PHYSICAL REVIEW A 75, 042302 共2007兲

Using a qubit to measure photon-number statistics of a driven thermal oscillator
A. A. Clerk and D. Wahyu Utami
Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8
共Received 11 December 2006; revised manuscript received 7 February 2007; published 2 April 2007兲
We demonstrate theoretically how photon-number statistics of a driven damped oscillator at finite temperature can be extracted by measuring the dephasing spectrum of a two-level system dispersively coupled to theoscillator; we thus extend the work of M. I. Dykman and M. A. Krivoglaz 关Sov. Phys. Solid State 29, 210
共1987兲兴 and J. Gambetta et al. 关Phys. Rev. A 74, 042318 共2006兲兴. We carefully consider the fidelity of thisscheme—to what extent does the measurement reflect the initial number statistics of the mode? We also derive
analytic results for the dephasing of a qubit by a driven thermal mode, and compare results obtained at different
levels of approximation. Our results have relevance both to experiments in circuit cavity QED, as well as tonanoelectromechanical systems.
DOI: 10.1103/PhysRevA.75.042302

PACS number共s兲: 03.67.Lx, 32.80.⫺t

I. INTRODUCTION

sively 关11–13兴 and is being actively pursued experimentally.
One could thus easily envisage a setup where a superconducting qubit is coupled to both a electromagnetic cavity anda nanomechanical oscillator 关10,14兴. The mechanical oscillator would be strongly coupled to the qubit and would generate the number splitting effect, while the electromagnetic
cavity would be weakly coupled and would be used simply
to read out the qubit dephasing spectrum. The idea of using
the approach of Ref. 关5兴 to detect number states in a nanomechanical oscillator is highly attractive. From an experimental perspective, it is not as demanding as previous proposals 共e.g., strong nonlinearities in the oscillator are notrequired兲. In addition, we will see that such an experimentoffers the possibility of providing an unambiguous signatureof the discrete number states of the oscillator; no extensive
postexperiment interpretation is required.
The above discussion notwithstanding, there remain unresolved theoretical issues with the number splitting effect thatshould be addressed before contemplating a NEMS versionof the experiment. These issues are also relevant to thecircuit-QED experiment of Ref. 关5兴. First, as NEMS oscillators do not typically have frequencies much larger than temperature, one should now consider both the effect of finite
temperature and a driving force; this regime is not addressedby existing theory 关6,7兴. In this paper, we calculate analytically the qubit dephasing spectrum for a qubit dispersively
coupled to a driven, finite temperature oscillator, using a conceptually simple Wigner function approach. We recover theprevious results in the appropriate limits. Our results also
provide analytic expression for the long-time oscillatorinduced dephasing rate of the qubit in the driven, thermalcase 关cf. Eqs. 共44兲 and 共69兲兴. Note that the dephasing rate in
the absence of a drive was considered recently by Serban etal. 关15兴.
A second important issue we address here is the questionof measurement fidelity: To what extent is the qubit dephasing spectrum a faithful measurement of the mode’s number
statistics? It was already pointed out in Refs. 关5,7兴 that for ageneric driving force, there was not a simple connection between the observed peaks in the qubit dephasing spectrum
and the photon-number distribution in the mode. We demon-

There has been a recent explosion of interest in the properties of quantum oscillators. On one hand, nanoelectromechanical systems 共NEMS兲 共systems comprising of a micronscale oscillator coupled to an electronic conductor兲 havebeen used for near quantum-limited position detection, andare thought to be good candidates for the preparation of exotic quantum states and for use in quantum control experiments 关1兴. On the other hand, there has been considerable
success in studying circuit QED systems, in which a superconducting qubit is strongly coupled to a single mode in anelectromagnetic resonator 关2–4兴. Experiments in these systems have been able to enter the strong coupling regime ofcavity QED, achieving such milestones as the resolution of
the vacuum Rabi splitting 关3兴.
A clear goal in both these classes of systems is to reach aregime where the quantum nature of the resonator is unambiguously manifest in measured experimental quantities. An
example of this would be the observation of features associated with the discrete energy states of the oscillator. This wasrecently achieved in spectacular fashion in the circuit QED
system with a so-called “number splitting” experiment 关5兴. It
was experimentally shown that for a strong dispersive coupling between the qubit and the mode, the qubit dephasing
spectrum 共i.e., the time dependence of the qubit’s offdiagonal density matrix element兲 could be used to probe thephoton number distribution in the cavity mode. Theoretically, this effect was first described by Dykman for the caseof a undriven, thermal mode 关6兴. More recently, Gambetta etal. 关7兴 described the effect for the case of a zero-temperaturedriven mode, a case more relevant to the experiment of Ref.
关5兴.
In the case of NEMS systems, there have been several
theoretical proposals for measuring discrete number states,
but to date these have been difficult to implement experimentally 关8–10兴. Given the success of the circuit QED number
splitting experiment 关5兴, it is natural to ask whether a similar
approach could be taken with NEMS systems. In principle,
the answer is yes. The coupling of nanomechanical oscillators to superconducting qubits has been discussed exten1050-2947/2007/75共4兲/042302共12兲

