PHYSICAL REVIEW A 83, 033803 (2011)

Quantum-limited amplification with a nonlinear cavity detector
C. Laflamme and A. A. Clerk
Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 6 December 2010; published 7 March 2011)
We consider the quantum measurement properties of a driven cavity with a Kerr-type nonlinearity that is used
to amplify a dispersively coupled input signal. Focusing on an operating regime that is near a bifurcation point,
we derive simple asymptotic expressions describing the cavity’s noise and response. We show that the cavity’s
backaction and imprecision noise allow for quantum-limited linear amplification and position detection only if
one is able to utilize the sizable correlations between these quantities. This is possible when one amplifies anonresonant signal but is not possible in quantum nondemolition qubit detection. We also consider the possibilityof using the nonlinear cavity’s backaction for cooling a mechanical mode.
DOI: 10.1103/PhysRevA.83.033803

PACS number(s): 42.60.Da, 03.65.Ta, 03.65.Yz

I. INTRODUCTION

A number of recent experiments has made use of driven
microwave transmission-line resonators for sensitive nearquantum-limited measurements. These include measurementsof the position of a nanomechanical oscillator near the
standard quantum limit [1,2] as well as measurements ofsingle and multiqubit systems in circuit QED setups [3,4].
Such experiments use the microwave cavity as an op-amp typeamplifier [5], where the signal to be detected (e.g., the position
x of a mechanical resonator or the σz operator of a qubit) isdispersively coupled to the microwave cavity, meaning thatthe cavity frequency depends on the signal. When the cavity
is driven, the resulting modulation of the cavity frequency by
the signal leads to a modulation of the phase of the reflectedbeam from the cavity. By monitoring this phase (e.g., via
homodyne interferometry), one has essentially amplified thesignal.
Most experiments using a cavity for such dispersive
measurements and amplification have not exploited nonlinearities in the cavity—the cavity is just a driven damped
harmonic oscillator. The resulting measurement and amplification properties of the system are well understood. In
particular, it is known that this system can be used forquantum-limited linear amplification, meaning that the total
added noise of the measurement can be as small as allowedby quantum mechanics (see, e.g., Ref. [5] for a pedagogicaldiscussion).
While the linear-cavity regime is certainly useful, it isalso interesting to consider another possibility afforded by
microwave circuit cavities: They can be engineered to havestrong Kerr-type nonlinearities through the use of Josephson
junctions [6–8]. The resulting nonlinear cavity can then be
used for amplification in ways not possible with a linear
cavity. Attention has largely focused on using such devices
in the scattering mode of operation, where the signal to be
amplified is incident on the cavity from a coupled transmission
line and where backaction effects are irrelevant. Experiments
using this mode have realized single-quadrature amplificationand squeezing [7,9–11]; this operation mode has also been thesubject of many theoretical treatments, e.g., Refs. [6,7,12,13].
For qubit detection, another possibility is to use the bifurcation
in a nonlinear cavity to give a latching-type measurement,
1050-2947/2011/83(3)/033803(13)where the final dynamical state of the cavity depends on theinitial state of the qubit [14,15]; this scheme has also received
theoretical attention [16].
Instead, in this paper, we will study theoretically thequantum measurement properties of a driven nonlinear cavity
in the operation mode most relevant to experiments in nanomechanics and quantum information, the so-called op-amp modeof operation described above. Unlike the scattering modestudied in Refs. [6,7,12], here backaction is indeed relevant andplays a crucial role in enforcing the quantum limit on the added
noise: To reach the quantum limit, the backaction noise must
be as small as allowed by quantum mechanics [5,17]. We will
focus exclusively on regimes where there is no multistability
in the cavity dynamics (in contrast to the bifurcation amplifier
setup). We note that experiments using a nonlinear microwave
cavity amplifier in the op-amp mode discussed here haverecently been performed. Hartridge et al. have constructeda nonlinear cavity formed by a superconducting quantum
interference device (SQUID) [18] and have used this to detecta dispersively coupled superconducting qubit [19]. A recent
experiment by Ong et al. [20] used a nonlinear microwave
formed from a transmission line resonator and a Josephson
junction to detect a dispersively coupled qubit; this experimentalso investigated backaction effects.
Our analysis focuses on operation points close (but notpast) the bifurcation in the cavity response, a regime thatyields extremely large small-signal low-frequency amplification gain. The approach we use is standard: We linearize
the cavity dynamics about its mean classical value and use
the resulting linear quantum Langevin equations to study thenoise properties of the cavity detector. This allows us to assess
its ability to reach the op-amp amplifier quantum limit; arelated analysis is presented in Ref. [8]. Despite this standard
approach, we find a number of surprising conclusions thatseem not to have been appreciated in the existing literature.
In particular, we show that the nonlinear cavity near thebifurcation is equivalent to a degenerate parametric amplifier(DPA) driven with a detuned pump [21]. The value of thiseffective detuning is not an independent parameter and tends
to a universal value as one approaches the bifurcation. This
mapping allows us to derive simple analytic asymptotic
expressions for the cavity’s noise and gain that are universally
valid as one approaches the bifurcation point.

033803-1©2011 American Physical Society

C. LAFLAMME AND A. A. CLERKPHYSICAL REVIEW A 83, 033803 (2011)

In the low-frequency limit, large-gain limit, we findthat the imprecision noise of the cavity is precisely fourtimes what would be expected of an ideal resonantly
pumped degenerate parametric amplifier with equivalent gain[cf. Eqs. (45) and (46)]. We also show, somewhat surprisingly,that the cavity’s backaction noise at low frequencies is always
given by the same simple expression valid for a linear cavity,
an expression that is usually interpreted as describing theoverlap of displaced coherent states [cf. Eqs. (49) and (50)].
We find that the nonlinear cavity amplifier is quantum limited
at low frequencies but only if one can make use of thelarge correlations between the backaction and imprecision
noises. Such correlations cannot be utilized simply in quantum
nondemolition (QND) qubit detection; hence, the cavity
amplifier misses the quantum limit on QND qubit detection
by a large factor.
We also use our approach to study the possibility of using
the backaction of a nonlinear-driven microwave cavity to cool ananomechanical resonator. Near the bifurcation point, we findan extremely simple expression for the effective temperatureof the nonlinear cavity’s backaction [cf. Eq. (60)]; surprisingly,
the only relevant cavity parameter is its damping rate κ. We
also show that a driven nonlinear cavity is far better at coolinga low-frequency mechanical oscillator than a corresponding
driven linear cavity with comparable parameters. This latterconclusion matches what was found by Nation et al. [8], who
studied backaction cooling by a nonlinear cavity numericallyover a wide range of cavity parameters, including regimes
where there is bistability in the cavity dynamics. Aspects ofcooling and heating using a driven nonlinear cavity were also
addressed by Dykman [22].
The remainder of this paper is organized as follows. In
Sec. II, we review the basics of how one uses a nonlinear cavity
as a linear op-amp style amplifier and review the formulationand origin of the quantum limit applicable here. In Sec. III,
we show how, near the bifurcation, the nonlinear cavity isequivalent to a DPA driven by a detuned pump. In Sec. IV,
we use this mapping to derive asymptotic expressions forthe cavity’s noise and amplifier gain near the bifurcation
point and assess its ability to reach the quantum limit forsmall signal frequencies. Section V extends this analysis tononzero signal frequencies. Finally, in Sec. VI, we consider
the asymmetric quantum backaction noise of the cavity andassess the possibility of using the cavity for backaction coolinga mechanical oscillator.
II. BASICS OF A NONLINEAR CAVITY AMPLIFIER

FIG. 1. (Color online) Schematic of a realization of the nonlinear
cavity amplifier. The cavity is formed by an LC circuit containing a
Josephson junction (energy EJ ). The cavity is damped and is driven
by a coupled transmission line; b̂in and b̂out denote the input andoutput fields in the transmission. The input signal ẑ is a flux that
controls the value of EJ and, hence, the frequency of the cavity ωc .
An experimental realization of this system is presented in Ref. [18].

