PRL 109, 123602 (2012)

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21 SEPTEMBER 2012PHYSICAL REVIEW LETTERS

Weak Qubit Measurement with a Nonlinear Cavity: Beyond Perturbation Theory
C. Laflamme and A. A. Clerk
Physics Department, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 25 June 2012; published 19 September 2012)
We analyze the use of a driven nonlinear cavity to make a weak continuous measurement of a dispersively
coupled qubit. We calculate the backaction dephasing rate and measurement rate beyond leading-order
perturbation theory using a phase-space approach which accounts for cavity noise squeezing. Surprisingly,
we find that increasing the coupling strength beyond the regime describable by leading-order perturbation
theory (i.e., linear response) allows one to come significantly closer to the quantum limit on themeasurement efficiency. We interpret this behavior in terms of the non-Gaussian photon number fluctuationsof the nonlinear cavity. Our results are relevant to recent experiments using superconducting microwave
circuits to study quantum measurement.
DOI: 10.1103/PhysRevLett.109.123602

PACS numbers: 42.50.Dv, 03.65.Yz, 42.50.Pq, 85.25.Cp

Introduction.—There is considerable interest in exploiting
continuous weak quantum measurements for the detectionof fundamental quantum behavior as well as for quantum
information processing [1–3]. By weak measurement, wemean the generic situation where the signal produced in thedetector by the measured system is small compared to intrinsic output noise, and thus information is obtained only gradually in time. Such measurements are ultimately constrainedby the Heisenberg uncertainty principle, which dictates thatthe backaction disturbance of the system by the detector
cannot be arbitrarily small, but is instead bounded by therate at which information is acquired [1–5]. Detectors
capable of yielding an optimally small ratio of backactionto-information gain are known as quantum limited. They areboth of fundamental interest, and are also necessary if one
wishes to implement continuous quantum feedback algorithms [6–8] or certain quantum error correction schemes [9].
Not surprisingly, weak measurements are usually analyzed in the limit of a system-detector coupling smallenough that leading-order perturbation theory in the coupling applies; in this standard regime, the quantum limit
reduces to a constraint on the noise properties of the detector[3,4]. Here, we focus on an alternate regime, where thedetector-system coupling is still weak enough that information is obtained gradually in time, but not so weak thatleading-order perturbation theory is sufficient. This regimeof a ‘‘weak-but-not-too-weak’’ measurement has recently
been achieved in experiments using a driven, nonlinear
superconducting microwave cavity to measure the state ofa superconducting qubit [10]. As with experiments using
linear microwave cavities [11,12], the qubit is dispersively
coupled to the cavity, meaning that the cavity frequency
depends on the qubit state; by monitoring the phase ofreflected microwaves from the cavity, one can monitor thequbit state. Introducing a nonlinearity in the cavity via a
Josephson junction (see Fig. 1) allows one to operate thecavity detector close to a point of bifurcation, where thestate of the driven cavity is an extremely sensitive (but still
0031-9007=12=109(12)=123602(5)single-valued) function of its frequency. The enhanced
sensitivity of this regime naturally leads to conditions where
the measurement is weak, but the qubit-detector couplingcannot be treated perturbatively. While information gain isenhanced here, the question remains whether this speedup
comes at the cost of deviating from the quantum limit(i.e., excess backaction dephasing).
In this Letter, we present an analytic theory describing
weak measurement of a qubit with a nonlinear cavity operated close to a point of bifurcation. Our nonperturbative

FIG. 1 (color online). (a) Inset: schematic showing one possible realization of a Josephson-junction circuit (i.e., nonlinear
cavity) dispersively coupled to a qubit. Main: Measurement rateand dephasing rate versus qubit coupling strength , in units ofthe cavity damping rate , using logarithmic axes. The blue (dotdashed) curve is the full measurement rate meas , while the lightblue (dotted) curve is the linear-response approximation to meas .
The red (solid) line is the dephasing rate ’ as obtained from thefull theory presented in the main text. The remaining lines are ’
calculated within less rigorous approximations: the grey (shortdashed) curve is the linear-cavity formula ’;0 [cf. Eq. (5)] andgreen (long-dashed) curve is leading-order perturbation theory.
Parameters are  ¼ 103 ; f0 ¼ 0:75fbif ; n  200;  ¼ bif
where fbif and bif are the driving force amplitude and detuningat the cavity bifurcation. The parametric photon number gain
is G  102 . One clearly sees that for moderate couplings, thedephasing rate is strongly suppressed compared to the perturbative result. (b) Same, but using linear axes.

