭I⬘关␻兴 is the reverse current
gain of the amplifier, Zout is the output impedance, andZin is the input impedance. The last term on the RHS of
Eq. 共6.4兲 describes the two familiar kinds of amplifierˆnoise. Ṽ is the usual voltage noise of the amplifier 共reˆferred back to the amplifier input兲, while Ĩ is the usual
current noise of the amplifier. Recall that in this standard description of a voltage amplifier 共see Sec. V.F兲 Ĩ
represents the back-action of the amplifier: the systemproducing the input signal responds to these current

Clerk et al.: Introduction to quantum noise, measurement, …

1200

fluctuations, resulting in an additional fluctuation in the
input signal going into the amplifier. Similarly, ␭VṼ represents the intrinsic output noise of the amplifier: thiscontribution to the total output noise does not dependon properties of the input signal. Note that we are using
a sign convention where a positive 具Îa典 indicates a current flowing into the amplifier at its input, while a positive 具Îb典 indicates a current flowing out of the amplifier
at its output. Also note that the operators V̂a and Îb on
the RHS of Eq. 共6.4兲 will have noise; this noise is entirely due to the systems attached to the input and output of the amplifier, and as such should not be included
in what we call the added noise of the amplifier.
Additional important properties of our amplifier follow immediately from quantities in the op-amp representation. As discussed in Sec. V.E, the most importantmeasure of gain in our amplifier is the dimensionless
power gain. This is the ratio between power dissipatedat the output to that dissipated at the input, taking theoutput current IB to be VB / Zout,

冉冊␭V␭I⬘ Zin −1
共␭V兲2 Zin
1+.
GP ⬅
4 Zout2 Zout共6.5兲

Another important quantity is the loaded input impedance: what is the input impedance of the amplifier inthe presence of a load attached to the output? In thepresence of reverse current gain ␭I⬘ ⫽ 0, the input impedance will depend on the output load. Taking the load
impedance to be Zload, some simple algebra yields
1/Zin,loaded = 1/Zin + ␭I⬘␭V/共Zload + Zout兲.

共6.6兲

It is of course undesirable to have an input impedance
which depends on the load. Thus, we see yet again that it
is undesirable to have appreciable reverse gain in our
amplifier 共cf. Sec. IV.A.2兲.
3. Converting between representations

After some straightforward algebra we now express
the op-amp parameters appearing in Eq. 共6.4兲 in termsof the scattering matrix appearing in Eq. 共6.3兲,␭V = 2冑冑

Zb s21,
Za D

共6.7a兲

Zb s12,Za D

共6.7b兲

共1 + s11兲共1 + s22兲 − s12s21,
D

共6.7c兲

1 共1 − s11兲共1 − s22兲 − s12s21
1,=
D
Zin Za

共6.7d兲

␭I⬘ = 2

Zout = Zb

where all quantities are evaluated at the same frequency
␻, and D is defined asRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

D = 共1 + s11兲共1 − s22兲 + s12s21 .

共6.8兲

Further, the voltage and current noises in the op-amp
representation are simple linear combinations of thenoises F̂a and F̂b appearing in the scattering representation,

冢 冣
ˆ
Ṽ

ˆ
Za · Ĩ= 冑2ប␻Za

冢

−

1
2

s22 − 1D

1 + s11
2s21
−

s12
D

冣冉 冊
F̂a

F̂b

.

共6.9兲

Again, all quantities above are evaluated at frequency ␻.
Equation 共6.9兲 immediately leads to an important conclusion and caveat: what one calls the ‘‘back-action’’ and‘‘added noise’’ in the scattering representation 共i.e., Fa
and Fb兲 are not the same as the ‘‘back-action’’ and‘‘added noise’’ defined in the usual op-amp representaˆ
tion. For example, the op-amp back-action Ĩ does not ingeneral coincide with F̂a, the back-action in the scattering picture. If we are indeed interested in using our amplifier as a voltage amplifier, we are interested in the
total added noise of our amplifier as defined in the opamp representation. As we saw in Sec. V.F 关cf. Eq.
ˆˆ
共5.71兲兴, this quantity involves both the noises Ĩ and Ṽ.
We thus see explicitly something discussed in Sec. V.D: it
is dangerous to make conclusions about how an amplifier behaves in the op-amp mode of operation based on
its properties in the scattering mode of operation. As wesee, even though an amplifier is ideal in the scatteringmode 共i.e., Fa as small as possible兲, it can nonethelessfail to reach the quantum limit in the op-amp mode ofoperation.
ˆ
In what follows, we calculate the op-amp noises Ṽ andˆ
Ĩ in a minimal bosonic voltage amplifier, and show explicitly how this description is connected to the moregeneral linear-response treatment of Sec. V.F. However,before proceeding, it is worth noting that Eqs. 共6.7a兲,
共6.7b兲, 共6.7c兲, and 共6.7d兲 are themselves completely consistent with linear-response theory. Using linear response, one would calculate the op-amp parameters ␭V,␭I⬘, Zin, and Zout using Kubo formulas 关cf. Eqs. 共5.84兲 and共5.85兲 and the discussion following Eq. 共5.79兲兴. These inturn would involve correlation functions of Îa and V̂bevaluated at zero coupling to the amplifier input andoutput. Zero coupling means that there is no input voltage to the amplifier 共i.e., a short circuit at the amplifier
input, V̂a = 0兲 and there is nothing at the amplifier output
drawing current 共i.e., an open circuit at the amplifier
output, Îb = 0兲. Equation 共6.4兲 tells us that in this case V̂bˆˆand Îa reduce to the noise operators ␭VṼ and Ĩ, respectively. Using the fact that the commutators of F̂a and F̂b
are completely determined by the scattering matrix 关cf.
Eq. 共6.3兲兴, we verify explicitly in Appendix I.5 that the
Kubo formulas yield the same results for the op-amp

Clerk et al.: Introduction to quantum noise, measurement, …

gains and impedances as Eqs. 共6.7a兲, 共6.7b兲, 共6.7c兲, and共6.7d兲 above.

1201
cold
load

vacuum noise uin

uout

aout

bin

B. Minimal two-port scattering amplifierain

1. Scattering versus op-amp quantum limit

In this section we demonstrate that an amplifier which
is ideal and minimally complex when used in the scattering operation mode fails, when used as a voltage opamp, to have a quantum-limited noise temperature. The
system we look at is similar to the amplifier consideredby Grassia 共1998兲, though our conclusions are somewhat
different than those found there.
In the scattering representation, one might guess that
an ideal amplifier would be one where there are no reflections of signals at the input and output, and no way
for incident signals at the output port to reach the input.
In this case, Eq. 共6.3兲 takes the form

冉 冊 冉冑 冊冉 冊 冉 冊
âout

b̂out

=

0

âin

G 0

b̂in

0

+

F̂a

F̂b

共6.10兲

,where we have defined 冑G ⬅ s21. All quantities above
should be evaluated at the same frequency ␻; for clarity,
we omit the explicit ␻ dependence of quantities
throughout this section.
Turning to the op-amp representation, the above
equation implies that our amplifier has no reverse gain,
and that the input and output impedances are simply
given by the impedances of the input and output transmission lines. From Eqs. 共6.7兲, we have␭V = 2冑ZbG/Za ,共6.11a兲␭I⬘ = 0,

共6.11b兲

Zout = Zb ,共6.11c兲1/Zin = 1/Za .

共6.11d兲We immediately see that our amplifier looks less ideal asan op-amp. The input and output impedances are thesame as those of the input and output transmission line.
However, for an ideal op-amp, we would have liked
Zin → ⬁ and Zout → 0.
Also of interest are the expressions for the amplifier
noises in the op-amp representation

冢 冣
ˆ
Ṽ

ˆ
Za · Ĩ= − 冑2ប␻Za

冢

1
1
−
2
2 冑G
1

0

冣冉 冊
F̂a

F̂b

.

共6.12兲

As s12 = 0, the back-action noise is the same in both the
op-amp and scattering representations: it is determined
completely by the noise operator F̂a. However, the voltage noise 共i.e., the intrinsic output noise兲 involves bothF̂a and F̂b. We thus have the unavoidable consequenceˆˆthat there will be correlations in Ĩ and Ṽ. Note that frombasic linear-response theory we know that there must be
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

output line

input lineboutcout =G1/2cin+(G-1)1/2v†in

cin
ideal
1-portamp.

vout

vin vacuum noise

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 coupled
op-amp. See text for further description.

ˆˆ
some correlations between Ĩ and Ṽ if there is to be gain关i.e., ␭V is given by a Kubo formula involving these operators, see Eq. 共4.3兲兴.
To make further progress, we note again that commutators of the noise operators F̂a and F̂b are completely
determined by Eq. 共6.10兲 and the requirement that theoutput operators obey canonical commutation relations.
We thus have关F̂a,F̂†a兴 = 1,

共6.13a兲关F̂b,F̂b† 兴 = 1 − 兩G兩,

共6.13b兲关F̂a,F̂b兴 = 关F̂a,F̂b† 兴 = 0.

共6.13c兲We will be interested in the limit of a large power
gain, which requires 兩G兩 Ⰷ 1. A minimal solution to theabove equations would be to have the noise operators
determined by two independent 共i.e., mutually commut†,ing兲 auxiliary input mode operators uin and vin
F̂a = ûin ,共6.14兲

†F̂b = 冑兩G兩 − 1v̂in.

共6.15兲

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 共1998兲: an
input and output line coupled to a negative resistancebox and an auxiliary “cold load” via a four-port circulator 共see Fig. 13兲. The negative resistance box is nothing
but the single-mode bosonic amplifier discussed in Sec.
V.B; an explicit realization of this element would be theparametric amplifier discussed in Sec. V.C. The ‘‘cold
load’’ is a semi-infinite transmission line which models
dissipation due to a resistor at zero temperature 共i.e., its
noise is vacuum noise, cf. Appendix D兲.

Clerk et al.: Introduction to quantum noise, measurement, …

1202

Note that within the scattering picture one would conclude that our amplifier is ideal: in the large-gain limit,
the noise added by the amplifier to b̂out corresponds to asingle quantum at the input†
具兵F̂b,F̂b† 其典/兩G兩 = 关共兩G兩 − 1兲/兩G兩兴具兵v̂in, v̂in其典 → 1.

共6.16兲

This, however, is not the quantity which interests us: as
we want to use this system as a voltage op-amp, we wantto know if the noise temperature defined in the op-amp
picture is as small as possible. We are also usually interested in the case of a signal that is weakly coupled to ouramplifier; here weak coupling means that the input impedance of the amplifier is much larger than the impedance of the signal source 共i.e., Zin Ⰷ Zs兲. In this limit, theamplifier only slightly increases the total damping of thesignal source.
To address whether we can reach the op-amp quantum limit in the weak-coupling regime, we can make useof the results of the general theory presented in Sec. V.F.
In particular, we need to check whether the quantum
noise constraint of Eq. 共5.88兲 is satisfied, as this is a prerequisite for reaching the 共weak-coupling兲 quantum
limit. Thus, we need to calculate the symmetrized spectral densities of the current and voltage noises, and theircross correlation. It is easy to confirm from the definitions of Eqs. 共6.1a兲 and 共6.1b兲 that these quantities takethe form
ˆˆS̄VV关␻兴 = 具兵Ṽ关␻兴,Ṽ†共␻⬘兲其典/4␲␦共␻ − ␻⬘兲,

共6.17a兲ˆˆS̄II关␻兴 = 具兵Ĩ关␻兴,Ĩ†共␻⬘兲其典/4␲␦共␻ − ␻⬘兲,

共6.17b兲ˆˆS̄VI关␻兴 = 具兵Ṽ关␻兴,Ĩ†共␻⬘兲其典/4␲␦共␻ − ␻⬘兲.

共6.17c兲

The expectation values here are over the operating stateof the amplifier; we have chosen this state to be thevacuum for the auxiliary mode operators ûin and v̂in tominimize the noise.
Taking 兩s21兩 Ⰷ 1, and using Eqs. 共6.14兲 and 共6.15兲, we
haveS̄VV关␻兴 = ប␻Za共␴uu + ␴vv兲/4 = ប␻Za/2,

共6.18a兲S̄II关␻兴 = ប␻␴uu/Za = ប␻/Za ,共6.18b兲

S̄VI关␻兴 = ប␻␴uu/2 = ប␻/2,

共6.18c兲where we have defined␴ab ⬅ 具âb̂† + b̂†â典

共6.19兲

and have used the fact that there cannot be any correlations between the operators u and v in the vacuum state
共i.e., 具ûv̂†典 = 0兲.
It follows immediately from the above equations that
our minimal amplifier does not optimize the quantum
noise constraint of Eq. 共5.88兲,Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

S̄VV关␻兴S̄II关␻兴 − 关Im S̄VI兴2 = 2 ⫻ 共ប␻/2兲2 .

共6.20兲

The noise product S̄VVS̄II is precisely twice the quantumlimited value. As a result, the general theory of Sec. V.F
tells us that if one couples an input signal weakly to thisamplifier 共i.e., Zs Ⰶ Zin兲, it is impossible to reach thequantum limit on the added noise. Thus, while our amplifier is ideal in the scattering mode of operation 关cf.
Eq. 共6.16兲兴, it fails to reach the quantum limit when usedin the weak-coupling, op-amp mode of operation. Our
amplifier’s failure to have ideal quantum noise alsomeans that if we tried to use it to do QND qubit detection, the resulting back-action dephasing would be twiceas large as the minimum required by quantum mechanics 共cf. Sec. IV.B兲.
One might object to the above conclusions based on
the classical expression for the minimal noise temperature, Eq. 共5.76兲. Unlike the quantum noise constraint of
Eq. 共5.88兲, this equation also involves the real part ofS̄VI, and is optimized by our minimal amplifier. However,this does not mean that one can achieve a noise temperature of ប␻ / 2 at weak coupling. Recall from Sec. V.F
that in the usual process of optimizing the noise temperature one starts by assuming the weak-coupling condition that the source impedance Zs is much smaller
than the amplifier input impedance Zin. One then findsthat to minimize the noise temperature, 兩Zs兩 should betuned to match the noise impedance of the amplifier

ZN ⬅ 冑S̄VV / S̄II. However, in our minimal bosonic amplifier, it follows from Eqs. 共6.18兲 that ZN = Zin / 冑2 ⬃ Zin: thenoise impedance is on the order of the input impedance.
Thus, it is impossible to match the source impedance tothe noise impedance while at the same time satisfying
the weak-coupling condition Zs Ⰶ 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 coupling to
the input signal. To see this explicitly, imagine we connected our amplifier to a signal source with source impedance Zs. The total output noise of the amplifier, referred back to the signal source, will now have the formˆ ˆˆṼtot = − 关ZsZa/共Zs + Za兲兴Ĩ + Ṽ .

共6.21兲

Note that this classical-looking equation can be rigorously justified within the full quantum theory if one
starts with a full description of the amplifier and thesignal source 共e.g., a parallel LC oscillator attached inparallel to the amplifier input兲. Plugging in the expresˆˆ
sions for Ĩ and Ṽ, we findˆṼtot ==

冑 冋冉 冊冉 冑 冊 冑冑 冋冉 冊 册
ប␻
2

ប␻Za
2

Z sZ aZs + Za2

Za†ûin − Za共ûin − v̂in兲Zs − Za†ûin − v̂in.
Zs + Za册

共6.22兲

Thus, if one tunes Zs to Za = Zin, the mode ûin does notcontribute to the total added noise, and one reaches theClerk et al.: Introduction to quantum noise, measurement, …

quantum limit. Physically speaking, by matching the signal source to the input line the back-action noise described by F̂a = ûin does not feed back into the input of
the amplifier. Note that to achieve this matching explicitly requires one to be far from weak coupling. Having
Zs = Za means that when we attach the amplifier to thesignal source, we dramatically increase the damping of
the signal source.

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

Returning to the more interesting case of a weak
amplifier-signal coupling, one might still be puzzled as towhy our seemingly ideal amplifier misses the quantum
limit. While the mathematics behind Eq. 共6.20兲 is fairly
transparent, it is also possible to understand this result
heuristically. To that end, note again that the amplifier
noise cross correlation S̄IV does not vanish in the largegain limit 关cf. Eq. 共6.18c兲兴. Correlations between the two
amplifier noises represent a kind of information, as bymaking use of them, we can improve the performance of
the amplifier. It is easy to take advantage of out-of-phasecorrelations between Ĩ and Ṽ 共i.e., Im S̄VI兲 by simply tuning the phase of the source impedance 关cf. Eq. 共5.75兲兴.
However, one cannot take advantage of in-phase noisecorrelations 共i.e., Re S̄VI兲 as easily. To take advantage ofthe information here, one needs to modify the amplifier
itself. By feeding back some of the output voltage to the
input, one could effectively cancel out some of the back-兩␭V兩2S̄VV/Zout = ZinS̄II ⬅ 2kBTeff .

1203

共6.23兲

In contrast, our minimal bosonic amplifier has very different input and output effective temperatures
2kBTeff,in = ZinS̄II = ប␻/2,

共6.24兲

2kBTeff,out = 兩␭V兩2 · S̄VV/Zout = 2兩G兩ប␻ .

共6.25兲

This large difference in effective temperatures meansthat it is impossible for the system to possess ideal quantum noise, and thus it cannot reach the weak-coupling
quantum limit.
While it implies that one is not at the quantum limit,
the fact that Teff,in Ⰶ Teff,out can nonetheless be viewed as
an asset. From a practical point of view, a large Teff,in canbe dangerous. Even though the direct effect of the large
Teff,in is offset by an appropriately weak coupling to theamplifier 关see Eq. 共5.70兲 and following discussion兴, thislarge Teff,in can also heat up other degrees of freedom ifthey couple strongly to the back-action noise of the amplifier. This can in turn lead to unwanted heating of the
input system. As Teff,in is usually constant over a broad
range of frequencies, this unwanted heating effect canbe quite bad. In the minimal amplifier discussed here,
this problem is circumvented by having a small Teff,in.
The only price that is paid is that the added noise will be冑2 the quantum limit value. We discuss this issue further
in Sec. VII.
VII. REACHING THE QUANTUM LIMIT IN PRACTICE
A. Importance of QND measurements

action current noise Ĩ and thus reduce the overall magnitude of S̄II. Hence, the unused information in the crosscorrelator Re S̄VI represents a kind of wasted information: had we made use of these correlations via a feedback loop, we could have reduced the noise temperatureand increased the information provided by our amplifier.
The presence of a nonzero Re S̄VI thus corresponds towasted information, implying that we cannot reach thequantum limit. Recall that within the linear-response approach we were able to prove rigorously that a largegain amplifier with ideal quantum noise must have
Re S̄VI = 0 关cf. the discussion following Eq. 共5.57兲兴; thus, anonvanishing Re S̄VI rigorously implies that one cannot
be at the quantum limit. In Appendix H, we give an
explicit demonstration of how feedback may be used toutilize these cross correlations to reach the quantum
limit.
Finally, yet another way of seeing that our amplifier
does not reach the quantum limit 共in the weak-coupling
regime兲 is to realize that this system does not have awell-defined effective temperature. Recall from Sec. V.F
that a system with ideal quantum noise 关i.e., one thatsatisfies Eq. 共5.88兲 as an equality兴 necessarily has thesame effective temperature at its input and output ports
关cf. Eq. 共5.55兲兴. Here that implies the requirement
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

The fact that QND measurements are repeatable is offundamental practical importance in overcoming detector inefficiencies 共Gambetta et al., 2007兲. A prototypicalexample is the electron shelving technique 共Nagourney
et al., 1986; Sauter et al., 1986兲 used to measure trapped
ions. A related technique is used in present implementations of ion-trap-based quantum computation. Herethe 共extremely long-lived兲 hyperfine state of an ion isread out via state-dependent optical fluorescence. With
properly chosen circular polarization of the exciting laser, only one hyperfine state fluoresces and the transition
is cycling; that is, after fluorescence the ion almost always returns to the same state it was in prior to absorbing the exciting photon. Hence the measurement is
QND. Typical experimental parameters 共Wineland et al.,1998兲 allow the cycling transition to produce N ⬃ 106
fluorescence photons. Given the photomultiplier quantum efficiency and typically small solid angle coverage,
only a small number n̄d will be detected on average. The
probability of getting zero detections 共ignoring darkcounts for simplicity兲 and hence misidentifying the hyperfine state is P共0兲 = e−n̄d. Even for a very poor overall
detection efficiency of only 10−5, we still have n̄d = 10 andnearly perfect fidelity F = 1 − P共0兲 ⬇ 0.999 955. It is important to note that the total time available for measurement is not limited by the phase coherence time 共T2兲 ofClerk et al.: Introduction to quantum noise, measurement, …

1204

the qubit or by the measurement-induced dephasing
共Korotkov, 2001a; Makhlin et al., 2001; Schuster et al.,2005; Gambetta et al., 2006兲, but rather only by the rate
at which the qubit makes real transitions between measurement 共␴ˆ z兲 eigenstates. In a perfect QND measurement there is no measurement-induced state mixing
共Makhlin et al., 2001兲 and the relaxation rate 1 / T1 isunaffected by the measurement process.

=

冉

冊

2
2
␭2vin
Zload/Zout
Zin.
Zout Zin + Zs 共1 + Zload/Zout兲2

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

冉

冊

2
2
␭2vin
Zin
.
4Zout Zin + Zs

共7.3兲The available power gain is now defined as
B. Power matching versus noise matching

GP,avail ⬅

In Sec. V, we saw that an important part of reaching
the quantum limit on the added noise of an amplifier
共when used in the op-amp mode of operation兲 is to optimize the coupling strength to the amplifier. For a position detector, this condition corresponds to tuning thestrength of the back-action damping ␥ to be muchsmaller than the intrinsic oscillator damping 关cf. Eq.
共5.69兲兴. For a voltage amplifier, this condition corresponds to tuning the impedance of the signal source tobe equal to the noise impedance 关cf. Eq. 共5.77兲兴, an impedance that is much smaller than the amplifier’s input
impedance 关cf. Eq. 共5.90兲兴.
In this section, we make the simple point that optimizing the coupling 共i.e., source impedance兲 to reach thequantum limit is not the same as what one would do to
optimize the power gain. To understand this, we need to
introduce another measure of power gain commonlyused in the engineering community, the available power
gain GP,avail. For simplicity, we discuss this quantity inthe context of a linear voltage amplifier, using the notations of Sec. V.F; it can be analogously defined for the
position detector of Sec. V.E. GP,avail tells us how much
power we are providing to an optimally matched output
load relative to the maximum power we could in principle have extracted from the source. This is in marked
contrast to the power gain GP, which was calculated using the 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 delivered to the input of the amplifier, assuming we optimized both the value of the input impedance Zin and theload impedance Zload while keeping Zs fixed. For simplicity, we take all impedances to be real in our discussion. In general, the power drawn at the input of the2
amplifier is Pin = vin
Zin / 共Zs + Zin兲2. Maximizing this overZin, we obtain the available input power Pin,avail,共7.1兲

2/4Zs .
Pin,avail = vin

The maximum occurs for Zin = Zs.
The output power supplied to the load Pout2
= vload/ Zload is calculated as before, keeping Zin and Zs
distinct. One has
Pout =

冉

冊

2
2
vout
Zload
Zload Zout + Zload

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

共7.2兲

冉

冊

2
Pout,max ␭2Zs
Zin=
Pin,avail Zout Zin + Zs

= GP

4Zs/Zin.
共1 + Zs/Zin兲2

共7.4兲

We see that GP,avail is strictly less than or equal to thepower gain GP; equality is achieved only when Zs = Zin
共i.e., when the source impedance is “power matched” tothe input of the amplifier兲. The general situation where
GP,avail ⬍ GP indicates that we are not drawing as much
power from the source as we could, and hence the actual
power 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 power
gain in this case is
GP,avail ⯝ ␭2ZN/Zout ⯝ 2冑GP Ⰶ GP .