the transmission line used to drive the cavity). Derivationsof this Hamiltonian for microwave circuits incorporating
Josephson junctions are presented in many places in theliterature, and we do not repeat them here (see, e.g., Refs.
[6–8,12,18]); a schematic is presented in Fig. 1. In writing
Eq. (1), we have assumed the relevant case of a high-Q cavity
and, thus, made use of the rotating wave approximation to write
the nonlinear term. We will also be interested throughout inthe case of a weak nonlinearity,   κ. For clarity, we focus
exclusively on the ideal case where there is no internal cavity
loss; we also focus on the case of a one-sided cavity. Our
analysis could be easily generalized to incorporate either atwo-sided cavity or internal loss (see, e.g., Ref. [12]).
Unlike a linear cavity, the nonlinear cavity described by
Eq. (1) can undergo a bifurcation as a function of its parametersfrom a regime where the average cavity photon number n̄ =â † â is a single-valued function of the drive frequency ωd , toa regime where it is multivalued. For drive strengths just below
the bifurcation threshold, n̄ is a single-valued function of drive
frequency but exhibits a very pronounced slope (see Fig. 2).
This extreme sensitivity to cavity frequency makes the cavity
an extremely sensitive dispersive detector and amplifier in thisregime. However, it is not a priori obvious whether the cavity’s
noise in this regime is small enough to allow quantum-limited
performance. Answering this question is our main goal.

A. System Hamiltonian

The Hamiltonian of a cavity detector with a Kerr-type
nonlinearity has the general form†

† †Ĥ = Ĥsys + Ĥκ = h̄ωc â â − h̄â â â â + Ĥκ ,(1)

where ωc is the cavity resonance frequency and  is the Kerr
constant. We take  > 0 in what follows, as is appropriate
for a microwave cavity incorporating Josephson junctions; theresults are easily generalized to  < 0. The term Ĥκ represents
the damping (at rate κ) and driving of the cavity due to its
coupling to extra cavity modes (e.g., in a microwave circuit, to

B. Quantum limit on amplification in the op-amp
mode of operation

We focus throughout on the op-amp mode of amplifier
operation, where the input signal to be detected (described by
an operator ẑ) is coupled directly to the cavity photon number:
Hint = Aâ † â ẑ ≡ F̂ ẑ.

(2)

The operator ẑ could represent (for example) the position of
a nanomechanical beam (as considered in Ref. [8]) or thesignal flux applied to a SQUID circuit (as in Ref. [18]). As
a result of this dispersive coupling, the cavity frequency and033803-2

QUANTUM-LIMITED AMPLIFICATION WITH A . . .

PHYSICAL REVIEW A 83, 033803 (2011)

50

1000
100

30

G

n

40

20

10

10

1
0

− 1.5

− 1.0

− 0.5

0.0

0.5

1.0

−2

1.5

−1

∆/κ
FIG. 2. (Color online) Average cavity photon number n̄ versus
drive detuning  = ωd − ωc for various driving strengths b̄in . The
curves show the evolution of the cavity response as one goes throughthe bifurcation: Solid curves are for b̄in < b̄in,bif , while the dashedcurve is for b̄in > b̄in,bif . The diverging slope of d n̄/d near thebifurcation allows for amplification with a large gain.

reflected-beam phase shift become z dependent; thus, one canamplify z(t) by monitoring this phase. We will focus here
on homodyne detection, where the output beam is interferedwith a classical reference beam, and the resulting intensity ismeasured using a square-law detector. Letting b̂out (t) denotethe output field from the cavity (as defined in standard inputoutput theory [23,24]), the measured homodyne intensity will
be described by an operator Iˆ(t),Iˆ = B κ/2[eiφ b̂out (t) + H.c.],(3)

where φ is the phase of the classical reference beam and B isa dimensionless constant proportional to the amplitude of thisbeam. Note that Iˆ has units corresponding to a photon flux. As
the value of B plays no role in what follows (it is just a scale
factor for the output), we set B = 1 without loss of generality
in what follows.
We will be interested in weak enough couplings that ourcavity acts as a linear amplifier. As such, we have a linear
relation between the input signal and the cavity output, ∞ˆ
dt  χI F (t − t  )ẑ(t  ),(4)
I (t) =−∞where χI F (t) ∝ A is the forward gain of the amplifier and isdetermined by a standard Kubo formula [5].
The amplifier output (i.e., I ) will have fluctuations even in
the absence of any coupling to the detector; these are describedby the symmetrized spectral density:1 ∞dt eiωt {Iˆ(t),Iˆ(0)}.
(5)
S̄I I [ω] =2 −∞Again, as we are interested in linear amplification, theexpectation value is taken in with respect to the state of theuncoupled detector (i.e., A = 0). It is useful and standard to
think of these intrinsic output fluctuations in terms of effective

0

1

2

∆/κ

FIG. 3. Parametric zero-frequency photon number gain G ≡G[0] versus drive detuning  for various driving strengths b̄in < b̄in,bif(same values as in Fig. 2). The gain is maximized near values of thedetuning where the slope d n̄/d is maximal.

signal (i.e., z) fluctuations; thus, we introduce the imprecision
noise spectral density:S̄zz [ω] = S̄I I [ω]/|χI F [ω]|2 .

(6)

In the op-amp mode of operation, a second crucial aspect of
the amplifier’s noise is its backaction. By virtue of the thedetector-signal coupling in Eq. (2), the operator F̂ = Aâ † â(i.e., the cavity photon number) acts as a noisy backaction force
on the signal. Extra fluctuations in ẑ due to this stochastic force
will necessarily increase the noise in the output of the amplifier
and, thus, are part of the total added noise of the amplifier. The
backaction force noise is characterized by a symmetrized noise
spectral density S̄F F [ω] defined analogously as Eq. (5).
Thus, we have that the total amplifier contribution to the
output noise has contributions from both imprecision andbackaction noises. It is convenient (and common) to think ofthis total added noise in terms of a noise temperature TN [ω]:The total amplifier added noise at frequency ω is equivalent
to the extra equilibrium noise we would get by raising thetemperature of the signal source by TN [ω].1 This quantity isrelevant no matter what the signal, be it the position of aharmonic oscillator or the voltage produced by some input
circuit; we also stress that achieving the quantum limit on thenoise temperature is equivalent to achieving the quantum limit
on continuous weak displacement detection [5].
Minimizing the noise temperature at a given frequency
requires one to first optimize the signal source’s susceptibilityχzz [ω]. This linear-response susceptibility tells us how theaverage value of ẑ changes in response to a perturbation that
couples to ẑ, i.e., ∞δẑ(t) =dt  χzz (t − t  )F̂ (t  ).
(7)
−∞

1
We use the standard convention in which the noise temperature
TN [ω] is defined by assuming the signal source is initially at atemperature much larger than h̄ω. This definition leads to the standard
bound given in Eq. (12).