042302-1©2007 The American Physical Society

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMI

strate that in general, back-action effects arise from the finite
damping of the oscillator; these imply that the dephasing
spectrum of the qubit will fail to perfectly reflect the initial
photon distribution of the oscillator. Nonetheless, we demonstrate that for a sufficiently strong oscillator-qubit coupling,
one can reliably extract the initial photon distribution of theoscillator.
The remainder of this paper is organized as follows. In
Sec. II, we provide a heuristic discussion of the numbersplitting effect and possible issues with back action. Section
III contains technical details of our calculation of theoscillator-qubit reduced density matrix, and may be skippedby the uninterested reader. In Sec. IV, we provide results fora thermal, nondriven oscillator; Sec. IV C discusses the feasibility of measuring the effect in a NEMS device. Finally,
Sec. V discusses the case of a thermal driven oscillator.
II. HEURISTIC DISCUSSION OF THE NUMBER
SPLITTING EFFECT

We consider a qubit which is dispersively coupled to aharmonic oscillator. Ignoring for the moment the damping
and driving of the oscillator, the Hamiltonian of the system isgiven by共1兲H = H0 + Hint ,
H0 =

冉

冊

ប⍀qb
1
␴ z + ប ⍀ a †a +
,
2
2

冉

Hint = ␭ ប ⍀ a†a +

冊

1
␴z .
2

been demonstrated for superconducting qubits 关16兴. An alternate method, as discussed in Refs. 关6,7兴 and used in theexperiment of Ref. 关5兴, is to measure ␳↑↓共t兲 via an absorption
experiment. In such a scheme, one attempts to excite thequbit from its ground state via a time-dependent field at frequency ␻rf which couples to ␴x. Within Fermi’s golden ruleand the quantum regression theorem 关17兴, the absorption rate
⌫abs共␻rf兲 is given by⌫abs共␻rf兲 = 兩A兩2

冕

⬁

−⬁

dt e+i␻rft具␴−共t兲␴+共0兲典冋冕⬁= 2兩A兩2Re0册

dt e+i␻rft␳↑↓共t兲 .

共6兲Here, the second line assumes the quantum regression theorem, and A is a matrix element proportional to the strength ofthe driving field.
A. Number splitting interpretation of ␳_`„t…

The simplest expectation for the behavior of ␳↑↓共t兲 follows from the observation that 共a兲 this quantity should oscillate at the qubit splitting frequency and 共b兲 the dispersive
coupling in Eq. 共3兲 implies qubit splitting frequency depends
on the number of quanta in the mode. This leads to the expected form⬁

共2兲␳↑↓共t兲 = e−i⍀qbt兺 P共n兲e−2i␭⍀共n+1/2兲t␳↑↓共0兲,

共7兲

n=0

共3兲

Here, ប⍀qb is the energy splitting of the qubit, ⍀ is theoscillator frequency, and ␭ is a dimensionless measure of theinteraction strength. As discussed, e.g., in Ref. 关2兴, an interaction of this type can arise from a Jaynes-Cumming type
coupling in the limit where the frequency difference 兩⍀qb
− ⍀兩 is large 共i.e., one transforms the starting Hamiltonian toeliminate the first-order qubit-mode interaction; the dispersive interaction above then results at second order in theoriginal coupling strength兲. It has recently been suggested
关15兴 that it is important to keep nonrotating terms in theinteraction Hamiltonian Hint 关i.e., terms proportional to a2
and 共a†兲2兴. We find that such terms play no role in the weakdamping, weak coupling limit of interest 共i.e.,␭  1 , ␥ / ⍀  1兲; we discuss this further in what follows.
Note that in this limit, the effects of the qubit-oscillator interaction may still be significant, as ␭⍀ / ␥ can be large.
The quantity we wish to study is the qubit’s off-diagonal
density matrix element. Letting 兩 ↑ 典 , 兩 ↓ 典 denote ␴z eigenstatesand ␳ˆ the density matrix of the oscillator-qubit system, we
define␳ˆ ↑↓ ⬅ 具↑兩␳ˆ 共t兲兩↓典,共4兲

␳↑↓共t兲 = Trosc␳ˆ ↑↓共t兲.