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PHYSICAL REVIEW LETTERS

PRL 109, 123602 (2012)

approach accounts for the nontrivial cavity noise physics
associated with the nonlinearity. We find that the
information-gain to state-disturbance ratio is a strong function of the qubit-detector coupling strength. In the limit ofan extremely weak qubit-detector coupling, we recoverprevious perturbative results [13–16], which indicate a large
deviation from the quantum limit: the backaction dephasingrate is a large factor G greater than the rate of informationacquisition (the measurement rate), where G  1 is theparametric photon-number gain associated with the driven
nonlinear cavity (i.e., a single signal photon incident on thedetector will result in G photons at its output). Increasing thecoupling beyond the perturbative regime, we find remarkably that the dephasing rate is greatly suppressed comparedto the leading-order prediction. This allows one to approachthe quantum limit to within a factor of order unity. Our
approach provides a general framework for investigating
quantum measurement with driven nonlinear systems
beyond weak coupling.
Note that the backaction of a nonlinear cavity detector
was also considered by Boissonneault and co-workers[10,15,16]. They described backaction dephasing beyond
lowest-order in the coupling by approximating the state ofthe driven cavity state conditioned on the qubit state to be asimple coherent state; this is only valid for operating points
far from a bifurcation, where the detector has a relatively
small gain. We show that close to a bifurcation (where thecavity exhibits parametric gain and squeezing), this approach does not accurately capture the coupling dependence of the backaction dephasing.
Model.—We consider a qubit coupled dispersively to asingle-sided nonlinear cavity. While our approach applies
to an arbitrary nonlinearity, we focus here on the typicalexperimental situation [10,17] where a Kerr-type nonlinearity dominates. Working in a frame rotating at the cavity
drive frequency !d , the Hamiltonian is (@ ¼ 1)
H^ sys ¼ a a  a a a a þH^  þ H^ qb þ H^ int ; (1)^y ^^y ^y ^ ^where  ¼ !d  !cav is the detuning between the cavity
drive and resonance frequency,  is the Kerr constant, H^ 
describes the cavity damping (with rate ) and driving
due to coupling to external modes, and H^ qb ¼ ^ z is thequbit Hamiltonian. The dispersive quantum nondemolition^
qubit-cavity coupling Hamiltonian is H^ int ¼ ^ z a^ y a,where the coupling strength  sets the qubit-dependent
cavity frequency shift. We will take the cavity to be atzero temperature, and ignore any intrinsic qubit dissipation, as we are interested only in the measurement
backaction.
As discussed, making a weak z measurement of thequbit involves strongly driving the cavity while monitoring
the reflected light from the cavity via a homodyne measurement; the two possible values of z will lead to two
different average homodyne currents, which as time progresses can be resolved above the intrinsic noise in theseweek ending
21 SEPTEMBER 2012

currents [3,12]. We will focus exclusively on a weak
nonlinearity    and a strong drive amplitude, such^that the stationary average cavity photon number ha^ y ai1
regardless of the initial qubit state. Apart from this constraint, we will not place any other restrictions on how
small the qubit-cavity coupling  must be.
Dephasing rate.—We first calculate the backaction
dephasing of the qubit that occurs during such a measurement; this dephasing is a direct consequence of the intracavity photon-number fluctuations. If the qubit is initiallyin a z eigenstate, it will remain in this state (due to thequantum nondemolition nature of the measurement); thus,from the cavity’s perspective, the two qubit eigenstates
simply correspond to a shift of the cavity resonancefrequency by either . In each case, the classically
expected cavity amplitude  ( ¼"; # ) will be givenby [14,16]



(2)
 þ ið   þ 2j"=# j2 Þ "=# ¼ if0 ;2
where f0 is the amplitude of the cavity drive. Note that we
focus on driving strengths small enough that we are below
the bifurcation; i.e., there is only one classical solution 
for a given . By now, writing a^ ¼ d^ þ  and using thefact that j j  1, we can approximate the cavity
Hamiltonian corresponding to each qubit eigenstate by^ d^y .
only keeping terms that are at most quadratic in d;
We thus obtain two linearized cavity Hamiltonians, corresponding to the two qubit states. Each has the generalform of a degenerate parametric amplifier driven by an
off-resonant pump [14,18,19]:~  d^y d^ þ i ð~^H^  ¼ 
g d^y d^y  g~ d^ dÞ;
2 