共7.5兲

We have used Eq. 共5.90兲, which tells us that the noise
impedance is smaller than the input impedance by alarge factor of 1 / 共2冑GP兲. Thus, as reaching the quantum
limit requires the use of a source impedance muchsmaller than Zin, it results in a dramatic drop in theavailable power gain compared to the case where we
power match 共i.e., take Zs = Zin兲. In practice, one must
decide whether it is more important to minimize theadded noise or maximize the power provided at the output of the amplifier: one cannot do both at the sametime.
VIII. CONCLUSIONSIn this review, we have given an introduction to quantum limits for position detection and amplification, limits that are tied to fundamental constraints on quantum
noise correlators. We end by emphasizing notable current developments and pointing out future perspectives
in the field.
As emphasized, much of our discussion has been directly relevant to the measurement of mechanical nanoresonators, a topic attracting considerable recent attention. These nanoresonators are typically studied bycoupling them either to electrical 共often superconducting兲 circuits or to optical cavities. A key goal is toachieve quantum-limited continuous position detection
共cf. Sec. V.E兲; current experiments are coming tantalizingly close to this limit 共cf. Table III兲. Although the ability to follow the nanoresonator’s motion with a precision

Clerk et al.: Introduction to quantum noise, measurement, …

set by the quantum limit is in principle independent
from being at low temperatures, it becomes interesting
only when the systems are near their ground state; one
could then, e.g., monitor the oscillator’s zero-point fluctuations 共cf. Sec. III.B.3兲. Given the comparatively smallvalues of mechanical frequencies 共mostly less than 1
GHz兲, this often calls for the application of nonequilibrium cooling techniques which exploit back-action to reduce the effective temperature of the mechanical device,a technique that has been demonstrated recently in bothsuperconducting circuits and optomechanical setups 关see
Marquardt and Girvin 共2009兲 for a review兴.
The ability to perform quantum-limited position detection will in turn open up many new interesting avenues of research. Among the most significant is thepossibility of quantum feedback control 共Wiseman and
Milburn, 1993, 1994兲, where one uses the continuously
obtained measurement output to tailor the state of themechanical resonator. The relevant theoretical framework is that of quantum conditional evolution and quantum trajectories 共see, e.g., Brun, 2002; Jacobs and Steck,
2006兲, where one tracks the state of a measured quantum system in a particular run of the experiment. Application of these ideas has only recently been explored incondensed matter contexts 共Korotkov, 1999, 2001b;
Goan and Milburn, 2001; Goan et al., 2001; Oxtoby et al.,2006, 2008; Bernád et al., 2008兲. Fully understanding thepotential of these techniques, as well as differences that
occur in condensed matter versus atomic physics contexts, remains an active area of research. Other important directions in nanomechanics include the possibilityof detecting quantum jumps in the state of a mechanicalresonator via QND measurement of its energy 共Santamore, Doherty, and Cross, 2004; Santamore, Goan, et al.,
2004; Jayich et al., 2008; Thompson et al., 2008兲, as wellas the possibility of making back-action-evading measurements 共cf. Sec. V.H兲. Back-action evasion using a
microwave cavity detector coupled to a nanomechanicalresonator was recently reported 共Hertzberg et al., 2010兲.
Another area distinct from nanomechanics whererapid progress is being made is the readout of solid state
qubits using microwave signals sent through cavities
whose transmission properties are controlled by the qubit. At the moment, one is close to achieving goodfidelity single-shot QND readout, which is a prerequisite
for a large number of applications in quantum information processing. The gradually growing information
about the qubit state is extracted from the measured
noisy microwave signal trace, leading to a corresponding
collapse of the qubit state. This process can also be described by conditional quantum evolution and quantum
trajectories.
A promising method for superconducting qubit readout currently employed is a so-called latching measurement, where the hysteretic behavior of a strongly driven
anharmonic system 共e.g., a Josephson junction兲 is exploited to toggle between two states depending on thequbit state 共Siddiqi et al., 2004; Lupaşcu et al., 2006兲.
Although this is then no longer a linear measurement
scheme and is therefore distinct from what was disRev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1205

cussed in this review, it can be turned into a linear amplifier for a sufficiently weak input signal. An interestingand important open question is whether such a setup canreach the quantum limit on linear amplification.
Both qubit detection and mechanical measurements
in electrical circuits would benefit from quantum-limited
on-chip amplifiers. Such amplifiers are now being developed using the tools of circuit quantum electrodynamics,
employing Josephson junctions or SQUIDs coupled tomicrowave transmission line cavities 共Bergeal et al.,2008; Castellanos-Beltran et al., 2008兲. Such an amplifier
has already been used to perform continuous position
detection with a measurement imprecision below the
SQL level 共Teufel et al., 2009兲.
ACKNOWLEDGMENTS

This work was supported in part by NSERC, the Canadian Institute for Advanced Research 共CIFAR兲, NSA
under ARO Contract No. W911NF-05-1-0365, the NSF
under Grant No. DMR-0653377 and No. DMR-0603369,
the David and Lucile Packard Foundation, the W. M.
Keck Foundation, and the Alfred P. Sloan Foundation.
We acknowledge the support and hospitality of the Aspen Center for Physics where part of this work was carried out. F.M. acknowledges support via DIP, GIF, andthrough SFB 631, SFB/TR 12, the Nanosystems Initiative Munich, and the Emmy-Noether program. S.M.G.
acknowledges the support of the Centre for Advanced
Study at the Norwegian Academy of Science and Letterswhere part of this work was carried out.
REFERENCES
Aassime, A., G. Johansson, G. Wendin, R. J. Schoelkopf, and
P. Delsing, 2001, Phys. Rev. Lett. 86, 3376.
Abramovici, A., 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, and A. Hedimann, 2006, Nature 共London兲 444, 71.
Armen, M. A., J. K. Au, J. K. Stockton, A. C. Doherty, and H.
Mabuchi, 2002, Phys. Rev. Lett. 89, 133602.
Averin, D. V., 2000a, e-print arXiv:cond-mat/0010052.
Averin, D. V., 2000b, Fortschr. Phys. 48, 1055.
Averin, D., 2003, in Quantum Noise in Mesoscopic Systems,
edited by Y. Nazarov 共Kluwer, Amsterdam兲, pp. 205–228.
Bergeal, N., R. Vijay, V. E. Manucharyan, I. Siddiqi, R. J.
Schoelkopf, S. M. Girvin, and M. H. Devoret, 2008, e-print
arXiv:0805.3452, Nature 共to be published兲.
Bernád, J. Z., A. Bodor, T. Geszti, and L. Diósi, 2008, Phys.
Rev. B 77, 073311.
Blais, A., R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J.
Schoelkopf, 2004, Phys. Rev. A 69, 062320.
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 共Dover, New York兲.

1206

Clerk et al.: Introduction to quantum noise, measurement, …

Boylestad, R., and L. Nashelsky, 2006, Electronic Devices andCircuit Theory, 9th ed. 共Prentice Hall, Engelwood Cliffs, NJ兲.
Braginsky, V. B., and F. Y. Khalili, 1992, Quantum Measurement 共Cambridge University Press, Cambridge兲.
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.
Brun, T. A., 2002, Am. J. Phys. 70, 719.
Buks, E., R. Schuster, M. Heiblum, D. Mahalu, and V. Umansky, 1998, Nature 共London兲 391, 871.
Callen, H. B., and T. A. Welton, 1951, Phys. Rev. 83, 34.
Castellanos-Beltran, M. A., K. D. Irwin, G. C. Hilton, L. R.
Vale, and K. W. Lehnert, 2008, Nat. Phys. 4, 929.
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. Gossard, 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., M. H. Devoret, S. M. Girvin, F. Marquardt, and
R. J. Schoelkopf, 2009, see supplementary material at http://
link.aps.org/supplemental/10.1103/RevModPhys.82.1155 foradditional pedagogical appendices which complement themain text of this article.
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.
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.
Danilov, V. V., K. K. Likharev, and A. B. Zorin, 1983, IEEE
Trans. Magn. 19, 572.
Devoret, M. H., and R. J. Schoelkopf, 2000, Nature 共London兲
406, 1039.
Doherty, A. C., S. Habib, K. Jacobs, H. Mabuchi, and S. M.
Tan, 2000, Phys. Rev. A 62, 012105.
Doiron, C. B., B. Trauzettel, and C. Bruder, 2007, Phys. Rev. B
76, 195312.
Duty, T., G. Johansson, K. Bladh, D. Gunnarsson, C. Wilson,
and P. Delsing, 2005, Phys. Rev. Lett. 95, 206807.
Dykman, M. I., 1978, Sov. Phys. Solid State 20, 1306.
Etaki, S., M. Poot, I. Mahboob, K. Onomitsu, H. Yamaguchi,
and H. S. J. V. der Zant, 2008, Nat. Phys. 4, 785.
Falferi, P., M. Bonaldi, M. Cerdonio, R. Mezzena, G. A. Prodi,
A. Vinante, and S. Vitale, 2008, Appl. Phys. Lett. 93, 172506.
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, and R. J.
Schoelkopf, 2007, Phys. Rev. A 76, 012325.
Gardiner, C. W., and P. Zoller, 2000, Quantum Noise 共Springer,
Berlin兲.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Gavish, U., Y. Levinson, and Y. Imry, 2000, Phys. Rev. B 62,
共R兲 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.
Geremia, J., J. K. Stockton, and H. Mabuchi, 2004, Science
304, 270.
Gerry, C. S., 1985, Phys. Rev. A 31, 2721.
Gigan, S., H. Böhm, M. Paternostro, F. Blaser, J. B. Hertzberg,
K. C. Schwab, D. Bauerle, M. Aspelmeyer, and A. Zeilinger,
2006, Nature 共London兲 444, 67.
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: Fundamentals 共Benjamin, Reading, MA兲.
Grassia, F., 1998, Ph.D. thesis 共Universite Pierre et Marie Curie兲.
Groblacher, S., J. B. Hertzberg, M. R. Vanner, G. D. Cole, S.
Gigan, K. C. Schwab, and M. Aspelmeyer, 2009, Nat. Phys. 5,
485.
Gurvitz, S. A., 1997, Phys. Rev. B 56, 15215.
Haroche, S., and J.-M. Raimond, 2006, Exploring the Quantum
Atoms, Cavities, and Photons 共Oxford University Press, Oxford兲.
Haus, H. A., 2000, Electromagnetic Noise and Quantum Optical Measurements 共Springer, New York兲.
Haus, H. A., and J. A. Mullen, 1962, Phys. Rev. 128, 2407.
Heffner, H., 1962, Proc. IRE 50, 1604.
Hertzberg, J. B., T. Rocheleau, T. Ndukum, M. Savva, A. A.
Clerk, and K. C. Schwab, 2010, Nat. Phys. 6, 213.
Il’ichev, E., N. Oukhanski, A. Izmalkov, T. Wagner, M. Grajcar, H.-G. Meyer, A. Y. Smirnov, A. Maassen van den Brink,
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, Contemp. Phys. 47, 279.
Jayich, A. M., J. C. Sankey, B. M. Zwickl, C. Yang, J. D.
Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and J. G.
E. Harris, 2008, New J. Phys. 10, 095008.
Johansson, G., P. Delsing, K. Bladh, D. Gunnarsson, T. Duty,
A. Kack, G. Wendin, and A. Aassime, 2003, in Quantum
Noise in Mesoscopic Physics, edited by Y. V. Nazarov 共Kluwer, Dordrecht兲, pp. 337–356.
Johansson, G., A. Käck, and G. Wendin, 2002, Phys. Rev. Lett.
88, 046802.
Johansson, G., L. Tornberg, and C. M. Wilson, 2006, Phys. Rev.
B 74, 100504共R兲.
Jordan, A. N., and M. Büttiker, 2005a, Phys. Rev. Lett. 95,
220401.
Jordan, A. N., and M. Büttiker, 2005b, 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 共London兲
444, 75.
Knight, P. L., and V. Buzek, 2004, in Quantum Squeezing, edited by P. D. Drummond, and Z. Ficek 共Springer, Berlin兲, pp.

Clerk et al.: Introduction to quantum noise, measurement, …

3–32.
Knobel, R. G., and A. N. Cleland, 2003, Nature 共London兲 424,
291.
Koch, R. H., D. J. van Harlingen, and J. Clarke, 1981, 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.
Levinson, Y., 1997, Europhys. Lett. 39, 299.
Louisell, W. H., A. Yariv, and A. E. Siegman, 1961, Phys. Rev.
124, 1646.
Lupaşcu, A., E. F. C. Driessen, L. Roschier, C. J. P. M. Harmans, and J. E. Mooij, 2006, Phys. Rev. Lett. 96, 127003.
Lupaşcu, A., C. J. P. M. Harmans, and J. E. Mooij, 2005, Phys.
Rev. B 71, 184506.
Lupaşcu, A., S. Saito, T. Picot, P. C. de Groot, C. J. P. M.
Harmans, and J. E. Mooij, 2007, Nat. Phys. 3, 119.
Lupaşcu, A., C. J. M. Verwijs, R. N. Schouten, C. J. P. M.
Harmans, and J. E. Mooij, 2004, Phys. Rev. Lett. 93, 177006.
Makhlin, Y., G. Schön, and A. Shnirman, 2000, Phys. Rev. Lett.
85, 4578.
Makhlin, Y., G. Schon, and A. Shnirman, 2001, Rev. Mod.
Phys. 73, 357.
Marquardt, F., and S. M. Girvin, 2009, Phys. 2, 40.
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.
Mück, 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 共London兲
443, 193.
Nogues, G., A. Rauschenbeutel, S. Osnaghi, M. Brune, J. M.
Raimond, and S. Haroche, 1999, Nature 共London兲 400, 239.
Oxtoby, N. P., J. Gambetta, and H. M. Wiseman, 2008, Phys.
Rev. B 77, 125304.
Oxtoby, N. P., H. M. Wiseman, and H.-B. Sun, 2006, Phys. Rev.
B 74, 045328.
Pathria, R. K., 1996, Statistical Mechanics, 2nd ed. 共Elsevier,
Amsterdam兲.
Peres, A., 1993, Quantum Theory: Concepts and Methods 共Kluwer, Dordrecht兲.
Pilgram, S., and M. Büttiker, 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, Nat.
Phys. 4, 635.
Regal, C. A., J. D. Teufel, and K. W. Lehnert, 2008, Nat. Phys.
4, 555.
Rugar, D., R. Budakian, H. J. Mamin, and B. W. Chui, 2004,
Nature 共London兲 430, 329.
Ruskov, R., and A. N. Korotkov, 2002, Phys. Rev. B 66,
041401.
Ruskov, R., K. Schwab, and A. N. Korotkov, 2005, Phys. Rev.
B 71, 235407.
Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

1207

Santamore, D. H., A. C. Doherty, and M. C. Cross, 2004, Phys.
Rev. B 70, 144301.
Santamore, D. H., H.-S. Goan, G. J. Milburn, and M. L.
Roukes, 2004, Phys. Rev. A 70, 052105.
Sauter, T., W. Neuhauser, R. Blatt, and P. E. Toschek, 1986,
Phys. Rev. Lett. 57, 1696.
Schliesser, A., O. Arcizet, R. Riviere, G. Anetsberger, and T. J.
Kippenberg, 2009, Nat. Phys. 5, 509.
Schliesser, A., P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J.
Kippenberg, 2006, Phys. Rev. Lett. 97, 243905.
Schliesser, A., R. Rivière, G. Anetsberger, O. Arcizet, and T. J.
Kippenberg, 2008, Nat. Phys. 4, 415.
Schoelkopf, R. J., A. A. Clerk, S. M. Girvin, K. W. Lehnert,
and M. H. Devoret, 2003, in Quantum Noise in Mesoscopic
Physics, edited by Y. V. Nazarov 共Kluwer, Dordrecht兲, pp.
175–203.
Schoelkopf, R. J., P. Wahlgren, A. A. Kozhevnikov, P. Delsing,and D. E. Prober, 1998, Science 280, 1238.
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.
Schwinger, J., 1961, J. Math. Phys. 2, 407.
Shnirman, A., and G. Schön, 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.
Sillanpää, 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., T. Donner, M. A. Castellanos-Beltran, J. W. Harlow, and K. W. Lehnert, 2009, e-print arXiv:0906.1212.
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 共London兲 452,
72.
Thorne, K. S., R. W. P. Drever, C. M. Caves, M. Zimmermann,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.
van den Brink, A. M., 2002, Europhys. Lett. 58, 562.
Vinante, A., R. Mezzena, G. A. Prodi, S. Vitale, M. Cerdonio,
P. Falferi, and M. Bonaldi, 2001, Appl. Phys. Lett. 79, 2597.
von Neumann, J., 1932, Mathematische Grundlagen der Quantenmechanik 共Springer, Berlin兲.
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 共London兲 431, 162.
Walls, D. F., and G. J. Milburn, 1994, Quantum Optics
共Springer, Berlin兲.
Weiss, U., 1999, Quantum Dissipative Systems, 2nd ed. 共World
Scientific, Singapore兲.
Wellstood, F. C., C. Urbina, and J. Clarke, 1994, Phys. Rev. B

1208

Clerk et al.: Introduction to quantum noise, measurement, …

49, 5942.
Wheeler, J. A., and W. H. Zurek, 1984, Quantum Theory andMeasurement 共Princeton University Press, Princeton兲.
Wigner, E. P., 1955, Phys. Rev. 98, 145.
Wineland, D. J., C. Monroe, W. M. Itano, D. Leibfried, B.
King, and D. M. Meekhof, 1998, J. Res. Natl. Inst. Stand.
Technol. 103, 259.

Rev. Mod. Phys., Vol. 82, No. 2, April–June 2010

Wiseman, H. M., and G. J. Milburn, 1993, Phys. Rev. Lett. 70,
548.
Wiseman, H. M., and G. J. Milburn, 1994, Phys. Rev. A 49,
1350.
Yurke, B., L. R. Corruccini, P. G. Kaminsky, L. W. Rupp, A. D.
Smith, A. H. Silver, R. W. Simon, and E. A. Whittaker, 1989,
Phys. Rev. A 39, 2519.

Introduction to Quantum Noise, Measurement and Amplification: Online
AppendicesA.A. Clerk,1 M.H. Devoret,2 S.M. Girvin,3 Florian Marquardt,4 and R.J. Schoelkopf2
1

Department of Physics, McGill University, 3600 rue University
Montréal, QC Canada H3A 2T8∗
2
Department of Applied Physics, Yale University
PO Box 208284, New Haven, CT 06520-8284
3
Department of Physics, Yale University
PO Box 208120, New Haven, CT 06520-8120
4Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München
Theresienstr. 37, D-80333 München, Germany(Dated: July 2, 2009)

Contents

2. Explicit examples
3. Op-amp with negative voltage feedback

A. Basics of Classical and Quantum Noise
1. Classical noise correlators
2. The Wiener-Khinchin Theorem
3. Square law detectors and classical spectrum analyzers

1
1
2
3

B. Quantum Spectrum Analyzers: Further Details
1. Two-level system as a spectrum analyzer
2. Harmonic oscillator as a spectrum analyzer
3. Practical quantum spectrum analyzers
a. Filter plus diodeb. Filter plus photomultiplierc. Double sideband heterodyne power spectrum

4
4
5
6
6
7
7

C. Modes, Transmission Lines and ClassicalInput/Output Theory
1. Transmission lines and classical input-output theory
2. Lagrangian, Hamiltonian, and wave modes for atransmission line
3. Classical statistical mechanics of a transmission line
4. Amplification with a transmission line and a negative
resistance13

D. Quantum Modes and Noise of a Transmission Line
1. Quantization of a transmission line
2. Modes and the windowed Fourier transform
3. Quantum noise from a resistor

15
15
16
17

References

40

Appendix A: Basics of Classical and Quantum Noise

1011

1. Classical noise correlators

24
28

F. Information Theory and Measurement Rate
1. Method I
2. Method II

29
29
30

G. Number Phase Uncertainty

30

H. Using feedback to reach the quantum limit
1. Feedback using mirrors

31
31∗ Electronic address: clerk@physics.mcgill.ca

I. Additional Technical Details
34
1. Proof of quantum noise constraint
34
2. Proof that a noiseless detector does not amplify
36
3. Simplifications for a quantum-limited detector
36
4. Derivation of non-equilibrium Langevin equation
37
5. Linear-response formulas for a two-port bosonic amplifier37
a. Input and output impedances
38
b. Voltage gain and reverse current gain38
6. Details for the two-port bosonic voltage amplifier withfeedback
39

8
8

E. Back Action and Input-Output Theory for Driven
Damped Cavities
1. Photon shot noise inside a cavity and back action
2. Input-output theory for a driven cavity
3. Quantum limited position measurement using a cavity
detector
4. Back-action free single-quadrature detection

18
19
20

32
33

Consider a classical random voltage signal V (t). The
signal is characterized by zero mean hV (t)i = 0, andautocorrelation function
GV V (t − t0 ) = hV (t)V (t0 )i(A1)

whose sign and magnitude tells us whether the voltage
fluctuations at time t and time t0 are correlated, anticorrelated or statistically independent. We assume thatthe noise process is stationary (i.e., the statistical properties are time translation invariant) so that GV V depends
only on the time difference. If V (t) is Gaussian distributed, then the mean and autocorrelation completely
specify the statistical properties and the probability distribution. We will assume here that the noise is due tothe sum of a very large number of fluctuating charges
so that by the central limit theorem, it is Gaussian distributed. We also assume that GV V decays (sufficiently
rapidly) to zero on some characteristic correlation timescale τc which is finite.
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 understand2
the spectral density of a random signal, it is useful todefine its ‘windowed’ Fourier transform as follows:
Z +T /2
1
dt eiωt V (t),(A2)
VT [ω] = √T −T /2
where T is the sampling time. In the limit T  τc theintegral is a sum of a large number N ≈ τTc of random
uncorrelated terms. We can think of the value of theintegral as the end point of a random walk in the complex
plane which starts at the√ origin. Because the distancetraveled will scale with T , our choice of normalization
makes the statistical properties of V [ω] independent of
the 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] ≡ lim h|VT [ω]|2 i = lim hVT [ω]VT [−ω]i (A3)
T →∞

T →∞

The second equality follows from the fact that V (t) isreal valued. The Wiener-Khinchin theorem (derived inAppendix A.2) tells us that the spectral density is equal
to the Fourier transform of the autocorrelation function
Z +∞SV V [ω] =dt eiωt GV V (t).
(A4)

with a large heat bath at temperature T via some infinitesimal coupling which we will ignore in considering
the dynamics. The solution of Hamilton’s equations of
motion are
1sin(Ωt)MΩ
p(t) = p(0) cos(Ωt) − x(0)M Ω sin(Ωt),x(t) = x(0) cos(Ωt) + p(0)where x(0) and p(0) are the (random) values of the position and momentum at time t = 0. It follows that theposition autocorrelation function isGxx (t) = hx(t)x(0)i

We thus see that a short auto-correlation time implies
a spectral density which is non-zero over a wide range of
frequencies. In the limit of ‘white noise’
GV V (t) = σ 2 δ(t)SV V [ω] = σ 2

Gxx (t) =SV V [ω] = SV V [−ω].

kB Tcos(Ωt)M Ω2Sxx [ω] = πkB T[δ(ω − Ω) + δ(ω + Ω)]M Ω2

As a prototypical example of these ideas, let us consider a simple harmonic oscillator of mass M and frequency Ω. The oscillator is maintained in equilibrium(A12)

which is indeed symmetric in frequency.

2. The Wiener-Khinchin Theorem

From the definition of the spectral density in Eqs.(A2A3) we have
SV V [ω] =
1
T

Z T

1
T

Z T

0

0

Z +2B(t)
dt

Z T
dt

0

dt0 eiω(t−t ) hV (t)V (t0 )i0dτ eiωτ hV (t + τ /2)V (t − τ /2)i−2B(t)

(A13)

(A7)

(A8)

(A11)

which leads to the spectral density=

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 naturally follows that GV V (t) is always real. Since V (t) is nota quantum operator, it commutes with its value at othertimes and thus, hV (t)V (t0 )i = hV (t0 )V (t)i. From this itfollows that GV V (t) is always symmetric in time and thepower spectrum is always symmetric in frequency

1
sin(Ωt).
MΩ

Classically in equilibrium there are no correlations between position and momentum. Hence the second term
vanishes. Using the equipartition theorem 12 M Ω2 hx2 i =1
2 kB T , we arrive at(A6)

the spectrum is flat (independent of frequency)(A10)= hx(0)x(0)i cos(Ωt) + hp(0)x(0)i−∞

The inverse transform relates the autocorrelation function to the power spectrum
Z +∞dω −iωt
GV V (t) =e
SV V [ω].
(A5)
−∞ 2π

(A9)

where
B (t) = t if t < T /2
= T − t if t > T /2.
If T greatly exceeds the noise autocorrelation time τcthen it is a good approximation to extend the bound B(t)in the second integral to infinity, since the dominant contribution is from small τ . Using time translation invariance gives
Z
Z +∞
1 T
SV V [ω] =dtdτ eiωτ hV (τ )V (0)i
T 0
−∞
Z +∞
dτ eiωτ hV (τ )V (0)i .
(A14)
=
−∞

3
This proves the Wiener-Khinchin theorem stated in
Eq. (A4).
A useful application of these ideas is the following.
Suppose that we have a noisy signal V (t) = V̄ + η(t)which we begin monitoring at time t = 0. The integrated
signal up to time t is given by
Z T
I(T ) =
dt V (t)(A15)
0and has meanhI(T )i = V̄ T.

