PHYSICAL REVIEW B 70, 245306 (2004)

Quantum-limited position detection and amplification: A linear response perspective
A. A. Clerk
Departments of Applied Physics and Physics, Yale University, New Haven, Connecticut 06511, USA(Received 6 July 2004; published 7 December 2004)
Using standard linear response relations, we derive the quantum limit on the sensitivity of a generic linearresponse position detector, and the noise temperature of a generic linear amplifier. Particular emphasis is placedon the detector’s effective temperature and damping effects; the former quantity directly determines the dimensionless power gain of the detector. Unlike the approach used in the seminal work of Caves [Phys. Rev. D
26 1817 (1982)], the linear-response approach directly involves the noise properties of the detector, and allows
one to derive simple necessary and sufficient conditions for reaching the quantum limit. Our results have direct
relevance to recent experiments on nanoelectromechanical systems, and complement recent theoretical studiesof particular mesoscopic position detectors.
DOI: 10.1103/PhysRevB.70.245306

PACS number(s): 85.85.⫹j, 05.30.⫺d, 73.50.Td

I. INTRODUCTION

Recent advances in the field of nanoelectromechanical(NEMS) systems have renewed interest in the question ofquantum limited detection and amplification;1–8 several recent experiments have even come close to achieving thisideal limit.5,7 For position measurements, the quantum limit
corresponds to the maximum sensitivity allowed by quantum
mechanics of a weak, continuous measurement.9,10 For amplifiers, the quantum limit refers to the minimum amount ofnoise that must be added by a high-gain linear amplifier tothe input signal.11,12 Despite the fact that quantum constraints
on amplifiers have been studied and understood for quite
some time, there still seems to be some confusion in the
NEMS and mesoscopics communities as to the precise definition of the quantum limit, and on its origin. It has evenbeen conjectured that it may be possible to beat the quantum
limit using a weakly coupled, continuously measuring mesoscopic detector.2 Much of the confusion here stems from thefact that the seminal work on quantum limited amplification
by Caves11 uses a description that is difficult to apply directly to the mesoscopic detectors presently in use (i.e., asingle-electron transistor or quantum point contact).
In this paper, we approach the question of quantum limited detection and amplification using nothing more thanstandard linear-response (Kubo) relations. This approach has
the advantage that it directly involves the noise properties ofthe detector, and allows one to derive simple conditions thata detector must satisfy in order to reach the quantum limit.
Achieving the quantum limit is seen to require a detector
with “ideal” noise properties, a requirement that many detectors (e.g., a SET in the sequential tunneling regime13–15) failto meet. We demonstrate that the quantum limit for displacement detection and amplification is analogous to the quantum limit constraining quantum nondemolition (QND) measurements of a qubit.13–20 This is despite the fact that in thepresent problem, the detector-system coupling is not QND–
the coupling Hamiltonian does not commute with the Hamiltonian of the input system, and thus back-action force noise
results in additional output noise at later times. We place a
special emphasis on the effective temperature and damping
effects of the detector; in amplifier language, the latter cor1098-0121/2004/70(24)/245306(9)/$22.50

responds to the amplifier’s input and output impedances. We
find that the detector’s effective temperature directly determines the dimensionless power gain of the detector, and also
constrains correlations between the detector’s intrinsic output
noise and back-action force noise. The approach presented
here sheds light on recent findings which show that the effects of an out-of-equilibrium detector on an oscillator can be
described via an effective damping coefficient andtemperature,1,4,6 we show that this is a generic feature of
weakly coupled linear response detectors.
Finally, turning to the specific case of position detectionof an oscillator, we find that to reach the quantum limit on
the displacement sensitivity with a large power gain, thedamping of the oscillator must be predominantly independent of the detector. We also show that optimizing the displacement sensitivity (the quantity measured in the experiments of Refs. 5 and 7) is not the same as minimizing thesmallest detectable force.6 Note that linear-response constraints on position measurements were also considered
briefly by Averin in Ref. 16, though that work did not consider the role of detector-dependent damping or the detector’s effective temperature, two crucial elements of the work
presented here.

II. BASICS

For definiteness, we start by considering the case of ageneric linear-response detector measuring the position of aharmonic oscillator; the almost equivalent case of a genericlinear amplifier will be discussed in Sec. V. Our generic position detector has an input and output port, each of which ischaracterized by an operator (F̂ and Î, respectively). The input operator F̂ is linearly coupled to the position x̂ of theoscillator
Hint = − AF̂ · x̂,

共1兲

where A is the dimensionless coupling strength and A · F̂ isnothing more than the back-action force associated with themeasurement. The expectation value of the output operator Î

245306-1©2004 The American Physical Society

PHYSICAL REVIEW B 70, 245306 (2004)

A. A. CLERK(e.g., current) responds to the motion of the oscillator; we
assume throughout that the coupling is weak enough that one
can use linear response, and thus we have
⌬具Î共t兲典 = A冕⬁dt⬘␭共t − t⬘兲具x̂共t⬘兲典,␥共␻兲 =

Re

共2兲

where ␭ is the detector gain given by the Kubo formula=
共3兲

The expectation value here is over the (stationary) zerocoupling density matrix ␳0 of the detector. Neither this state
nor the Hamiltonian of the detector need to be specified inwhat follows.

冕⬁

dt具关F̂共t兲,F̂共0兲兴典ei␻t

0

⬅

−⬁

i
␭共t − t⬘兲 = − ␪共t − t⬘兲具关Î共t兲,F̂共t⬘兲兴典.
ប− Im ␭F共␻兲␻ប␻
1 SF共␻兲 − SF共− ␻兲
,
2␻
ប

共7兲

where ␭F is the linear response susceptibility describing the
response of F to a change in x, and SF共␻兲 is the (unsymmetrized) detector F-noise, calculated at zero coupling:
S F共 ␻ 兲 =冕

⬁

dt具F̂共t兲F̂共0兲典ei␻t .