(3)

~ "=# ¼    þ 4j"=# j2  is the effective pumpwhere detuning, and g~ ¼ 2i2  is the parametric strength. As
one approaches a point of bifurcation in the cavity (e.g., byincreasing the drive strength f0 ), the corresponding degenerate parametric amplifier Hamiltonian approaches thethreshold of self-oscillation [14] (see inset of Fig. 2). For
such operating points, incident waves on the cavity in theappropriate quadrature will be strongly amplified; thisamplification is described by a photon number gain G
which diverges as one approaches pthe
ﬃﬃﬃﬃ bifurcation, as well
as a narrow bandwidth slow  = G [14,17,19].
While the cavity evolution is easy to understand whenthe qubit is initially in a z eigenstate, to calculate thedephasing rate we need to understand the cavity dynamics
when the qubit is in a superposition of its eigenstates. We
focus on the long-time qubit dephasing rate, which isdefined as usual in terms of the decay of the qubit’s offdiagonal density matrix elements in the long-time limit:
j lnTrðj#ih"^
jÞj=t ! ’ , where ^ is the density matrix
describing the full system.

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PRL 109, 123602 (2012)PHYSICAL REVIEW LETTERS

FIG. 2 (color online). Measurement efficiency ratio meas =’ as a function of coupling strength for the same
parameters as Fig. 1 (G ¼ 102 , red solid curve), and for an
operating point closer to the bifurcation yielding G  103 (bluedashed curve); the quantum limit is  ¼ 1 (dashed black line).
Vertical lines indicate the maximum  for which the measurement timep1=
ﬃﬃﬃﬃ meas is longer than the detector response time
1=slow  G=. For  ! 0, one misses the quantum limit by alarge amount:   1=G. However, a small increase in coupling
greatly improves this efficiency ratio. Inset: Average cavity
photon number versus drive detuning  (parameters as inFig. 1). The black point indicates the chosen working point.
The two qubit states lead to two different effective values of ;
these are shown as white circles for  ¼ 0:05.

To proceed, we first introduce ^ "# ¼ Trqb;bath ðj^ #ih"jÞ,where the trace is over the qubit and cavity bath degrees of
freedom. This is an operator acting in the cavity Hilbert
space; its trace yields the off-diagonal element of the qubit
density matrix, and hence can be used to obtain ’ . We
further transform ^ "# by displacing away the two stationaryclassical cavity amplitudes  associated with each qubit
state. We thus obtain an operator ~^ "# :^
~^ "# ðtÞ  Dð^ "# ðtÞD^ y ð# Þ;
" Þ

(4)

^
where DðÞ
¼ expða^ y  H:c:Þ is the cavity displacement
operator. One can show that in the long-time limit, theexponential decay of Tr
~^ "# ðtÞ also yields the dephasing rate
’ [20].
It is now straightforward to rigorously derive the evolution equation of ~^ "# , starting from the standard Linbladmaster equation describing the evolution of the cavityplus-qubit density matrix [20] (see also Ref. [16]):@ ^
~ ¼ D½d^ ~^ "#  ~^ "#~^ "#  iðH^ " ~^ "# H^ # Þ  ’;0 @t "#þ ½ð"  # Þ
~^ "# d^y  ð"  # Þd^~^ "# :

(5)^~^ "# d^y  ðd^y d^~^ "# d^y dÞ=2
Here D½d^ ~^ "# ¼ d^~^ "# þ is thestandard Lindblad superoperator describing cavity damping. The second and third terms in Eq. (5) correspond to the
Hamiltonian evolution of the cavity in our doublydisplaced frame, where we have used j j  1 to linearize