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C. LAFLAMME AND A. A. CLERKPHYSICAL REVIEW A 83, 033803 (2011)

Optimizing the total added noise over the coupling strengthand phase of the signal source’s susceptibility χzz [ω] yields a
standard bound on TN [ω] [5],1kB TN [ω]S̄zz [ω]S̄F F [ω] − [Re(S̄zF [ω])]2h̄ωh̄−ImS̄zF [ω] ,(8)meas = (2S̄zz [0])where the inequality becomes an equality for an optimal source
susceptibility satisfying
|χzz [ω]| = S̄zz [ω]/S̄F F [ω],(9a)Re S̄zF [ω]Re χzz [ω]=.
|χzz [ω]|S̄zz [ω]S̄F F [ω](9b)We have introduced the correlator S̄zF , which describes
possible correlations between backaction and imprecision
noises:∞dteiωt {Iˆ(t),F̂ (0)}
S̄I F [ω]S̄zF [ω] =
= −∞
.
(10)
χI F [ω]2χI F [ω]Consider the simple case where the signal frequency ω is muchsmaller than the relevant frequency scales of the cavity; thus,
we may focus on the noise temperature in the ω → 0 limit.
Using the fact that there cannot be any out-of-phase noisecorrelations at zero frequency (i.e., S̄zF [0] = Re S̄zF [0]), thezero-frequency form of the fundamental Heisenberg inequality
on detector noise [5],S̄zz [0]S̄F F [0] − (S̄zF [0])  h̄ /42

2

kB TN  h̄ω/2,(12)i.e., the added noise amplifier must at least be as large as thezero-point noise of the signal source [5,17].2 We stress that,while the conclusion may appear similar, the op-amp quantum
limit considered here is not identical to the quantum limit on
the scattering mode described in the seminal works by Haus
and Mullen [25] and Caves [26]: the scattering-mode quantum
limit does not involve backaction. Moreover, an amplifier may
reach the quantum limit in the scattering mode but not in the
op-amp mode [5].
The case where the input signal ẑ is the spin operator
of a qubit is also interesting. Here, the quantum limit
on QND qubit detection involves the measurement ratemeas and the measurement-induced backaction dephasingrate ϕ [27,28],
ϕ meas ,(13)

where for weak coupling:2
ϕ = 2S̄F F [0]/h̄ ,

(14a)

.

III. BEHAVIOR NEAR BIFURCATION
A. Mapping to a degenerate parametric amplifier

We begin our analysis by using standard input-output theory[23,24] to derive the Heisenberg equation of motion for thecavity field, in the absence of any coupling to the signal:√iκdâ = − [â,Hsys ] − â − κ b̂in (t).
dth̄2Note that we have used the form of the quantum noise inequality
corresponding to a vanishing reverse gain, i.e., coupling to Iˆ cannotchange the average value of F̂ . The vanishing of the reverse gain
for our nonlinear cavity amplifier is explicitly demonstrated in the
Appendix.

(15)Here, b̂in (t) = b̄in e−iωd t + ξ̂ (t) describes the input field incident on the cavity from the transmission line; its averagevalue b̄in describes the coherent drive applied to the cavity
at frequency ωd = ωc + , while ξ̂ (t) describes quantum andclassical noises entering the cavity from the drive port. Without
loss of generality, we take the drive amplitude b̄in to be realand positive.
We are interested in driving strengths that result in a large
average number of quanta n̄ in the cavity but at the sametime, are not so strong that there is multistability in theclassical cavity dynamics. Thus, it is useful to write the cavity
annihilation operator â as the sum of a classical and quantum
part: This takes the form√â(t) = e−iωd t eiφa ( n̄ + eiπ/4 d̂(t)).
(16)√
The complex number eiφa n̄ ≡ â is simply determinedby the classical equations of motion, whereas, d̂ describes
the influence of noise (and eventually, the coupling to the
input signal). We have chosen the phase of the second term in
Eq. (16) to simplify the following analysis. From
√ Eq. (15), we
find that the average cavity photon number n̄ is determinedby the classical equation,n̄[(κ/2)2 + (2n̄ + )2 ] = κ(b̄in )2 .

2

(14b)

Thus, reaching the quantum limit on QND qubit detection
places more stringent requirements on the detector than thoserequired to have a quantum-limited noise temperature: Not
only must the quantum noise bound of Eq. (11) be satisfied as
an equality, but also there must be no backaction-imprecision
correlations (e.g., S̄zF [0] = 0).
In Secs. III– VI, we will calculate the nonlinear cavity’s
noise and response functions and determine whether theyreach the quantum limit on their noise temperature andon QND detection. We note in passing that Ref. [8] also
addresses the quantum limit on amplification (specifically
position detection) using a nonlinear cavity in a similar regime
to that considered here. Their analysis is based on alternative
formulation of the quantum limit, which is not equivalent to theone discussed here; in particular, they did not address whether
the nonlinear cavity optimizes the quantum noise inequalityof Eq. (11) or consider its noise temperature as definedin Eq. (8).

(11)

implies that

−1

(17)

We can now use Eq. (15) to write an equation for d̂; retaining
only leading terms in n̄
1 yields a linear equation,

033803-4√iκdd̂ = − [d̂,Hdpa ] − d̂ − κ ξ̂ (t),
dt
h̄
2

(18)

QUANTUM-LIMITED AMPLIFICATION WITH A . . .

PHYSICAL REVIEW A 83, 033803 (2011)

where
g̃
 d̂ † d̂ + ih̄ (d̂ † d̂ † − d̂ d̂).
Ĥdpa = −h̄(19)2Equation (19) is simply the Hamiltonian of a DPA driven by anonresonant pump, where a single pump mode photon can beconverted into two signal mode photons and vice versa (see,e.g., Ref. [24]). Here, the classical cavity field ā plays the roleof the pump mode, while the displaced cavity field d̂ plays therole of the signal mode. The effective parametric interaction
 are given bystrength g̃ and effective pump detuning g̃ = 2n̄,(20a) =  + 4n̄.


(20b)

The above mapping of the driven nonlinear cavity to a
detuned DPA is general and only relies on n̄
1. We willbe especially interested in operating points near the point ofbifurcation, as these allow a maximal amplifier gain. As oneapproaches the bifurcation, the effective DPA parameters g̃,approach universal values. To see this, note first that a standard
analysis of the classical equations of motion shows that thebifurcation occurs at a critical drive amplitude b̄in,bif satisfying
1 κ2[b̄in,bif ] = √
.
6 3
2

(21)

For b̄in < b̄in,bif , n̄ is a single-valued function of . For b̄in =b̄in,bif , the slope of n̄ versus  is infinite at a single point = bif ; one finds from Eq. (17),
√
3
κ,
(22a)
bif = −2
1 κn̄bif = √
.
(22b)
2 3
Thus, it follows from Eqs. (20) that the parameters of theeffective DPA attain universal values at the bifurcation:
κg̃bif = √ ,
(23a)
3
κ
 bif = √ .

(23b)
2 3
Note crucially that, for cavity operating points near the is nonzero. Asbifurcation, the effective DPA pump detuning 
we will see in Sec. III B, this will have a pronounced impact:
The amplified and squeezed quadratures of the DPA are notorthogonal. This, in turn, has a significant effect on the noiseproperties of the nonlinear cavity detector.
B. Amplified cavity quadrature

To appreciate the implications of pump detuning in our
effective paramp model, we consider the equations of motion
corresponding to Eq. (19). We will be interested throughout inparameter regimes where this effective paramp has a photon
 |. In
number gain larger than 1; this necessarily requires g̃ > |such regimes, the analysis is most conveniently presented byfirst introducing canonically conjugate quadrature operators X̂
and P̂ ,1
X̂ = √ (e−iθ/2 d̂ + eiθ/2 d̂ † ),
(24a)
2

−i
P̂ = √ (e−iθ/2 d̂ − eiθ/2 d̂ † ),
2

(24b)

where the angle θ (−π/2  θ  π/2) is given by /g̃.
sin θ = (25)

As we will see, the above definition ensures that X̂ is thequadrature amplified by the cavity. We also define corresponding quadratures X̂in and P̂in of the operator ξ̂ associatedwith noise entering the drive port [e.g., these are defined bysubstituting d̂ → ξ̂ in Eqs. (24)].
With these definitions, the equations of motion are easily
solved upon Fourier transforming (see Appendix),√X̂[ω] = − κ{χ1 [ω]X̂in [ω] − tan θ (χ1 [ω] − χ2 [ω])P̂in [ω]},(26a)√P̂ [ω] = − κχ2 [ω]P̂in [ω],(26b)where the susceptibilities χ1 ,χ2 are given by 2 )−1 ,χ1 [ω] = (−iω + κ/2 − g̃ 2 −  2 )−1 .
χ2 [ω] = (−iω + κ/2 + g̃ 2 − 

(27a)
(27b)

 = θ = 0), these
For the case of a resonant pump (i.e., 
equations take a simple form and describe the usual behaviorof a DPA: As g̃ approaches κ/2 from below (the parametricthreshold), κχ1 [0] → ∞, κχ2 [0] → 1, and X̂ (P̂ ) is theamplified (squeezed) quadrature. By considering quadraturesof the output field leaving the cavity, one finds that the photon
number gain for the X quadrature is given byG[ω] ≡ |1 − κχ1 [ω]|2 .