共8兲

−⬁

A. Effective environment for oscillator

Turning to the oscillator, we assume that it is coupled bothto the detector and to an equilibrium Ohmic bath with temperature Tbath. The bath models intrinsic (i.e., detectorindependent) dissipation of the oscillator. For a weak coupling to the detector 共A → 0兲, one can calculate the oscillator
Keldysh Green functions using lowest-order-in-A perturbation theory. One finds that at this level of approximation, thefull quantum dynamics of the oscillator is described by aclassical-looking Langevin equation (see Appendix A and
Ref. 20):mẍ共t兲 = − m⍀2x共t兲 − ␥0ẋ共t兲 − A2冕⬘

dt ␥共t − t⬘兲ẋ共t⬘兲 + F0共t兲

冉

coth

冊

SF共␻兲 + SF共− ␻兲ប␻.
⬅
2kBTeff共␻兲SF共␻兲 − SF共− ␻兲In this Langevin equation, x共t兲 is a classically fluctuating
quantity, not an operator. Its average value, as determinedfrom Eq. (4), corresponds to the full quantum-mechanicalexpectation of the operator x̂共t兲. Similarly, the noise in x共t兲
calculated from Eq. (4) corresponds precisely to S̄x共t兲
= 具兵x̂共t兲 , x̂共0兲其典 / 2, the symmetrized noise in the quantum operator x̂ (see Appendix A for the details of this correspondence). Here, ⍀ is the renormalized frequency of the oscillator, m its renormalized mass, ␥0 describes damping due tothe equilibrium bath, and F0共t兲 is the corresponding fluctuating force. The spectrum of the F0 fluctuations are given by
the standard equilibrium relation

冉 冊

ប␻
.
S̄F0共␻兲 = ␥0ប␻ coth
2kBT

共5兲

The remaining terms in Eq. (4) describe the influence of thedetector—A · F共t兲 is the random back-action force producedby the detector, while A2␥ describes damping due to thedetector. The spectrum of the F共t兲 fluctuations is given by thesymmetrized force noise of the detector
12while for ␥共t兲, one has

冕

⬁

−⬁

dtei␻t具兵F̂共t兲,F̂共0兲其典

共6兲

共9兲

In the ␻ → 0 limit this reduces to

共4兲+ A · F共t兲.

S̄F共␻兲 ⬅

Thus, though the detector is not assumed to be in equilibrium, by treating its coupling to the oscillator to lowest order,
we obtain a simple Langevin equation in which the detector
provides both damping and a fluctuating force. Note that ingeneral, the detector force noise S̄F共␻兲 will not be related to␥共␻兲 via the temperature, as would hold for an equilibrium
system [i.e., Eq. (5)]. However, for a given frequency ␻ we
can define the effective temperature Teff共␻兲 via

2kBTeff =

冏冏
S̄F
␥

.
␻=0

共10兲

Note that Teff is by no means equal to the physical temperature of the detector, nor does it correspond to the “noise
temperature” of the detector (see Sec. V); the effective temperature only serves as a measure of the asymmetry of thedetector’s quantum noise. For example, it has been found for
SET and tunnel junction detectors that kBTeff ⯝ eV, where V
is the source-drain voltage of the detector.3,4,6
B. Detector output noise

Next, we link fluctuations in the position of the oscillator
to noise in the output of the detector. On a strictly classicallevel, we would treat both the oscillator position x共t兲 and thedetector output I共t兲 as classically fluctuating quantities. Using the linearity of the detector’s response, we could thenwrite␦Itotal共␻兲 = ␦I0共␻兲 + A␭共␻兲 · ␦x共␻兲.

共11兲

The first term 共␦I0兲 describes the intrinsic (oscillatorindependent) fluctuations in the detector output, while thesecond term corresponds to the amplified fluctuations of theoscillator. These are in turn given by Eq. (4):245306-2

PHYSICAL REVIEW B 70, 245306 (2004)

QUANTUM-LIMITED POSITION DETECTION AND…

␦x共␻兲 = −

冋

册

1/m
关F0共␻兲 + A · F共␻兲兴22
共␻ − ⍀ 兲 + i␻⍀/Q共␻兲

⬅ − g共␻兲关F0共␻兲 + A · F共␻兲兴,S̄I共␻兲S̄F共␻兲 =共12兲

where Q共␻兲 = m⍀ / 关␥0 + ␥共␻兲兴 is the oscillator quality factor.
It follows that the total noise in the detector output is givenclassically by
SI,tot共␻兲 = SI共␻兲 + 兩g共␻兲兩2兩␭共␻兲兩2关A4SF共␻兲 + A2SF0共␻兲兴− 2A2 Re关g共␻兲SIF共␻兲兴.

共13兲

Here, SI, SF, and SIF are the (classical) detector noise correlators calculated in the absence of any coupling to the oscillator.
To apply the classically derived Eq. (13) to our quantum
detector-plus-oscillator system, we interpret SI,tot as the total
symmetrized quantum-mechanical output noise of the detector, and simply substitute in the right-hand side the symmetrized quantum-mechanical detector noise correlators S̄F, S̄I,
and S̄IF, defined as in Eq. (6). Though this may seem rather
ad hoc, one can easily demonstrate that Eq. (13) thus interpreted would be completely rigorous, quantum mechanically,
if the detector correlation functions obeyed Wick’s theorem.
Thus, quantum corrections to Eq. (13) will arise from the
non-Gaussian nature of the detector noise correlators. We
expect from the central limit theorem that such corrections
will be small in the relevant limit where ␻ is much smallerthat the typical detector frequency ⬃kBTeff / ប, and neglectthese corrections in what follows. Note that the validity ofEq. (13) for a specific model of a tunnel junction position
detector has been verified in Ref. 21.