(A16)

Provided that the integration time greatly exceeds the
autocorrelation time of the noise, I(T ) is a sum of a large
number of uncorrelated random variables. The central
limit theorem tells us in this case that I(t) is gaussian
distributed even if the signal itself is not. Hence theprobability distribution for I is fully specified by its mean
and its variance
Z T2h(∆I) i =dtdt0 hη(t)η(t0 )i.
(A17)
0From the definition of spectral density above we have thesimple result that the variance of the integrated signal
grows linearly in time with proportionality constant givenby the noise spectral density at zero frequencyh(∆I)2 i = SV V [0] T.

(A18)

As a simple application, consider the photon shot noiseof a coherent laser beam. The total number of photons
detected in time T is
Z T
N (T ) =dt Ṅ (t).
(A19)
0

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

3. Square law detectors and classical spectrum analyzers

Now that we understand the basics of classical noise,
we can consider how one experimentally measures a classical noise spectral density. With modern high speed
digital sampling techniques it is perfectly feasible to directly measure the random noise signal as a function oftime and then directly compute the autocorrelation function in Eq. (A1). This is typically done by first performing an analog-to-digital conversion of the noise signal, and then numerically computing the autocorrelation
function. One can then use Eq. (A4) to calculate thenoise spectral density via a numerical Fourier transform.
Note that while Eq. (A4) seems to require an ensemble
average, in practice this is not explicitly done. Instead,one uses a sufficiently long averaging time T (i.e. much
longer than the correlation time of the noise) such thata single time-average is equivalent to an ensemble average. This approach of measuring a noise spectral density
directly from its autocorrelation function is most appropriate for signals at RF frequencies well below 1 MHz.
For microwave signals with frequencies well above 1
GHz, a very different approach is usually taken. Here, thestandard route to obtain a noise spectral density involves
first shifting the signal to a lower intermediate frequency
via a technique known as heterodyning (we discuss thismore in Sec. B.3.c). This intermediate-frequency signal
is then sent to a filter which selects a narrow frequency
range 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 ‘video
bandwidth’). It is this final output which is then taken
to 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]

(A24)

(A21)

be the voltage at the output of the filter and the inputof the square law detector. Here, f [ω] is the (amplitude) transmission coefficient of the filter and V [ω] is theFourier transform of the noisy signal we are measuring.
From Eq. (A5) it follows that the output of the squarelaw detector is proportional to
Z +∞
dω
hIi =
|f [ω]|2 SV V [ω].
(A25)
−∞ 2π

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

Approximating the narrow band filter centered on frequency ±ω0 as1h(∆N )2 i = hN (T )i,|f [ω]|2 = δ(ω − ω0 ) + δ(ω + ω0 )hN (T )i = Ṅ T

(A20)

will be large and the photon number distribution will begaussian with varianceh(∆N )2 i = SṄ Ṅ T.

(A22)

(A26)

it follows that the shot noise power spectral density is
SṄ Ṅ (0) = Ṅ .

(A23)

Since the noise is white this result happens to be valid at
all frequencies, but the noise is gaussian distributed only
at low frequencies.

1 A

linear passive filter performs a convolution Vout (t) =R +∞ 000−∞ 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 transform of F , obeys f [−ω] = f ∗ [ω] and hence |f [ω]|2 is symmetricin frequency.

4
we obtain
hIi = SV V (−ω0 ) + SV V (ω0 )(A27)showing as expected that the classical square law detector
measures the symmetrized noise power.
We thus have two very different basic approaches forthe measurement of classical noise spectral densities: for
low RF frequencies, one can directly measure the noise
autocorrelation, whereas for high microwave frequencies,
one uses a filter and a square law detector. For noise
signals in intermediate frequency ranges, a combinationof different methods is generally used. The whole story
becomes even more complicated, as at very high frequencies (e.g. in the far infrared), devices such as the socalled ‘Fourier Transform spectrometer’ are in fact basedon a direct measurement of the equivalent of an autocorrelation function of the signal. In the infrared, visible
and ultraviolet, noise spectrometers use gratings followed
by a slit acting as a filter.
Appendix B: Quantum Spectrum Analyzers: Further Details
1. Two-level system as a spectrum analyzer

In this sub-appendix, we derive the Golden Rule transition rates Eqs. (2.6) describing a quantum two-level system coupled to a noise source (cf. Sec. II.B). Our derivation is somewhat unusual, in that the role of the continuum as a noise source is emphasized from the outset. We
start by treating the noise F (t) in Eq. (2.5) as being aclassically noisy variable. We assume that the coupling
A is under our control and can be made small enoughthat the noise can be treated in lowest order perturbation theory. We take the state of the two-level system tobeαg (t)|ψ(t)i =.
(B1)αe (t)In the interaction representation, first-order timedependent perturbation theory givesZi t|ψI (t)i = |ψ(0)i −dτ V̂ (τ )|ψ(0)i.
(B2)~ 0
If we initially prepare the two-level system in its groundstate, the amplitude to find it in its excited state at time
t is from Eq. (B2)ZiA tαe = −dτ he|σ̂x (τ )|giF (τ ),
~ 0
Z t
iA
= −
dτ eiω01 τ F (τ ).
(B3)~ 0
Since the integrand in Eq. (B3) 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 discussed above in defining the spectral density of classicalnoise). As a result, for times exceeding the autocorrelation 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 compute the probability
Z Z
A2 t tdτ1 dτ2 e−iω01 (τ1 −τ2 ) F (τ1 )F (τ2 )pe (t) ≡ |αe |2 = 2
~ 0 0
(B4)
which we expect to grow quadratically for short times
t < τc , but linearly for long times t > τc . Ensemble
averaging the probability over the random noise yields
Z Z
A2 t tp̄e (t) = 2dτ1 dτ2 e−iω01 (τ1 −τ2 ) hF (τ1 )F (τ2 )i
~ 0 0
(B5)
Introducing the noise spectral density
Z +∞dτ eiωτ hF (τ )F (0)i,(B6)
SF F (ω) =−∞and utilizing the Fourier transform defined in Eq. (A2)and the Wiener-Khinchin theorem from Appendix A.2,
we find that the probability to be in the excited stateindeed increases linearly with time at long times,2p̄e (t) = t

A2
SF F (−ω01 )
~2

(B7)

The time derivative of the probability gives the transition
rate from ground to excited statesΓ↑ =A2
SF F (−ω01 )
~2

(B8)

Note that we are taking in this last expression the spectral density on the negative frequency side. If F were astrictly classical noise source, hF (τ )F (0)i would be real,and SF F (−ω01 ) = SF F (+ω01 ). However, because as we
discuss below F is actually an operator
actingion the enh
vironmental degrees of freedom, F̂ (τ ), F̂ (0) 6= 0 and
SF F (−ω01 ) 6= SF F (+ω01 ).
Another possible experiment is to prepare the two-levelsystem in its excited state and look at the rate of decay
into the ground state. The algebra is identical to thatabove except that the sign of the frequency is reversed:
Γ↓ =A2
SF F (+ω01 ).
~2

(B9)

We now see that our two-level system does indeed act as aquantum spectrum analyzer for the noise. Operationally,

2 Note that for very long times, where there is a significant de-pletion of the probability of being in the initial state, first-order
perturbation theory becomes invalid. However, for sufficiently
small A, there is a wide range of times τc  t  1/Γ for which
Eq. B7 is valid. Eqs. (2.6a) and (2.6b) then yield well-defined
rates which can be used in a master equation to describe the fulldynamics including long times.

5
we prepare the system either in its ground state or in its
excited state, weakly couple it to the noise source, andafter an appropriate interval of time (satisfying the above
inequalities) simply measure whether the system is now
in its excited state or ground state. Repeating this protocol over and over again, we can find the probabilityof making a transition, and thereby infer the rate andhence the noise spectral density at positive and negative frequencies. Naively one imagines that a spectrometers measures the noise spectrum by extracting a small
amount of the signal energy from the noise source andanalyzes it. This is not the case however. There must
be energy flowing in both directions if the noise is to be
fully characterized.
We now rigorously treat the quantity F̂ (τ ) as a quantum Heisenberg operator which acts in the Hilbert spaceof the noise source. The previous derivation is unchanged(the ordering of F̂ (τ1 )F̂ (τ2 ) having been chosen correctly in anticipation of the quantum treatment), andEqs. (2.6a,2.6b) are still valid provided that we interpretthe angular brackets in Eq. (B5,B6) as representing aquantum expectation value (evaluated in the absence ofthe coupling to the spectrometer):Z +∞dτ eiωτSF F (ω) =−∞X

ραα hα|F̂ (τ )|γihγ|F̂ (0)|αi.

α,γ

(B10)
Here, we have assumed a stationary situation, where
the density matrix ρ of the noise source is diagonal in
the energy eigenbasis (in the absence of the coupling to
the spectrometer). However, we do not necessarily assume that it is given by the equilibrium expression. This
yields the standard quantum mechanical expression forthe spectral density:
Z +∞SF F (ω) =dτ eiωτ

−∞

= 2π~

X

X

i

ραα e ~ (α −γ )τ |hα|F̂ |γi|2α,γραα |hα|F̂ |γi|2 δ(γ − α − ~ω).(B11)α,γSubstituting this expression into Eqs. (2.6a,2.6b), we derive 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 continuum
as described by Fermi’s Golden Rule can (and indeed
should!) be viewed as resulting from the continuum acting as a quantum noise source which causes the amplitudes of the different components of the wave function to undergo random walks. The derivation presented
here hopefully provides a motivation for this interpretation. In particular, thinking of the perturbation (i.e. thecoupling to the continuum) as quantum noise with asmall but finite autocorrelation time (inversely related
to the bandwidth of the continuum) neatly explains why
the transition probability increases quadratically for very
short times, but linearly for very long times.

It it is important to keep in mind that our expressions
for the transition rates are only valid if the autocorrelation time of our noise is much shorter that the typicaltime we are interested in; this typical time is simply the
inverse of the transition rate. The requirement of a short
autocorrelation time in turn implies that our noise sourcemust have a large bandwidth (i.e. there must be large
number 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 at
the transition frequency (a consequence of energy conservation).
One standard model for the continuum is an infinite
collection of harmonic oscillators. The electromagnetic
continuum in the hydrogen atom case mentioned above is
a prototypical example. The vacuum electric field noise
coupling to the hydrogen atom has an extremely short
autocorrelation time because the range of mode frequencies ωα (over which the dipole matrix element coupling~ α is significant) isthe atom to the mode electric field E
extremely large, ranging from many times smaller than
the transition frequency to many times larger. Thus, theautocorrelation time of the vacuum electric field noise isconsiderably less than 10−15 s, whereas the decay time ofthe hydrogen 2p state is about 10−9 s. Hence the inequalities needed for the validity of our expressions are very
easily satisfied.

2. Harmonic oscillator as a spectrum analyzer

We now provide more details on the system describedin Sec. II.B, where a harmonic oscillator acts as a spectrometer of quantum noise. We start with the coupling
Hamiltonian givein in Eq. (2.9). In analogy to the TLS
spectrometer, noise in F̂ at the oscillator frequency Ω
can cause transitions between its eigenstates. We assume both that A is small, and that our noise source
has a short autocorrelation time, so we may again use
perturbation theory to derive rates for these transitions.
There is a rate for increasing the number of quanta inthe oscillator by one, taking a state |ni to |n + 1i:A2 (n + 1)x2ZPF SF F [−Ω] ≡ (n + 1)Γ↑
2
~
(B12)
As expected, this rate involves the noise at −Ω, as energy is being absorbed from the noise source. Similarly,
there is a rate for decreasing the number of quanta in theoscillator by one:
Γn→n+1 =Γn→n−1 =
A2
nx2ZPF SF F [Ω] ≡ nΓ↓
2
~

(B13)

This rate involves the noise at +Ω, as energy is being
emitted to the noise source.
Given these transition rates, we may immediately write
a simple master equation for the probability pn (t) that6
there are n quanta in the oscillator:dpn = [nΓ↑ pn−1 + (n + 1)Γ↓ pn+1 ]dt− [nΓ↓ + (n + 1)Γ↑ ] pn

(B14)

The first two terms describe transitions into the state |nifrom the states |n + 1i and |n − 1i, and hence increase
pn . In contrast, the last two terms describe transitions
out of the state |ni to the states |n + 1i and |n − 1i, andhence decrease pn . The stationary state of the oscillatordpn = 0, yielding:is given by solving Eq. (B14) for dt


(B15)
pn = e−n~Ω/(kB Teff ) 1 − e−~Ω/(kB Teff )where the effective temperature Teff [Ω] is defined in
Eq. (2.8). Eq. (B15) describes a thermal equilibrium
distribution of the oscillator, with an effective oscillator
temperature Teff [Ω] determined by the quantum noise
spectrum of F̂ . This is the same effective temperaturethat emerged in our discussion of the TLS spectrum analyzer. As we have seen, if the noise source is in thermal
equilibrium at a temperature Teq , then Teff [Ω] = Teq .
In the more general case where the noise source is notin thermal equilibrium, Teff only serves to characterize
the asymmetry of the quantum noise, and will vary withfrequency 3 .
We can learn more about the quantum noise spectrumof F̂ by also looking at the dynamics of the oscillator.
In particular, as the average
P∞energy hEi of the oscillator
is just given by hE(t)i = n=0 ~Ω n + 21 pn (t), we canuse the master equation Eq. (B14) to derive an equation for its time dependence. One thus finds Eq. (2.10).
By demanding dhEi/dt = 0 in this equation, we findthat the combination of damping and heating effects
causes the energy to reach a steady state mean valueof hEi = P/γ. This implies that the finite ground stateenergy hEi = ~Ω/2 of the oscillator is determined via thebalance between the ‘heating’ by the zero-point fluctuations of the environment (described by the symmetrized
correlator at T = 0) and the dissipation. It is possible totake an alternative but equally correct viewpoint, where
only the deviation hδEi = hEi − ~Ω/2 from the groundstate energy is considered. Its evolution equationdhδEi = hδEi(Γ↑ − Γ↓ ) + Γ↑ ~Ω
dt

(B16)

only contains a decay term at T = 0, leading to hδEi → 0.
3. Practical quantum spectrum analyzers

As we have seen, a ‘quantum spectrum analyzer’ canin principle be constructed from a two level system (or3 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.

a harmonic oscillator) in which we can separately measure the up and down transition rates between states
differing by some precise energy ~ω > 0 given by thefrequency of interest. The down transition rate tells us
the noise spectral density at frequency +ω and the up
transition rate tells us the noise spectral density at −ω.
While we have already discussed experimental implementation of these ideas using two-level systems and oscillators, similar schemes have been implemented in other systems. A number of recent experiments have made use of
superconductor-insulator-superconductor junctions (Billangeon et al., 2006; Deblock et al., 2003; Onac et al.,2006) to measure quantum noise, as the current-voltage
characteristics of such junctions are very sensitive tothe absorption or emission of energy (so-called photonassisted transport processes). It has also been suggested
that tunneling of flux in a SQUID can be used to measurequantum noise (Amin and Averin, 2008).
In this subsection, we discuss additional methods forthe detection of quantum noise. Recall from Sec. A.3 that
one of the most basic classical noise spectrum analyzers
consists of a linear narrow band filter and a square law
detector such as a diode. In what follows, we will consider
a simplified quantum treatment of such a device where we
do not explicitly model a diode, but instead focus on theenergy of the filter circuit. We then turn to various noise
detection schemes making use of a photomultiplier. We
will show that depending on the detection scheme used,
one can measure either the symmetrized quantum noise
spectral density S̄[ω], or the non-symmetrized spectral
density S[ω].

a. Filter plus diode

Using the simple treatment we gave of a harmonic oscillator as a quantum spectrum analyzer in Sec. B.2, one
can attempt to provide a quantum treatment of the classical ‘filter plus diode’ spectrum analyzer discussed inSec. A.3. This approach is due to Lesovik and Loosen(1997) and Gavish et al. (2000). The analysis starts bymodeling the spectrum analyzer’s resonant filter circuit
as a harmonic oscillator of frequency Ω weakly coupled
to some equilibrium dissipative bath. The oscillator thus
has an intrinsic damping rate γ0  Ω, and is initially ata finite temperature Teq . One then drives this damped
oscillator (i.e. the filter circuit) with the noisy quantum
force F̂ (t) whose spectrum at frequency Ω is to be measured.
In the classical ‘filter plus diode’ spectrum analyzer,
the output of the filter circuit was sent to a square law
detector, whose time-averaged output was then taken as
the measured spectral density. To simplify the analysis, 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

7
hEi to increase an amount directly proportional to theclassical spectrum SF F [Ω]. We now consider hEi in thecase of a quantum noise source, and ask how it relates tothe quantum noise spectral density SF F [Ω].
The quantum case is straightforward to analyze using
the approach of Sec. B.2. Unlike the classical case, thenoise will both lead to additional fluctuations of the filter
circuit and increase its damping rate by an amount γ(c.f. Eq. (2.12)). To make things quantitative, we let neq
denote the average number of quanta in the filter circuit
prior to coupling to F̂ (t), i.e.
1

neq =
exp



~Ω
kB Teq



,

(B17)

−1

and let neff represent the Bose-Einstein factor associatedwith the effective temperature Teff [Ω] of the noise sourceF̂ (t),
1

neff =exp~ΩkB Teff [Ω]



.

(B18)

−1

One then finds (Gavish et al., 2000; Lesovik and Loosen,1997):∆hEi = ~Ω ·γ(neff − neq )γ0 + γ

(B19)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 by
an amount γ. Re-expressing this result in terms of thesymmetric and anti-symmetric in frequency parts of thequantum noise spectral density SF F [Ω], we have:S̄F F (Ω) − neq + 12 (SF F [Ω] − SF F [−Ω])∆hEi =2m (γ0 + γ)
(B20)
We see that ∆hEi is in general not simply proportional
to the symmetrized noise S̄F F [Ω]. Thus, the ‘filter plus
diode’ spectrum analyzer does not simply measure thesymmetrized quantum noise spectral density. We stressthat there is nothing particularly quantum about this result. The extra term on the RHS of Eq. (B20) simply
reflects the fact that coupling the noise source to the 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. (B20) will be minimal, and our spectrum analyzer will (to a good approximation) measure
the symmetrized noise. Quantitatively, this requires:
neff  neq .

(B21)

We now see where quantum mechanics enters: if the noise
to 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. (B20)) will
always be greater than or equal to the expected heating
effect of the noise (i.e. first term in Eq. (B20)).
b. Filter plus photomultiplier

We now turn to quantum spectrum analyzers involving
a square law detector we can accurately model– a photomultiplier. 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
Z +∞
hIi =
dω |f [ω]|2 r[ω]SVV [ω],
(B22)
−∞

where f is the filter (amplitude) transmission function
defined previously and r[ω] is the response of the photodetector at frequency ω, and SVV represents the electric field spectral density incident upon the photodetector. Naively one thinks of the photomultiplier as a squarelaw detector with the square of the electric field representing the optical power. However, according to the
Glauber 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 photoelectron. Glauber describes this in terms of normal ordering of the photon operators in the electric field autocorrelation function. In our language of noise power at
positive and negative frequencies, this requirement becomes simply that r[ω] vanishes for ω > 0. Approximating the narrow band filter centered on frequency ±ω0 as
in Eq. (A26), we obtain
hIi = r[−ω0 ]SVV [−ω0 ](B23)which shows that this particular realization of a quantum
spectrometer only measures electric field spectral density
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. Of
course real photomultipliers suffer from imperfect quantum efficiencies and have non-zero dark current. Notethat we have assumed here that there are no additional
fluctuations associated with the filter circuit. Our result thus coincides with what we found in the previous
subsection for the ‘filter plus diode’ spectrum analyzer(c.f. Eq. (B20), in the limit where the filter circuit is initially at zero temperature (i.e. neq = 0).
c. Double sideband heterodyne power spectrum

At RF and microwave frequencies, practical spectrometers often contain heterodyne stages which mix the initial

8
frequency down to a lower frequency ωIF (possibly in theclassical regime). Consider a system with a mixer and local oscillator at frequency ωLO that mixes both the upper
sideband input at ωu = ωLO +ωIF and the lower sideband
input at ωl = ωLO − ωIF down to frequency ωIF . Thiscan be achieved by having a Hamiltonian with a 3-wave
mixing term which (in the rotating wave approximation)is given by
V = λ[âIF âl â†LO + â†IF â†l âLO ] + λ[â†IF âu â†LO + âIF â†u âLO ](B24)
The interpretation of this term is that of a Raman process. Notice that there are two energy conserving processes that can create an IF photon which could thenactivate the photodetector. First, one can absorb an LO
photon and emit two photons, one at the IF and one at
the lower sideband. The second possibility is to absorb
an upper sideband photon and create IF and LO photons.
Thus we expect from this that the power in the IF channel detected by a photomultiplier would be proportional
to the noise power in the following way
I ∝ S[+ωl ] + S[−ωu ](B25)

since creation of an IF photon involves the signal source
either absorbing a lower sideband photon from the mixer
or the signal source emitting an upper sideband photon
into the mixer. In the limit of small IF frequency thisexpression would reduce to the symmetrized noise power
I ∝ S[+ωLO ] + S[−ωLO ] = 2S̄[ωLO ](B26)

which is the same as for a ‘classical’ spectrum analyzer
with a square law detector (c.f. Appendix A.3). For equilibrium noise spectral density from a resistance R0 derived in Appendix D we would then have
SVV [ω] + SVV [−ω] = 2R0 ~|ω|[2nB (~|ω|) + 1],(B27)

Assuming our spectrum analyzer has high input
impedance so that it does not load the noise source, thisvoltage spectrum will determine the output signal of theanalyzer. This symmetrized quantity does not vanish atzero temperature and the output contains the vacuum
noise from the input. This vacuum noise has been seen
in experiment. (Schoelkopf et al., 1997)

XZPF = YZPF .

(C2)Thus the canonical commutation relation becomes
2[X̂, Ŷ ] = iXZPF.

(C3)

We will see that the fact that X and Y are canonically conjugate has profound implications both classicallyand quantum mechanically. In particular, the action of
any circuit element (beam splitter, attenuator, amplifier,etc.) must preserve the Poisson bracket (or in the quantum case, the commutator) between the signal quadratures. This places strong constraints on the propertiesof these circuit elements and in particular, forces every
amplifier to add noise to the signal.
1. Transmission lines and classical input-output theory

We begin by considering a coaxial transmission line
modeled as a perfectly conducting wire with inductance
per unit length of ` and capacitance to ground per unit
length c as shown in Fig. 1. If the voltage at position x
at 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∂t q + ∂x I = 0.

(C4)

The constitutive relation (essentially Newton’s law) gives
the acceleration of the charges`∂t I = −∂x V.

(C5)

We can decouple Eqs. (C4) and (C5) by introducing left
and right propagating modes

Appendix C: Modes, Transmission Lines and ClassicalInput/Output Theory

In this appendix we introduce a number of importantclassical concepts about electromagnetic signals which
are essential to understand before moving on to the studyof their quantum analogs. A signal at carrier frequency
ω 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).

We will see in the following that the physical oscillationsof this signal in a transmission line are precisely the sinusoidal oscillations of a simple harmonic oscillator. Comparison of Eq. (C1) with x(t) = x0 cos ωt+(p0 /M ω) sin ωtshows that we can identify the quadrature amplitude X
with the coordinate of this oscillator and thus the quadrature amplitude Y is proportional to the momentum conjugate to X. Quantum mechanically, X and Y become
operators X̂ and Ŷ which do not commute. Thus theirquantum fluctuations obey 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(C1)

V (x, t) = [V → + V ← ]1 →I(x, t) =[V − V ← ]
Zc

(C6)
(C7)

p
where Zc ≡ `/c is called the characteristic impedanceof the line. In terms of the left and right propagating
modes, Eqs. (C4) and C5 become
vp ∂x V → + ∂t V → = 0
vp ∂x V ← − ∂t V ← = 0

(C8)
(C9)

9
√
where vp ≡ 1/ `c is the wave phase velocity. Theseequations have solutions which propagate by uniform
translation without changing shape since the line is dispersionlessx
V → (x, t) = Vout (t − )
(C10)
vp
x
V ← (x, t) = Vin (t + ),(C11)
vp
where Vin and Vout are arbitrary functions of their arguments. For an infinite transmission line, Vout and Vin arecompletely independent. However for the case of a semiinfinite line terminated at x = 0 (say) by some system S,
these two solutions are not independent, but rather related by the boundary condition imposed by the system.
We have
V (x = 0, t) = [Vout (t) + Vin (t)]1
I(x = 0, t) =[Vout (t) − Vin (t)],Zc

indicating that the transmission line acts as a simple resistor which, instead of dissipating energy by Joule heating, carries the energy away from the system as propagating waves. The fact that the line can dissipate energy
despite containing only purely reactive elements is a consequence of its infinite extent. One must be careful withthe order of limits, taking the length to infinity before
allowing time to go to infinity. In this way the outgoingwaves never reach the far end of the transmission line andreflect back. Since this is a conservative Hamiltonian system, we will be able to quantize these waves and make aquantum theory of resistors (Caldeira and Leggett, 1983)
in Appendix D. The net power flow carried to the rightby the line isP =1 2[V (t) − Vin2 (t)].
Zc out

(C16)

(C12)
(C13)

from which we may derive
Vout (t) = Vin (t) + Zc I(x = 0, t).