共5兲Note that ␳↑↓共t兲 is indeed a measurable quantity; one could
imagine extracting it using state tomography, as has recently

where P共n兲 is the number distribution of the harmonic mode.
Equation 共7兲 would of course be exact in the absence of
damping or forcing of the oscillator. It tells us that the timedependence of ␳↑↓ at a fixed coupling strength contains information about the oscillator photon number distribution at
t = 0: If one Fourier transforms ␳↑↓共t兲, one expects a set of
discrete peaks with a spacing ⌬␻ = 2␭⍀, with the weight ofeach peak yielding the probability P共n兲 of the corresponding
number state.
In the more general case where we have to contend with
damping and forcing of the oscillator, Eq. 共7兲 is no longer apriori valid. The interaction Hint now fails to commute withthe total oscillator Hamiltonian, and thus there will be a backaction associated with the measurement. Heuristically, turning on the coupling to the qubit can now itself cause achange in the number statistics of the oscillator. For example,
the coupling to the qubit causes the oscillator’s frequency tochange; at nonzero damping, this will cause it to re-equilibrate with the bath and thus change its number distribution. We can nonetheless attempt to use Eq. 共7兲 to extract
the number statistics of the oscillator 共i.e., use the weights ofpeaks in the spectrum of ␳↑↓ to extract the distribution of
number states in the oscillator兲. We will thus refer to Eq. 共7兲
as the “number splitting” interpretation of ␳↑↓; in what follows, we will assess the validity of such an interpretation.
B. Full counting statistics interpretation of ␳_`„␭…

A second, somewhat different way to connect ␳↑↓共t兲 tonumber statistics is to use the coupling dependence of this042302-2

USING A QUBIT TO MEASURE PHOTON-NUMBER …

PHYSICAL REVIEW A 75, 042302 共2007兲

quantity 共as opposed to its time dependence兲 to extract theoscillator number statistics. This is the procedure used indefining the full counting statistics of transmitted charge inmesoscopic electron conductors 关18兴. The application of thisapproach to the present system will be discussed in a forthcoming work 关21兴.

W共x,p;t兲 ⬅

H = H0 + Hint − ប ⍀F共t兲共a + a 兲 + 共a + a 兲 兺†冉+ 兺 ប ␻ j b†j b j +j†

再⳵tW共x,p;t兲 = −−i

冊v j共b j + b†j 兲

j

1
.
2

共8兲

F共t兲 is the external drive on the oscillator. The b j modes
correspond to a thermal bath responsible for the intrinsic
dissipation of the oscillator; we choose the density of statesof these modes and the couplings v j to yield a standard
ohmic dissipation. Note we are considering an ideal casewhere the qubit has no intrinsic 共i.e., oscillator-independent兲relaxation or dephasing. Such intrinsic processes can be accounted for by multiplying ␳↑↓共t兲 for the ideal case by an
extra decaying factor exp关−t共1 / T2 + 1 / 2T1兲兴 关7兴.
We are interested in the evolution of ␳ˆ ↑↓共t兲, which is anoperator in the oscillator’s Hilbert space. We are also interested in the weak damping limit ␥  ⍀ which is relevant toexperiment. In this limit, one can start with an exact pathintegral description and then rigorously derive a local-intime equation of motion for ␳↑↓; this is in complete analogy
to the derivation of the usual weak-coupling CaledeiraLeggett master equation 关19兴. This equation can also be rigorously justified in the high temperature kBT  ប ⍀ limit.
Setting ⍀qb = 0 共as it plays no role in what follows兲, one has

⳵
F共t兲
i
i␥
关x,兵p, ␳ˆ ↑↓其兴␳ˆ ↑↓ = − 关Hosc, ␳ˆ ↑↓兴 + i⍀关x, ␳ˆ ↑↓兴 −
⌬x0
⳵t
ប
2ប
−

D
i
ˆ
ˆ
2 †x,关x, ␳↑↓兴‡ − ␭兵Hosc, ␳↑↓其.
ប
ប

共9兲

冉

冊

ប⍀
p
⳵x + m⍀2x −F共t兲 ⳵ p + ␥⳵p · p + D⳵2p
m
⌬x0

冋

冉

ប2 1 2
␭ p2
⳵ + m⍀2⳵2p+ m⍀2x2 −
4 m x
ប m

冊册冎

共12兲

The ␭-independent terms above are the usual terms in a classical Fokker-Planck equation for a damped, driven oscillator,
while the terms proportional to ␭ yield distinctly nonclassicalcontributions. As the additional quantum terms are at mostquadratic in x , p and their derivatives, we may solve thisequation in the same way one would solve the classical
Fokker-Planck equation: one makes a Gaussian ansatz. Notethat a similar approach was used recently by Serban et al.
关15兴 to study the dephasing of a qubit by a thermal, damped
mode.
Thus we write
W共x,p兲 =