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the two cavity Hamiltonians H^  [cf. Eq. (3)]. The remaining terms on the rhs of Eq. (5) describe decoherence of thequbit resulting from the combination of the cavity driveand cavity dissipation, with ’;0 ¼ ð=2Þj"  # j2 .
For a linear cavity, the terms on the last line of Eq. (5) play
no role, and the backaction dephasing rate is given completely by ’;0 (i.e., by the distinguishability of the twoclassical cavity amplitudes) [21]. For our case of a nonlinear
cavity, the same is true if one neglects the parametric amplification terms in H^  (proportional to d^2 and ðd^y Þ2 ), as then~^ "# ðtÞ ¼ C expð’;0 tÞj0ih0j (where j0i is the vacuum state
and C a constant) trivially solves Eq. (5). This is equivalent
to finding that (for long times) the cavity state conditionedon the qubit is a coherent state j i. In this approximation,
the backaction dephasing rate is given completely by thelinear-cavity formula ’;0 [10,15,16]. However, such an
approximation completely neglects the squeezing of noiseby the nonlinear cavity. It is thus only valid for cavity
parameters extremely far from any bifurcation, in regimes
where the nonlinear cavity closely resembles a linear cavity.
Given the importance of noise squeezing, we go beyond
the above approximation by retaining all terms in Eq. (5).
Defining ðtÞ ¼  lnTr
~^ "# ðtÞ, the long-time backaction
dephasing rate will be given by ’ ¼ limt!1 ReðtÞ=t.
Setting  ¼ "  # , the trace of Eq. (5) yields
_ ¼ ’;0  ihH^ "  H^ # i"# þ h d^y  H:c:i"# ; (6)^ ¼
where we have defined the quasiexpectation value hOi
"#
^
^
^
^
TrðO
~"# Þ=Trð~"# Þ. As ~"# is not a true density matrix, thequasiexpectation of a Hermitian operator can be complex,
and hence the second and third terms above can contributeto the backaction dephasing.
We now use the fact that Eq. (5) only involves terms that^ d^y , and hence can be solved
are at most quadratic in d,exactly by a ~^ "# which has a Gaussian form (i.e., its phasespace representation is Gaussian [20]). Equation (5) thus
reduces to a closed set of evolution equations for the^ d^y (see Ref. [20] for
quasimeans and covariances of d;details). Solving these and substituting into Eq. (6) directly
gives _ and thus the dephasing rate. We stress that thisapproach is not perturbative in the coupling , and it does
not neglect the noise squeezing expected near a cavity
bifurcation. A similar procedure can be used to calculate
the backaction dephasing of a linear cavity subject to bothquantum and thermal noise [22].
To gain insight on the effect of increasing , we first use
the above approach to calculate ’ to order 4 . For a cavity
detuning  and drive f0 chosen to be close to the bifurcation
point, one finds


 3
G 2 2
2 G3=2 32 4 nG
5 4 G9=2
 n þ
’ ’


; (7)
 3
27 2
81 2
9
where n ¼ j0 j2 , 0 is the zero-coupling classical cavity
amplitude (i.e., solution to Eq. (2) at  ¼ 0), and G  1 is123602-3

PRL 109, 123602 (2012)PHYSICAL REVIEW LETTERS

the parametric photon-number gain (see Ref. [14]). At eachorder in , we have retained the leading terms in n and G.
 reproduces the results of
The first term here ( / 2 n)
Refs. [10,14,16] and arises solely from the linear-cavity
dephasing rate ’;0 in Eq. (6). The second term in Eq. (7)(also order 2 ) is missed if one linearizes the qubit-cavity
interaction, or makes the approximation ’ ¼ ’;0 . Moreinteresting are the 4 terms in Eq. (7). Surprisingly, these
corrections are negative, suggesting the possibility of a
relative suppression of dephasing with increased coupling(relative to the lowest-order-in- expression). The leading4 corrections are completely due to the last line of Eq. (5),terms that would vanish if one ignored the squeezing ofnoise near the bifurcation. As such, the approximation
’ ’ ’;0 would predict both an incorrect sign and scaling
with G of this term.
To see the full consequence of this behavior, we numerically solve Eq. (6) and the ordinary differential equations
determining the needed averages; we use parameters corresponding to a weakly nonlinear cavity operated near abifurcation point, similar to those realizable in experiment[10,17], and well within the regime of validity of ourtheory. In Fig. 1, one sees clearly that the backaction
dephasing rate as a function of coupling (red) drops markedly below both the expectations from lowest-order perturbation theory (green), and below the linear-cavity
formula ’;0 (grey). We have confirmed that this behavior
is generic whenever one is close to a bifurcation in thecavity: higher-order-in-  terms yield a marked suppression of the dephasing rate.
For further insight into the above behavior, we return to thephysical picture that qubit dephasing is due to the photonnumber fluctuations of the driven cavity. Treating
R thesefluctuations classically and defining mðtÞ  t0 dt0 nðt0 Þ(where n is the cavity photon number), one finds that theoff-diagonal qubit density matrix is directly proportional tothe characteristic function of the probability distribution ofm [23,24]. As such, the long-time qubit dephasing rate can be
expressed in terms of the even cumulants of m, hhm2j ii:
1
1X
ð2Þ2jhhm2j ii:ð1Þj1
t!1 t
ð2jÞ!
j¼1