S̄F共␻兲 = ␲ប共14兲Here, ␭⬘ is the reverse gain of the detector (i.e., the gain in
an experiment where we couple x to I and attempt to measure F). Equation (14) tells us that if our detector has gain
and no positive feedback 共Re␭ · Re␭⬘ 艋 0兲, then it must havea minimum amount of back-action and output noise. Thisequation was proved rigorously in Ref. 18 for ␻ = 0 using a
Schwartz inequality; the proof given there may be straightforwardly generalized to finite ␻ if one now uses
symmetrized-in-frequency noise correlators. Note that almost all mesoscopic detectors that have been studied in detail (i.e., a SET or generalized quantum point contact) havebeen found to have ␭⬘ = 0.18
We now define a quantum-limited detector at frequency ␻
as having a minimal amount of noise at ␻, that is, it satisfies

具i兩␳0兩i典兩具f兩F兩i典兩2关␦共E f − Ei + ␻兲兺i,f+ ␦共E f − Ei − ␻兲兴,

共16兲

where ␳0 is the stationary detector density matrix, and 兩i / f典 isa detector eigenstate with energy Ei/f . To achieve the “ideal
noise” condition of Eq. (15) at frequency ␻, there must exist
a complex factor ␣ (having dimensions 关I兴 / 关F兴) such that
具f兩I兩i典 = ␣具f兩F兩i典

共17兲

for each pair of initial and final states 兩i典, 兩f典 contributing toS̄F共␻兲 and S̄I共␻兲 [cf. Eq. (16)]. Note that this not the same as
requiring Eq. (17) to hold for all possible states 兩i典 and 兩f典. In
Ref. 18, the requirement of Eq. (17) was further interpreted
as implying that there is no additional information regarding
the input signal that is available in the detector but not revealed in its output Î.
For a quantum limited detector with ␭⬘ = 0, the coefficient␣ in Eq. (17) can be found from兩␣共␻兲兩2 = S̄I共␻兲/S̄F共␻兲

We turn now to the fundamental constraint on the detector
noise correlators and gain ␭ which, in the present approach,serves as the basis of the quantum limit. We have
ប2
兵Re关␭共␻兲 − ␭⬘共␻兲兴其2 + 关Re S̄IF共␻兲兴2 .
4

共15兲

This is the same condition that arose in the study ofquantum-limited qubit detection,18,19 with the exception thatin that case, one also required that Re S̄IF = 0. We will showthat Eq. (15) must indeed be satisfied in order to achieve thequantum limit on position sensitivity, or the quantum limit on
the noise temperature of an amplifier.
As discussed in Ref. 18, having a quantum-limited detector [i.e., satisfying Eq. (15)] implies a tight connection between the input and output ports of the detector. Note firstthat we may write each symmetrized noise correlators as asum over transitions 兩i典 → 兩f典, e.g.,

C. Quantum constraint on detector noiseS̄I共␻兲S̄F共␻兲 艌

ប2
兵Re关␭共␻兲兴其2 + 关Re S̄IF共␻兲兴2 .
4tan关arg ␣共␻兲兴 = −ប Re ␭共␻兲/2
Re S̄IF共␻兲

共18兲
.

共19兲

Thus, a nonvanishing gain implies that Im ␣ ⫽ 0. It then follows from Eq. (17) and the hermiticity of Î, F̂ that for aquantum limited detector, the set of all initial states 兩i典 contributing to the noise has no overlap with the set of all final
states 兩f典. This immediately implies that a quantum-limited
detector cannot be in equilibrium.
D. Power gain

To be able to say that our detector truly amplifies the
motion of the oscillator, the power delivered by the detector
to a following amplifier must be much larger than the power
used to drive the oscillator—i.e., the detector must have adimensionless power gain G P共␻兲 much larger than one. As
we now show, this requirement places further constraints on
the effective temperature and noise properties of the detector.
In what follows, we consider the simple (and usual) casewhere there is no reverse gain ␭⬘ = 0. The power gain G P共␻兲245306-3

PHYSICAL REVIEW B 70, 245306 (2004)

A. A. CLERKof our generic position detector may be defined as follows.
Imagine first that we drive the oscillator with a force
FD cos ␻t; this will cause the output of our detector 具I共t兲典 toalso oscillate at frequency ␻. To optimally detect this signal
in the detector output, we further couple I to a second oscillator with natural frequency ␻ and position y: Hint
⬘ = BÎ · ŷ.
The oscillations in I will now act as a driving force on theauxiliary oscillator y. G P共␻兲 is then defined as the maximum
ratio between the power provided to the output oscillator yfrom the detector, versus the power fed into the input of theamplifier. This ratio is maximized if there is no intrinsic (i.e.,
detector independent) damping of either the input or output
oscillators. The damping of of the input oscillator is thencompletely given by A2␥ [see Eq. (7)], whereas the dampingof the output oscillator B2␥out is given by

Im ␭I共␻兲
␥out共␻兲 = −
⬅
␻

Re

冕

⬁

dt具关Î共t兲,Î共0兲兴典ei␻t0ប␻. 共20兲

III. QUANTUM LIMIT ON ADDED DISPLACEMENTNOISE

The sensitivity of a position detector is determined by theadded displacement noise Sx共␻兲, which is simply the total
detector contribution to the noise in the detector’s output,
referred back to the oscillator. It is this quantity which has
been probed in recent experiments,5,7 and which has a fundamental quantum constraint, as we derive below.
To define Sx共␻兲, we first introduce Sx,tot共␻兲, which is simply the total noise in the output of the detector [SI,tot共␻兲, cf.
Eq. (13)] referred back to the oscillator
Sx,tot共␻兲 ⬅

Pout
Pin

=

=

共21兲

A2␥共␻兲 · ẋ2

B2␥out共␻兲= 2
A ␥共␻兲

S x共 ␻ 兲 =

册

冊

2
␻
关B
·
A兩␭共
␻
兲兩
·
兩g共
␻
兲兩F
兴
ext
␻B2␥out共␻兲关␻兩g共␻兲兩Fext兴2兩␭共␻兲兩2兩␭共␻兲兩2=.
␻2␥out共␻兲 · ␥共␻兲 Im ␭F共␻兲 · Im ␭I共␻兲