冕 冕
dk
2␲

dq ikx iqp
e e W̃共k,q兲
2␲

共13兲

and make the Gaussian ansatz

冉1= ei␯eik⌬x0·x̄+iq⌬p0·p̄exp − 关k2共⌬x0兲2␴x2W̃共0,0;0兲W̃共k,q;t兲冊+ q2共⌬p0兲2␴ p + 2kq共⌬x0兲共⌬p0兲␴xp兴 . 共14兲
The Gaussian is parametrized by six time-dependent parameters: the means x̄共t兲, p̄共t兲, the variances ␴x共t兲 and ␴ p共t兲, thecross-correlation ␴xp共t兲, and the overall phase ␯共t兲; note that
we have scaled all quantities by the zero-point variances ofthe oscillator. Substituting the above ansatz into the appropriately transformed version of Eq. 共9兲, we find that it issatisfies the equation as long as the Gaussian parameters satisfy the following equations 共we set ⍀ = 1 from this point
onwards兲:

2 2ប⍀coth共␤ ប ⍀/2兲D = m␥
2冊

冉yydy e−ipy/ប␳↑↓ x + ,x − ;t . 共11兲22

⫻W共x,p;t兲.

Here, Hosc = p / 共2m兲 + m⍀ x / 2, ␥ is the damping rate of theoscillator due to the bath, and 关␤ = 1 / 共kBT兲兴
2

冕Writing Eq. 共9兲 in terms of the Wigner function, one finds

III. EQUATION OF MOTION FOR ␳_`

We now wish to calculate ␳↑↓共t兲 in the presence of bothoscillator damping and driving. The Hamiltonian of our system H thus takes the form

1
2␲ ប

共10兲

is the corresponding momentum diffusion constant. We alsouse ⌬x0 and ⌬p0 to denote the zero-point variances of theoscillator. Note that unlike the standard quantum optics master equation 共used in Refs. 关6,7兴兲, the above equation has
counter-rotating terms 关i.e., proportional to a2 and 共a†兲2兴 andis hence not in Lindblad form. We will comment more on
this difference in what follows.
To proceed, we will represent ␳↑↓ by its Wigner function:
042302-3dx̄= p̄ − i␭共␴xx̄ + ␴xp p̄兲,
dt

共15兲

dp̄
= − x̄ + 2F共t兲 − ␥ p̄ − i␭共␴ p p̄ + ␴xpx̄兲,
dt

共16兲

d␴x
2
= + 2␴xp − i␭共␴2x + ␴xp
− 1兲,
dt

共17兲

d␴ p
2
− 1兲,
= − 2␴xp − 2␥共␴ p − T̃兲 − i␭共␴2p + ␴xp
dt

共18兲

d␴xp
= ␴ p − ␴x − ␥␴xp − i␭␴xp共␴x + ␴ p兲,
dt

共19兲

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMId␯␭␭= − 共␴x + ␴ p兲 − 共x̄2 + p̄2兲.
dt22

␴ p = 1 + s̄ − ⌬s.

共20兲The equations for the Gaussian parameters thus become

Here, we have definedT̃ ⬅ coth ␤ ប ⍀/2 = 2n̄eq + 1.

d␯
= ␭共1 + s̄兲,
dt

共21兲

The above equations have a simple form. The
␭-independent terms describe the usual dynamics of a driven
dissipative oscillator, and have a completely classical form.
The ␭-dependent terms act as quantum source terms, andcause the means and variances of the Gaussian to acquire
imaginary parts. From Eq. 共20兲, we find that the decay of ␳↑↓
is determined from these quantities:␳↑↓共t兲 W̃共k = 0,q = 0;t兲=
= ⌳1共t兲⌳2共t兲␳↑↓共0兲 W̃共k = 0,q = 0;0兲
with

冉 冕
冉 冕

⌳1共t兲 = exp − i␭2⌳2共t兲 = exp − i

␭
2

t

冊
冊

dt⬘关x̄2共t⬘兲 + p̄2共t⬘兲兴 .

共24兲

0

The two factors above correspond to two distinct contributions to the decay of ␳↑↓, and hence to the number splitting
effect. The factor ⌳1共t兲 describes a purely thermal effect:
From Eqs. 共17兲 and 共18兲, we see that ␴x and ␴ p acquire
negative imaginary parts, causing ⌳1共t兲 to decay from its
initial value of 1. This effect only occurs at finite temperature, as at zero temperature ␴x共0兲 = ␴y共0兲 = T̃ = 1 and arehence time independent. Note that ⌳1共t兲 is completely independent of the means of the Gaussian 共i.e., x̄ and p̄兲 andhence of the driving force F共t兲.
In contrast, the factor ⌳2共t兲 in Eq. 共22兲 describes the effect of a finite drive on ␳↑↓: If the means x̄ and p̄ are nonzero,Eqs. 共15兲 and 共16兲 indicate that they will acquire an imaginary part and cause this factor to decay.
IV. DEPHASING SPECTRUM IN THE PURELY
THERMAL CASE