’ ¼ lim(8)This expansion also holds in the quantum case where m^ is anoperator, if one now interprets the cumulants above using
the standard Keldysh operator ordering [23,24].
Equation (8) implies that terms of order 4 and higher in
’ are due to the non-Gaussian nature of intracavity photon
number fluctuations. As shown in Eq. (7), the relative suppression of ’ with  is already presaged by the negative
sign of the 4 term. From Eq. (8), we see that this implies apositive kurtosis, hhm4 ii > 0. Even a driven linear cavity inthe classical limit has non-Gaussian intracavity photon
number fluctuations and a positive kurtosis; this is because
n is the square of a Gaussian random variable, the cavity
amplitude [24]. A positive kurtosis signals a distribution

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which is more peaked than a Gaussian, and hence noisethat generates less dephasing than Gaussian noise. In our
nonlinear resonator, the non-Gaussian nature of the photon
number fluctuations is strongly enhanced near bifurcationby the intrinsic nonlinearity of the system. hhm4 ii remains
positive (like a linear resonator), but is much larger than eventhat of a degenerate parametric amplifier near threshold [20].
Measurement rate and quantum limit.—For a measurement that occurs slowly on detector time scales, we cancharacterize the information gain of the measurement by asingle measurement rate meas : how quickly do the distributions of the output homodyne current corresponding toeach qubit eigenstate ( " or # ) become distinguishable.
Generalizing the standard weak-coupling expression [1–3]to a situation where the coupling is not perturbative but
the measurement is still slow compared to internal detector
timescales, we find (see Ref. [20] for details)
meas ¼ ðI"  I# Þ2 =½4ðSII;" þ SII;# Þ :(9)Here I is the average stationary homodyne current when thequbit is in the state  ¼"; # , and similarly, SII; is the zerofrequency spectral density of homodyne current fluctuations
when the qubit is frozen in the state . One can rigorously
show that for arbitrary , the measurement efficiency ratio ¼ meas =’ 1 [20].
Using the linearized Hamiltonians H^  given in Eq. (3)along with standard input-output theory [25] lets us evaluate Eq. (9) for an arbitrary value of the coupling; comparing against the dephasing rate then allows us to investigate
the behavior of  as a function of coupling. One finds that,similar to the linear-cavity dephasing rate ’;0 , the measurement rate is largely determined by the classical amplitudes  , and is hence a far weaker function of  than thedephasing rate. The result is that there is a range of  where
higher-order terms significantly suppress the dephasingrate (over the perturbative expression), whereas the measurement rate is still determined by the leading-order
expression [see Fig. 1(a)].
Shown in Fig. 2 is  versus  for the same parameters as inFig. 1. In the limit of a vanishing coupling strength, one
deviates strongly from the quantum limit [14]. However,
the effective suppression of dephasing that occurs with amodest increase of coupling brings one within a factor oforder unity of the ultimate quantum limit bound  ¼ 1. We
find that this behavior is generic for cavity operating points
near bifurcation: increasing the coupling  beyond thevalidity of leading-order perturbation theory allows one to
make a weak measurement with a much higher efficiency
than in the extreme weak coupling limit. On a physical level,
this is a direct result of the non-Gaussian nature of photon
number fluctuations in the driven cavity (namely, the large
positive kurtosis).
Conclusions.—We have described a general method to
calculate the measurement and dephasing rate of a qubit
coupled to a nonlinear resonator that is not perturbative in

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PRL 109, 123602 (2012)PHYSICAL REVIEW LETTERS

the qubit-detector coupling and which accounts for cavity
noise squeezing. By increasing the coupling to a regime
where higher-order corrections are relevant, one can comesignificantly closer to the fundamental quantum limit on
weak continuous qubit measurement.
This work was supported by NSERC and CIFAR.

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