(C14)

0

d

x

The fact that the transmission line presents a dissipative impedance to the system means that it causes damping of the system. It also however opens up the possibilityof controlling the system via the input field which partially determines the voltage driving the system. From
this point of view it is convenient to eliminate the output
field by writing the voltage as
V (x = 0, t) = 2Vin (t) + Zc I(x = 0, t).

(C17)V (x,t)a)

As we will discuss in more detail below, the first term
drives the system and the second damps it. From
Eq. (C14) we see that measurement of the outgoing field
can be used to determine the current I(x = 0, t) injectedby the system into the line and hence to infer the system
dynamics that results from the input drive field.
As a simple example, consider the system consisting ofan LC resonator shown in Fig. (1 c). This can be viewed
as a simple harmonic oscillator whose coordinate Q is the
charge on the capacitor plate (on the side connected toL0 ). The current I(x = 0, t) = Q̇ plays the role of thevelocity of the oscillator. The equation of motion for theoscillator is readily obtained fromI(x,t)
V(x,t)

Zc , vp
V (x,t)
L

b)

L
C

L
C

C

a

I
Vout

c)
V

Vin

˙ = 0+ , t)].
Q = C0 [−V (x = 0+ , t) − L0 I(xFIG. 1 a) Coaxial transmission line, indicating voltages andcurrents as defined in the main text. b) Lumped element
representation of a transmission line with capacitance per unit
length c = C/a and inductance per unit length ` = L/a. c)
Discrete LC resonator terminating a transmission line.

Using Eq. (C17) we obtain a harmonic oscillator dampedby the transmission line and driven by the incomingwaves
2
Vin (t),(C19)
L0√
where the resonant frequency is Ω20 ≡ 1/ L0 C0 . Notethat the term Zc I(x = 0, t) in Eq. (C17) results in thelinear viscous damping rate γ ≡ Zc /L0 .
If we solve the equation of motion of the oscillator, we
can 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 the
input field. We can characterize any such system by aQ̈ = −Ω20 Q − γ Q̇ −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 wave
reflecting from the open circuit termination. In general
however, there is an additional outgoing wave radiatedby the current I that is injected by the system dynamics
into the line. In the absence of an incoming wave we haveV (x = 0, t) = Zc I(x = 0, t),

(C15)

(C18)

10
complex, frequency dependent impedance Z[ω] definedbyZ[ω] = −V (x = 0, ω).
I(x = 0, ω)(C20)Note the peculiar minus sign which results from our definition of positive current flowing to the right (out of thesystem and into the transmission line). Using Eqs. (C12,
C13) and Eq. (C20) we have
Vout [ω] = r[ω]Vin [ω],(C21)

where the reflection coefficient r is determined by the
impedance mismatch between the system and the line
and is given by the well known resultr[ω] =

Z[ω] − Zc.
Z[ω] + Zc(C22)

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 consistent
with Eq. (C16) and the energy conservation requirementof no net power flow into the lossless system. For example, for the series LC oscillator we have been considering,
we haveZ[ω] =

1+ jωL0 ,
jωC0

(C23)

where, to make contact with the usual electrical engineering 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 resonance will be high and we need only consider
frequencies√
near the resonance frequency Ω0 ≡ 1/ L0 C0 where theimpedance has a zero. In this case we may approximate
2
Z[ω] ≈[Ω0 − ω] = 2jL0 (ω − Ω0 )jC0 Ω20ω − Ω0 + jγ/2
ω − Ω0 − jγ/2

(C24)Prior to moving on to the case of quantum noise it
is useful to review the classical statistical mechanics oftransmission lines. To do this we need to write down
the Lagrangian and then determine the canonical momenta and the Hamiltonian. Very conveniently, the system 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 the
one used by Caldeira and Leggett (Caldeira and Leggett,1983) in their seminal studies of the effects of dissipation
on tunneling. The only difference between this modeland the vacuum fluctuations in free space is that the relativistic 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) ≡dτ V (x, τ ),

(C26)

−∞

(C25)

showing that indeed |r| = 1 and that the phase of thereflected signal winds by 2π upon passing through theresonance. 4
Turning to the more general case where the system
also contains lossy elements, one finds that Z[ω] is nolonger purely imaginary, but has a real part satisfying
Re Z[ω] > 0. This in turn implies via Eq. (C22) that|r| < 1. In the special case of impedance matchingZ[ω] = Zc , all the incident power is dissipated in the

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

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

Z twhich yields for the reflection coefficientr[ω] =

system and none is reflected. The other two limits of interest are open circuit termination with Z = ∞ for whichr = +1 and short circuit termination Z = 0 for whichr = −1.
Finally, if the system also contains an active device
which has energy being pumped into it from a separate
external source, it may under the right conditions be described by an effective negative resistance Re Z[ω] < 0over a certain frequency range. Eq. (C22) then gives|r| ≥ 1, implying |Vout | > |Vin |. Our system will thus act
like the one-port amplifier discussed in Sec. V.D: it amplifies signals incident upon it. We will discuss this ideaof negative resistance further in Sec. C.4; a physical realization is provided by the two-port reflection parametricamplifier discussed in Appendix V.C.

where V (x, t) = ∂t ϕ(x, t) is the local voltage on the transmission line at position x and time t. Each segment of
the line of length dx has inductance ` dx and the voltage
drop along it is −dx ∂x ∂t ϕ(x, t). The flux through thisinductance is thus −dx ∂x ϕ(x, t) and the local value ofthe current is given by the constitutive equation
1
I(x, t) = − ∂x ϕ(x, t).
`

(C27)

The Lagrangian for the system is


Z ∞
Z ∞
1
c
Lg ≡
dx L(x, t) =dx(∂t ϕ)2 −(∂x ϕ)2 ,
2
2`
0
0
(C28)
The Euler-Lagrange equation for this Lagrangian is simply the wave equation
vp2 ∂x2 ϕ − ∂t2 ϕ = 0.

(C29)

11
The momentum conjugate to ϕ(x) is simply the charge
densityq(x, t) ≡δL= c∂t ϕ = cV (x, t)
δ∂t ϕ

(C30)
H=

(C31)

We know from our previous results that the charge
density consists of left and right moving solutions of arbitrary fixed shape. For example we might have for theright moving caseq(t−x/vp ) = αk cos[k(x−vp t)]+βk sin[k(x−vp t)]. (C32)
A confusing point is that since q is real valued, we seethat it necessarily contains both eikx and e−ikx terms
even if it is only right moving. Note however that for
k > 0 and a right mover, the eikx is associated with thepositive frequency term e−iωk t while the e−ikx term is
associated with the negative frequency term e+iωk t whereωk ≡ vp |k|. For left movers the opposite holds. We canappreciate this better if we define
(
)
r
Z
1
k2
1
−ikx
√ q(x, t) − idx eϕ(x, t)Ak ≡ √
2`
2c
L
(C33)

r
V

=

1X ∗
(Ak Ak + Ak A∗k ) .
2

(C34)

k

and so the Hamiltonian is given by


Z
1 2
1
2
H = dxq + (∂x ϕ) .
2c
2`

→

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

The classical equation of motion (C29) yields the simple
result∂t Ak = −iωk Ak .

(C35)

Thus

=

q(x, t)
r
c X
2L
r

=



eikx Ak (0)e−iωk t + A∗−k (0)e+iωk t (C36)

k

c Xh
2L

i
Ak (0)e+i(kx−ωk t) + A∗k (0)e−i(kx−ωk t) .

k

(C37)
We see that for k > 0 (k < 0) the wave is right (left)
moving, and that for right movers the eikx term is associated with positive frequency and the e−ikx term is associated with negative frequency. We will return to thisin the quantum case where positive (negative) frequency
will refer to the destruction (creation) of a photon. Notethat the right and left moving voltages are given byi
1 Xh
Ak (0)e+i(kx−ωk t) + A∗k (0)e−i(kx−ωk t)
2Lc

(C38)

i
1 Xh
Ak (0)e+i(kx−ωk t) + A∗k (0)e−i(kx−ωk t)
2Lc

(C39)

k>0

r
V

←

=

k<0

The voltage spectral density for the right moving waves is thus
SV→V [ω] =2π X{hAk A∗k iδ(ω − ωk ) + hA∗k Ak iδ(ω + ωk )}
2Lc

(C40)

k>0

The left moving spectral density has the same expression but k < 0.
Using Eq. (C16), 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
vp X
P = P→ − P← =sgn(k) [G[ωk ]hAk A∗k i + G[−ωk ]hA∗k Ak i] .
(C41)
2L
k

3. Classical statistical mechanics of a transmission line

Now that we have the Hamiltonian, we can consider
the classical statistical mechanics of a transmission line inthermal equilibrium at temperature T . Since each modek is a simple harmonic oscillator we have from Eq. (C34)
and the equipartition theoremhA∗k Ak i = kB T.

(C42)

12
Using this, we see from Eq. (C40) that the right moving
voltage signal has a simple white noise power spectrum.
Using Eq. (C41) we have for the right moving power ina bandwidth B (in Hz rather than radians/sec) the very
simple result
vp X
P→ =
hG[ωk ]A∗k Ak + G[−ωk ]Ak A∗k i
2L
k>0
Z
kB T +∞ dω
=
G[ω]
2
−∞ 2π
= kB T B.
(C43)a) in which a real resistor has in parallel a random current generator representing thermal noise fluctuations of
the electrons in the resistor. This is the essence of thefluctuation dissipation theorem.
In order to make a quantitative analysis in terms of
the power flowing in the two lines, voltage is not the best
variable to use since we are dealing with more than one
value of line impedance. Rather we define incoming andoutgoing fields via

where we have used the fact mentioned in connection
with Eq. (A26) and the discussion of square law detectors that all passive filter functions are symmetric in frequency.
One of the basic laws of statistical mechanics is Kirchhoff’s law stating that the ability of a hot object to emit
radiation is proportional to its ability to absorb. This
follows from very general thermodynamic arguments concerning the thermal equilibrium of an object with its radiation environment and it means that the best possible
emitter is the black body. In electrical circuits this principle is simply a form of the fluctuation dissipation theorem
which states that the electrical thermal noise produced
by a circuit element is proportional to the dissipation it
introduces into the circuit. Consider the example of a terminating resistor at the end of a transmission line. If theresistance R is matched to the characteristic impedance
Zc of a transmission line, the terminating resistor acts
as a black body because it absorbs 100% of the powerincident upon it. If the resistor is held at temperature T
it will bring the transmission line modes into equilibrium
at 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 final
temperature will not.
A good way to understand the fluctuation-dissipation
theorem is to represent the resistor R which is terminating the Zc line in terms of a second semi-infinite transmission line of impedance R as shown in Fig. (2). First
consider the case when the R line is not yet connected to
the Zc line. Then according to Eq. (C22), the open termination at the end of the Zc line has reflectivity |r|2 = 1
so that it does not dissipate any energy. Additionallyof course, this termination does not transmit any signals from the R line into the Zc . However when the
two lines are connected the reflectivity becomes less thanunity meaning that incoming signals on the Zc line seea 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 line
which is entirely equivalent. Having opened up this port
for energy to escape from the Zc system, we have also
allowed noise energy (thermal or quantum) from the Rline to be transmitted into the Zc line. This is completely equivalent to the effective circuit shown in Fig. (3

(C45)

1
Ain = √ Vc←
Zc
1
Aout = √ Vc→
Zc
1 →Bin = √ VRR
1 ←Bout = √ VR
R

(C44)

(C46)
(C47)

Normalizing by the square root of the impedance allows
us to write the power flowing to the right in each line in
the simple form
Pc = (Aout )2 − (Ain )2
PR = (Bin )2 − (Bout )2

(C48)
(C49)

The out fields are related to the in fields by the s matrix




Aout
Ain
=s
(C50)
Bout
Bin
Requiring continuity of the voltage and current at theinterface between the two transmission lines, we can solve
for the scattering matrix s:


+r t
s=
(C51)
t −r
where
R − ZcR + Zc
√
2 RZc
t =
.
R + Zc

r =

(C52)
(C53)

Note that |r|2 + |t|2 = 1 as required by energy conservation and that s is unitary with det (s) = −1. By moving
the point at which the phase of the Bin and Bout fields
are determined one-quarter wavelength to the left, we
can put s into different standard form


+r it
s0 =
(C54)
it +r
which has det (s0 ) = +1.
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 easy

13

R

VR

Vc

VR

Vc

Zc

voltage noise at the open termination of a semi-infinite
transmission line with Zc = R. For an open termination
V → = V ← so that the voltage at the end is given by
V = 2V ← = 2V →FIG. 2 (Color online) Semi-infinite transmission line of
impedance Zc terminated by a resistor R which is represented
as a second semi-infinite transmission line.

(C59)

and thus using Eqs. (C40) and (C42) we find
SV V = 4SV→V = 2RkB T(C60)

which is equivalent to Eq. (C57).
to understand. The energy absorbed from the Zc line byR, and the energy transmitted into it by thermal fluctuations in the R line are both proportional to the absorption
coefficient
A = 1 − |r|2 = |t|2 =4RZc.
(R + Zc )2

IN

(C55)

R

SII

R

VN

a)

SVV
b)

The requirement that the transmission line Zc come toequilibrium with the resistor allows us to readily computethe spectral density of current fluctuations of the random
current source shown in Fig. (3 a). The power dissipatedin Zc by the current source attached to R is
Z +∞R 2 Zcdω
P =SII [ω](R + Zc )2
−∞ 2π
(C56)
For the special case R = Zc we can equate this to theright moving power P → in Eq. (C43) because left moving
waves in the Zc line are not reflected and hence cannot
contribute to the right moving power. Requiring P =P → yields the classical Nyquist result for the current
noise of a resistor
2
kB T
R

(C57)

or in the electrical engineering convention
SII [ω] + SII [−ω] =

4
kB T.
R

We close our discussion of transmission lines by further expanding upon the idea mentioned at the end of
App. C.1 that one can view a one-port amplifier as atransmission line terminated by an effective negative resistance. The discussion here will be very general: we will
explore what can be learned about amplification by simply extending the results we have obtained on transmission lines to the case of an effective negative resistance.
Our general discussion will not address the important issues of how one achieves an effective negative resistanceover some appreciable frequency range: for such questions, one must focus on a specific physical realization,such as the parametric amplifier discussed in Sec. V.C.
We start by noting that for the case −Zc < R < 0 thepower gain G is given by
G = |r|2 > 1,

FIG. 3 Equivalent circuits for noisy resistors.

SII [ω] =4. Amplification with a transmission line and a negative
resistance(C58)

We can derive the equivalent expression for the voltage noise of a resistor (see Fig. 3 b) by considering the(C61)

and the s0 matrix introduced in Eq. (C54) becomes
√


√
G
± √G − 1
0
√
s =−
(C62)
± G−1
G
where the sign choice depends on the branch cut chosen
in the analytic continuation of the off-diagonal elements.
This transformation is clearly no longer unitary (because
there is no energy conservation since we are ignoring thework done by the amplifier power supply). Note howeverthat we still have det (s0 ) = +1. It turns out that thisnaive analytic continuation of the results from positive tonegative resistance is not strictly correct. As we will show
in 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 thetwo quadrature amplitudes of a mode are canonicallyconjugate. Following the complex amplitudes definedin Eqs. (C44-C47), let us define a vector of real-valued
quadrature amplitudes for the incoming and outgoing
fields in 
 out 
XA
XA
in 
out

X
XB


out
B , ~.
(C63)~q in = 
q
=

out
in
 YB 
YB 
out
in
YA
YA

14
The Poisson brackets amongst the different quadratureamplitudes is given by

the phases of the various input and output waves aremeasured.

{qiin , qjin } ∝ Jij ,Another allowed form of the scattering matrix is:(C64)

or equivalently the quantum commutators are2[qiin , qjin ] = iXZPF
Jij ,

(C65)


0 0 0 +1
 0 0 +1 0 .
J ≡
0 −1 0 0 −1 0 0 0

(C66)

where


In order for the transformation to be canonical, the same
Poisson bracket or commutator relations must hold forthe outgoing field amplitudes
2[qiout , qjout ] = iXZPF
Jij .

(C67)

In the case of a non-linear device these relations would
apply to the small fluctuations in the input and output
fields around the steady state solution. Assuming a linear
device (or linearization around the steady state solution)
we can define a 4 × 4 real-valued scattering matrix s̃ inanalogy to the 2 × 2 complex-valued scattering matrix s
in Eq. (C51) which relates the output fields to the input
fieldsqiout = s̃ij qjin .

(C68)

Eq. (C67) puts a powerful constraint on on the s̃ matrix,
namely that it must be symplectic. That is, s̃ and its
transpose must obeys̃J s̃T = J.

(C69)

From this it follows thatdet s̃ = ±1.

(C70)This in turn immediately implies Liouville’s theorem
that Hamiltonian evolution preserves phase space volume(since det s̃ is the Jacobian of the transformation which
propagates the amplitudes forward in time).
Let us further assume that the device is phase preserving, that is that the gain or attenuation is the same forboth quadratures. One form for the s̃ matrix consistent
with all of the above requirements is+ cos θ sin θ
0
0
0
0

 sin θ − cos θs̃ = 
.
00− cos θ sin θ 00
sin θ + cos θ


(C71)

This simply corresponds to a beam splitter and is theequivalent of Eq. (C51) with r = cos θ. As mentioned inconnection with Eq. (C51), the precise form of the scattering matrix depends on the choice of planes at which+ cosh θ + sinh θ0000 + sinh θ + cosh θs̃0 = − .
00
+ cosh θ − sinh θ 00− sinh θ + cosh θ(C72)√G, this scattering matrix is
If one takes cosh θ =essentially the canonically correct formulation of thenegative-resistance scattering matrix we tried to write in
Eq. (C62). Note that the off-diagonal terms have changed
sign for the Y quadrature relative to the naive expressionin Eq. (C62) (corresponding to the other possible analytic continuation choice). This is necessary to satisfy
the symplecticity condition and hence make the transformation canonical. The scattering matrix s̃0 can describe
amplification. Unlike the beam splitter scattering matrix s̃ above, s̃0 is not unitary (even though det s̃0 = 1).
Unitarity would correspond to power conservation. Here,power is not conserved, as we are not explicitly tracking
the power source supplying our active system.
The form of the negative-resistance amplifier scatteringmatrix s̃0 confirms many of the general statements wemade about phase-preserving amplification in Sec. V.B.
First, note that the requirement of finite gain G > 1 andphase preservation makes all the diagonal elements of s̃0(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 quantum
mechanically to build a linear phase-preserving amplifier
whose only effect is to amplify the desired signal. The
presence of the sinh θ term above means that the output
signal 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 B
are 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 here
gives us another way to think about the noise added byan amplifier: crudely speaking, we can view it as being
directly analogous to the fluctuation-dissipation theorem
simply continued to the case of negative dissipation. Just
as dissipation can occur only when we open up a newchannel and thus we bring in new fluctuations, so amplification can occur only when there is coupling to an additional channel. Without this it is impossible to satisfy
the requirement that the amplifier perform a canonicaltransformation.

15
Appendix D: Quantum Modes and Noise of a Transmission
Line
1. Quantization of a transmission line

Recall from Eq. (C30) and the discussion in Appendix
C that the momentum conjugate to the transmission line
flux variable ϕ(x, t) is the local charge density q(x, t).
Hence in order to quantize the transmission line modes
we simply promote these two physical quantities to quantum operators obeying the commutation relation[q̂(x), ϕ̂(x0 )] = −i~δ(x − x0 )(D1)from which it follows that the mode amplitudes definedin Eq. (C33) become quantum operators obeying[Âk0 , Â†k ] = ~ωk δkk0

(D2)

and we may identify the usual raising and lowering operators byp(D3)
Âk = ~ωk b̂kwhere b̂k destroys a photon in mode k. The quantum
form of the Hamiltonian in Eq. (C34) is thus


X
1
.
(D4)
H=
~ωk b̂†k b̂k +2
kFor the quantum case the thermal equilibrium expression
then becomeshÂ†k Âk i = ~ωk nB (~ωk ),(D5)which reduces to Eq. (C42) in the classical limit ~ωk 
kB T .
We have seen previously in Eqs. (C6) that the voltage fluctuations on a transmission line can be resolved
into right and left moving waves which are functions of a
combined space-time argumentxx
V (x, t) = V → (t − ) + V ← (t + ).
(D6)
vp
vp
Thus in an infinite transmission line, specifying V → everywhere in space at t = 0 determines its value for alltimes. Conversely specifying V → at x = 0 for all times
fully specifies the field at all spatial points. In preparation for our study of the quantum version of inputoutput theory in Appendix E, it is convenient to extend
Eqs. (C38-C39) to the quantum case (x = 0):
r
h
i
1 Xp
V̂ → (t) =~ωk b̂k e−iωk t + h.c.
2Lc
k>0
r
Z ∞
i
dω ~ωZc h →b̂ [ω]e−iωt + h.c. (D7)=2π
20In the second line, we have defined:r
vp X→b̂ [ω] ≡ 2πb̂k δ(ω − ωk )L
k>0

In a similar fashion, we have:r
Z ∞idω ~ωZc h ←←b̂ [ω]e−iωt + h.c. (D9)V̂ (t) =
2π
2
0
r
X
vp
b̂← [ω] ≡ 2πb̂k δ(ω − ωk )
(D10)
L
k<0

One can easily verify that among the b̂→ [ω], b̂← [ω] operators and their conjugates, the only non-zero commutators
are given by:†  † = 2πδ(ω − ω 0 )b̂→ [ω], b̂→ [ω 0 ]= b̂← [ω], b̂← [ω 0 ](D11)
We have taken the continuum limit L → ∞ here, allowing
us to change sums on k to integrals. We have thus obtained 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. VI (seeEqs. 6.2). Note that if the right-moving modes are further taken to be in thermal equilibrium, one finds (again,in the continuum limit):


†
→
→ 0
b̂ [ω] b̂ [ω ] = 2πδ(ω − ω 0 )nB (~ω) (D12a)


† 
→
→ 0
b̂ [ω] b̂ [ω ]= 2πδ(ω − ω 0 ) [1 + nB (~ω)].

(D12b)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 5 Eqs. (D8) and (D10), one finds:r
vp X −i(ωk −Ω0 )t→eb̂k (0), (D13a)b̂ (t) =L
k>0r
vp X −i(ωk −Ω0 )t←b̂ (t) =eb̂k (0). (D13b)
L
k<0

These represent temporal right and left moving modes.
Note that the normalization factor in Eqs. (D13) has beenchosen so that the right moving photon flux at x = 0 andtime t is given byhṄ i = hb̂†→ (t)b̂→ (t)i(D14)

In the same rotating frame, and within the approximation that all relevant frequencies are near Ω0 , Eq. (D7)becomes simply:ri~Ω0 Zc h →→V̂ (t) ≈b̂ (t) + b̂†→ (t)(D15)
2

5 As in the main text, we use in this appendix a convention which(D8)

differs from the one commonly used in quantum optics: â[ω] =R +∞R+iωt â(t) and â† [ω] = [â[−ω]]† = +∞ dt e+iωt â† (t).
−∞ dt e−∞16
We have already seen that using classical statisticalmechanics, the voltage noise in equilibrium is white.
The corresponding analysis of the temporal modes using
Eqs. (D13) shows that the quantum commutator obeys[b̂→ (t), b̂†→ (t0 )] = δ(t − t0 ).