We first consider the case of zero drive 关i.e., F共t兲 = 0兴, andtake the oscillator to initially be in a thermal state: x̄ = p̄ = 0,
␴x = ␴ p = T̃, and ␴xp = 0. Note that this corresponds to an oscillator in equilibrium with frequency ⍀; one might argue
that for the absorption experiment, the initial condition
should correspond to an equilibrium oscillator with frequency ⍀ − ␭. This difference is easy to include; in the weakcoupling limit of interest 共␭  1兲, the corrections are lower
order in ␭ and hence not significant.
As there is no drive, Eqs. 共15兲 and 共16兲 imply x̄ = 0, p̄
= 0 for all time, and thus the factor ⌳2共t兲 = 1. The evolution of␳↑↓ is the completely determined by the factor ⌳1共t兲. To proceed with the analysis, it is convenient to write

␴x = 1 + s̄ + ⌬s,ds̄2兴 − ␥共s̄ − ⌬s兲 + 2␥n̄eq ,= − i␭关2s̄ + s̄2 + 共⌬s兲2 + ␴xp
dt
共28兲
d⌬s
= − 2i␭共1 + s̄兲共⌬s兲 + ␥共s̄ − ⌬s兲 − 2␥n̄eq + 2␴xp ,
dt
共29兲
d␴xp
= − ␥␴xp − 2⌬s − 2i␭␴xp关1 + s̄兴.
dt

共23兲

t

共27兲

共22兲

dt⬘关␴x共t⬘兲 + ␴ p共t⬘兲兴 ,

0

共26兲

共30兲

As we start in a thermal state, both ⌬s共0兲 = 0 and ␴xp共0兲
= 0; further, if we took the limit ␥ → 0, these quantities would
remain zero for all time. Thus, in the weak damping 共␥  1兲limit of interest, it will be an excellent approximation to set⌬s = ␴xp = 0 in Eq. 共28兲; this results in the approximate equationds̄
= − i␭关2s̄ + s̄2兴 − ␥共s̄ − 2n̄eq兲.
dt

共31兲

Within this approximation, ⌳1共t兲 is completely determinedby Eq. 共31兲 and Eq. 共27兲. Note crucially that our approximation does not amount to doing perturbation theory in ␭ or ␥,as terms to all orders in the ratio ␭ / ␥ are retained. Also notethat if one had started by using the quantum optics version of
Eq. 共9兲 关i.e., drop all a2 and 共a†兲2 terms兴, one would immediately find Eq. 共31兲 without any additional approximation.
We thus see that the differences between the Lindblad andnon-Lindblad forms of the master equation play no role in
the weak coupling, small damping limit of interest. In what
follows, we will explicitly discuss the correction terms thatarise when one uses the full equation, Eq. 共28兲, versus theapproximate version, Eq. 共31兲.
A. Results for dephasing spectrum

We can now solve directly Eq. 共31兲 for s̄. We find共␴x + ␴ p兲/2 =
where

共25兲
042302-4␣=冑冉

1 + Me−2i␭␣ti␥,+␣
1 − Me−2i␭␣t
2␭

1−

共32兲

冊

i␥ 2 2i␥n̄eq
−
,
2␭
␭

冋 冉 冊册
冋 册

i␥
2␭
␥
i
2 + 共e−␤⍀ − 1兲 1 − ␣ +
2␭

共33兲

2e−␤⍀ + 共e−␤⍀ − 1兲 ␣ − 1 −
M=

.

共34兲

USING A QUBIT TO MEASURE PHOTON-NUMBER …

PHYSICAL REVIEW A 75, 042302 共2007兲

Note that at T = 0, we have simply ␣ = 1 − i␥ / 共2␭兲 and M = 0.
Combining Eqs. 共23兲 and 共32兲 we find␳↑↓共t兲
= ⌳1共t兲␳↑↓共0兲
=e␥t/2e−i␭␣t

1−M
1 − Me−2i␭␣t

共35兲

makes the oscillator appear somewhat colder than it should.
On a heuristic level, it is not difficult to understand why wehave a correspondence. The time scale for back-action effects to set in is ⬃1 / ␥—this is how long the oscillator takes
to re-equilibrate its number distribution given the change infrequency brought on by the coupling to the qubit. For␭  ␥, the qubit effectively “measures” the number statistics
well before they have a chance to be changed by the backaction.

⬁

=e

␥t/2

共1 − M兲 兺 M ne−i共2n+1兲␭␣t .