共22兲

Thus, the power gain is a simple dimensionless ratio formedby the three different response coefficients characterizing thedetector. It is completely analogous to the power gain of a
voltage amplifier [see Eq. (48) of Sec. V].
Turning now to the important case of a quantum limited
detector, that is a detector satisfying the ideal noise conditionof Eq. (15), we find that the expression for the power gain
can be further simplified using Eq. (17). One finds冉共Im ␣兲2coth
G P共 ␻ 兲 =冊

ប␻+ 共Re ␣兲2
2kBTeff.
兩␣兩2/4

共23兲

It thus follows that to have G P Ⰷ 1, one needs kBTeff Ⰷ ប␻: alarge power gain implies a large effective detector temperature. In the large G P limit, we have
GP ⯝

冋

␥0
S 共␻,T兲,␥0 + A2␥ x,eq

Sx,tot共␻兲 ⬅ Sx共␻兲 +

共26兲

where

B2␥out共␻兲 · ẏ 2

冋冉

共25兲

We then separate out detector-dependent contributions to
Sx,tot共␻兲 from the intrinsic equilibrium fluctuations of the oscillator; the added displacement noise Sx共␻兲 is defined as theformer quantity

Using an overbar to denote a time average, we then have
G P共 ␻ 兲 ⬅

SI,tot共␻兲.
A 2␭ 2册Im ␣ 4kBTeff 2.
兩␣兩 ប␻

共24兲

Finally, an additional consequence of the large G P共␻兲,large Teff limit is that the imaginary parts of the gain ␭共␻兲and cross correlator S̄IF共␻兲 become negligible; they are suppressed by the small factor ប␻ / kBTeff. This is shown explicitly in Appendix B.

2Re关␭*g*共␻兲S̄IF兴
S̄I
2
2
+
A
兩g共
␻
兲兩
S̄
−
, 共27兲
F
兩␭兩2A2
兩␭兩2

冉 冊

Sx,eq共␻,T兲 = ប cothប␻关− Im g共␻兲兴
2kBT

共28兲

and where we have omitted writing the ␻ dependence of thenoise correlators and ␭. Here, Sx,eq共␻ , T兲 represents the equilibrium fluctuations that would result if all the damping were
due to the equilibrium bath; its contribution to Sx,tot共␻兲 isreduced, as only part of the damping results from the bath.
We now proceed to derive the quantum limit on Sx共␻兲.
Examining Eq. (27) for Sx共␻兲, and ignoring for a moment thedetector-dependent damping of the oscillator, we see that thefirst term (i.e., the intrinsic detector output noise referredback to the detector input) is proportional to 1 / A2, while thesecond term (i.e., the back-action of the detector) scales as
A2. We would thus expect Sx共␻兲 to attain a minimum value at
an optimal choice of coupling A = Aopt, where both theseterms make equal contributions. Using the inequality X2+ Y 2 艌 2XY we see that this value serves as a lower bound on
Sx even in the presence of detector-dependent damping. Defining ␾ = arg g共␻兲, we thus have the boundSx共␻兲 艌 2兩g共␻兲兩冋冑S̄IS̄F/兩␭兩2 −Re关␭*e−i␾共␻兲S̄IF兴
兩␭兩2

册

共29兲

where the minimum value is achieved when
2
=
Aopt

冑

S̄I共␻兲
兩␭共␻兲g共␻兲兩2S̄F共␻兲

.

共30兲

In the case where the detector-dependent damping is negligible, the RHS of this equation is independent of A, and thus
Eq. (30) can be satisfied by simply tuning the coupling

245306-4

PHYSICAL REVIEW B 70, 245306 (2004)

QUANTUM-LIMITED POSITION DETECTION AND…

strength A; in the more general case where there is detectordependent damping, the RHS is also a function of A, and it
may no longer be possible to achieve Eq. (30) by simply
tuning A.
While Eq. (29) is certainly a bound on the added displacement noise Sx共␻兲, it does not in itself represent the quantum
limit. Reaching the quantum limit requires more than simply
balancing the detector back-action and intrinsic output noises[i.e., the first two terms in Eq. (27)]; one also needs a detector with ideal noise properties, that is a detector which satisfies Eq. (15) and thus the proportionality condition of Eq.
(17). Using the quantum noise constraint of Eq. (14) to further bound Sx共␻兲, we obtain
S x共 ␻ 兲 艌 2

兩g共␻兲兩
兩␭兩

−

冋冑冉 冊
册

ប Re ␭ 2+ 共Re S̄IF兲2
2

Re关␭*e−i␾共␻兲SIF兴
.
兩␭兩2

共31兲

The minimum value of Sx共␻兲 in Eq. (31) is now achieved
when one has both an optimal coupling [i.e., Eq. (30)] and aquantum limited detector, that is one which satisfies Eq. (15).
Note again that an arbitrary detector will not satisfy the ideal
noise condition of Eq. (15).
Next, we consider the relevant case where our detector isa good amplifier and has a power gain G P共␻兲 Ⰷ 1 over thewidth of the oscillator resonance. As discussed in Appendix
B, this implies that we may neglect the imaginary parts of ␭and S̄IF, as they are suppressed by ប⍀ / kBTeff Ⰶ 1. We thenhave
Sx共␻兲 艌 2兩g共␻兲兩冋冑冉 冊 冉 冊
ប 2
S̄IF
+
2
␭