(28)

We will refer to G[ω] as the parametric gain of our system
in what follows. For a resonant pump, the amplified andsqueezed quadratures are clearly orthogonal (i.e., canonicallyconjugate). Note that G[ω] has a Lorentzian form, implyingthat there is only appreciable gain for frequency in a bandwidthB ,B ≡ 1/χ1 [0].

(29)

We refer to B as the parametric bandwidth√in what follows;in the large parametric gain limit, B ∼ κ/ G[0].
The situation is more involved in the case of interest here,
where the effective pump is not resonant [cf. Eqs. (23b)], and,hence, θ = 
0. We still have a parametric threshold when g̃
 2 from below; as before, κχ1 [0] → ∞approaches κ 2 /4 + in this limit, while κχ2 [0] → 1. It is easy to verify from
Eqs. (23) that the parametric threshold coincides with thecavity bifurcation. It also follows from Eqs. (26) that, for any
 , X̂ is the amplified quadrature: noise (orpump detuning 
signal) incident in the X quadrature (i.e., X̂in ) only drives theX cavity quadrature and is multiplied by the large susceptibility
χ1 . The photon number gain for signals in the X quadrature
continues to be described by Eq. (28); as expected, this gain
diverges as one approaches the bifurcation (see Fig. 3). As one

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C. LAFLAMME AND A. A. CLERKPHYSICAL REVIEW A 83, 033803 (2011)

approaches the bifurcation, Eqs. (23) imply that the angle θ ,which defines X̂, takes the universal value:θbif = π/6.

(30)

is essentially one quadrature of the cavity output field and may
be written√(35)Iˆ[ω] = κ(cos ϕh X̂out [ω] + sin ϕh P̂out [ω]), = 0, are the dynamics of theMore troublesome, when 
P quadrature, the quadrature orthogonal to the amplified
quadrature. For a nonzero detuning, P is not the squeezed
quadrature. Noise or signals incident on the cavity in the P
quadrature (i.e., P̂in ) appear both in the cavity P quadrature(where it is multiplied by the small susceptibility χ2 ), as wellas in the cavity X quadrature, where it is also amplified (i.e.,
multiplied by the large susceptibility χ1 ).
To summarize, we have shown that, near the bifurcation,
the driven nonlinear cavity of Eq. (1) maps onto a DPA with a . This nonzero detuning means that
nonzero pump detuning 
the dynamics does not correspond to the simple situation of
canonically conjugate amplified and squeezed quadratures. As
we will see, this lack of orthogonality will have pronounced
implications on the cavity noise properties near the bifurcation.

Thus, it follows from Eqs. (26) and (33) that the linear-response
gain of the cavity amplifier will have the general form√
(37)
χI F [ω] = h̄A 2n̄κ(λ1 χ1 [ω] + λ2 χ2 [ω]),C. Coupling to signal and cavity gain

IV. AMPLIFIER NOISE IN THE LARGE GAIN
LOW-FREQUENCY LIMIT

To complete our mapping of the nonlinear cavity detector
to a DPA, we need to restore the signal-detector coupling
Hamiltonian and consider the forward gain χI F (t) of thesystem [cf. Eq. (4)]. This forward gain tells us how strongly
the input signal ẑ influences the output homodyne current
and will not be identical to the parametric photon number
gain G discussed above. While one could calculate χI F [ω]directly using a Kubo formula, it is simpler here to simply
rederive the equations of motion for the cavity field including
the coupling to ẑ. Retaining only leading-order terms in n̄, thesignal-cavity coupling Hamiltonian Hint in Eq. (2) retains theform Ĥint = F̂ ẑ, with the generalized force operator F̂ taking
the form:√(31)
F̂ ≡ Aâ † â ( 2n̄A)(sin ν X̂ + cos ν P̂ ),where ϕh is determined by the phase of the reference beam
used in the homodyne measurement and the output operators
are given by the standard input-output relations [23,24], e.g.,√X̂out [ω] = X̂in [ω] + κ X̂[ω].
(36)

where

Note that, at the bifurcation, the angle ν takes on the universal
value:
νbif = π/12 + 3π/4 = 5π/6 = π − θbif .

(34)

Having characterized the signal-detector coupling, we now
turn to the output homodyne current [cf. Eq. (3)]. This current

λ2 = − sin ν(cos ϕh tan θ + sin ϕh ).

(38b)

G∼4

(32)

We have dropped a constant term in F̂ , which can be absorbed
into the Hamiltonian of the signal source. We see that, ingeneral, the input signal ẑ couples to both X̂ and P̂ and, thus,
will enter the linearized cavity equations of motion as a driving
term for both these quadratures. To be explicit, one shouldmake the following replacements in Eqs. (26):2n̄X̂in [ω] → X̂in [ω] −(33a)
A cos ν ẑ[ω],κ2n̄P̂in [ω] → P̂in [ω] +(33b)
A sin ν ẑ[ω].
κ

(38a)

We are most interested in the properties of our cavity
amplifier close to the bifurcation, where the parametric gain
defined in Eq. (28) satisfies G[0] ≡ G
1. In this regime,
we expect amplification of√input signals in a narrow band of
frequencies ω < B ∼ κ/ G. Thus, we begin our analysis
by considering the amplifier noise to leading order in the large
parameter G and for ω  B ; the latter condition allows us
to take the zero-frequency limit of cavity noise and response
functions. We also assume the ideal case where the cavity is
only driven by vacuum noise. Note that it is straightforward
to use Eq. (17) to determine how G behaves as a function ofdriving strength as one approaches the bifurcation from below.
Assuming that the drive detuning  is always chosen in orderto maximize G, one finds that, near the bifurcation,whereν ≡ θ/2 + 3π/4.

λ1 = cos ϕh (cos ν + tan θ sin ν),|b̄in,bif |2|b̄in,bif |2 − |b̄in |2

3

.

(39)

We start with the amplifier’s forward gain. In the limit,
G → ∞. Equation (37) yields√√(40)
χI F [0] ∼ 2n̄κAλ1 G.
As expected, the forward gain is (to leading order) proportional
to the square root of the DPA photon number gain.
Turning to the cavity output noise, we note that, for G
1
and for small frequencies, Eqs. (26) yields that the cavity P̂
quadrature is negligible in comparison to the X̂ quadrature. As
such, we can drop the second term in Eq. (35) and treat thehomodynecurrent operator Iˆ as being proportional to X̂out ∝√X̂ ∝ G, Thus, for G → ∞,Iˆ[ω] ∼ κ cos ϕh X̂[ω].

(41)

Looking at Eq. (31) for the backaction force operator F̂ ,we see a similar argument holds.
√ Thus, to leading order in G,
F̂ is also proportional to X̂ ∝ G:√
(42)F̂ [ω] ∼ 2n̄A sin ν X̂[ω].

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Thus, to leading order in G, backaction and imprecision noises
are perfectly correlated with one another, as they only differ by
a constant. Their spectral densities will simply be proportional
to the spectral density of the amplified cavity quadrature X̂.