共36兲

n=0

Note that the result above coincides exactly with Eq. 共11兲 in
Ref. 关6兴. Unlike that work, we did not start by making arotating-wave approximation in the bath-oscillator interaction, but instead took a weak-damping limit.
Consider first the ␥ → 0 limit. In this limit, one finds M= e−␤⍀ and ␣ = 1. Equation 共36兲 thus yields ␳↑↓共t兲 in the formof Eq. 共7兲 with a thermal number distribution. As expected,
␳↑↓ is a completely faithful measure of the oscillator number
statistics in this limit, as there is no back action: At zero
damping, the mode has no way to re-equilibrate its number
distribution in response to a change in its frequency.
In the case of small but finite ␥, we see that 兩M兩 will play
the role of the Boltzman weight e−␤␻, and that ␣ acts torenormalize the coupling ␭. For small damping, the Fourier
spectrum of ␳↑↓共t兲 will again consist of a set of equally
spaced discrete peaks occurring at frequencies ␻n. The peaks
will now have a Lorentzian form, each having a nonzero
width ␥n. To leading order in ␥ we have

冋 冉冊

册

1 ␥ 2
␻n = 共2n + 1兲␭ 1 +n̄eq共1 + n̄eq兲 ,
2 ␭

共37兲

␥n = ␥关n̄eq + n共1 + 2n̄eq兲兴.

共38兲

Further, to order ␥2, the weighting factor 兩M兩 is given by冋兩M兩 ⯝ e−␤⍀ 1 −册

␥2
共1 + 2n̄eq兲 .
4␭2

共39兲

B. Results for oscillator-induced dephasing rate

It is also interesting to consider the limiting behavior of␳↑↓共t兲 in the long-time limit, t  1 / ␥. We can define the longtime dephasing rate ⌫␸,th and frequency shift ⌬th of the qubit
via

冉

冊

␥ 2 k BT
共1 + 2n̄eq兲 .
4␭2 ប⍀

共40兲

Thus, in the weak but finite damping limit, the probability
distribution extracted from the time dependence of ␳↑↓ atfixed coupling strength ␭  ␥ 关i.e., using Eq. 共7兲兴 is very
close to the initial, equilibrium photon number distribution of
the oscillator. Note that there is a one-to-one correspondence
between the peaks in the spectrum and the peaks expectedfrom Eq. 共7兲. The main effect of the damping is to give thepeaks associated with each number state a finite width infrequency. There is also deviation between the weight ofeach peak and a pure thermal weight; the finite damping

共41兲

log ␳↑↓共t兲
.
t
t→⬁

共42兲

⌫␸,th ⬅ − Re lim
This yields⌬th = Re ␭␣ =⌫␸,th = −

冋冑冉 冊
冋冑冉 冊

␥
Im
2

␥
␥
− Im ␭␣ = Re
2
2

1+

册

2i␭ 2 8i␭
+
n̄ ,
␥
␥ eq

1+

共43兲

册

2i␭ 2 8i␭+n̄ − 1 .
␥
␥ eq
共44兲

We find that the thermal oscillator-induced dephasing rate
may be written as ␥ times a function which depends only on
the ratio ␭ / ␥ and the temperature; in typical experimental
situations, one has both ␥  1 and ␭  1, but the ratio ␭ / ␥
can be arbitrary. Note also that the thermal dephasing rate of
the qubit vanishes at zero temperature. Expanding in ␥ / ␭ wehave
⌬th = ␭ +We can use 兩M兩 to define an effective temperature Teff via
兩M 兩 ⬅ e−ប⍀/共kbTeff兲. This is the temperature one would associate with the oscillator based on using Eq. 共7兲 to interpret thepeaks in the spectrum of ␳↑↓共t兲. To leading order in ␥ / ␭, we
have
Teff = T 1 −log ␳↑↓共t兲,tt→⬁⌬th ⬅ − Im lim⌫␸,th = ␥n̄eq −␥2n̄eq共1 + n̄eq兲 + O共␥4兲,
2␭

共45兲

冉 冊

␥3
␥5
. 共46兲
2 n̄eq共1 + n̄eq兲共1 + 2n̄eq兲 + O
4␭
␭4

In contrast, in the high-temperature limit n̄eq  1, we find forthe dephasing rate
⌫␸,th → 冑␥␭n̄eq .