2

−

册

cos关␾共␻兲兴S̄IF.
␭

oscillator, and the minimum value of Sx共␻兲 in Eq. (34) corresponds precisely to the zero-point noise of the damped oscillator.
Though it reaches a similar conclusion, the linearresponse approach has several advantages over the approachof Caves. First, we do not have to model our detector as a setof bosonic modes and a scattering matrix, something that isdifficult to do for many mesoscopic detectors. More significantly, the linear-response approach makes explicitly clear
what is needed to reach the quantum limit. We find that toreach the quantum limit on the added displacement noise
Sx共␻兲, one needs(i) A quantum limited detector, that is a detector which
satisfies the “ideal noise” condition of Eq. (15), and hence
the proportionality condition of Eq. (17).
(ii) A coupling A which satisfies Eq. (30).
(iii) A detector cross-correlator S̄IF which satisfies Eq.
(33).
Note that condition (i) is identical to what is required forquantum-limited detection of a qubit; it is rather demanding,and requires that there is no “wasted” information about the
input signal in the detector which is not revealed in the
output.18 Also note that cot ␾ changes quickly as a functionof frequency across the oscillator resonance, whereas S̄IF will
be roughly constant; condition (ii) thus implies that it will
not be possible to achieve a minimal Sx共␻兲 across the entire
oscillator resonance. A more reasonable goal is to optimize
Sx at resonance, ␻ = ⍀. As g共⍀兲 is imaginary, Eq. (33) tells
us that S̄IF should be zero. Assuming we have a quantumlimited detector with a large power gain 共kBTeff Ⰷ ប⍀兲, theremaining condition on the coupling A [Eq. (30)] may be
written as
2
␥
Aopt

=
2
␥0 + Aopt␥

共32兲
Finally, as there is no further constraint on S̄IF / ␭, we canminimize the expression over its value. The minimum Sx共␻兲
is achieved for a detector whose cross-correlator satisfies

冏 冏
S̄IF共␻兲␭

=
optimalប
cot ␾共␻兲,2

共33兲

with the minimum value being given by兩Sx共␻兲兩min = ប兩Im g共␻兲兩 = lim Sx,eq共␻,T兲,
T→0

共34兲

where Sx,eq共␻ , T兲 is the equilibrium contribution to Sx,tot共␻兲
defined in Eq. (28). Thus, in the limit of a large power gain,
we have that at each frequency, the minimum displacement
noise due to the detector is precisely equal to the noise arising from a zero temperature bath. This conclusion is irrespective of the strength of the intrinsic (detectorindependent) oscillator damping.
The result of Eq. (34) is essentially identical to the conclusion of Caves,11 who found that a high-gain amplifier(modelled as a set of bosonic modes and a scattering matrix)
must add at least ប␻ / 2 of noise to an input signal at frequency ␻. Here, our input signal corresponds to the damped

冏 冏冑
Im ␣
␣

1
G P共⍀兲

=ប⍀.
4kBTeff

共35兲

As A2␥ is the detector-dependent damping of the oscillator,
we thus have that to achieve the quantum-limited value of
Sx共⍀兲 with a large power gain, one needs the intrinsic damping of the oscillator to be much larger than the detectordependent damping. The detector-dependent damping mustbe small enough to compensate the large effective temperature of the detector; if the bath temperature satisfies ប⍀ / kB
Ⰶ Tbath Ⰶ Teff, Eq. (35) implies that at the quantum limit, thetemperature of the oscillator will be given by
Tosc ⬅

A2␥ · Teff + ␥0 · Tbathប⍀→+ Tbath .
2
A ␥ + ␥04kB

共36兲Thus, at the quantum limit and for large Teff, the detector
raises the oscillator’s temperature by ប⍀ / 4kB. As expected,
this additional heating is only half the zero point energy; incontrast, the quantum-limited value of Sx共␻兲 corresponds tothe full zero point result, as it also includes the contributionof the intrinsic output noise of the detector.
Finally, we remark that if one did not assume kBTeffⰇ ប⍀ as is needed for a large power gain, we would have tokeep the imaginary parts of ␭ and S̄IF. One can show that for
kBTeff / ប⍀ → 0, it is possible to have a perfect anti-

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A. A. CLERKPHYSICAL REVIEW B 70, 245306 (2004)

correlation between the intrinsic detector output noise ␦I0
and the back-action force ␦F, and thus have Sx共␻兲 = 0. Thus,
similar to the results of Caves,11 in the limit of unit power
gain (i.e., small detector effective temperature), there is noquantum limit on Sx, as perfect anticorrelations between thetwo kinds of detector noise (i.e., in Î and F̂) are possible.

␥0
kBTeffⰆ Qdet .
2 ⰆA␥ប⍀If this condition is satisfied, it follows that ⍀̃ ⯝ ⍀, and we
have
S/N 艋

IV. QUANTUM LIMIT ON FORCE SENSITIVITY

In this section, we now ask a different question: what is
the smallest magnitude force acting on the oscillator that canbe detected with our displacement detector?6 This force sensitivity is also subject to a quantum limit; as with the quantum limit on Sx共␻兲, one again needs a quantum-limited detector [i.e., one satisfying the ideal noise condition of Eq.
(15)] to reach the maximal sensitivity. However, the conditions on the coupling A (i.e., the detector-dependent damping) are quite different than what is needed to optimize
Sx共␻ = ⍀兲; in particular, one can reach the quantum limit on
the force sensitivity even if there is no intrinsic oscillator
damping.
We start by imagining that a force Fext共t兲 = ⌬p · ␦共t兲 acts on
our oscillator, producing a change ⌬I共t兲 in the output of ourdetector. The corresponding signal-to-noise ratio is defined
as6,9
S/N =

冕

d␻ 兩⌬I共␻兲兩2= 共⌬p兲2
2␲ SI,tot共␻兲冕d␻ 兩␭共␻兲g共␻兲兩2. 共37兲
2␲ SI,tot共␻兲In what follows, we make the reasonable assumption that thedetector noise correlators and gain are frequency independent over the width of the oscillator resonance. We also takethe relevant limit of a large power gain (i.e., kBTeff Ⰷ ប⍀),and assume a best-case scenario where Tbath = 0 and where
the total oscillator quality factor Q Ⰷ 1. Using Eq. (13), one
finds that the maximal S / N is indeed obtained for a quantumlimited detector [i.e., one satisfying Eq. (15)]:
S/N 艋