The leading-order-in-G intrinsic output noise of the amplifier (i.e., noise in the homodyne current at A = 0), thus,
follows easily from Eqs. (41) and (26a) (see the Appendix). In
the low-frequency limit, we haveκS̄I I [0] ∼ cos2 ϕh G(1 + tan2 θ ).
(43)
2
The two terms in the last factor represent two distinct physicalcontributions to the output noise. The θ -independent termarises from X-quadrature input noise (X̂in ) being amplified
and appearing in X̂. In contrast, the term proportional to tan2 θ
is a direct consequence of the nonzero effective pump detuning
 . The resulting nonorthogonality of amplified and squeezedquadratures causes P -quadrature input noise (P̂in ) also to be
amplified and to appear in the amplifier output Iˆ. Thus, we = 0 causes the output noise to be larger than what
see that 
would be expected for a resonant pump DPA with equivalent
photon number gain G.
Combing the above expression with Eq. (40) for the gain,
we find that the low-frequency imprecision noise [cf. Eq. (6)]near the bifurcation (i.e., G → ∞) is given byS̄zz [0] ∼ S̄zz,ideal [0]himp (θ,ν),

(44)

8

S zz /S zz,ideal

A. Imprecision noise

10

6
4
2
100

104

106

G
FIG. 4. (Color online) Imprecision noise Szz [0], scaled by theimprecision noise of an ideal DPA, S̄zz,ideal versus parametric photon
number gain G. For large G, the imprecision noise is four times theideal value. (Solid blue) Increase G by increasing the drive amplitudeb̄in toward b̄in,bif , using an optimal  for each drive strength. (Dashedred) Fix (b̄in /b̄in,bif )2 = 0.995; increase G by tuning  to approach
the optimal value from below. (Long dash-short dash, green) Same asprevious, but increase G by tuning  to approach the optimal value
from above.

behavior of the imprecision noise relative to the ideal value
is shown in Fig. 4.
B. Backaction noise and backaction-imprecision productwhereS̄zz,ideal =himp (θ,ν) =

h̄2 κ
4n̄A2

,

1 + tan2 θ.
(cos ν + sin ν tan θ )2

(45a)
(45b)

Here, S̄zz,ideal is the imprecision noise of an ideal DPA in thelarge gain limit. By ideal, we mean a DPA that was pumped
on resonance, and where the input signal z only drives theamplified X quadrature [i.e., the angle ν in Eq. (31) would bezero]. himp (θ,ν) > 1 describes the increase of S̄zz due to thefact that our nonlinear cavity does not realize a DPA in thisideal fashion. The numerator of himp describes the extra output
noise due to the nonresonant effective pump as discussed after
Eq. (43). The denominator describes the reduction in gain
coming from the fact that the signal drives both the X̂ and theP̂ quadratures.
Finally, we can further simplify our result by using the fact
that, near the point of bifurcation, the effective DPA parameters
approach universal values [cf. Eq. (23)]. For leading order, we
can simply replace θ by its value at the bifurcation θbif = π/6.
Thus, we obtain our final expression for the imprecision near
the bifurcation:S̄zz [0] ∼ 4S̄zz,ideal .

(46)

In the large gain limit, the imprecision of the nonlinear cavity
amplifier is a factor of 4 times what would be expected froma theoretically ideal degenerate parametric amplifier. The

From Eq. (42), we see that to leading order in G, the lowfrequency backaction noise spectral density S̄F F [0] will just be
proportional to the low-frequency output noise spectral density
S̄I I [0]. Using the universality of the DPA parameters near thebifurcation, we find that, in the G → ∞ limit,
SF F [0] ∼1 A2 n̄
G.
3 κ

(47)

We see the backaction diverges as the parametric photon
number gain G; this is a simple consequence of the factthat our dispersive coupling unavoidably leads the signal ẑ
to be coupled to the amplified cavity quadrature X. The full
expression (valid for arbitrary G) is not too unwieldy andis given in the Appendix as Eq. (A14). Note that similar
spectral densities for a nonlinear cavity were calculated using
a linearized Fokker-Planck approach in Refs. [16] and (in theclassical high-temperature regime) [29].
Combining our results, we see that, near the bifurcation,
the backaction-imprecision product will be much larger thanthe minimum value of h̄2 /4 allowed by quantum mechanics.
In the large-G limit, we haveS̄F F [0]S̄zz [0] ∼

G 2
h̄ .
3

(48)

Figure 5 shows the scaling of S̄F F [0]S̄zz [0] versus parametricgain G as one approaches the bifurcation by either tuning thedrive detuning  or tuning the drive strength b̄in ; the universal
asymptotic behavior described by Eq. (48) is clear.

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0.60
0.55

S zz S FF / h̄2 G

0.50
0.45
0.40
0.35
0.30
0.25
1

100

104

106

G
FIG. 5. (Color
online)Backaction-imprecision
product
S̄zz [0]S̄F F [0] (scaled by h̄2 G) versus parametric photon number
gain G, demonstrating the universal scaling predicted for large G.
Individual curves correspond to the same parameters as in Fig. 4.

The above result implies that, if one cannot make useof backaction-imprecision noise correlations, one is very far
from having a quantum-limited device. In particular, near thebifurcation, the nonlinear cavity detector cannot be used for
QND qubit detection: In such an experiment, the backaction
dephasing rate will be a factor G
1 larger than the minimum
rate dictated by quantum mechanics [cf. Eq. (13)]. Note thatthe situation is very different for a linear cavity: there, as longas one drives the cavity on resonance, the S̄F F [0]S̄zz [0] productattains the minimum possible value of h̄2 /4 [5,30].
It is tempting to think that, by simply changing the cavity
operating point slightly, one could achieve a situation where
the input signal is only coupled to the cavity quadrature P̂
and, thus, avoid the problematic diverging backaction foundabove. From Eqs. (31) and (32), we see that this would require
an operating point for which the angle θ = π/2. However,from Eqs. (25) and (27), this, in turn, implies that the cavity
would have no parametric gain: G = 1. Thus, one cannot solve
the problem of large backaction by simply changing the drive
detuning without simultaneously getting rid of the amplifiergain.
C. Comparison with linear-cavity backaction formula

For a linear cavity, one can directly connect the backaction
noise spectral density at zero frequency to how strongly theaverage cavity amplitude â changes in response to a change
in the signal. One finds
S̄F F,lin [0] = A2dâ 2.
d

describing the cavity. Equation (49) implies that the backaction
dephasing ϕ (and, hence, S̄F F ) is directly determined by theoverlap between this displaced coherent state and the original
coherent state describing the cavity.
One would not expect Eq. (49) to apply, in general, to
our nonlinear cavity detector, as now the intracavity statecorresponding to a given fixed value of the input signal is not acoherent state or even a pure state [32]. This is a direct result of
the squeezing and amplification of the cavity noise that occurs
as one approaches the bifurcation. However, if one is far fromthe bifurcation, these effects should be minimal, and one might
expect Eq. (49) to remain valid. This idea was put forward
recently in Ref. [20] and was derived within an approximationthat neglects noise squeezing of the cavity. Our approach fully
accounts for the squeezing of the intracavity fluctuations andallows us to test the general validity of Eq. (49). Surprisingly,
we find that this expression exactly captures the full backaction
noise, even close to the bifurcation:S̄F F,lin [0] = S̄F F [0].