共47兲

Finally, expanding ⌫␸,th as a function of the coupling yields
the usual Fermi’s golden rule result⌫␸,th → ␭24n̄eq共1 + n̄eq兲
= ␭2Sn共␻ = 0兲,␥共48兲

where Sn共␻ = 0兲 is the zero-frequency charge noise of theoscillator at zero coupling 共i.e., ␭ = 0兲. Note that the hightemperature limit of the golden rule result and the full an-

042302-5

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMI
1.6

swer are not the same. This can be understood from the factthat higher order terms in ␭ become increasingly important at
higher temperatures; the golden rule result is only valid if␭n̄eq / ␥  1 is satisfied.
Equations 共44兲–共47兲 for the thermal dephasing rate arevalid within the small-damping approximation made in Eq.
共31兲, or equivalently, within a rotating-wave description of
the bath-oscillator coupling. One can also find ⌫␸,th without
making these approximations by solving the full Eqs.
共27兲–共30兲 for a stationary s̄. Results are given in the Appendixes. One again finds that the thermal dephasing rate vanishes at zero temperature, and has the same high-temperature
limit as Eq. 共47兲. One also finds that corrections to the result
Eq. 共44兲 are higher-order in ␥. If we keep the ratio ␥ / ␭ fixedand expand in ␥, we find关⌫␸,th兴full = ⌫␸,th + O共␥3兲.

共49兲This confirms our prior claim that the error introduced by theapproximation of Eq. 共31兲 is small in the weak damping␥  1 limit.
Finally, we have also calculated the thermal dephasingrate for a qubit-oscillator interaction Hamiltonian which contains counter-rotating wave terms, i.e.,

Hint,2 =␭ប⍀
共a + a†兲2 .
2

共50兲

The thermal dephasing rate of the qubit for the interaction
Hamiltonian Hint,2 was previously considered by Serban etal. 关15兴. Using a completely analogous procedure as theabove, we can find the oscillator induced dephasing rate of
the qubit with this modified interaction; results are provided
in the Appendixes. As expected, we find that corrections to
the result of Eq. 共44兲 are negligible in the ␥ , ␭  1 limit ofinterest.
C. Practical considerations for NEMS number splitting

Before leaving this section, we comment on the feasibilityof measuring the thermal number splitting effect using current NEMS devices. In particular, we consider a metalized
nanomechanical resonator coupled to a superconducting
charge qubit, a device very similar to the system studied in
the experiment of Naik et al. 共Ref. 关20兴兲. As discussed inSec. II 关cf. Eq. 共6兲兴, the qubit’s absorption spectrum in sucha system could be used to probe the number splitting effect.
This absorption spectrum could be measured by further coupling the qubit weakly to a superconducting stripline resonator as in Ref. 关5兴.
A key question is whether the qubit absorption spectrum⌫abs共␻兲 关cf. Eq. 共6兲兴 will have distinct, resolvable peaks corresponding to each number state in the oscillator. Thesepeaks will in general be broadened both by finite temperature关cf. Eq. 共38兲兴, as well as by the intrinsic dephasing rate of thequbit 1 / ␶␸ 共i.e., the dephasing rate of the qubit in the absenceof any coupling to the oscillator兲. In Fig. 1, we show a calculated absorption spectrum 关obtained from Eqs. 共36兲 andΓabs(ω+Ωqb)Ω /|A|2

1.4

a)

1.2
1

b)

0.8
0.6
0.4

c)

0.2
0

0

2

4

6

8

10

ω /(2λ)

12

14

16

18

20

FIG. 1. Plot of the thermal absorption spectrum 关cf. Eq. 共6兲兴using oscillator parameters from the experiment of Naik et al. 关20兴
共⍀ = 2␲ ⫻ 21.9 MHz, ␥ = 10−5 ⍀, and m = 0.86⫻ 1015 kg兲. 共a兲 Spectrum for a temperature of 15 mK and using a coupling关⍀ / 共2␲兲兴␭ = 0.71 MHz, consistent with Ref. 关20兴. This couplingcorresponds to applying a voltage of 10 V to the oscillator, andhaving a qubit frequency ⍀qb = 共2␲兲 ⫻ 2 GHz. We have also taken
the intrinsic qubit dephasing rate 1 / ␶␸ = 106 s−1, corresponding to abest case scenario for a superconducting qubit. 共b兲 Same as 共a兲, but␭ has been increased by a factor of 2. 共c兲 Same as 共b兲, but thetemperature has been increased to 30 mK.