2共⌬p兲2
បm⍀̃

冕

dx
⌫
,
2
2
␲
共x
−
1兲
+
⌫2共1 + ⌳2x兲0

冉 冊冉 冊冉
冊
冉 冊冉 冊冉 冊

1⍀̃ = ⍀ −
Qdet
2

⌫=

1
Qdet

⌳2 =

共⌬p兲2
.
បM⍀

共43兲

Demanding now S / N 艌 1, we find that the minimum detectable ⌬p is 冑បM⍀, 冑2 times the zero point value. Note thatthe requirement of Eq. (42) on the coupling A is very different from what is needed to reach the quantum limit on Sx共⍀兲[see Eq. (35)]. In the present case, it is possible to reach thequantum limit on the force sensitivity even if the damping
from the detector dominates 共A2␥ Ⰷ ␥0兲; in contrast, it is impossible to achieve the quantum limit on Sx共⍀兲 in this regime.
V. QUANTUM LIMIT ON THE NOISE TEMPERATURE OF
A VOLTAGE AMPLIFIER

In this final section, we generalize the discussion of theprevious sections to the case of a generic linear voltage amplifier (see, e.g., Ref. 12); the quantum limit on the noise
temperature of the amplifier is seen to be analogous to thequantum limit on the added displacement noise Sx共␻兲.
As with the position detector, the voltage amplifier ischaracterized by an input operator Q̂ and an output operator
V̂; these play the role, respectively, of F̂ and Î in the position
detector. V̂ 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
Hint = vin共t兲 · Q̂.

共44兲

⬁

共38兲

where equality occurs for a quantum-limited detector. For
such a detector, we have
2

共42兲

4kBTeff
ប⍀

4kBTeff
ប⍀

冉 冊冉
␥0
A 2␥

ប⍀̃
4kBTeff

Im ␣ Re ␣,兩␣兩2Im ␣ ⍀兩␣兩 ⍀̃

冊冉 冊

共39兲

2

兩␣兩
,
Im ␣

,

共40兲

共41兲

where Qdet = m⍀ / 共A2␥兲 is the quality factor corresponding to
the detector-dependent damping. It is clear from Eq. (38) thatthe signal to noise ratio can be further maximized by having
both ⌫ Ⰶ 1 and ⌳ Ⰶ 1. This requires

In more familiar terms, Ĩin = −dQ̂ / dt represents the current
flowing into the amplifier. We also assume that the output of
the amplifier (V̂) is connected to an external circuit via a term

⬘ = qout共t兲 · V̂,
Hint

共45兲

where ĩout = dqout / dt is the current in the external circuit. In
what follows, we will use quantities defined in the previous
sections for a position detector, simply substituting in Î → V̂and F̂ → Q̂.
Similar to the position detector, there are three response
coefficients for the amplifier: the voltage gain coefficient ␭[see Eq. (3)], the Q-Q susceptibility ␭Q which determines
damping at the input [see Eq. (7)], and the V-V susceptibility
␭V which determines damping at the output [see Eq. (20)].
The diagonal susceptibilities determine the input and output
impedances:

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Zin共␻兲 = 关i␻␭Q共␻兲兴−1 ,共46兲

PHYSICAL REVIEW B 70, 245306 (2004)

QUANTUM-LIMITED POSITION DETECTION AND…

Zout共␻兲 =␭ V共 ␻ 兲
,
− i␻

共47兲

i.e., 具Ĩin典␻ = 关1 / Zin共␻兲兴vin共␻兲 and 具V典␻ = Zout共␻兲ĩout共␻兲, where
the subscript ␻ indicates the Fourier transform of a timedependent expectation value.
We will consider throughout this section the case of noreverse gain, ␭⬘ = 0. We may then use Eq. (22) for the power
gain; it has the expected form
G P = ␭2

Re Zin 具V典2/Re Zout=.
Re Zout 共vin兲2/Re Zin

共48兲

Finally, we may again define the effective temperature
Teff共␻兲 of the amplifier via Eq. (9), and define a quantumlimited amplifier as one which satisfies the ideal noise condition of Eq. (15). For such an amplifier, the power gain willagain be determined by the effective temperature via Eq.
(23).
Turning to the noise, we introduce Sv,tot共␻兲, the total noiseat the output port of the amplifier referred back to the input.
Assuming the voltage source producing the input signal vin
has an impedance ZS and a temperature TS Ⰷ ប␻ / kB, we may
write
Sv,tot共␻兲 = 2kBTs Re ZS共␻兲 + Sv共␻兲.

关wS̄VQ兴S̄V
2
2,
2 + 兩Z̃兩 关␻ S̄Q兴 + 2 Im Z̃
␭
␭
Z̃ =

ZSZin
,
ZS + Zin

冋

册

1
␻S̄VQ
S̄V
.
+ 兩Zs兩共␻2S̄Q兲 + 2 tan ␾␭cos ␾ 兩Zs兩␭2
共53兲

To derive a bound on TN, we first perform the classicalstep of optimizing over the the magnitude and phase of thesource impedance ZS共␻兲; this is in contrast to the optimization of Sx共␻兲, where one would optimize over the strength ofthe coupling. We findk BT N 艌 ␻冑S̄V共␻兲S̄Q共␻兲 − 关S̄VQ共␻兲兴2,关␭共␻兲兴2

共51兲

共52兲

In what follows, we assume that 兩Zs 兩 Ⰶ 兩Zin兩, which meansthat Zin drops out of Eq. (50); we will test the validity of this

共54兲

where the minimum is achieved for an optimal source impedance satisfying
兩Zs共␻兲兩opt =冑S̄V共␻兲/关␭共␻兲兴2

兩sin ␾共␻兲兩opt = −

,

共55兲

冑S̄V共␻兲S̄Q共␻兲 .