Here, S̄F F [0] is the full expression for the backaction noise
spectral density that follows from Eqs. (26) [see Eq. (A14)]. We
see that, despite the fact that the cavity is not in a coherent stateor even a pure state, Eq. (49) remains valid for the nonlinear
cavity amplifier; that this should be so is by no means a priori
obvious.
Reference [20] also suggests that the nonlinear cavity
detector reaches the quantum limit on QND detection [cf.
Eq. (13)], implying that the backaction-imprecision productS̄F F S̄zz attains its minimum possible value of h̄2 /4. In contrast,
we find that the backaction noise S̄F F (and, hence, backaction
dephasing rate) is a factor G
1 larger than the quantumlimited value [cf. Eq. (48) and Fig. 5]. Here, the discrepancy
arises from the fact that Ref. [20] does not explicitly calculate
the measurement rate (i.e., 1/S̄zz ) for a specific optimized
cavity readout scheme but rather assumes that it also begiven (up to a prefactor) by overlap expression in Eq. (49).
This would imply the measurement imprecision noise S̄zz [0]scales like 1/G in the large-G limit. In contrast, we explicitly
consider homodyne detection of the cavity output. We findthat the imprecision noise (and, hence, measurement rate) are
independent of G in the large gain limit, in agreement with
Ref. [18]. This is a simple consequence of the fact that thenonlinear cavity’s parametric gain amplifies both the signal
and√ the vacuum fluctuations driving it by the same factor ofG.

D. Quantum limit on the amplifier added noise(49)

This elegant result was first derived in Ref. [31] in thecase where ẑ is a spin operator for a qubit; S̄F F [0], inthis case, is directly proportional to the qubit dephasingrate [cf. Eq. (14a)]. Heuristically, it expresses the fact thatthe backaction disturbance of the measurement is directly
related to the distinguishability of cavity states associated withdifferent values of the input signal. A small change in the
input signal causes a small displacement of the coherent state(50)

While in the low-frequency large gain limit, the nonlinear
cavity system cannot function as a quantum-limited QND
qubit detector; nonetheless, it may be a quantum-limited
linear amplifier (i.e., have the minimum noise temperature TN
allowed by quantum mechanics). This difference stems fromthe fact that, when used as an amplifier (in the op-amp mode),
one can take advantage of correlations between backaction andimprecision noises by tuning the susceptibility of the signal
source (e.g., in a voltage amplifier, the source impedance).

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1.00

S zF /

S zz S FF

0.98
0.96
0.94
0.92
0.90

5

10

50 100500 1000

G
FIG. 6. (Color online)
Backaction-imprecision correlations asmeasured by S̄zzF [0]/ S̄zz [0]S̄F F [0] versus parametric photon number gain G. As expected, the curves all tend to the universal value of
1 as G → ∞. Individual curves correspond to the same parameters
as in Fig. 4.

To leading order in G and at low frequencies, we haveshown that the backaction and output noise operators areproportional to one another, implying perfect correlation:
S̄zF [0] ∼ S̄zz [0]S̄F F [0].
(51)
Figure 6 shows the behavior of these correlations versus G,
where G is tuned is various ways; the asymptotic perfectcorrelation behavior is clear.
Turning to Eq. (8) for the optimized noise temperature TN ,we see that perfectly correlated backaction and imprecision
noises do not contribute. This implies that our leading-orderin-G analysis is insufficient to determine whether TN isquantum√ limited: This analysis only tells us that there is noorder- G term in TN . To determine whether the quantum
limit is reached near the bifurcation, one must go beyond
leading-order expressions, although we are interested in the
low-frequency limit. Such an analysis is straightforward,
although tedious; details are presented in the Appendix.
Obtaining the cavity noise correlators and forward gain exactly
from Eqs. (26) with no large-G assumption, we find that, atzero frequency, the nonlinear cavity detector always optimizes
the quantum noise inequality of Eq. (11) (i.e., it is satisfied
as an equality). As such, the minimal low-frequency noise
temperature given by Eq. (8) is indeed the quantum-limited
value of h̄ω/2. We stress that this result is completely
independent of the choice of homodyne phase ϕh .
E. Utility of backaction-imprecision correlations

As always, achieving a quantum-limited noise temperature
is not simply a question of having an amplifier that saturates
the fundamental quantum noise inequality of Eq. (11)—one
also needs to optimally tune the susceptibility χzz [ω] of thesignal source (i.e., the source impedance). This optimization
results in two conditions, cf. Eqs. (9). The magnitude condition[cf. Eq. (9a)] can always be achieved by an appropriate
tuning of the signal-detector coupling A; it corresponds toproperly balancing the relative contributions of backaction andimprecision noises to the total added noise. In contrast, thephase condition [cf. Eq. (9b)] cannot be achieved by simply
tuning A. It corresponds to optimizing χzz [ω] to optimally
make use of in-phase backaction-imprecision correlations
described by ReS̄zF .
Consider the nonlinear cavity detector in the low-frequency
large gain regime considered above. We found that it has amaximal value of correlations S̄zF , Eq. (51). Equation (9b)then implies that reaching the quantum limit on the noise
temperature requires Imχzz [ω] = 0. This is in sharp contrast
to the more common situation where S̄zF vanishes, and the
optimal source susceptibility χzz must be purely imaginary.
This has interesting consequences. For concreteness, consider the case where our input system is a mechanical oscillator
and z represents a position χzz [ω] is simply given byχzz [ω] =−1/m.
2
ω2 − ωM
+ iωγ

(52)

Here, ωM is the resonance frequency of the mechanicaloscillator, m is its mass, and γ is its damping rate. We see thatχzz [ω] is purely real if one is far from resonance, i.e., |ω −ωM |γ . Thus, the nonlinear cavity detector is ideally suited
to applications where one is interested in nonresonant position
detection. For example, standard interferometric gravitational
wave detectors require sensitive position detection of a test
mass in the free-mass limit, i.e., ωM → 0 [33]. In this case, aslong as ωγ , one always has a nonresonant situation, andχzz is real. For such frequencies, the nonlinear cavity amplifier
would be able to achieve a quantum-limited noise temperature.
In contrast, if one used a detector with S̄zF = 0 in this regime,
the noise temperature is, at best, a factor ωM /γ
1 larger than
the quantum-limited value. The utility of using correlations
between backaction and imprecision noises is well known
in the gravitational wave community [34], although it is not
usually discussed in terms of the general noise temperature
language used here.
Finally, we note that, if the input signal ẑ was a voltage
and we think of our cavity amplifier as a voltage amplifier,the requirement that the input susceptibility be purely real
to optimize the noise temperature translates into requiring asignal source with a purely imaginary source impedance [5].
V. AMPLIFIER NOISE AT NONZERO FREQUENCIES

It is straightforward to extend our analysis to describe theamplification of signals with frequencies ω that are nonzero
but still small enough that the parametric gain G[ω]1. It
follows from Eq.
(28)
that,
in
the
G[0]
1
limit,
thisrequires√ω  B ∼ κ/ G[0]. Simple analytic expressions are easily
obtained in the limit where G[0] → ∞ while ω/ B stays
finite. For leading order in G[0], one finds (as expected) thatthe photon number gain G[ω] and forward gain χI F [ω] have a
Lorentzian frequency dependence on a scale set by B . Lettingω̃ = ω/ B , we have
G[0]
,
(53a)
1 + ω̃2
χI F [0]χI F [ω] =.
(53b)
1 − i ω̃In the same limit, we find that the imprecision noise isfrequency independent, whereas, the remaining correlators

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C. LAFLAMME AND A. A. CLERKPHYSICAL REVIEW A 83, 033803 (2011)

also decay with frequency on a scale set by B :S̄zz [ω] = S̄zz [0],(54a)S̄F F [0],1 + ω̃2(54b)

S̄F F [ω] =S̄zF [0].
(54c)
1 + i ω̃It immediately follows that, for finite frequencies, the noise
temperature behaves as