(D16)

In deriving this result, we have converted summations
over 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:hb̂†→ (t0 )b̂→ (t)i = nB (~Ω0 )δ(t − t0 )(D17a)

hb̂→ (t)b̂†→ (t0 )i = [1 + nB (~Ω0 )] δ(t − t0 ). (D17b)
Equations (D15) to (D17b) indicate that V̂ → (t) can betreated as the quantum operator equivalent of white
noise; a similar line of reasoning applies mutatis mutandis to the left moving modes. We stress that these results rely crucially on our assumption that we are dealing
with a relatively narrow band of frequencies in the vicinity of Ω0 ; the resulting approximations we have madeare known as the Markov approximation. As one can already see from the form of Eqs. (D7,D9), and as will be
discussed further, the actual spectral density of vacuum
noise on a transmission line is not white, but is linear infrequency. The approximation made in Eq. (D16) treats
it as a constant within the narrow band of frequencies
of interest. If the range of frequencies of importance islarge then the Markov approximation is not applicable.
2. Modes and the windowed Fourier transform

While delta function correlations can make the quantum noise relatively easy to deal with in both the timeand frequency domain, it is sometimes the case that it
is easier to deal with a ‘smoothed’ noise variable. The
introduction of an ultraviolet cutoff regulates the mathematical singularities in the noise operators evaluated at
equal times and is physically sensible because every realmeasurement apparatus has finite time resolution. A second 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 ‘video
bandwidth’). The mathematical tool of choice for dealing
with such situations in which time and frequency both
appear is the ‘windowed Fourier transform’. The windowed transform uses a kernel which is centered on somefrequency window and some time interval. By summation over all frequency and time windows it is possible
to 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 representing
the finite integration time of our detector. In the frame
rotating at Ω0 we can define
Z tj+11dτ b̂→ (τ )(D18)
B̂j→ = √∆t tj
where tj = j(∆t) denotes the time of arrival of the jth
temporal mode at the point x = 0. Recall that b̂→ hasa photon flux normalization and so B̂j→ is dimensionless.
From Eq. (D16) we see that these smoothed operators
obey the usual bosonic commutation relations[B̂j→ , B̂k†→ ] = δjk .

(D19)

The state Bj† |0i has a single photon occupying basis
mode j, which is centered in frequency space at Ω0 and intime space on the interval j∆t < t < (j + 1)∆t (i.e. this
temporal mode passes the point x = 0 during the jthtime interval.) This basis mode is much like a note in amusical score: it has a certain specified pitch and occurs
at a specified time for a specified duration. Just as we
can play notes of different frequencies simultaneously, we
can define other temporal modes on the same time interval and they will be mutually orthogonal provided theangular frequency spacing is a multiple of 2π/∆t. The
result is a set of modes Bm,p labeled by both a frequency
index m and a time index p. p labels the time interval as
before, while m labels the angular frequency:
ωm = Ω0 + m

2π
∆t

(D20)

The result is, as illustrated in Fig. (4), a complete lattice of possible modes tiling the frequency-time phasespace, each occupying area 2π corresponding to the timefrequency 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 defineXΨ=ψj B̂j→ .
(D21)jThis is also a canonical bosonic mode operator obeying[Ψ, Ψ† ] = 1(D22)

provided that the coefficients obey the normalization conditionX|ψj |2 = 1.
(D23)
j

We might for example want to describe a mode which iscentered at a slightly higher frequency Ω0 + δΩ (obeying

17
Δt

ω

m

area = 2π

Bmp

2π/Δt
t

ω=Ω0

-m

B-mp

p

FIG. 4 (Color online) Schematic figure indicating how thevarious modes defined by the windowed Fourier transform tile
the time-frequency plane. Each individual cell corresponds toa different mode, and has an area 2π.

(δΩ)(∆t) << 1) and spread out over a large time interval
T centered at time T0 . This could be given for examplebyψj = N e−(j∆t−T0 )2
4T 2

e−i(δΩ)(j∆t)(D24)

where N is the appropriate normalization constant.
The state having n photons in the mode is simply
n1√
Ψ† |0i.
n!

(D25)

The concept of ‘wave function of the photon’ is fraught
with dangers. In the very special case where we restrict attention solely to the subspace of single photon
Fock states, we can usefully think of the amplitudes {ψj }as the ‘wave function of the photon’ (Cohen-Tannoudji
et 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 of
coupled LC oscillators with an infinite number of degrees
of freedom. Let us simplify the argument by consideringa single LC oscillator. We can perfectly well write a wave
function for the system as a function of the coordinate(say the charge q on the capacitor). The ground state
wave function χ0 (q) is a gaussian function of the coordinate. The one photon state created by Ψ† has a wave
function χ1 (q) ∼ qχ0 (q) proportional to the coordinate
times the same gaussian. In the general case χ is a wave
functional of the charge distribution q(x) over the entire
transmission line.
Using Eq. (D17a) we havehB̂j†→ B̂k→ i = nB (~Ω0 )δjk

(D26)

independent of our choice of the coarse-graining time window ∆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. Equivalently
the flux in bandwidth B isX
(B∆t) = XB.
(D27)
Ṅ =
∆t
The interpretation of this is that X photons in a temporal mode of duration ∆t pass the origin in time ∆t. Each1mode has bandwidth ∼ ∆tand so there are B∆t independent temporal modes in bandwidth B all occupying
the 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 following
elementary question: What is the mode occupancy of aP?
laser beam of power P and hence photon flux Ṅ = ~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–it
has essentially no amplitude fluctuations. The frequency
is nominally set by the physical properties of the oscillator, but there is nothing to pin the phase which consequently undergoes slow diffusion due to unavoidable noise
perturbations. This leads to a finite phase coherence timeτ and corresponding frequency spread 1/τ of the laser
spectrum. (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 = Ṅ τ . A convenient approximate 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 stateY √ iϕj †→e Xe B̂j |0i(D28)jwhich obeys
hB̂k→ i =

√

Xeiϕk

(D29)

hB̂k†→ B̂k→ i = X.

(D30)

and

3. Quantum noise from a resistor

Let us consider the quantum equivalent to Eq. (C60),
SV V = 2RkB T , for the case of a semi-infinite transmission line with open termination, representing a resistor.
From Eq. (C27) we see that the proper boundary condition for the ϕ field is ∂x ϕ(0, t) = ∂x ϕ(L, t) = 0. (Wehave temporarily made the transmission line have a large
but finite length L.) The normal mode expansion thatsatisfies these boundary conditions isr ∞2 X
ϕ(x, t) =ϕn (t) cos(kn x),
(D31)
L n=1

18
where ϕn is the normal coordinate and kn ≡ πn
L . Substitution of this form into the Lagrangian and carryingout the spatial integration yields a set of independent
harmonic oscillators representing the normal modes.

∞ 
X
c 2
1 2 2
Lg =
ϕ̇ − k ϕ .
(D32)
2 n 2` n n
n=1

Again, the step function tells us that the resistor can only
absorb energy, not emit it, at zero temperature.
If we use the engineering convention and add the noiseat positive and negative frequencies we obtain

From this we can find the momentum operator p̂n canonically conjugate to the coordinate operator ϕ̂n and quantize the system to obtain an expression for the operator
representing the voltage at the end of the transmission
line in terms of the mode creation and destruction operators
∞ r
X
~Ωn †
V̂ =
i(b̂n − b̂n ).
(D33)
Lc
n=1

for the symmetric part of the noise, which appears in thequantum fluctuation-dissipation theorem (cf. Eq. (2.16)).
The antisymmetric part of the noise is simply

The spectral density of voltage fluctuations is then foundto beThis quantum treatment can also be applied to any
arbitrary dissipative network (Burkhard et al., 2004; Devoret, 1997). If we have a more complex circuit containing capacitors and inductors, then in all of the above
expressions, Zc should be replaced by Re Z[ω] where Z[ω]is the complex impedance presented by the circuit.
In the above we have explicitly quantized the standing wave modes of a finite length transmission line. We
could instead have used the running waves of an infinite
line and recognized that, as the in classical treatment in
Eq. (C59), the left and right movers are not independent.
The open boundary condition at the termination requires
V ← = V → and hence b→ = b← . We then obtain∞

SVV [ω] =2π X ~Ωn nB (~Ωn )δ(ω + Ωn )L n=1 c+[nB (~Ωn ) + 1]δ(ω − Ωn ) ,(D34)

where nB (~ω) is the Bose occupancy factor for a photonwith energy ~ω. Taking the limit L → ∞ and converting
the summation to an integral yields

SVV (ω) = 2Zc ~|ω| nB (~|ω|)Θ(−ω)+[nB (~|ω|)+1]Θ(ω) ,(D35)
where Θ is the step function. We see immediately thatat zero temperature there is no noise at negative frequencies because energy can not be extracted from zero-point
motion. However there remains noise at positive frequencies indicating that the vacuum is capable of absorbing
energy from another quantum system. The voltage spectral density at both zero and non-zero temperature isplotted in Fig. (1).
Eq. (D35) for this ‘two-sided’ spectral density of a resistor can be rewritten in a more compact form
SVV [ω] =2Zc ~ω
,1 − e−~ω/kB T(D36)

which reduces to the more familiar expressions in various
limits. For example, in the classical limit kB T  ~ω thespectral density is equal to the Johnson noise result6
SVV [ω] = 2Zc kB T,(D37)

in agreement with Eq. (C60). In the quantum limit it
reduces to
SVV [ω] = 2Zc ~ωΘ(ω).

6 Note(D38)

again that in the engineering convention this would be
SVV [ω] = 4Zc kB T .

SVV [ω] + SVV [−ω] = 2Zc ~ω coth

~ω
2kB T

(D39)

SVV [ω] − SVV [−ω] = 2Zc ~ω,

(D40)

~ω
SVV [ω] − SVV [−ω]= tanh.
SVV [ω] + SVV [−ω]
2kB T

(D41)

yielding

SV V [ω] = 4SV→V [ω](D42)

and from the quantum analog of Eq. (C40) we have4~|ω|{Θ(ω)(nB + 1) + Θ(−ω)nB }2cvp= 2Zc ~|ω| {Θ(ω)(nB + 1) + Θ(−ω)nB }(D43)

SV V [ω] =in agreement with Eq. (D35).

Appendix E: Back Action and Input-Output Theory for
Driven Damped Cavities

A high Q cavity whose resonance frequency can beparametrically controlled by an external source can act
as a very simple quantum amplifier, encoding information about the external source in the phase and amplitude of the output of the driven cavity. For example,
in an optical cavity, one of the mirrors could be moveable and the external source could be a force acting onthat mirror. This defines the very active field of optomechanics, which also deals with microwave cavities coupled to nanomechanical systems and other related setups(Arcizet et al., 2006; Brown et al., 2007; Gigan et al.,19
2006; Harris et al., 2007; Höhberger-Metzger and Karrai, 2004; Marquardt et al., 2007, 2006; Meystre et al.,
1985; Schliesser et al., 2006; Teufel et al., 2008; Thompson et al., 2008; Wilson-Rae et al., 2007). In the case of a
microwave cavity containing a qubit, the state-dependent
polarizability of the qubit acts as a source which shifts
the frequency of the cavity (Blais et al., 2004; Schuster
et 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 equation of motion method to exactly solve for this quantum
noise 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 of
the cavity field noise spectrum, deferring the detailed
microscopic derivation to subsequent subsections. We
then provide a review of the input-output theory for
driven cavities, and employ this theory to analyze theimportant example of a dispersive position measurement,
where we demonstrate how the standard quantum limit
can be reached. Finally, we analyze an example wherea modified dispersive scheme is used to detect only one
quadrature of a harmonic oscillator’s motion, such that
this quadrature does not feel any back-action.
1. Photon shot noise inside a cavity and back action

x̂
xZPF

.

(E1)

(E2)

We have subtracted the hâ† âi term so that the mean
force on the degree of freedom is zero. To obtain the full
Hamiltonian, we would have to add the cavity damping
and driving terms, as well as the Hamiltonian governing
the intrinsic dynamics of the system ẑ. From Eq. (3.18)
we know that the back action noise force acting on ẑ isproportional to the quantum fluctuations in the numberof photons n̂ = â† â in the cavity,
Snn (t) = hâ† (t)â(t)â† (0)â(0)i − hâ† (t)â(t)i2 .

(E3)For the case of continuous wave driving at frequency
ωL = ωc + ∆ detuned by ∆ from the resonance, thecavity is in a coherent state |ψi obeying
ˆâ(t) = e−iωL t [ā + d(t)](E5)ˆ
so the coherent state is annihilated by d:ˆ = 0.
d|ψi(E6)

ˆ the coherent state
That is, in terms of the operator d,looks like the undriven quantum ground state . The displacement transformation in Eq. (E4) is canonical sinceˆ dˆ† ] = 1.
[â, â† ] = 1 ⇒ [d,

(E4)

(E7)

Substituting the displacement transformation into
Eq. (E3) and using Eq. (E6) yields
ˆ dˆ† (0)i,Snn (t) = n̄hd(t)(E8)

where n̄ = |ā|2 is the mean cavity photon number. If we
set the cavity energy damping rate to be κ, such that theamplitude damping rate is κ/2, then the undriven stateobeysκˆ dˆ† (0)i = e+i∆t e− 2 |t| .
hd(t)

(E9)

This expression will be justified formally in the subsequent subsection, after introducing input-output theory.
We thus arrive at the very simple resultκwhere following Eq. (3.12), we have taken A to be dimensionless, and use ẑ to denote the dimensionless systemvariable that we wish to probe. For example, ẑ could
represent the dimensionless position of a mechanical oscillatorẑ ≡â|ψi = ψ|ψi

Snn (t) = n̄ei∆t− 2 |t| .

Consider a degree of freedom ẑ coupled parametricallywith strength A to the cavity oscillator
Ĥint = ~ωc (1 + Aẑ) [â† â − hâ† âi]where the first term is the ‘classical part’ of the mode amplitude ψ(t) = āe−iωL t determined by the strength of thedrive field, the damping of the cavity and the detuning∆, and d is the quantum part. By definition,(E10)

The power spectrum of the noise is, via the WienerKhinchin theorem (Appendix A.2), simply the Fouriertransform of the autocorrelation function given in
Eq. (E10)Z +∞κ
Snn [ω] =dt eiωt Snn (t) = n̄.
2(ω + ∆) + (κ/2)2−∞(E11)
As can be seen in Fig. 5a, for positive detuning ∆ =ωL − ωc > 0, i.e. for a drive that is blue-detuned withrespect to the cavity, the noise peaks at negative ω. Thismeans that the noise tends to pump energy into the degree of freedom ẑ (i.e. it contributes negative damping).
For negative detuning the noise peaks at positive ω corresponding to the cavity absorbing energy from ẑ. Basically, the interaction with ẑ (three wave mixing) tries to
Raman scatter the drive photons into the high density ofstates at the cavity frequency. If this is uphill in energy,
then ẑ is cooled.
As discussed in Sec. B.2 (c.f. Eq. (2.8)), at each frequency ω, we can use detailed balance to assign the noise
an effective temperature Teff [ω]:Snn [ω]= e~ω/kB Teff [ω] ⇔Snn [−ω]~ωhikB Teff [ω] ≡nn [ω]log SSnn
[−ω]

(E12)

20

(a)

(b)

4

0

0
-6

-4
-6

Noise power

2

(c) 3

Noise temperature

40Frequency

6

2

0

6

5

6

-

1
0
0

1

3
2
Frequency

4

FIG. 5 (Color online) (a) Noise spectrum of the photon number 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) Effective temperature Teff of the low frequency noise, ω → 0,
as a function of the detuning ∆ of the drive from the cavity
resonance. (c) Frequency-dependence of the effective noise
temperature, for different values of the detuning.

or equivalently
Snn [ω] − Snn [−ω]= tanh(β~ω/2).
Snn [ω] + Snn [−ω](E13)

If ẑ is the coordinate of a harmonic oscillator of frequency
ω (or some non-conserved observable of a qubit with level
splitting ω), then that system will acquire a temperature
Teff [ω] in the absence of coupling to any other environment. In particular, if the characteristic oscillation frequency of the system ẑ is much smaller than κ, then we
have the simple result
1
2 Snn [ω] − Snn [−ω]= lim+kB Teff
ω→0 ~ω Snn [ω] + Snn [−ω]d ln Snn [ω]
= 2
d~ω
1
−4∆
=
.
~ ∆2 + (κ/2)2(E14)

As can be seen in Fig. 5, the asymmetry in the noisechanges sign with detuning, which causes the effective
temperature to change sign.
First we discuss the case of a positive Teff , where thismechanism can be used to laser cool an oscillating mechanical cantilever, provided Teff is lower than the intrinsic equilibrium temperature of the cantilever. (Arcizet et al., 2006; Brown et al., 2007; Gigan et al., 2006;
Harris et al., 2007; Höhberger-Metzger and Karrai, 2004;
Marquardt et al., 2007; Schliesser et al., 2006; Thompson
et al., 2008; Wilson-Rae et al., 2007). A simple classicalargument helps us understand this cooling effect. Suppose 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 frequency
will 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. At
the end of the cycle it ends up having done net positive
work 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 intrinsic mechanical losses, negative values of the effective
temperature indicate a dynamical instability of the cantilever (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 interpreted
as negative damping introduced by the optomechanicalcoupling and can be used to create parametric amplification of mechanical forces acting on the oscillator.
Finally, we mention that cooling towards the quantum
ground 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 contrast
to the ω  κ regime discussed above).

2. Input-output theory for a driven cavity

The results from the previous section can be more formally and rigorously derived in a full quantum theoryof a cavity driven by an external coherent source. The
theory relating the drive, the cavity and the outgoingwaves radiated by the cavity is known as input-output
theory and the classical description was presented in Appendix C. The present quantum discussion closely follows standard references on the subject (Walls and Milburn, 1994; Yurke, 1984; Yurke and Denker, 1984). The
crucial feature that distinguishes such an approach frommany other treatments of quantum-dissipative systems

21is the goal of keeping the bath modes instead of tracing them out. This is obviously necessary for the situations we have in mind, where the output field emanatingfrom the cavity contains the information acquired during
a measurement of the system coupled to the cavity. As
we learned from the classical treatment, we can eliminate the outgoing waves in favor of a damping term forthe system. However we can recover the solution for theoutgoing modes completely from the solution of the equation of motion of the damped system being driven by the
incoming waves.
In order to drive the cavity we must partially open oneof its ports which exposes the cavity both to the external
drive and to the vacuum noise outside which permits energy in the cavity to leak out into the surrounding bath.
We will formally separate the degrees of freedom into internal cavity modes and external bath modes. Strictly
speaking, once the port is open, these modes are not distinct and we only have ‘the modes of the universe’ (GeaBanacloche et al., 1990a,b; Lang et al., 1973). However
for high Q cavities, the distinction is well-defined and we
can model the decay of the cavity in terms of a spontaneous emission process in which an internal boson isdestroyed and an external bath boson is created. We
assume a single-sided cavity. For a high Q cavity, thisphysics is accurately captured in the following HamiltonianĤ = Ĥsys + Ĥbath + Ĥint .

X~ωq b̂†q b̂qi˙b̂q = [Ĥ, b̂q ] = −iωq b̂q + fq∗ â
~

(E16)

t0

(E21)
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 isiâ˙ = [Ĥsys , â] −~[b̂q , b̂†q0 ] = δq,q0 .

q

(E17)

Note that since the bath terminates at the system, there
is no translational invariance, the normal modes are
standing not running waves, and the quantum numbers
q are not necessarily wave vectors.
The coupling Hamiltonian is (within the rotating wave
approximation)i
XhĤint = −i~fq â† b̂q − fq∗ b̂†q â .
(E18)
q

For the moment we will leave the system (cavity) Hamiltonian to be completely general, specifying only that it
consists of a single degree of freedom (i.e. we concentrateon only a single resonance of the cavity with frequency
ωc ) obeying the usual bosonic commutation relation[â, â† ] = 1.

(E19)

(N.B. this does not imply that it is a harmonic oscillator. We will consider both linear and non-linear cavities.)Xfq b̂q .

(E22)

qSubstituting Eq. (E21) into the last term above yields
XX
fq b̂q =fq e−iωq (t−t0 ) b̂q (t0 )qwhere q labels the quantum numbers of the independent
harmonic oscillator bath modes obeying(E20)

We see that this is simply the EOM of a harmonic oscillator 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. (E20) is
Z t−iωq (t−t0 )b̂q (t) = eb̂q (t0 ) +dτ e−iωq (t−τ ) fq∗ â(τ ).

(E15)

The bath Hamiltonian isĤbath =Note that the most general linear coupling to the bathmodes would include terms of the form b̂†q â† and b̂q a butthese are neglected within the rotating wave approximation because in the interaction representation they oscillate at high frequencies and have little effect on the
dynamics.
The Heisenberg equation of motion (EOM) for thebath variables is+X|fq |2Z tqdτ e−i(ωq −ωc )(t−τ ) [e+iωc (τ −t) â(τ )], (E23)t0qwhere the last term in square brackets is a slowly varying
function of τ . To simplify our result, we note that if
the cavity system were a simple harmonic oscillator of
frequency ωc then the decay rate from the n = 1 single
photon excited state to the n = 0 ground state would begiven by the following Fermi Golden Rule expressionXκ(ωc ) = 2π
|fq |2 δ(ωc − ωq ).
(E24)
qFrom this it follows that
Z +∞
X
dν
κ(ωc + ν)e−iν(t−τ ) =|fq |2 e−i(ωq −ωc )(t−τ ) .
−∞ 2π
q
(E25)
We now make the Markov approximation which assumes
that κ(ν) = κ is a constant over the range of frequencies
relevant to the cavity so that Eq. (E25) may be represented asX|fq |2 e−i(ωq −ωc )(t−τ ) = κδ(t − τ ).
(E26)
q

22
Using
Z x0

1
2

(E27)

fq e−iωq (t−t0 ) b̂q (t0 ).

(E28)

dx δ(x − x0 ) =−∞we obtain for the cavity EOMiκâ˙ = [Ĥsys , â] − â −
~
2

X
q

The second term came from the part of the bath motion
representing the wave radiated by the cavity and, within
the Markov approximation, has become a simple linear
damping 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
p approximation it isfurther convenient to treat f ≡ |fq |2 as a constant anddefine the density of states (also taken to be a constant)byXρ=δ(ωc − ωq )(E29)
q

so that the Golden Rule rate becomesκ = 2πf 2 ρ.

(E30)

We can now define the so-called ‘input mode’b̂in (t) ≡ √1 X −iωq (t−t0 )eb̂q (t0 ) .
2πρ q

(E31)

For the case of a transmission line treated in Appendix
D, this coincides with the field b̂→ moving towards thecavity [see Eq. (D13a)]. We finally have for the cavityEOMiκâ˙ = [Ĥsys , â] − â −~2√

κ b̂in (t).

(E32)

Note that when a wave packet is launched from the bathtowards the cavity, causality prevents it from knowing
about the cavity’s presence until it reaches the cavity.
Hence the input mode evolves freely as if the cavity were
not present until the time of the collision at which point
it begins to drive the cavity. Since b̂in (t) evolves under
the free bath Hamiltonian and acts as the driving term in
the cavity EOM, we interpret it physically as the input
mode. Eq. (E32) is the quantum analog of the classical equation (C19), for our previous example of an LCoscillator driven by a transmission line. The latter would
also have been first order in time if as in Eq. (C35) we
had worked with the complex amplitude A instead of thecoordinate Q.
Eq. (E31) for the input mode contains a time label
just as in the interaction representation. However it isbest interpreted as simply labeling the particular linear
combination of the bath modes which is coupled to thesystem at time t. Some authors even like to think of

the bath modes as non-propagating while the cavity flies
along the bath (taken to be 1D) at a velocity v. The
system then only interacts briefly with the local mode
positioned at x = vt before moving on and interactingwith the next local bath mode. We will elaborate on thisview further at the end of this subsection.
The expression for the power Pin (energy per time)
impinging on the cavity depends on the normalization
chosen in our definition of b̂in . It can be obtained, forexample, by imagining the bath modes b̂q to live on a onedimensional waveguide with propagation velocity v andlength L (using periodic boundary conditions). In thatcase we have to sum over all photons to get the average
power flowing throughof the waveguide,D a cross-sectionE
P
Pin = q ~ωq (vp /L) b̂†q b̂q . Inserting the definition forb̂in , Eq. (E31), the expression for the input power carried
by a monochromatic beam at frequency ω is
D
E
(E33)
Pin (t) = ~ω b̂†in (t)b̂in (t)Note that this has the correct dimensions due√to our
choice of normalization for b̂in (with dimensions ω). In
the general case, an integration over frequencies is needed(as will be discussed further below). An analogous formula holds for the power radiated by the cavity, to be
discussed now.
The output mode b̂out (t) is radiated into the bath andevolves freely after the system interacts with b̂in (t). If
the cavity did not respond at all, then the output mode
would simply be the input mode reflected off the cavity 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 dynamics of the cavity. To analyze this output field, let
t1 > t be a time in the distant future after the input
field has interacted with the cavity. Then we can write
an alternative solution to Eq. (E20) in terms of the final
rather than the initial condition of the bath
Z t1−iωq (t−t1 )b̂q (t) = eb̂q (t1 ) −dτ e−iωq (t−τ ) fq∗ â(τ ).
t(E34)Note the important minus sign in the second term associated with the fact that the time t is now the lower
limit of integration rather than the upper as it was in
Eq. (E21).
Definingb̂out (t) ≡ √1 X −iωq (t−t1 )eb̂q (t1 ),
2πρ q

(E35)

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.