共56兲

␻2S̄Q共␻兲
S̄VQ共␻兲As with the displacement sensitivity, simply performing aclassical optimization (here, over the choice of source impedance) is not enough to reach the quantum limit on TN.
One also needs to have an amplifier which satisfies the ideal
noise condition of Eq. (15). Using this equation, we obtainthe final bound

共50兲

where for clarity, we have dropped the ␻ dependence of thenoise correlators, gain, and impedances. The first term in Eq.
(50) represents the intrinsic output noise of the amplifier,while the second term represents the back-action of the amplifier: fluctuations in the input current Îin = −dQ̂ / dt lead tofluctuations in the voltage drop across the parallel combination of the source impedance Zs and the input impedance Zin,
and hence in the signal going into the amplifier. The last termin Eq. (50) represents correlations between these two sourcesof noise. We see that the general form of Eq. (50) is completely analogous to that for the added displacement noise
Sx共␻兲 of a displacement detector, see Eq. (27).
We are now ready to introduce the noise temperature of
our amplifier: TN is defined as the amount we must increase
the temperature TS of the voltage source to account for thenoise added by the amplifier,22, i.e., we wish to rewrite Eq.
(49) asSv,tot共␻兲 ⬅ 2kB共TS + TN兲Re Zs共␻兲.

2kBTN =

共49兲The first term is the equilibrium noise associated with thesignal source, while the second term is the total amplifiercontribution to the noise at the output port, referred back to
the input. Taking the limit of a large power gain (which
ensures ␭, S̄VQ are real), we have [see Eq. (27) and Ref. 12]:S v共 ␻ 兲 =assumption at the end. Using Eq. (50), and writing Zs= 兩Zs 兩 ei␾, we have immediatelyk BT N 艌

ប␻
,
2

共57兲

where the minimum corresponds to both having optimized Zs
and having an amplifier whose noise satisfies Eq. (15).
Finally, we need to test our initial assumption that 兩ZS 兩Ⰶ 兩Zin兩. Using the proportionality condition of Eq. (17) andthe fact that we are in the large power gain limit [G P共␻兲Ⰷ 1], we find

冏

冏冏 冏

1
Z S共 ␻ 兲
␣
ប␻
Ⰶ 1.
=
=
Re Zin共␻兲Im ␣ 4kBTeff 冑G P共␻兲

共58兲It follows that 兩ZS 兩 Ⰶ 兩Zin兩 in the large power gain, large effective temperature regime of interest, and our neglect of兩Zin兩 in Eq. (53) is justified. Eq. (58) is analogous to the caseof the displacement detector, where we found that reaching
the quantum limit on resonance required the detectordependent damping to be much weaker than the intrinsic
damping of the oscillator [see Eq. (35)].
Thus, similar to the situation of the displacement detector,
the linear response approach allows us both to derive rigorously the quantum limit on the noise temperature TN of anamplifier, and to state conditions that must be met to reachthis limit. To reach the quantum-limited value of TN with alarge power gain, one needs both a tuned source impedance
ZS, and an amplifier which possesses ideal noise properties[see Eqs. (15) and (17)].

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PHYSICAL REVIEW B 70, 245306 (2004)

A. A. CLERK
VI. CONCLUSIONSIn this paper we have derived the quantum limit on position measurement of an oscillator by a generic linear response detector, and on the noise temperature of a genericlinear amplifier. The approach used makes clear what must
be done to reach the quantum limit; in particular, one needs adetector or amplifier satisfying the ideal noise constraint of
Eq. (15), a demanding condition which is not satisfied bymost detectors. Our treatment has emphasized both thedamping effects of the detector and its effective temperature
Teff; in particular, we have found that the requirement of alarge detector power gain translates into a requirement of alarge detector effective temperature.

关Ǧ共␻兲兴−1 = 关Ǧ0共␻兲兴−1 −

⬅

冊

共A1兲

where GR共t − t⬘兲 = −i␪共t − t⬘兲具关x共t兲 , x共t⬘兲兴典, GA共t − t⬘兲 = i␪共t⬘ − t兲
⫻具关x共t兲 , x共t⬘兲兴典, and GK共t − t⬘兲 = −i具兵x共t兲 , x共t⬘兲其典. At zero coupling to the detector 共A = 0兲, the oscillator is only coupled tothe equilibrium bath, and thus Ǧ0 has the standard equilibrium form
Ǧ0共␻兲 =冉冢ប␻ប − 2 Im g0共␻兲coth
2kBTbathm
g0共␻兲*冊

g 0共 ␻ 兲
0

冣

,

A2
ប

冕

dtei␻t1
␻ − ⍀ + i␻␥0/m220i␪共− t兲具关F̂共t兲,F̂共0兲兴典− i␪共t兲具关F̂共t兲,F̂共0兲兴典− i具兵F̂共t兲,F̂共0兲其典

冊

.

ប
,
m共␻ − ⍀ 兲 − A Re DR共␻兲 + i␻关␥0 + ␥共␻兲兴2

2

2

GA共␻兲 = 关GR共␻兲兴 * ,GK共␻兲 = − 2i Im GR共␻兲冉␥0 coth
⫻

冊

共A8兲

冉

冊

ប␻
ប␻
+ ␥共␻兲coth
2kBTbath
2kBTeff,
␥0 + ␥共␻兲
共A9兲

where ␥共␻兲 is given by Eq. (7), and Teff共␻兲 is defined by Eq.
(9). The main effect of the real part of the retarded F Greenfunction DR共␻兲 in Eq. (A7) is to renormalize the oscillator
frequency ⍀ and mass m; we simply incorporate these shifts
into the definition of ⍀ and m in what follows.
If Teff共␻兲 is frequency independent, then Eqs. (A7)–(A9)for Ĝ corresponds exactly to an oscillator coupled to twoequilibrium baths with damping kernels ␥0 and ␥共␻兲. The
correspondence to the Langevin equation (4) is then immediate. In the more general case where Teff共␻兲 has a frequency
dependence, the correlators GR共␻兲 and GK共␻兲 are in exact
correspondence to what is found from the Langevin equation
Eq. (4): GK共␻兲 corresponds to symmetrized noise calculated
from Eq. (4), while GR共␻兲 corresponds to the response coefficient of the oscillator calculated from Eq. (4). This again
proves the validity of using the Langevin equation (4) tocalculate the oscillator noise in the presence of the detector
to lowest order in A.