2
ω̃
1 G
G ω̃
kB TN [ω] h̄ω++.
4
3 1 + ω̃2
3 1 + ω̃2
S̄zF [ω] =(55)Thus, we see that, at finite frequencies, the reduced noise
temperature 2kB TN /h̄ω rapidly increases as a function of
frequency from the quantum-limited value of 1; in particular, it
is already much greater than 1 for frequencies small enough to
not appreciably reduce the gain. The leading correction at finite
ω comes from the imaginary part of the noise cross correlator
S̄zF [ω]. As discussed extensively in Ref. [5], such out-ofphase backaction-imprecision correlations cannot be taken
advantage of by simply tuning the susceptibility of the source;
as a result, their existence represents unused information and,thus, leads to a departure from the quantum limit. In principle,
such correlations can be utilized via feedback techniques.
VI. BACKACTION COOLING

In the preceding analysis, we have seen that, near thebifurcation point, the backaction noise of the nonlinear cavity
amplifier diverges; this prevents quantum-limited amplification unless one can make use of noise correlations. In thissection, we change focus somewhat and consider the specificcase where the input signal ẑ is the position of a mechanicalresonator. In this case, the large backaction of the nonlinear
cavity may actually be useful: It has the potential to strongly
cool the mechanical resonator toward its quantum ground state.
The topic of backaction cooling has received considerable
attention in the optomechanics and electromechanics communities [35]. It has been shown that the backaction of alinear cavity dispersively coupled to a mechanical resonatorcan be used to ground-state cool the mechanical resonator if
one is in the so-called good cavity limit, where the mechanicalfrequency ωM is much larger than the cavity damping rate κ[36,37]. This regime has been exploited in recent experimentswith linear microwave cavities [38,39] and optical cavities[40–42].
In the opposite regime of a low-frequency mechanicalresonator (ωM  κ), cooling using a linear cavity is still
possible, but the lowest achievable temperature is on the orderof TBA ∼ κ/kB . A crucial parameter is the backaction damping(or optical damping) rate γBA : this is the enhanced dampingof the mechanical resonator resulting from a net energy loss tothe driven cavity. The cooling power of the cavity backaction
will be directly proportional to γBA . In the low-frequency
regime, a simple classical linear-response argument yields that,for a mechanical resonator dispersively coupled to a cavity,γBA ∝ d n̄/d [37,43]. As discussed in Ref. [8], thus, one

expects that a nonlinear cavity will be capable of much stronger
backaction damping than a linear cavity, given the enhanced
slope of the cavity response curve (cf. Fig. 3 ).
A large backaction damping is not, however, in itself enoughto ensure good cooling. One needs that the cavity acts as asource of cold damping for the mechanical resonator. Thus, onemust also consider the effective temperature of the backaction
noise TBA . Reference [8] examined this quantity numerically;in contrast, our approach allows us to obtain simple analytic
expressions in the interesting regime where one is near thebifurcation point and the parametric gain G[0]1.
Our analysis is based on the unsymmetrized backaction
noise spectral density, defined as ∞SF F [ω] ≡dtF̂ (t)F̂ (0).
(56)−∞

The symmetrized noise considered in Secs. II– V is given by
S̄F F [ω] = (SF F [ω] + SF F [−ω])/2. The frequency asymmetry of SF F [ω] describes the asymmetry between emission andabsorption of energy by the cavity; a standard perturbative
calculation shows that it directly determines the backaction
damping of the mechanical resonator [5]:γBA [ω] =1(SF F [+ω] − SF F [−ω]),2mh̄ωm(57)

where m is the oscillator mass. Using the linearized-Langevinapproach described in Secs. II– V, we find a particular simple
form for γBA near the bifurcation, in the limit where ωM / B
remains constant as the parametric gain G → ∞:G[0]1 A2 n̄
1 A2 n̄≡G[ω].
γBA ∼ √√
3 h̄mκ 2 1 + (ω/ B )2
3 h̄mκ 2(58)

In the ωM → 0 limit, this reproduces the classical expression
γBA ∝ d n̄/d, while for nonzero frequency, we see that thebackaction damping decays rapidly on the scale of the parametric amplification bandwidth B . The full expression (valid
even for small G) is given in the Appendix. It is instructive
to compare this result for γBA against the corresponding
expression for a linear cavity, in the relevant limit ωM  κ,
and for an optimized detuning [36,37]. As expected, one findsthat the nonlinear cavity’s γBA is enhanced by a factor of theparametric gain G[ω].
As already discussed, we must also consider the effective
temperature TBA [ω] of the backaction, a quantity that is, ingeneral, frequency dependent and is defined as [5]
SF F [−ω]
h̄ω
≡
.
(59)
exp −
kB TBA [ω]SF F [+ω]Using our linearized-Langevin approach, we find a particularly
simple asymptotic expression for the Bose-Einstein factor nBA
associated with TBA [ω] in the large-G limit relevant near thebifurcation:
1 + 2nBA [ω] ≡ cothh̄ω
2kB TBA [ω]∼1 + 3(ω/κ)2.
√3(ω/κ)(60)

The full expression for nBA is given in the Appendix. The expression for nBA [ω] is remarkably similar to the corresponding
expression for a linear cavity [36,37]. In particular, the relevant
frequency scale is κ and not the much smaller scale set by033803-10

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the parametric bandwidth B . Thus, one finds that, in the
low-frequency limit ωM  κ,κTBA [0] ∼ √ .
(61)
2 3

and defining x̂in ,p̂in to be the corresponding quadratures of thenoise operator ξ̂ , the equations of motion take the form
√ x̂
d x̂
x̂
(A3)
=M
− κ in .
p̂in
p̂
p̂
dt

In contrast, in the low-frequency limit, the effective backaction
temperature of an optimally driven linear cavity is κ/2.
Thus, the effective backaction temperature of our nonlinear
cavity near the bifurcation only differs by a numerical prefactor
from that of a linear cavity. For low frequencies, the final
oscillator temperature Tosc is given by [37]Here, M is the matrix defined asg̃ − κ2 − κ  .
M=− g̃ + 2

Tosc =

γ0 T0 + γBA TBA.
γ0 + γBA(62)

Here, γ0 is the oscillator damping resulting from its intrinsic(i.e., nonbackaction) sources of dissipation, and T0 is thetemperature of this bath. Thus, we have established that, for a
low-frequency mechanical resonator, the nonlinear cavity is a
far better way to cool than the linear cavity. One has a muchgreater backaction damping rate as well as a slightly smaller
effective backaction temperature.
VII. CONCLUSIONSIn this paper, we have given a theoretical treatment of
the quantum measurement properties of a driven nonlinear
cavity used as a linear detector or amplifier. By using theequivalence between this system near its bifurcation point
and a degenerate parametric amplifier driven by a detuned
pump, we were able to give a relatively simple descriptionof the physics. We find that quantum-limited amplification isindeed possible, but only if one is able to make use of thelarge correlations between backaction and imprecision noises.
Such correlations are ideally suited for position detection of amechanical system far from resonance; however, they cannot
be utilized in QND qubit detection, and, hence, one is far
from reaching the relevant quantum limit on this task. We
also examined the possibility of backaction cooling using
this system, demonstrating that the nonlinearity is particularly
useful in the case where one wants to cool a mechanicalresonator whose frequency is ωM  κ.
ACKNOWLEDGMENTS

We thank K. Lehnert and R. Vijay for useful discussions.
This work was supported by NSERC, FQRNT, and theCanadian Institute for Advanced Research.