23
Proceeding as before we obtainiκâ˙ = [Ĥsys , â] + â −~2√κ b̂out (t).

(E36)

Subtracting Eq. (E36) from Eq. (E32) yieldsb̂out (t) = b̂in (t) +√κ â(t)(E37)

which is consistent with our interpretation of the outgoing field as the reflected incoming field plus the fieldradiated by the cavity out through the partially reflecting
mirror.
The above results are valid for any general cavity
Hamiltonian. The general procedure is to solve Eq. (E32)
for â(t) for a given input field, and then solve Eq. (E37)
to obtain the output field. For the case of an empty cavity we can make further progress because the cavity mode
is a harmonic oscillatorĤsys = ~ωc â† â.

(E38)

In this simple case, the cavity EOM becomesκâ˙ = −iωc â − â −2√

κ b̂in (t).

(E39)

and
ω − ωc − iκ/2b̂in [ω]ω − ωc + iκ/2

(E42)which is the result for the reflection coefficient quoted in
Eq. (3.13). For brevity, here and in the following, we will
sometimes use the susceptibility of the cavity, defined asχc [ω − ωc ] ≡1−i(ω − ωc ) + κ/2

P = ~ωκ2 /(4κL ) â† â .

(E46)

Here for κL = κR the interference effect completely eliminates the reflected beam and we have in contrast to
Eq. (E45)
κ
P = ~ω â† â .
(E47)
2Eq. (E39) can also be solved in the time domain to
obtainâ(t) = e−(iωc +κ/2)(t−t0 ) â(t0 )
Z t
√
− κ
dτ e−(iωc +κ/2)(t−τ ) b̂in (τ ).

(E48)

t0

Eq. (E39) can be solved by Fourier transformation, yielding√κb̂in [ω](E40)â[ω] = −i(ωc − ω) + κ/2√(E41)= − κχc [ω − ωc ]b̂in [ω]b̂out [ω] =incoming wave which is promptly reflected from the cavity and the field radiated by the cavity. The naive expression becomes correct after the drive has been switched off(where ignoring the effect of the incoming vacuum noise,√
we would have b̂out = κâ). We note in passing that fora driven two-sided cavity with coupling constants κL andκR (where κ = κL + κR ), the incoming power sent into
the left port is related to the photon number by(E43)

For the case of steady driving on resonance where ω = ωc ,the above equations yield√κb̂out [ω] =â[ω].
(E44)
2
In steady state, the incoming power equals the outgoingpower, and both are related to the photon number inside
the single-sided cavity byDEκP = ~ω b̂†out (t)b̂out (t) = ~ω â† (t)â(t)(E45)
4Note that this does not coincide with the naive expectation, which would be P = ~ωκ â† â . The reason for thisdiscrepancy is the the interference between the part of the

If we take the input field to be a coherent drive at frequency ωL = ωc + ∆ so that its amplitude has a classicaland a quantum partˆb̂in (t) = e−iωL t [b̄in + ξ(t)](E49)

and if we take the limit t0 → ∞ so that the initial transient in the cavity amplitude has damped out, then thesolution of Eq. (E48) has the form postulated in Eq. (E4)with√κā = −b̄in(E50)−i∆ + κ/2and (in the frame rotating at the drive frequency)
Z t
√
ˆ
ˆ ).
d(t) = − κdτ e+(i∆−κ/2)(t−τ ) ξ(τ
(E51)
−∞

Even in the absence of any classical drive, the input
field delivers vacuum fluctuation noise to the cavity. Notice that from Eqs. (E31, E49)ˆ ξˆ† (t0 )][b̂in (t), b̂†in (t0 )] = [ξ(t),1 X −i(ωq −ωL )(t−t0 )
=
e
2πρ q
= δ(t − t0 ),(E52)which is similar to Eq. (D16) for a quantum transmission
line. This is the operator equivalent of white noise. Using
Eq. (E48) in the limit t0 → −∞ in Eqs. (E4,E51) yields
ˆ dˆ† (t)][â(t), â† (t)] = [d(t),
Z t
Z t
= κ
dτ
dτ 0 e−(−i∆+κ/2)(t−τ )
−∞

−∞
0

e−(+i∆+κ/2)(t−τ ) δ(τ − τ 0 )
= 1

(E53)

24
as is required for the cavity bosonic quantum degree of
freedom. We can interpret this as saying that the cavity
zero-point fluctuations arise from the vacuum noise thatenters through the open port. We also now have a simple
physical interpretation of the quantum noise in the number of photons in the driven cavity in Eqs. (E3,E8,E11).
It is due to the vacuum noise which enters the cavity
through the same ports that bring in the classical drive.
The interference between the vacuum noise and the classical drive leads to the photon number fluctuations in thecavity.
In thermal equilibrium, ξˆ also contains thermal radiation. If the bath is being probed only over a narrow
range of frequencies centered on ωc (which we have assumed in making the Markov approximation) then we
have to a good approximation (consistent with the above
commutation relation)
ˆ 0 )i = N δ(t − t0 )hξˆ† (t)ξ(tˆ ξˆ† (t0 )i = (N + 1)δ(t − t0 )
hξ(t)

(E54)
(E55)

where N = nB (~ωc ) is the thermal equilibrium occupation number of the mode at the frequency of interest. We
can gain a better understanding of Eq. (E54) by Fouriertransforming it to obtain the spectral density
Z +∞S[ω] =

ˆ 0 )ieiω(t−t0 ) = N.
dt hξˆ† (t)ξ(t

(E56)

−∞

As mentioned previously, this dimensionless quantity is
the spectral density that would be measured by a photomultiplier: it represents the number of thermal photons
passing a given point per unit time per unit bandwidth.
Equivalently the thermally radiated power in a narrow
bandwidth B is
P = ~ωN B.

(E57)

One often hears the confusing statement that the noise
added by an amplifier is a certain number N of photons(N = 20, say for a good cryogenic HEMT amplifier operating at 5 GHz). This means that the excess output
noise (referred back to the input by dividing by the powergain) produces a flux of N photons per second in a 1 Hz
bandwidth, or 106 N photons per second in 1 MHz of
bandwidth (see also Eq. (D27)).
We can gain further insight into input-output theory
by using the following picture. The operator b̂in (t) represents the classical drive plus vacuum fluctuations which
are just about to arrive at the cavity. We will be able
to show that the output field is simply the input fielda short while later after it has interacted with the cavity. Let us consider the time evolution over a short time
period ∆t which is very long compared to the inverse
bandwidth of the vacuum noise (i.e., the frequency scale
beyond which the vacuum noise cannot be treated as constant due to some property of the environment) but very
short compared to the cavity system’s slow dynamics. In

this circumstance it is useful to introduce the quantum
Wiener increment related to Eq. (D18)
Z t+∆tc≡dWˆ )dτ ξ(τ

(E58)twhich obeys
c , dWc † ] = ∆t.
[dW

(E59)

In the interaction picture (in a displaced frame inwhich the classical drive has been removed) the Hamiltonian term that couples the cavity to the quantum noiseof the environment is from Eq. (E18)√V̂ = −i~ κ(â† ξˆ − âξˆ† ).
(E60)
Thus the time evolution operator (in the interaction picture) on the jth short time interval [tj , tj + ∆t] is
√

c†

†

Ûj = e κ(â dW −â dW )
c

(E61)

Using this we can readily evolve the incoming temporalmode forward in time by a small step ∆t√
c 0 = Û † dW
c Û ≈ dW
c + κ∆t â.
dW(E62)
Recall that in input-output theory we formally defined
the outgoing field as the bath field far in the future propagated back (using the free field time evolution) to thepresent, which yielded√(E63)b̂out = b̂in + κâ.
Eq. (E62) is completely equivalent to this. Thus we confirm 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 especially
useful in the quantum trajectory approach to conditional
quantum evolution of a system subject to weak continuous measurement (Gardiner et al., 1992; Walls and Milburn, 1994). On top of the classical drive b̄in (t), the bath
supplies to the system a continuous stream of “fresh” harmonic oscillators, each in their ground state (if T = 0).
c interEach oscillator with its quantum fluctuation dW
acts briefly for a period ∆t with the system and thenis disconnected to propagate freely thereafter, never interacting with the system again. Within this picture it
is useful to think of the oscillators arrayed in an infinite
stationary line and the cavity flying over them at speed
vp and touching each one for a time ∆t.
3. Quantum limited position measurement using a cavity
detector

We will now apply the input-output formalism introduced in the previous section to the important example

25
of a dispersive position measurement, which employs acavity whose resonance frequency shifts in response to
the motion of a harmonic oscillator. This physical system was considered heuristically in Sec. III.B.3. Here wewill present a rigorous derivation using the (linearized)
equations of motion for the coupled cavity and oscillator
system.
Let the dimensionless position operator
ẑ =

1
xZPF

x̂ = [ĉ† + ĉ](E64)

and similarly for the cavity field following Eq. (E4)ˆâ = ā + d.

(E70)

Substituting these expressions into the equation of motion, we find that the constant classical fields obey√κā = −b̄in(E71)
κ/2 − i∆and the new quantum equation of motion is, after neˆ
glecting a small term dẑ:be the coordinate of a harmonic oscillator whose energy
is

√
κ
˙
ˆ
dˆ = +i∆dˆ − iAωc āẑ − dˆ − κξ.
2

HM = ~ωM ĉ† ĉ

The quantum limit for position measurement will bereached only at zero detuning, so we specialize to thecase ∆ = 0. We also choose the incoming field amplitude
and phase to obey
q

(E65)

and whose positionp uncertainty in the quantum groundstate is xZPF = h0|x̂2 |0i.
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 frequency shifts,b̄in = −i

(E72)

Ṅ ,

(E73)

Ṅ
,
κ

(E74)

so thatsω̃c = ωc [1 + Aẑ(t)](E66)

where for a cavity of length L, A = −xZPF /L.
Assuming that the mirror moves slowly enough for thecavity 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 discussed
in Sec. III.B, and from this phase shift we can determine
the position of the mechanical oscillator. In addition to
the actual zero-point fluctuations of the oscillator, ourmeasurement will suffer from shot noise in the homodyne
signal and from additional uncertainty due to the backaction noise of the measurement acting on the oscillator.
All of these effects will appear naturally in the derivation
below.
We begin by considering the optical cavity equationof motion based on Eq. (E32) and the optomechanicalcoupling Hamiltonian in Eq. (E1). These yieldκâ˙ = −iωc (1 + Aẑ)â − â −2√κb̂in .

(E67)

Let the cavity be driven by a laser at a frequency ωL =
ωc + ∆ detuned from the cavity by ∆. Moving to a frame
rotating at ωL we haveκâ˙ = +i(∆ − Aωc ẑ)â − â −2√κb̂in .

(E68)

and we can write the incoming field as a constant plus
white noise vacuum fluctuations (again, in the rotating
frame)b̂in = b̄in + ξˆ(E69)ā = +2i

where Ṅ is the incoming photon number flux. The quantum equation of motion for the cavity then becomes√κ˙ˆdˆ = +gẑ − dˆ − κξ,
2

(E75)

where the opto-mechanical coupling constant is proportional to the laser drive amplitudes√Ṅ
= Aωc n̄.
g ≡ 2Aωc
(E76)
κ
and
n̄ = |ā|2 = 4

Ṅ
κ

(E77)

is the mean cavity photon number. Eq. (E75) is easily
solved by Fourier transformation
n
o
√
1
ˆ =
ˆ
d[ω]
gẑ[ω] − κξ[ω].
(E78)[κ/2 − iω]Let us assume that we are in the limit of low mechanical 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 good
approximation
o
√
2n
ˆ
ˆ
d[ω]
=
gẑ[ω] − κξ[ω]
(E79)
κ
n
o
√
2
dˆ† [ω] =gẑ[ω] − κξˆ† [ω]
(E80)
κ

26
The mechanical oscillator equation of motion which is
identical in form to that of the optical cavity∂t ĉ = −[iγ0√+ iΩ]ĉ − γ0 η̂(t) + [Ĥint , ĉ(t)],
2
~

(E81)

where Ĥint is the Hamiltonian in Eq. (E1) and η̂ isthe mechanical vacuum noise from the (zero temperature) bath which is causing the mechanical damping at
rate γ0 . Using Eq. (E70) and expanding to first orderin small fluctuations yields the equation of motion linearized about the steady state solution∂t ĉ = −[

g ˆ
γ0
√
+ iΩ]ĉ − γ0 η̂(t) + 2 √ [ξ(t)− ξˆ† (t)]. (E82)2κ

It is useful to consider an equivalent formulation inwhich we expand the Hamiltonian in Eq. (E1) to second
order in the quantum fluctuations about the classical solutionĤint ≈ ~ωc dˆ† dˆ + x̂F̂ ,(E83)

where the force (including the coupling A) is (up to asign)F̂ = −i~g ˆ ˆ†[d − d ].
xZPF

(E84)

Note that the radiation pressure fluctuations (photon
shot noise) inside the cavity provide a forcing term. The
state of the field inside the cavity in general depends on
the 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 for
the force obtained from Eq. (E75)κ~g √ ˆ ˆ†˙
κ[ξ − ξ ].
F̂ = − F̂ + i
2
xZPF

FIG. 6 (Color online) Phasor diagram for the cavity amplitude showing that (for our choice of parameters) the imaginary quadrature of the vacuum noise ξˆ interferes with theclassical drive to produce photon number fluctuations while
the real quadrature produces phase fluctuations which lead tomeasurement imprecision. The quantum fluctuations are illustrated in the usual fashion, depicting the Gaussian Wigner
density of the coherent state in terms of color intensity.

phase noise which determines the measurement imprecision (shot noise in the homodyne signal). This will be
discussed further below.
The solution for the cantilever position can again be
obtained by Fourier transformation. For frequencies
small on the scale of κ the solution of Eq. (E85) is(E85)

Within our linearization approximation, the position of
the mechanical oscillator has no effect on the radiationpressure (photon number in the cavity), but of course it
does affect the phase of the cavity field (and hence theoutgoing field) which is what we measure in the homodyne detection.
Thus for this special case ẑ does not appear on the
RHS of either Eq. (E85) or Eq. (E82), which means that
there is no optical renormalization of the cantilever frequency (‘optical spring’) or optical damping of the cantilever. The lack of back-action damping in turn implies
that the effective temperature Teff of the cavity detector
is infinite (cf. Eq. (2.8)). 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 number fluctuations). This is illustrated in the cavity amplitude phasor diagram of Fig. (6). We see that the vacuum noise quadrature ξˆ + ξˆ† conjugate to F̂ controls theF̂ [ω] =o
2i~g n ˆ√ξ[ω] − ξˆ† [ω]xZPF κ

(E86)

and hence the back action force noise spectral density is
at low frequencies
SF F [ω] =4~2 g 2x2ZPF κ(E87)

in agreement with Eq. (3.18).
Introducing a quantity proportional to the cantilever(mechanical) susceptibility (within the rotating wave approximation we are using)χM [ω − Ω] ≡1,−i(ω − Ω) + γ20(E88)

we find from Eq. (E82)ẑ[ω] = ẑ0 [ω] −i
xZPF {χM [ω − Ω] − χM [ω + Ω]} F̂ [ω],
~
(E89)

27
where the equilibrium fluctuations in position are givenby√ ẑ0 [ω] ≡ − γ0 χM [ω − Ω]η̂[ω] + χM [ω + Ω]η̂ † [ω] .
(E90)
We can now obtain the power spectrum Szz describing
the total position fluctuations of the cantilever driven by
the mechanical vacuum noise plus the radiation pressure
shot noise. From Eqs. (E89, E90) we findSxx [ω]= Szz [ω]x2ZPF

in agreement with Eq. (3.10).
Notice also from Eq. (E94) that the quadrature of thevacuum noise which leads to the measurement imprecision is conjugate to the one which produces the backaction force noise as illustrated previously in Fig. (6).
Recall that the two quadratures of motion of a harmonic
oscillator in its ground state have no classical (i.e., symmetrized) correlation. Hence the symmetrized cross correlator
S̄IF [ω] = 0= γ0 |χ[ω − Ω]|2
(E91)
2
x
2
|χM [ω − Ω] − χM [ω + Ω]| SF F .
+ ZPF~2Note that (assuming high mechanical Q, i.e. γ0  Ω) theequilibrium part has support only at positive frequencies
while 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


0
[Ω]
S̄xx
0
S̄xx [ω] ≈ S̄xx [ω] 1 +S̄F F ,(E92)~20[ω] is the symmetrized spectral density for posiwhere S̄xx
tion fluctuations in the ground state given by Eq. (3.54).
Now that we have obtained the effect of the back action
noise on the position fluctuations, we must turn our attention to the imprecision of the measurement due to shotnoise in the output. The appropriate homodyne quadrature variable to monitor to be sensitive to the output
phase shift caused by position fluctuations isIˆ = b̂out + b̂†out ,(E93)which, using the input-output results above, can be writtenIˆ = −(ξˆ + ξˆ† ) + λx̂.

(E94)

We see that the cavity homodyne detector system acts
as a position transducer with gain
λ=

4g
√ .
xZPF κ

(E95)

The first term in Eq. (E94) represents the vacuum noisethat mixes with the homodyne local oscillator to produce
the shot noise in the output. The resulting measurement
imprecision (symmetrized) spectral density referred backto the position of the oscillator is
I
S̄xx
=

1
.
λ2

(E96)

Comparing this to Eq. (E87) we see that we reach thequantum limit relating the imprecision noise to the backaction noiseI
S̄xxS̄F F =

~2
4

(E97)

(E98)

vanishes. Because there is no correlation between the
output imprecision noise and the forces controlling theposition fluctuations, the total output noise referred backto the position of the oscillator is simplyIS̄xx,tot [ω] = S̄xx [ω] + S̄xx
(E99)


0
2
S̄ [Ω]~
0= S̄xx[ω] 1 + xx2 S̄F F +.
~4S̄F F

This expression again clearly illustrates the competition
between the back action noise proportional to the drive
laser intensity and the measurement imprecision noise
which is inversely proportional. We again emphasize that
all 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 frequency
will be a minimum at some optimal drive intensity. The
precise optimal value depends on the frequency chosen.
Typically this is taken to be the mechanical resonancefrequency where we find that the optimal coupling leads
to an optimal back action noiseS̄F F,opt =

~2 γ0
~2
=
.
0 [Ω]
4x2ZPF
2S̄xx(E100)

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 density
at frequency Ω referred to the position is simply twice
the vacuum value0S̄xx,tot [Ω] = 2S̄xx[Ω],(E101)

in agreement with Eq. (3.62). Evaluation of Eq. (E100)
at the optimal coupling yields the graph shown in
Fig. (6). The background noise floor is due to thefrequency independent imprecision noise with value
1 0
2 S̄xx [Ω]. The peak value at ω = Ω rises a factor of three
above this background.
We derived the gain λ in Eq. (E95) by direct solutionof the equations of motion. With the results we have derived above, it is straightforward to show that the Kubo
formula in Eq. (4.3) yields equivalent results. We havealready seen that the classical (i.e. symmetrized) correlations between the output signal Iˆ and the force F̂ which
couples to the position vanishes. However the Kubo formula evaluates the quantum (i.e. antisymmetric) correlations for the uncoupled system (A = g = 0). Hence we
have

28

i
χIF (t) = − θ(t)~ˆ − δt) + ξˆ† (t − δt) ,− ξ(twhere δt is a small (positive) time representing the delay
between the time when the vacuum noise impinges on
the cavity and when the resulting outgoing wave reaches
the homodyne detector. (More precisely it also compensates for certain small retardation effects neglected in thelimit ω  κ used in several places in the above derivations.) 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 limitω δt  1 the desired result
χIF [ω] = λ.

(E103)

Similarly we readily find that the small retardation
causes the reverse gain to vanish. Hence all our results
are consistent with the requirements needed to reach thestandard quantum limit.
Thus with this study of the specific case of an oscillator
parametrically coupled to a cavity, we have reproduced
all of the key results in Sec. V.E derived from completely
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. V.H.2. We again consider a high-Q cavity whose resonance frequency is modulated by a high-Q mechanical oscillator with co-ordinatex̂ (cf. Eqs. (E1) and (E64)). To use this system for amplification of a single quadrature, we will consider the typical case of a fast cavity (ωc  Ω), and take the “good cavity” limit, where Ω  κ. As explained in the main text,
the crucial ingredient for single-quadrature detection is
to take an amplitude-modulated cavity drive describedby the classical input field b̄in given in Eq. (5.101). As
before (cf. Eq. (E4)), we may write the cavity annihilation operator â as the sum of a classical piece ā(t) and aˆ only dˆ is influenced by the mechanicalquantum piece d;oscillator. ā(t) is easily found from the classical (noisefree) equations of motion for the isolated cavity; making
use of the conditions ωc  Ω  κ, we havepṄ κā(t) 'cos (Ωt + δ) e−iωc t
2Ω(E104)

To proceed with our analysis, we work in an interaction picture with respect to the uncoupled cavity andoscillator Hamiltonians. Making standard rotating-wave
approximations, the Hamiltonian in the interaction pic-

2i~g√

xZPF κˆ − ξˆ† (0)
ξ(0)


,

(E102)

0ture takes the simple form corresponding to Eq. (5.102b):Hint = ~Ã dˆ + dˆ† eiδ ĉ + e−iδ ĉ† X̂δ= ~Ã dˆ + dˆ†,
xZPF(E105)wherepÃ = A · ωc

Ṅ κ
,
4Ω

(E106)and in the second line, we have made use of the definitionof the quadrature operators X̂δ , Ŷδ given in Eqs. (5.92).
The form of Hint was discussed heuristically in the main
text in terms of Raman processes where photons are removed from the classical drive b̄in and either up or down
converted to the cavity frequency via absorption or emission of a mechanical phonon. Alternatively, we can thinkof the drive yielding a time-dependent cavity-oscillator
coupling which “follows” the Xδ quadrature. Note that
we made crucial of use of the good cavity limit (κ  Ω)to drop terms in Ĥint 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 so
far off resonance and these processes are suppressed.
Similar to Eqs. (E39) and (E81), the Heisenberg equations of motion (in the rotating frame) follow directlyfrom Hint and the dissipative terms in the total Hamiltonian:√κˆ iωc t − iÃ eiδ ĉ + e−iδ ĉ† (E107a)∂t dˆ = − dˆ − κξ(t)e
2

i
h 
γ0
√
∂t ĉ = − ĉ − γ0 η̂(t)eiΩt − ie−iδ Ã dˆ + dˆ† − f (t)2(E107b)ˆ represents the unavoidable noise in the
As before, ξ(t)cavity drive, and η̂(t), γ0 are the noisy force and damping resulting from an equilibrium bath coupled to themechanical oscillator. Note from Eq. (E107a) that as anticipated, the cavity is only driven by one quadrature of
the oscillator’s motion. We have also included a driving
force F (t) on the mechanical oscillator which has somenarrow bandwidth centered on the oscillator frequency;
this force is parameterized as:
F (t) =


2~
Re f (t)e−iΩt e−iδxZPF

(E108)

where f (t) is a complex function which is slowly varying
on the scale of an oscillator period.

29
The equations of motion are easily solved upon Fouriertransformation, resulting in:"X̂δ [ω] = −xZPF · χM [ω] i (f ∗ [−ω] − f [ω])(E109a)√+ γ0 eiδ η̂(ω + Ω) + e−iδ η̂ † (ω − Ω)#"Ŷδ [ω] = ixZPF · χM [ω] (−i) (f [ω] + f ∗ [−ω]) (E109b)
√+ γ0 eiδ η̂(ω + Ω) − e−iδ η̂ † (ω − Ω)
#

√ 
†
ˆ
ˆ
−2iÃχc [ω] κ ξ(ω + ωc ) + ξ (ω − ωc )be the homodyne signal integrated up to time t as inSec. III.B. We would like to understand the relationshipbetween the signal-to-noise ratio defined in Eq. (3.23),
and the rate at which information about the state of thequbit is being gained. The probability distribution for I
conditioned on the state of the qubit σ = ±1 is1−(I − σθ0 t)2p(I|σ) = √
exp
.
(F1)
2Sθθ t2πSθθ tBased on knowledge of this conditional distribution, we
now present two distinct but equivalent approaches togiving an information theoretic basis for the definition of
the measurement rate.
1. Method I

where the cavity and mechanical susceptibilities χc , χM
are defined in Eqs. (E43) and (E88).
As anticipated, the detected quadrature X̂δ is completely unaffected by the measurement: Eq. (E109a) is
identical to what we would have if there were no coupling
between the oscillator and the cavity. In contrast, theconjugate quadrature Ŷδ experiences an extra stochastic
force due to the cavity: this is the measurement backaction.
Turning now to the output field from the cavity b̂out ,we use the input-output relation Eq. (E37) to find in thelab (i.e. non-rotating) frame:−i(ω − ωc ) − κ/2 ˆb̂out [ω] = b̄out [ω] +ξ[ω]−i(ω − ωc ) + κ/2√Ã κχc [ω − ωc ] · X̂δ (ω − ωc )−i
xZPF
(E110)
The first term on the RHS simply represents the output
field from the cavity in the absence of the mechanicaloscillator and any fluctuations. It will yield sharp peaks
at the two sidebands associated with the drive, ω = ωc ±Ω. The second term on the RHS of Eq. (E110) represents
the reflected noise of the incident cavity drive. This noise
will play the role of the “intrinsic ouput noise” of thisamplifier.
Finally, the last term on the RHS of Eq. (E110) is theamplified signal: it is simply the amplified quadratureX̂δ of the oscillator. This term will result in a peak in
the output spectrum at the resonance frequency of thecavity, ωc . As there is no back-action on the measuredX̂δ quadrature, the added noise can be made arbitrarily
small by simply increasing the drive strength Ṅ (andhence Ã).