共A2兲

APPENDIX B: SUPPRESSION OF IMAGINARY PARTS
OF ␭ AND S̄IQ⬁
Defining SIF共␻兲 = 兰−⬁
dt具I共t兲F共0兲典, one has the relations

共A3兲

and where ␥0 is the intrinsic damping coefficient, and Tbath isthe bath temperature.
We next treat the effects of the coupling to the detector in
perturbation theory. Letting ⌺̌ denote the corresponding selfenergy, the Dyson equation for Ǧ has the form

冉

共A5兲

共A7兲

where
g 0共 ␻ 兲 =

共A4兲

.

Using this lowest-order self energy, Eq. (A4) yields

APPENDIX A: DERIVATION OF LANGEVIN EQUATION

GK共t兲 GR共t兲
,
0
GA共t兲

冊

共A6兲

We thank Steve Girvin, Douglas Stone, and especiallyMichel Devoret for numerous useful conversations. This
work was supported by the W. M. Keck Foundation, and by
the NSF under Grants No. NSF-ITR 0325580 and DMR0084501.

冉⌺ R共 ␻ 兲 ⌺ K共 ␻ 兲⌺̌共␻兲 = A2Ď共␻兲ACKNOWLEDGMENTS

Ǧ共t兲 =

⌺ A共 ␻ 兲0

To lowest order in A, ⌺̌共␻兲 is given by

G R共 ␻ 兲 =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. (4), an equation which associates an effective temperature and damping
kernel to the detector; a similar perturbative approach for the
problem of a qubit coupled to a detector was considered by
Shnirman et al. in Ref. 20
We start by defining the oscillator matrix Keldysh Greenfunction冉ប关␭共␻兲 − ␭⬘共␻兲 * 兴 = − i关SIF共␻兲 − SIF共− ␻兲兴,

共B1兲S̄IF共␻兲 = 21 关SIF共␻兲 + SIF共− ␻兲 * 兴

共B2兲which follow directly from the definitions of S̄IF and ␭. Assuming now that we have a quantum limited detector [i.e.,

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PHYSICAL REVIEW B 70, 245306 (2004)

QUANTUM-LIMITED POSITION DETECTION AND…

Eq. (15) is satisfied], a vanishing reverse gain 共␭⬘ = 0兲, andkBTeff Ⰷ ប␻, we can use Eqs. (9) and (17) in conjunction withthe above equations to showប␭共␻兲 = − ␥共␻兲关4kBTeffIm ␣ + i2ប⍀ Re ␣兴,

1

共B3兲

M. P. Blencowe and M. N. Wybourne, Appl. Phys. Lett. 77, 3845
(2000).
2
Y. Zhang and M. P. Blencowe, J. Appl. Phys. 91, 4249 (2002).
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125313 (2004).
4 D. Mozyrsky and I. Martin, Phys. Rev. Lett. 89, 018301 (2002).
5
R. G. Knobel and A. N. Cleland, Nature (London) 424, 291(2003).
6 D. Mozyrsky, I. Martin, and M. B. Hastings, Phys. Rev. Lett. 92,
018303 (2004).
7 M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science
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8 N. M. Chtchelkatchev, W. Belzig, and C. Bruder, cond-mat/
0401486 (unpublished).
9 V. B. Braginsky, and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992).
10 C. M. Caves, K. S. Thorne, W. P. Drever, V. D. Sandberg, and N.
Zimmermann, Rev. Mod. Phys. 52, 341 (1980).
11
C. M. Caves, Phys. Rev. D 26, 1817 (1982).
12 M. H. Devoret and R. J. Schoelkopf, Nature (London) 406, 1039
(2000).

S̄IF共␻兲 = ␥共␻兲关2kBTeff Re ␣ − iប⍀ Im ␣兴.

共B4兲

We see immediately that the imaginary parts of ␭ and S̄IF aresuppressed compared to the corresponding real parts by asmall factor ប⍀ / 2kBTeff.

A. Shnirman and G. Schön, Phys. Rev. B 57, 15 400 (1998).
A. N. Korotkov, Phys. Rev. B 63, 115403 (2001).
15
Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73,
357 (2001).
16 D. V. Averin, cond-mat/0004364 (unpublished).
17 S. Pilgram and M. Büttiker, Phys. Rev. Lett. 89, 200401 (2002).
18
A. A. Clerk, S. M. Girvin, and A. D. Stone, Phys. Rev. B 67,
165324 (2003).
19 D. V. Averin, cond-mat/0301524 (unpublished).
20 A. Shnirman, D. Mozyrksy, and I. Martin, cond-mat/0311325(unpublished).
21 A. A. Clerk and S. M. Girvin, Phys. Rev. B 70, 121303(R)(2004).
22
Note that our definition of the noise temperature TN conforms
with that of Devoret and Schoelkopf (Ref. 12), but is slightly
different than that of Caves (Ref. 11). Caves assumes the source
is initially at zero temperature, and consequently uses the fullquantum expression for its equilibrium noise. In contrast, wehave assumed that kBTS Ⰷ ប␻. The different definition used by
Caves leads to the result kBTN 艌 ប␻ / 共ln 3兲 as opposed to our Eq.
(57).
13
14

245306-9