Equation (A3) can conveniently be solved by first diagonalizing M. The only subtlety is that, due to the nonzero
 , M is non-Hermitian; as a result, its
effective drive detuning eigenvectors are not orthogonal to one another. Defining θ asper Eq. (25), we letcos(θ/2) sin(θ/2)V=(A5)
sin(θ/2) cos(θ/2)
denote the matrix whose columns are the eigenvectors of M.
One then has0(χ1 [0])−1V−1 ,M = −V(A6)0(χ2 [0])−1
where the eigenvalues of M are just the inverses of thesusceptibilities χ1 [0],χ2 [0] defined in Eq. (27).
The rotation defining the quadratures X̂ and P̂ introducedin Eq. (24a) can now be written as


X̂
x̂
cos (θ/2) sin (θ/2)
x̂
≡T=. (A7)p̂− sin (θ/2) cos (θ/2)
p̂
P̂The form of M makes it clear that X̂, as defined in Eq. (24a),
is indeed the amplified eigenquadrature of the cavity: It
corresponds to the first eigenvector and eigenvalue of M. In
contrast, the orthogonal quadrature P̂ defined in Eq. (24b)does not correspond to an eigenvector of M.
Finally, Fourier transforming the equations of motion
Eq. (A7) using the convention,
 ∞Â[ω] ≡
dt Â(t)e−iωt ,(A8)−∞and making the above rotation, we find
√ X̂in [ω]X̂[ω](iω1 + TMT−1 )= κ,P̂in [ω]P̂ [ω](A9)

where the identity matrix is 1ij = δij . Solving for X̂[ω] andP̂ [ω] directly yields Eqs. (26).
2. Backaction force

Using the definition of the backaction force operator F̂
given in Eq. (31) and solutions to the cavity equations of
motion, Eqs. (26), we find

APPENDIX
1. Mapping to the Detuned DPA

Using Eqs. (18) and (19), the equation of motion for the
displaced cavity annihilation operator d̂ takes the form√ )d̂ + g̃ d̂ † − κ ξ̂ (t).
d̂˙ = (−κ/2 + i (A1)Introducing the canonical quadratures,


1
x̂
1 1
d̂
=√
,
p̂
d̂ †
2 −i i(A4)(A2)

F̂ [ω] = Fx [ω]X̂in [ω] + Fp [ω]P̂in [ω],where√Fx [ω] ≡ −A 2n̄κ sin νχ1 [ω],(A11a)√
Fp [ω] ≡ −A 2n̄κ{(χ2 [ω] − χ1 [ω]) sin ν tan θ+χ2 [ω] cos ν},(A11b)and the angle ν is defined in Eq. (32).

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C. LAFLAMME AND A. A. CLERKPHYSICAL REVIEW A 83, 033803 (2011)

The unsymmetrized force noise spectral density SF F [ω]defined in Eq. (56) can be written in terms of F̂ [ω] as
2π δ(ω + ω )SF F [ω] = F̂ [ω]F̂ [ω ].

(A12)Thus, we can use Eq. (A10) to calculate SF F [ω] if we know
the correlation functions of the input noise operators X̂in , P̂in .
From standard input-output theory and our assumption thatξ̂ (t) describes vacuum noise, one easily finds

4. Imprecision-backaction correlation

The symmetrized imprecision-backaction noise correlator
S̄I F [ω] may be written

X̂in [ω]X̂in [ω ] = π δ(ω + ω ),(A13a)S̄I F [ω] = 12 (SI F [ω] + SI F [−ω]∗ ),P̂in [ω]P̂in [ω ] = π δ(ω + ω ),(A13b)

where the unsymmetrized correlator is given by

X̂in [ω]P̂in [ω ] = iπ δ(ω + ω ).

(A13c)

2π δ(ω + ω )SI F [ω] = Iˆ[ω]F̂ [ω ].

Explicitly computing the symmetrized spectral density
S̄F F [ω] = (SF F [ω] + SF F [−ω])/2, we findS̄F F [ω] = 4A2 n̄κκ 2 + 6g̃ 2 + 4ω2 − 2g̃ 2 (cos 2θ + 4 sin θ ).
×(κ 2 + 4ω2 )2 − 8g̃ 2 cos2 θ (κ 2 − 4ω2 − 2g̃ 2 cos2 θ )(A14)
We have used the value of the angle ν given in Eq. (32). We
stress that this expression only involves our initial linearizationof the dynamics and does involve any further assumptionof being close to the bifurcation. In the limit where one
approaches the bifurcation (i.e., κχ1 [0] → ∞, θ → π/6), one
obtains the asymptotic form given in Eq. (47).
One can use Eq. (A14) to verify that S̄F F [0] is indeed
related to the derivative of â with respect to √as per
Eqs. (49) and (50). This is easily done using â = n̄eiφa ,where n̄ is given by Eq. (17), and the phase φa is given byκ(A15)
tan φa = −2 + 4n̄
(as follows from the classical equations of motion).
Finally, we note that the above results are easily generalized
for finite temperature. The input noise correlators in Eqs. (A13)and S̄F F [ω] are simply multiplied by (1 + n̄th ), where n̄th isa Bose-Einstein factor evaluated at the temperature of theincident thermal noise and the frequency of the cavity.
3. Imprecision noise

The intensity Iˆ[ω] of the homodyne measurement is given
by Eq. (35). Using the solutions to the equation of motion in
Eq. (26), we obtain
Iˆ[ω] = Ix [ω]X̂in + Ip [ω]P̂in ,where

SI I [ω] = SI I [−ω], and, thus, SI I [ω] = S̄I I [ω]. The imprecision noise spectral density S̄zz [ω] then follows using
Eqs. (6) and (37).

−κ cos ϕh tan θ (χ2 [ω] − χ1 [ω])].

(A19)

Thus, we may calculate S̄I F [ω] using Eqs. (A10), (A16),and (A13); dividing by χI F [ω] as given in Eq. (37) then yields
the desired correlator S̄zF [ω].
5. Cooling

Using Eqs. (A11), one finds that the full expression for theasymmetric-in-frequency part of SF F [ω] is given by
SF F [+ω] − SF F [−ω]ω64A2 n̄g̃κ(sin θ − 1).
= 2
(κ + 4ω2 )2 − 8g̃ 2 cos2 θ (κ 2 − 4ω2 − 2g̃ 2 cos2 θ )(A20)This expression then directly gives the backaction damping
via Eq. (57).
Combining the above expression with Eq. (A14) forS̄F F [ω], we can find a general expression for nBA [ω], theeffective temperature of the backaction expressed as a numberof quanta [cf. Eq.(60)]. We have
1 + 2nBA [ω] =κ 2 + 6g̃ 2 + 4ω2 − 2g̃ 2 (cos 2θ + 4 sin θ ).
8g̃ω(sin θ − 1)
(A21)

We stress that Eqs. (A20) and (A21) do not involve anassumption of being near the bifurcation.
6. Reverse gain

The forward gain in our system was defined in Eq. (4),which, upon Fourier transforming, takes the form

(A16)Iˆ[ω] = χI F [ω]ẑ[ω].

√Ix [ω] = κ cos ϕh (1 − κχ1 [ω]),√Ip [ω] = κ[sin ϕh (1 − κχ2 [ω])(A18)

(A17a)(A17b)

It is now a straightforward exercise to compute the spectral density SI I [ω] from Eqs. (A16) and (A13), in complete analogy to our calculation of SF F [ω]. One finds this
output noise to be completely symmetric in frequency:(A22)

We derived χI F [ω] in the main text by accounting for the
coupling to ẑ in the cavity equations of motion, resulting in
Eq. (37).
In general, an amplifier may also have a reverse gain
χF I [ω]; this describes how signals coupled to the output
operator Iˆ could affect the average value of the backaction
force operator F̂ [5]. In general, reverse gain is undesirable,
as it implies that measuring the detector output (by coupling
to it) can lead to enhanced backaction fluctuations. The forms

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QUANTUM-LIMITED AMPLIFICATION WITH A . . .

PHYSICAL REVIEW A 83, 033803 (2011)of the fundamental quantum noise inequality of Eq. (11) arealso modified in the presence of reverse gain.
To show that the reverse gain of our cavity amplifier
vanishes, we make use of the equation [5],χI F [ω] − χF I [ω]∗ = −(i/h̄)(SI F [ω] − SI F [−ω]∗ ).

Using the solution of the cavity equations of motion to
calculate SI F [ω] and using the expression for χI F [ω] from
Eq. (37), Eq. (A23) directly yields that there is no reverse gain
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