Suppose we start with an initial qubit density matrix

 1
0
(F2)
ρ0 = 2 1 .
0 2
After measuring for a time t, the new density matrix
conditioned on the results of the measurement is


p+ 0
(F3)
ρ1 =
0 p−
where it will be convenient to parameterize the two probabilities by the polarization m ≡ Tr(σz ρ1 ) by
p± =

1±m
.
2

(F4)

The information gained by the measurement is the entropy loss7 of the qubit
I = Tr(ρ1 ln ρ1 − ρ0 ln ρ0 ).

(F5)

We are interested in the initial rate of gain of information
at short times θ02 t  Sθθ where m will be small. In thislimit we have
I≈

m2
.
2

(F6)

We must now calculate m conditioned on the measurement result IX
mI ≡σp(σ|I).
(F7)
σ

From Bayes theorem we can express this in terms ofp(I|σ), which is the quantity we know,p(I|σ)p(σ).
00
σ 0 p(I|σ )p(σ )p(σ|I) = P(F8)Appendix F: Information Theory and Measurement Rate

Suppose that we are measuring the state of a qubit
via the phase shift ±θ0 from a one-sided cavity. Let I(t)7 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.

30
Using Eq. (F1) the polarization is easily evaluated


Iθ0
.
(F9)
mI = tanhSθθThe information gain is thus



2
1
Iθ0
I 2 θ0II = tanh2
≈
2
Sθθ
2 Sθθ

Appendix G: Number Phase Uncertainty(F10)

where the second equality is only valid for small |m|.
Ensemble averaging this over all possible measurement
results yields the mean information gain at short times
I≈

1 θ02
t
2 Sθθ

exactly the same result as Eq. (F11). (Here hI(t)iσ is themean value of I given that the qubit is in state σ.)(F11)which justifies the definition of the measurement rate
given in Eq. (3.24).

In this appendix, we briefly review the number-phase
uncertainty relation, and from it we derive the relationship between the spectral densities describing the photon
number fluctuations and the phase fluctuations. Consider a coherent state labeled by its classical amplitude
α


|α|2
|αi = exp −exp{αâ† }|0i.
(G1)2This is an eigenstate of the destruction operatorâ|αi = α|αi.

2. Method II

An alternative information theoretic derivation is toconsider the qubit plus measurement device to be a signaling channel. The two possible inputs to the channel
are the two states of the qubit. The output of the channel is the result of the measurement of I. By toggling
the qubit state back and forth, one can send information
through the signal channel to another party. The channel is noisy because even for a fixed state of the qubit,
the measured values of the signal I have intrinsic fluctuations. Shannon’s noisy channel coding theorem (Coverand Thomas, 1991) tells us the maximum rate at which
information can be reliably sent down the channel by toggling the state of the qubit and making measurements of
I. It is natural to take this rate as defining the measurement rate for our detector.
The reliable information gain by the receiver on a noisy
channel is a quantity known as the ‘mutual information’of the communication channel (Clerk et al., 2003; Coverand Thomas, 1991))
Z +∞ (X
R=−
dI p(I) ln p(I) −p(σ) [p(I|σ) ln p(I|σ)]
−∞

(G2)

It is convenient to make the unitary displacement transformation which maps the coherent state onto a new vacuum state and the destruction operator ontoâ = α + dˆ(G3)

where d annihilates the new vacuum. Then we haveˆN̄ = hN̂ i = h0|(α∗ + dˆ† )(α + d)|0i= |α|2 ,(G4)(∆N )2 = h(N̂ − N̄ )2 i = |α|2 h0|dˆdˆ† |0i = N̄ .

(G5)

and

Now define the two quadrature amplitudes
1X̂ = √ (â + â† )2iŶ = √ (â† − â).
2

(G6)
(G7)

Each of these amplitudes can be measured in a homodyne
experiment. For convenience, let us take α to be real andpositive. Then√hX̂i = 2α
(G8)

σ

(F12)
The first term is the Shannon entropy in the signal I
when 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 over
the two possible input values). Thus the first term issignal plus noise, the second is just the noise. Subtracting
the two gives the net information gain. Expanding thisexpression for short times yields
1 (hI(t)i+ − hI(t)i− )2
8
Sθθ t
θ02
=
t
2Sθθ
= Γmeas t

andhŶ i = 0.

(G9)

If the phase of this wave undergoes a small modulationdue for example to weak parametric coupling to a qubit
then one can estimate the phase by
hθi =

hŶ i
hX̂i

.

(G10)

This result is of course only valid for small angles, θ  1.
For N̄  1, the uncertainty will be

R =

(F13)

(∆θ)2 =

hŶ 2 i(hX̂i)21= 2

h0|dˆdˆ† |0i
1
=
.
2N̄
4N̄

(G11)

31
Thus using Eq. (G5) we arrive at the fundamental quantum uncertainty relation∆θ∆N =

ˆ is the vacuum noise obeying
where ξ(t)ˆ ξˆ† (t0 )] = δ(t − t0 ).
[ξ(t),

1.
2

Using the input-output theory described in Appendix E we can restate the results above in terms ofnoise spectral densities. Let the amplitude of the fieldcoming in to the homodyne detector beˆb̂in = b̄in + ξ(t)(G13)

We are using a flux normalization for the field operatorssoṄ = hb̂†in b̂in i = |b̄in |2

2

From this it follows that the shot noise spectral density
isSṄ Ṅ = Ṅ .

(G17)

Similarly the phase can be estimated from the quadrature operatori(b̂†in − b̂in )hb̂†in + b̂in i= hθ̂i + iˆ(ξˆ† − ξ)
2b̄in

(G18)

which has noise correlator1hδ θ̂(t)δ θ̂(0)i =

δ(t)

(G19)

4Ṅ
corresponding to the phase imprecision spectral density
Sθθ =

1

.

(G20)

(G15)

and

ˆ
ˆ
b̄∗in + ξˆ† (0))(b̄in + ξ(0))|0i− b̄inhṄ (t)Ṅ (0)i − Ṅ = h0|(b̄∗in + ξˆ† (t))(b̄in + ξ(t))(

θ̂ =

(G14)

(G12)

4

= Ṅ δ(t).

(G16)

able to reach the weak-coupling (i.e. op-amp) quantum
limit. Note that quantum amplifiers with feedback arealso treated in Courty et al. (1999); Grassia (1998).
On a heuristic level, we can understand the need for
either reflections or reverse gain to reach the quantum
limit. A problem with the “minimal” amplifier of thelast subsection was that its input impedance was too low
in comparison to its noise impedance ZN ∼ Za . From
general expression for the input impedance, Eq. (6.7d),
we see that having non-zero reverse gain (i.e. s12 6= 0)
and/or non-zero reflections (i.e. s11 6= 0 and/or s22 6= 0)
could lead to Zin  Za . This is exactly what occurs
when feedback is used to reach the quantum limit. Keepin mind that having non-vanishing reverse gain is dangerous: as we discussed earlier, an appreciable non-zero λ0I
can lead to the highly undesirable consequence that theamplifier’s input impedance depends on the impedanceof the load connected to its output (cf. Eq. (6.6)).

4Ṅ
We thus arrive at the fundamental quantum limit relationp

Sθθ SṄ Ṅ =

1
.
2

(G21)

Appendix H: Using feedback to reach the quantum limit

In Sec. (VI.B), we demonstrated that any two port amplifier whose scattering matrix has s11 = s22 = s12 = 0
will fail to reach quantum limit when used as a weakly
coupled op-amp; at best, it will miss optimizing the quantum noise constraint of Eq. (5.88) 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 “minimal” amplifier of the previous subsection, one can takeadvantage of noise correlations to reduce the back-action
current noise SII by a factor of two. As a result, one is

1. Feedback using mirrors

To introduce reverse gain and reflections into the “minimal” two-port bosonic amplifier of the previous subsection, we will insert mirrors in three of the four arms leading from the circulator: the arm going to the input line,
the arm going to the output line, and the arm goingto the auxiliary “cold load” (Fig. 7). Equivalently, one
could imagine that each of these lines is not perfectly
impedance matched to the circulator. Each mirror will
be described by a 2 × 2 unitary scattering matrix:âj,outb̂j,in(H1)= Uj ·b̂j,outâj,incos θj − sin θj
Uj =
(H2)
sin θj cos θjHere, the index j can take on three values: j = z for the
mirror in the input line, j = y for the mirror in the arm

32
cold
load

input linebz,out
bz,in

Z

by,in

by,outmirrorYay,outay,inoutput line
bx,in

az,in

ax,outaz,outax,inXbx,outcout =G1/2cin+(G-1)1/2v†in

cin
ideal
1-port
amp.

vin

vout

FIG. 7 Schematic of a modified minimal two-port amplifier,where partially reflecting mirrors have been inserted in the
input and output transmission lines, as well as in the line
leading to the cold load. By tuning the reflection coefficientof the mirror in the cold load arm (mirror Y ), we can induce negative feedback which takes advantage of correlations
between current and voltage noise. This then allows this system to reach the quantum limit as a weakly coupled voltage
op amp. See text for further description

going to the cold load, and j = x for the mirror in the
output line. The mode aj describes the “internal” mode
which 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 convenience. Note that θj = 0 corresponds to the case of no
mirror (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 matrix
s appearing in Eq. (6.3), as well as the noise operators
Fj . These may then in turn be used to obtain the op-amp
description of the amplifier, as well as the commutatorsof the added noise operators. These latter commutators
determine the usual noise spectral densities of the amplifier. Details and intermediate steps of these calculations
may be found in Appendix I.6.
As usual, to see if our amplifier can reach the quantum
limit when used as a (weakly-coupled) op-amp , we needto see if it optimizes the quantum noise constraint of
Eq. (5.88). We consider the optimal situation where both†the auxiliary modes of the amplifier (ûin and v̂in) are inthe vacuum state. The surprising upshot of our analysis(see Appendix I.6) is the following: if we include a small

amount of reflection in the cold load line with the correct
phase, 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. (5.88) 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 voltage
gain λV and power gain GP to decrease by a factor of
two compared to their θy = 0 values (cf. Eqs. (I60) and(I64)). This choice for θy also leads to Zin  Za ∼ ZN
(cf. (I62)), in keeping with our general expectations.
Physically, what does this precise tuning of θy correspond to? A strong hint is given by the behaviour of theamplifier’s cross-correlation noise S̄V I [ω] (cf. Eq. (I65c)).
In general, we find that S̄V I [ω] √is real and non-zero.
However, the tuning sin θy = −1/ G is exactly what isneeded to have S̄V I vanish. Also note from Eq. (I65a)
that this special tuning of θy decreases the back-action
current noise precisely by a factor of two compared toits value at θy = 0. A clear physical explanation now
emerges. Our original, reflection-free amplifier had correlations between its back-action current noise and output voltage noise (cf. Eq. (6.18c)). By introducing negative 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 Ṽ par˜ For antially cancel the original current fluctuations I).
optimal feedback (i.e. optimal choice of θy ), the current
noise is reduced by a half, and the new current noise is
not correlated with the output voltage noise. Note that
this is indeed negative (as opposed to positive) feedback–it results in a reduction of both the gain and the power
gain. To make this explicit, in the next section we will
map the amplifier described here onto a standard op-amp
with negative voltage feedback.
2. Explicit examples

To obtain a more complete insight, it is useful to go
back and consider what the reduced scattering matrix ofour system looks like when θy has been tuned to reachthe quantum limit. From Eq. (I58), it is easy to see thatat the quantum limit, the matrix s satisfies:
s11 = −s22
1
s12 =
s21
G

(H3a)
(H3b)

The second equation also carries over to the op-amp picture; at the quantum limit, one has:
λ0I =

1
λV
G

(H4)

One particularly simple limit is the case where there
are no mirrors in the input and output line (θx = θz = 0),33
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:√ 
0 1/ G
√
s =
(H5)
G
0
In this case, the principal effect of the weak mirror in
the cold-load line is to introduce a small amount of reverse gain. The amount of this reverse gain is exactly
what is needed to have the input impedance diverge(cf. Eq. (6.7d)). It is also what is needed to achieve an
optimal, noise-canceling feedback in the amplifier. To
see this last point explicitly, we can re-write the ampli˜ in terms of its original
fier’s back-action current noise (I)˜noises I0 and Ṽ0 (i.e. what the noise operators would havebeen in the absence of the mirror). Taking the relevant
limit of small reflection (i.e. r̃ ≡ sin θy goes to zero as|G| → ∞), we find that the modification of the current
noise operator is given by:√
2 Gr̃ Ṽ0˜
˜√
I ' I0 +(H6)
1 − Gr̃ Za
As claimed, the presence of a small amount of reflectionr̃ ≡ sin θy in the cold load arm “feeds-back” the original
voltage noise of√the amplifier Ṽ0 into the current. The
choice r̃ = −1/ G corresponds to a negative feedback,and optimally makes use of the fact that I˜0 and Ṽ0 are
˜
correlated 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 output
arms, this case is not really of practical interest. The
reversecurrent gain in this case may be small (i.e. λ0I ∝√1/ G), but it is not small enough: one finds that becauseof the non-zero λ0I , the amplifier’s input impedance isstrongly reduced in the presence of a load (cf. Eq. (6.6)).
There is a second simple limit we can consider which ismore practical. This is the limit where reflections in the
input-line mirror and output-line mirror are both strong.
Imagine we take θz√= −θx = π/2 − δ/G1/8 . If again
we set sin θy = −1/ G to reach the quantum limit, thescattering matrix now√ takes the form (neglecting terms
which are order 1/ G):


+1
0
s =
(H7)
δ 2 G1/4−1
2
In this case, we see that at the quantum limit, the reflection coefficients s11 and s22 are exactly what is needed
to have the input impedance diverge, while the reverse
gain coefficient s12 plays no role. For this case of strong
reflections in input and output arms, the voltage gain is
reduced compared to its zero-reflection value:
r
 2
Zb δ
λV →
G1/4
(H8)
Za 2
The power gain however is independent of θx , θz , and is
still given by G/2 when θy is tuned to be at the quantum
limit.

Va

+
G
- f

R1

R2

Vb

FIG. 8 Schematic of a voltage op-amp with negative feedback.

3. Op-amp with negative voltage feedback

We now show that a conventional op-amp with feedback can be mapped onto the amplifier described in
the 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 subsection, where an optimal tuning of the mirrors also lets
one reach the quantum limit.
More precisely, we consider a scattering description ofa non-inverting op-amp amplifier having negative voltage
feedback. The circuit for this system is shown in Fig. 8.
A fraction B of the output voltage of the amplifier is fed
back 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. The
op-amp with zero feedback is described by the “ideal”
amplifier of Sec. VI.B: at zero feedback, it is describedby Eqs. (6.11a)- (6.11d). For simplicity, we consider the
relevant case where:
Zb  R1 , R2  Za

(H9)In this limit, R1 and R2 only play a role through the
feedback fraction B, which is given by:
B=

R2
R1 + R2(H10)

Letting Gf denote the voltage gain at zero feedback(B = 0), an analysis of the circuit equations for our opamp system yields:
Gf
1 + B · Gf
B
λ0I =1 + B · Gf
Zb
Zout =
1 + B · Gf
Zin = (1 + B · Gf )Za
λV =

GP

G2f /2=
B · Gf + 2Zb /Za(H11a)(H11b)(H11c)(H11d)(H11e)

Again, Gf represents the gain of the amplifier in the absence of any feedback, Za is the input impedance at zero

34
feedback, and Zb is the output impedance at zero feedback.
Transforming this into the scattering picture yields ascattering matrix s satisfying:
BGf (Za − (2 + BGf )Zb )(H12a)BGf Za + (2 + BGf )2 Zb√
2 Za Zb Gf (1 + BGf )s21 = −
(H12b)BGf Za + (2 + BGf )2 Zb
B
s12 =
s21
(H12c)
Gf

s11 = −s22 = −Note the connection between these equations and the necessary 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. These
are given in general by:
p
†
F̂a =
1 − |s11 |2 − |s12 |2 + |l|2 · ûin + l · v̂in
(H13)
p
†
F̂b =
|s21 |2 + |s22 |2 − 1 · v̂in
(H14)
∗
∗
s11 s21 + s12 s22
(H15)
l = p
|s21 |2 + |s22 |2 − 1
Applying this to the s matrix for our op-amp, and then†taking the auxiliary modes ûin and v̂into be in the vacuum state, we can calculate the minimum allowed S̄V V
and S̄II for our non-inverting op-amp amplifier. One canthen calculate the product S̄V V S̄II and compare against
the quantum-limited value (S̄IV 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 limited
value. However, if one takes the large Gf limit while

S̄F F [ω] = π~Xkeeping B non-zero but finite, one obtains:
2S̄V V S̄II → (~ω)

2B
1−
+O
Gf



1
Gf

2 !
(H16)

Thus, for a fixed, non-zero feedback ratio B, it is possible
to reach the quantum limit. Note that if B does not tend
to zero as Gf tends to infinity, the voltage gain of thisamplifier will be finite. The power gain however will be
proportional to Gf and will be large. If one wants alarge voltage gain, one could set B to go to zero with
Gf i.e. B ∝ √1 . In this case, one will still reachGfthe quantum limit in the large Gp
f limit, and the voltage
gain will also be large (i.e. ∝ Gf ). Note that in allthese limits, the reflection coefficients s11 and s22 tendto −1 and 1 respectively, while the reverse gain tends
to 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 input
and at the output (cf. Eq. (H7)). We thus see yet again
how the use of feedback allows the system to reach thequantum limit.
Appendix I: Additional Technical Details

This appendix provides further details of calculations
presented in the main text.

1. Proof of quantum noise constraint

Note first that we may write the symmetrized Iˆ and F̂
noise correlators defined in Eqs. (4.4a) and (4.4b) as sums
over transitions between detector energy eigenstates:hi|ρ̂0 |ii · |hf |F̂ |ii|2 [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)](I1)

ˆ 2 [δ(Ef − Ei + ~ω) + δ(Ef − Ei − ~ω)]hi|ρ̂0 |ii · |hf |I|ii|

(I2)

i,f

S̄II [ω] = π~Xi,fHere, ρ̂0 is the stationary density matrix describing thestate of the detector, and |ii (|f i) is a detector energy
eigenstate with energy Ei (Ef ). Eq. (I1) expresses thenoise at frequency ω as a sum over transitions. Eachtransition starts with an an initial detector eigenstate|ii, occupied with a probability hi|ρ0 |ii, and ends with afinal detector eigenstate |f i, where the energy difference
between the two states is either +~ω or −~ω . Further,each transition is weighted by an appropriate matrix element.

To proceed, we fix the frequency ω > 0, and let theindex ν label each transition |ii → |f i contributing to thenoise. More specifically, ν indexes each ordered pair ofdetector energy eigenstates states {|ii, |f i} which satisfy
Ef − Ei ∈ ±~[ω, ω + dω] and hi|ρ0 |ii =6 0. We can now
consider the matrix elements of Iˆ and F̂ which contributeto S̄II [ω] and S̄F F [ω] to be complex vectors ~v and w.
~

35
Letting δ be any real number, let us define:[w]~ ν = hf (ν)|F̂ |i(ν)i(I3)(ˆ
e−iδ hf (ν)|I|i(ν)i
if Ef (ν) − Ei(ν) = +~ω,[~v ]ν =iδˆe hf (ν)|I|i(ν)i
if Ef (ν) − Ei(ν) = −~ω.
(I4)
Introducing an inner product h·, ·iω via:X∗
h~a, ~biω = πhi(ν)|ρ̂0 |i(ν)i · (aν ) bν ,

(I5)

ν

we see that the noise correlators S̄II and S̄F F may be
written as:
S̄II [ω]dω = h~v , ~v iω
S̄F F [ω]dω = hw,
~ wi
~ ω

(I6)
(I7)

We may now employ the Cauchy-Schwartz inequality:2

h~v , ~v iω hw,~ wi~ ω ≥ |h~v , wi
~ ω|

(I8)

A straightforward manipulation shows that the realpart of h~v , wi
~ ω is determined by the symmetrized crosscorrelator S̄IF [ω] defined in Eq. (4.4c):
Re h~v , wi
~ ω = Re eiδ S̄IF [ω] dω(I9)In contrast, the imaginary part of h~v , wi
~ ω is independentof S̄IF ; instead, it is directly related to the gain χIF andreverse gain χF I of the detector:~∗ dω (I10)Im h~v , wi
~ ω = Re eiδ χIF [ω] − [χF I [ω]]2
Substituting Eqs. (I10) and (I9) into Eq. (I8), one immediately finds the quantum noise constraint given in
Eq. (4.16). As in the main text, we let χ̃IF = χIF − χ∗F I .
Maximizing the RHS of this inequality with respect tothe phase δ, one finds that the maximum is achieved forδ = δ0 = − arg(χ̃IF ) + δ̃0 withtan 2δ̃0 = −|S̄IF | sin 2φ(~/2)|χ̃IF | + |S̄IF | cos 2φ(I11)

for each pair of initial and final states |ii, |f i contributing
to S̄F F [ω] and S̄II [ω] (cf. Eq. (I1)). Note that this not
the same as requiring Eq. (I13) to hold for all possible
states |ii and |f i. This proportionality condition in turnimplies a proportionality between the input and output(unsymmetrized) quantum noise spectral densities:
SII [ω] = |α|2 SF F [ω](I14)It thus also follows that the imaginary parts of the input
and output susceptibilities are proportional:
Im χII [ω] = |α|2 Im χF F [ω],(I15)

as well as the symmetrized input and output noise (i.e.
Eq. (4.18)). Finally, one can also use Eq. (I13) to relate
the unsymmetrized I-F quantum noise correlator SIF [ω]to SF F [ω]: (cf. Eq. (4.7)):(e−iδ α∗ SF F [ω] if ω > 0,
SIF [ω] =(I16)eiδ α∗ SF F [ω]if ω < 0Note that SF F [ω] is necessarily real and positive.
Finally, for a detector with quantum-ideal noise properties, the magnitude of the constant α can be found from
Eq. (4.18). The phase of α can also be determined from:−Im α~|χ̃IF |/2= p
cos δ̃0|α|
S̄II S̄F F(I17)

For zero frequency or for a large detector effective temperature, this simplifies to:~χ̃IF /2
−Im α
= p
|α|
S̄II S̄F F(I18)

Note importantly that to have a non-vanishing gain
and power gain, one needs Im α 6= 0. This in turn places
a very powerful constraint on a quantum-ideal detectors:
all transitions contributing to the noise must be to final
states |f i which are completely unoccupied. To see this,imagine a transition taking an initial state |ii = |ai toa final state |f i = |bi makes a contribution to the noise.
For a quantum-ideal detector, Eq. (I13) will be satisfied:

where φ = arg(S̄IF χ̃IF ∗). At δ = δ0 , Eq. (4.16) becomesthe final noise constraint of Eq. (4.11).
The proof given here also allows one to see what must
be done in order to achieve the “ideal” noise conditionof Eq. (4.17): one must achieve equality in the CauchySchwartz inequality of Eq. (I8). This requires that thevectors ~v and w~ be proportional to one another; there
must exist a complex factor α (having dimensions [I]/[F ])such that:

where the plus sign corresponds to Eb > Ea , the minus to Ea > Eb . If now the final state |bi was also occupied (i.e. hb|ρ̂0 |bi =6 0), then the reverse transition|i = bi → |f = ai) would also contribute to the noise.
The proportionality condition of Eq. (I13) would now require:~v = α · w~

ˆ = e∓iδ αha|F̂ |bi
ha|I|bi

Equivalently, we have that(eiδ αhf |F |ii
if Ef − Ei = +~ω
hf |I|ii =−iδe αhf |F |ii if Ef − Ei = −~ω.

(I12)

(I13)

ˆ = e±iδ αhb|F̂ |ai
hb|I|ai

(I19)

(I20)

As Iˆ and F̂ are both Hermitian operators, and as α musthave an imaginary part in order for there to be gain, we
have a contradiction: Eq. (I19) and (I20) cannot both betrue. It thus follows that the final state of a transition

36
contributing to the noise must be unoccupied in orderfor Eq. (I13) to be satisfied and for the detector to have
ideal noise properties. Note that this necessary asymmetry in the occupation of detector energy eigenstates
immediately tells us that a detector or amplifier cannot
reach the quantum limit if it is in equilibrium.

α(SF F [ω] − SF F [−ω])~= 2α [−Im χF F [ω]]χ̃IF =(I22)

Using this result and the relation between χF F and χII
in Eq. (I15), we can write the power gain in the absenceof reverse gain, GP (c.f. Eq. (5.52)), as

2. Proof that a noiseless detector does not amplify

With the above results in hand, we can now prove assertions made in Sec. IV.A.4 that detectors which evadethe quantum noise constraint of Eq. (4.11) and simply
satisfyS̄F F S̄II = |S̄IF |2

proportionality relation in Eq. (4.8b) for χ̃IF yields:(I21)

are at best transducers, as their power gain is limited tobeing at most one.
The first way to make the RHS of Eq. (4.11) vanish is
to have χIF = χ∗F I . We have already seen that whenever this relation holds, the detector power gain cannot
be any larger than one (c.f. Eq. (5.53)). Now, imagine
that the detector also has a minimal amount of noise,
i.e. Eq. (I21) also holds. This latter fact implies that the
proportionality condition of Eq. (I13) also must hold. In
this situation, the detector must have a power gain ofunity, and is thus a transducer. There are two possibilities to consider here. First, S̄F F and S̄II could bothbe non-zero, but perfectly correlated: |S̄IF |2 = S̄F F S̄II .
In this case, the proportionality constant α must be real(c.f. Eq. (I17)). Using this fact along with Eqs. (I16) and(4.8b), one immediately finds that SF F [ω] = SF F [−ω].
This implies the back-action damping γ associated withthe detector input vanishes (c.f. Eq. (2.12)). It thus follows immediately from Eq. (5.52) and Eq. (5.53) that thepower gain GP,rev (defined in a way that accounts for thereverse gain) is exactly one. The detector is thus simply atransducer. The other possibility here is that χIF = χ∗F I
and one or both of S̄II , S̄F F are equal zero. Note that if
the symmetrized noise vanishes, then so must the asymmetric part of the noise. Thus, it follows that either thedamping induced by the detector input, γ, or that induced by the output, γout (c.f. Eq. (5.48)) (or both) mustbe zero. Eqs. (5.52) and (5.53) then again yield a power
gain GP,rev = 1. We thus have shown that any detector
which has χF I = χ∗IF and satisfies S̄F F S̄II = |S̄IF |2 mustnecessarily be a transducer, with a power gain precisely
equal to one.
A second way to make the RHS of Eq. (4.11) vanish is
to have S̄IF /χ̃IF be purely imaginary and larger in magnitude than ~/2. Suppose this is the case, and that thedetector also satisfies the minimal noise requirement of
Eq. (I21). Without loss of generality, we take χ̃IF to bereal, implying that S̄IF is purely imaginary. Eqs. (I10)
and (I11) then imply that the phase factor eiδ appearing in the proportionality relation of Eq. (I16) is purely
imaginary, while the constant α is purely real. Using thisGP = 1/ |1 − χF I /χIF |

2

(I23)

If the reverse gain vanishes (i.e. χF I = 0), we immediately find that GP = 1: the detector has a power gain of
one, and is thus simply a transducer. If the reverse gain
is non-zero, we must take the expression for GP above
and plug it into Eq. (5.53) for the power gain with reverse gain, GP,rev . Some algebra again yields that thefull power gain is at most unity. We again have the conclusion that the detector does not amplify.

3. Simplifications for a quantum-limited detector

In this appendix, we derive the additional constraints
on the property of a detector that arise when it satisfies
the quantum noise constraint of Eq. (4.17). We focus on
the ideal case where the reverse gain χF I vanishes.
To start, we substitute Eq. (I16) into Eqs. (4.8b)(4.8a); writing SF F [ω] in terms of the detector effective
temperature Teff (cf. Eq. (2.8)) yields:~λ[ω]= −e−iδ ~ [−Im χF F [ω]]
(I24)
2




~ω
(Im α) coth
+ i (Re α)
2kB Teff
S̄IF [ω] = e−iδ ~ [−Im χF F [ω]]
(I25)




~ω
(Re α) coth
− i (Im α)
2kB Teff
To proceed, let us write:e−iδ =

λ −iδ̃
e
|λ|

(I26)

The condition that |λ| is real yields the condition:tan δ̃ =

Re α
tanh
Imα



~ω
2kB Teff(I27)

We now consider the relevant limit of a large detector
power gain GP . GP is determined by Eq. (5.56); the only
way this can become large is if kB Teff /(~ω) → ∞ while
Im α does not tend to zero. We will thus take the large
Teff limit in the above equations while keeping both α andthe phase of λ fixed. Note that this means the parameterδ̃ must evolve; it tends to zero in the large Teff limit. In

37
Using this lowest-order self energy, Eq. (I33) yields:
this limit, we thus find for λ and S̄IF :
"
"
2 ##
~
~ω
~λ[ω]
GR [ω] =222= −2e−iδ kB Teff γ[ω] (Im α) 1 + O
m(ω − Ω ) − A Re DR [ω] + iω(γ0 + γ[ω])2kB Teff(I36)(I28)
 R ∗AG [ω] = G [ω](I37)~ωS̄IF [ω] = 2e−iδ kB Teff γ[ω] (Re α) 1 + O
KR
G [ω] = −2iIm G [ω] ×kB Teff




~ω
(I29)
+
γ[ω]
coth
γ0 coth 2kB~ω
Tbath
2kB Teff
Thus, in the large power-gain limit (i.e. large Teff
limit), the gain λ and the noise cross-correlator S̄IF havethe same phase: S̄IF /λ is purely real.
4. Derivation of non-equilibrium Langevin equation

In this appendix, we prove that an oscillator weakly
coupled to an arbitrary out-of-equilibrium detector is described by the Langevin equation given in Eq. (5.41), an
equation which associates an effective temperature anddamping kernel to the detector. The approach taken here
is directly related to the pioneering work of Schwinger(Schwinger, 1961).
We start by defining the oscillator matrix Keldysh
green function:
 KG (t) GR (t)Ǧ(t) =(I30)
GA (t)
0where GR (t − t0 ) = −iθ(t − t0 )h[x̂(t), x̂(t0 )]i, GA (t − t0 ) =iθ(t0 − t)h[x̂(t), x̂(t0 )]i, and GK (t − t0 ) = −ih{x̂(t), x̂(t0 )}i.
At zero coupling to the detector (A = 0), the oscillator
is only coupled to the equilibrium bath, and thus Ǧ0 has
the standard equilibrium form:!
~−2Im g0 [ω] coth 2kB~ω
g
[ω]
0
T
bath
Ǧ0 [ω] =m
g0 [ω]∗
0(I31)
where:
g0 [ω] =1
ω 2 − Ω2 + iωγ0 /m(I32)

and where γ0 is the intrinsic damping coefficient, andTbath is the bath temperature.
We next treat the effects of the coupling to the detector
in perturbation theory. Letting Σ̌ denote the corresponding self-energy, the Dyson equation for Ǧ has the form:



−1 
−1
0
ΣA [ω]Ǧ[ω]= Ǧ0 [ω]−(I33)ΣR [ω] ΣK [ω]To lowest order in A, Σ̌[ω] is given by:Σ̌[ω] = A2 Ď[ω]
(I34)
2 Z
A
≡
dt eiωt
(I35)
~


0
iθ(−t)h[F̂ (t), F̂ (0)]i−iθ(t)h[F̂ (t), F̂ (0)]i −ih{F̂ (t), F̂ (0)}iγ0 + γ[ω](I38)where γ[ω] is given by Eq. (2.12), and Teff [ω] is defined by
Eq. (2.8). The main effect of the real part of the retarded
F̂ Green function DR [ω] in Eq. (I36) is to renormalize theoscillator frequency Ω and mass m; we simply incorporate
these shifts into the definition of Ω and m in what follows.
If Teff [ω] is frequency independent, then Eqs. (I36)- (I38) for Ǧ corresponds exactly to an oscillator coupled to two equilibrium baths with damping kernels γ0and γ[ω]. The correspondence to the Langevin equation Eq. (5.41) is then immediate. In the more generalcase where Teff [ω] has a frequency dependence, the correlators GR [ω] and GK [ω] are in exact correspondence
to what is found from the Langevin equation Eq. (5.41):
GK [ω] corresponds to symmetrized noise calculated from
Eq. (5.41), while GR [ω] corresponds to the response coefficient of the oscillator calculated from Eq. (5.41). Thisagain proves the validity of using the Langevin equation
Eq. (5.41) to calculate the oscillator noise in the presenceof the detector to lowest order in A.
5. Linear-response formulas for a two-port bosonic amplifier

In this appendix, we use the standard linear-response
Kubo formulas of Sec. V.F to derive expressions forthe voltage gain λV , reverse current gain λ0I , input
impedance Zin and output impedance Zout of a two-portbosonic voltage amplifier (cf. Sec. VI). We recover thesame expressions for these quantities obtained in Sec. VI
from the scattering approach. We stress throughout thisappendix the important role played by the causal structure of the scattering matrix describing the amplifier.
In applying the general linear response formulas, wemust bear in mind that these expressions should be applied 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 output (i.e. Ib = 0, no load at the output drawing current).
These two conditions define the uncoupled amplifier. Using the definitions of the voltage and current operators(cf. Eqs. (6.2a) and (6.2b)), they take the form:âin [ω] = −âout [ω](I39a)b̂in [ω] = b̂out [ω](I39b)

38
The scattering matrix equation Eq. (6.3) then allows us
to solve for âin and âout in terms of the added noise
operators F̂a and F̂b .
s12
1 − s22F̂a [ω] −F̂b [ω](I40a)âin [ω] = −DDs21
1 + s11b̂in [ω] = −F̂a [ω] +F̂b [ω](I40b)
D
D
where D is given in Eq. (6.8), and we have omitted writing the frequency dependence of the scattering matrix.
Further, as we have already remarked, the commutatorsof the added noise operators is completely determined by
the scattering matrix and the constraint that output operators have canonical commutation relations. The nonvanishing commutators are thus given by:hiF̂a [ω], F̂a† (ω 0 ) = 2πδ(ω − ω 0 ) 1 − |s11 |2 − |s12 |2hF̂b [ω], F̂b† (ω 0 )ihF̂a [ω], F̂b† (ω 0 )i(I41a)
= 2πδ(ω − ω ) 1 − |s21 | − |s22 |202(I41b)=0−2πδ(ω − ω ) (s11 s∗21 + s12 s∗22 )(I41c)

The above equations, used in conjunction with
Eqs. (6.2a) and (6.2b), provide us with all all the information needed to calculate commutators between current
and voltage operators. It is these commutators which enter 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 before
they propagate to the output. Causality manifests itself
in the energy dependence of the scattering matrix: as afunction of energy, it is an analytic function in the upper
half complex plane.
a. Input and output impedances

Eq. (5.84) is the linear response Kubo formula for
the input impedance of a voltage amplifier. Recall
that the input operator Q̂ for a voltage amplifier is related to the input current operator Iˆa via −dQ̂/dt =Iˆa (cf. Eq. (5.83)). The Kubo formula for the input
impedance may thus be re-written in the more familiar
form:


Z
i
i ∞
Yin,Kubo [ω] ≡−dth[Iˆa (t), Iˆa (0)]ieiωt
(I42)
ω
~ 0
where Yin [ω] = 1/Zin [ω],Using the defining equation for Iˆa (Eq. (6.1b)) andEq. (I39a) (which describes an uncoupled amplifier), we
obtain:Yin,Kubo [ω] =
(I43)
Z ∞
Z ∞


0 0
0
0
2
dω ω
dteiωtΛaa (ω 0 ) e−iω t − eiω t
Za 0
2π ω
0

where we have defined the real function Λaa [ω] for ω > 0via:hiâin [ω], â†in (ω 0 ) = 2πδ(ω − ω 0 )Λaa [ω](I44)
It will be convenient to also define Λaa [ω] for ω < 0
via Λaa [ω] = Λaa [−ω]. Eq. (I43) may then be written as:
Z ∞ Z ∞
02
dω 0 ω 0
Yin,Kubo [ω] =dt
Λaa (ω 0 )ei(ω−ω )t
Za 0
2π
ω
−∞
Z ∞
i
ω 0 Λaa (ω 0 )/ZaΛaa [ω]
+
P
dω 0
=
Za
πω
ω − ω0−∞(I45)
Next, by making use of Eq. (I40a) and Eqs. (I41) forthe commutators of the added noise operators, we can
explicitly evaluate the commutator in Eq. (I44) to calculate Λaa [ω]. Comparing the result against the result
Eq. (6.7d) of the scattering calculation, we find:Λaa [ω]= Re Yin,scatt [ω]
Za

(I46)

where Yin,scatt [ω] is the input admittance of the amplifier obtained from the scattering approach. Returning to
Eq. (I45), we may now use the fact that Yin,scatt [ω] isan analytic function in the upper half plane to simplify
the second term on the RHS, as this term is simply a
Kramers-Kronig integral:Z ∞1
ω 0 Λaa (ω 0 )/Za
P
dω 0
πω
ω − ω0−∞Z ∞1
ω 0 Re Yin,scatt (ω 0 )
=
P
dω 0
πω
ω − ω0−∞= Im Yin,scatt [ω](I47)
It thus follows from Eq. (I45) that input impedance calculated 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 given
in Eq. (5.85). As the steps are completely analogous to
the above calculation, we do not present it here. One
again recovers Eq. (6.7c), as found previously within the
scattering approach.

b. Voltage gain and reverse current gain

Within linear response theory, the voltage gain of theamplifier (λV ) is determined by the commutator between
the “input operator” Q̂ and V̂b (cf. Eq. (4.3); recall thatQ̂ is defined by dQ̂/dt = −Iˆa . Similarly, the reverse
current gain (λ0I ) is determined by the commutator between Iˆa and Φ̂, where Φ̂ is defined via dΦ̂/dt = −V̂b(cf. Eq. (4.6)). Similar to the calculation of the input
impedance, to properly evaluate the Kubo formulas for39
the gains, we must make use of the causal structure of
the scattering matrix describing our amplifier.
Using the defining equations of the current and voltage operators (cf. Eqs. (6.1a) and (6.1b)), as well as
Eqs. (I39a) and (I39b) which describe the uncoupled amplifier, the Kubo formulas for the voltage gain and reverse
current gain become:r
Zb
λV,Kubo [ω] = 4
(I48)
Za
Z ∞

Z ∞
0
dω 0
dt eiωt Re×Λba (ω 0 )e−iω t
2π
0
0
r
Zb
λ0I,Kubo [ω] = −4
(I49)
Za
Z ∞

Z ∞
0
dω 0
×
dt eiωt Re
Λba (ω 0 )eiω t
2π00where we define the complex function Λba [ω] for ω > 0via:hib̂in [ω], â†in (ω 0 ) ≡ (2πδ(ω − ω 0 )) Λba [ω] (I50)
We can explicitly evaluate Λba [ω] by using Eqs. (I40a)(I41) to evaluate the commutator above. Comparing theresult against the scattering approach expressions for thegain and reverse gain (cf. Eqs. (6.7a) and (6.7b)), one
finds:
∗Λba [ω] = λV,scatt [ω] − λ0I,scatt [ω](I51)
Note crucially that the two terms above have different
analytic 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] forω < 0 via Λba [−ω] = Λ∗ba [ω], we can re-write Eqs. (I48)
and (I49) in terms of principle part integrals.


Z ∞
0
i
Zb
0 Λba (ω )Λba [ω] + Pdω
λV,Kubo [ω] =Zaπω − ω0
−∞


Z ∞
0
i
Zb
0 Λba (ω )0λI,Kubo [ω] = −Λba [ω] + PdωZaπω + ω0−∞(I52)
Using the analytic properties of the two terms in Eq. (I51)for Λba [ω], we can evaluate the principal part integrals
above as Kramers-Kronig relations. One then finds that
the Kubo formula expressions for the voltage and current gain coincide precisely with those obtained from thescattering 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 energy

dependence 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 output lines). Because of these stubs, a wavepacket incident
on the amplifier will take a time τ = 2a/v to be either
reflected or transmitted, where v is the characteristic velocity of the transmission line. This situation is describedby a scattering matrix which has the form:s[ω] = e2iωa/v · s̄(I53)

where s̄ is frequency-independent and real. To further
simplify things, let us assume that s̄11 = s̄22 = s̄12 = 0.
Eqs. (I51) then simplifies toΛba [ω] = = s21 [ω] = s̄21 eiωτ(I54)

where the propagation time τ = 2a/v We then have:Z ∞Zb
λV [ω0 ] = 2s̄21
dt eiω0 t δ(t − τ )
(I55)
Za
0
r
Z ∞
Zb
s̄21
λI [ω0 ] = −2
dt eiω0 t δ(t + τ ) (I56)Za0r

If we now do the time integrals and then take the limit
τ → 0+ , we recover the results of the scattering approach(cf. Eqs. (6.7a) and (6.7b)); 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!

6. Details for the two-port bosonic voltage amplifier withfeedback

In this appendix, we provide more details on the calculations for the bosonic-amplifier-plus-mirrors system discussed in Sec. H. Given that the scattering matrix foreach of the three mirrors is given by Eq. (H2), and that
we know the reduced scattering matrix for the mirror-freesystem (cf. Eq. (6.10)), we can find the reduced scatteringmatrix and noise operators for the system with mirrors.
One finds that the reduced scattering matrix s is now
given by:
s =

1
×
(I57)
M
√


sin θ√
G sin θx sin θy− cos θ√z +x cos θz sin θy
G cos θx cos θzsin θx + G sin θy sin θz

where the denominator M describes multiple reflection
processes:√
M = 1 + G sin θx sin θz sin θy(I58)
Further, the noise operators are given by:

40



Fa
Fb


=

1
M



√ cos θy cos θz− G cos θx cos θy sin θz

The next step is to convert the above into the op-amp
representation, and find the gains and impedances of theamplifier, along with the voltage and current noises. The
voltage gain is given by:
r
λV =

√

G −√1 cos θz sin θx sin θyG − 1 cos θxsin θyλ0I = − √ λV
GS̄II =√
1 − G sin θy 1 + sin θz√
Zin = Za·
1 + G sin θy 1 − sin θz

Similarly, the output impedance only depends on theamount of reflection in the output line and the in the
cold-load line:√
1 + G sin θy 1 + sin θx√·
Zout = Zb
1 − G sin θy 1 − sin θx

(I63)

√
Note that as sin θy tends to −1/ G , both the input
admittance and output impedance tend to zero.
Given that we now know the op-amp parameters of ouramplifier, we can use Eq. (6.5) to calculate the amplifier’s
power gain GP . Amazingly, we find that the power gain
is completely independent of the mirrors in the input andoutput lines:
GP =G
1 + G sin2 θy

S̄V V

S̄V I

(I62)

(I64)

√
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 reflection
free case (i.e. θy = 0).
Turning to the noise spectral densities, we assume theoptimal situation where both the auxiliary modes ûin and†v̂inare in the vacuum state. We then find that both Iˆ˜
and Ṽˆ are independent of the amount of reflection in the


(I59)



2~ω 1 − sin θz
×
Za 1 + sin θz




2

(I61)

The input impedance is determined by the amount ofreflection in the input line and in the line going to thecold load:

uin
†
vin

output line (e.g. θx ):√ZB
1 + sin θx 1 − sin θz
2 G
√
·
(I60)
ZA 1 − G sin θycos θxcos θzwhile the reverse gain is related to the voltage gain by
the simple relation: G sin θy + cos(2θy ) 
 √
2 
G sin θy − 1
1 + sin θz= ~ωZa×
1 − sin θz
3 + cos(2θy )
1
−
4
2G
√
G(1 − 1/G) sin θy + cos2 θy√=
1 − G sin θy(I65a)(I65b)(I65c)

As could be expected, introducing reflections in the input
line (i.e. θz 6= 0) has the opposite effect on S̄II versus
S̄V V : if one is enhanced, the other is suppressed.
It thus follows that the product of noise spectral
densities appearing in the quantum noise constraint of
Eq. (5.88) is given by (taking the large-G limit):
S̄II S̄V V
2

(~ω)

=


1 + G sin2 θy
2 − sin2 θy · 2 (I66)√
1 − G sin θyNote that somewhat amazingly, this product (and hence
the amplifier noise temperature) is completely independent of the mirrors in the input and output arms (i.e. θzand θx ). This is a result of both S̄V V and S̄II having
no dependence on the output mirror (θx ), and their having opposite dependencies on the input mirror (θz ). Also
note that Eq. (I66) does indeed reduce to the result of
the last subsection: if θy = 0 (i.e. no reflections in theline going to the cold load), the product S̄II S̄V V is equal2
to precisely twice√ the quantum limit value of (~ω) . For
sin(θy ) = −1/ G, the RHS above reduces to one, implying that we reach the quantum limit for this tuning of
the mirror in the cold-load arm.

References
Amin, M. H. S., and D. V. Averin, 2008, Phys. Rev. Lett.
100, 197001.
Arcizet, O., P.-F. Cohadon, T. Briant, M. Pinard, and
A. Hedimann, 2006, Nature 444, 71.
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.

41
Brown, K. R., J. Britton, R. J. Epstein, J. Chiaverini,
D. Leibfried, and D. J. Wineland, 2007, Phys. Rev. Lett.
99, 137205.
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.
Clerk, A. A., S. M. Girvin, and A. D. Stone, 2003, Phys. Rev.
B 67, 165324.
Cohen-Tannoudji, C., J. Dupont-Roc, and G. Grynberg, 1989,
Photons and Atoms - Introduction to Quantum Electrodynamics (Wiley, New York).
Courty, J. M., F. Grassia, and S. Reynaud, 1999, Europhys.
Lett. 46, 31.
Cover, T., and J. Thomas, 1991, Elements of Information
Theory (Wiley, New York).
Deblock, R., E. Onac, L. Gurevich, and L. P. Kouwenhoven,
2003, Science 301, 203.
Devoret, M., 1997, in Quantum Fluctuations (Les Houches
Session LXIII) (Elsevier, Amsterdam), pp. 351–86.
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.
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.
Scully, and K. Wodkiewicz, 1990b, Phys. Rev. A 41, 381.
Gigan, S., H. Böhm, M. Paternostro, F. Blaser, J. B.
Hertzberg, K. C. Schwab, D. Bauerle, M. Aspelmeyer, and
A. Zeilinger, 2006, Nature 444, 67.
Glauber, R. J., 2006, Rev. Mod. Phys. 78, 1267.
Grassia, F., 1998, Fluctuations quantiques et thermiques dansles transducteurs electromecaniques, Ph.D. thesis, Universite Pierre et Marie Curie.
Harris, J. G. E., B. M. Zwickl, and A. M. Jayich, 2007, Review
of Scientific Instruments 78, 013107.
Höhberger-Metzger, C., and K. Karrai, 2004, Nature 432,
1002.
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.
Mallat, S., 1999, A Wavelet Tour of Signal Processing (Academic, 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.
Onac, E., F. Balestro, B. Trauzettel, C. F. J. Lodewijk, andL. P. Kouwenhoven, 2006, Phys. Rev. Lett. 96, 026803.
Schliesser, A., P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J.
Kippenberg, 2006, Phys. Rev. Lett. 97, 243905.
Schoelkopf, R. J., P. J. Burke, A. A. Kozhevnikov, D. E.
Prober, and M. J. Rooks, 1997, Phys. Rev. Lett. 78, 3370.
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.
Schwinger, J., 1961, J. Math. Phys. 2, 407.
Teufel, J. D., C. A. Regal, and K. W. Lehnert, 2008, New J.
Phys. 10, 095002.

Thompson, J. D., B. M. Zwickl, A. M. Jayich, F. Marquardt,
S. M. Girvin, and J. G. E. Harris, 2008, Nature (London)
452, 06715.
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).
Wilson-Rae, I., N. Nooshi, W. Zwerger, and T. J. Kippenberg,
2007, Phys. Rev. Lett. 99, 093901.
Yurke, B., 1984, Phys. Rev. A 29, 408.
Yurke, B., and J. S. Denker, 1984, Phys. Rev. A 29, 1419.