共6兲兴 using experimentally-achievable parameter values. We
have taken oscillator and oscillator-qubit coupling parameters from the experiment of Ref. 关20兴 共see inset for values兲.
We have also assumed a typical qubit splitting frequency anda best case value for the intrinsic qubit coherence time ␶␾
共⍀qb = 2␲ ⫻ 2 GHz, ␶␾ = 10−6 s兲. For these parameter values
and a temperature of 15 mK 共which corresponds to n̄eq
⯝ 13.8兲, we see that peaks in the absorption spectrum arealready beginning to be resolvable. Also shown in the figureis the absorption spectrum in a case where ␭ has been increased by 2 关i.e., ␭⍀ / 共2␲兲 = 1.43 MHz兴, but all other parameters have been kept the same. This could be achieved
by, e.g., reducing the oscillator-qubit distance by a factor of
冑2. For this modest increase in coupling, the number splitting peaks are clearly resolvable.
We close by noting that for experimentally achievable oscillator quality factors and temperatures, the main source ofpeak broadening in the number splitting spectrum is not the
intrinsic thermal width ␥n given in Eq. 共38兲, but rather the
intrinsic qubit dephasing rate 1 / ␶␸. Thus key challenges inexperimentally realizing the effect discussed here will be toachieve a sufficient coupling strength and have a sufficiently
long intrinsic qubit coherence time ␶␸.
V. DEPHASING SPECTRUM IN THE PRESENCE
OF AN OSCILLATOR DRIVE

We now consider the additional effect of a driving force
F共t兲 on the oscillator. The thermal factor ⌳1共t兲 considered in
the previous section will be unchanged by the presence ofthe force, assuming the initial state of the oscillator has vari-042302-6

USING A QUBIT TO MEASURE PHOTON-NUMBER …

PHYSICAL REVIEW A 75, 042302 共2007兲

g1共t,t⬘兲 = e−i共t−t⬘兲e−␥共t−t⬘兲/2ances consistent with being in equilibrium. However, because of the drive, the means of the Gaussian 共i.e., x̄ and p̄兲will be time dependent; as a result, the factor ⌳2共t兲 关cf. Eq.
共24兲兴 will also contribute to the suppression of ␳↑↓共t兲 and to
the number splitting effect. To characterize this, we must
calculate x̄共t兲 and p̄共t兲.
We begin by writing1
a1共t兲 = 关x̄共t兲 + ip̄共t兲兴,
2

共51兲

1
a2共t兲 = 关x̄共t兲 − ip̄共t兲兴.
2

共52兲

Note that in general, a1共t兲 ⫽ 关a2共t兲兴* as x̄ and p̄ have complex
parts; we only have a1共t兲 = 关a2共t兲兴* at t = 0, when both x̄ and p̄
are real. Also note that with our rescaling, a1 and a2 correspond to the usual a for a harmonic oscillator. As x̄2 + p̄2= 4a2a1, we have

冉 冕

冊

t

dt⬘a2共t⬘兲a1共t⬘兲 .

⌳2共t兲 = exp − 2i␭

0

共53兲

Note the similarity to Eq. 共5.16兲 in Ref. 关7兴. In the positive
P-function approach used in that work, it was important tohave a different coherent states associated with the two statesof the qubit 共兩 ↑ 典 , 兩 ↓ 典兲. In our approach, this “doubling” ofdegrees of freedom manifests itself in the fact that x̄ and p̄have both real and imaginary parts 关i.e., a2 ⫽ 共a1兲*兴.
We use Eqs. 共15兲 and 共16兲 to obtain equations for a1 anda2. Assuming the drive is near the oscillator frequency ⍀,and that ␭  1, it is safe to make a rotating wave approximation to these equations. Writing F共t兲 = Re关Fe−i␻dt兴, we thus
find 共recall we have set ⍀ = 1兲冋冋冉␴x + ␴ p␥da1 = − i − − i␭
dt
2
2

冉

␴x + ␴ p␥da2 = i − − i␭
dt
2
2

冊册

冊册

i
a1 + F+共t兲,
2

共54兲

i
a2 − 关F+共t兲兴* ,2

共55兲

where F+共t兲 = Fe−i共␻d−1兲t. We see that due to the coupling ␭,
the average variance ␴x共t兲 + ␴ p共t兲 acts as a time-dependent
modulation of the oscillator frequency. This average varianceis again completely determined from Eq. 共28兲 of the previous
section, and is completely independent of the force F共t兲 orthe initial value of a共t兲. In addition, as we have seen, it will
have an imaginary part.
The above equations are easily solved to yield

冕 ⬘

共56兲

冕 ⬘

共57兲

i
a1共t兲 = g1共t,0兲a1共0兲 +2

a2共t兲 = g2共t,0兲a2共0兲 −i
2

tdt g1共t,t⬘兲F+共t⬘兲,0冋 冕 ⬙冉 ⬙t⫻exp − i␭0␴x共t 兲 + ␴ p共t⬙兲
2冊册

␴x共t⬙兲 + ␴ p共t⬙兲2冊册

g2共t,t⬘兲 = ei共t−t⬘兲e−␥共t−t⬘兲/2

冋 冕 冉
t

⫻exp − i␭

t⬘

dt⬙

,

共58兲

.

共59兲

Equations 共53兲 and 共56兲–共59兲 are enough to completely
determine the drive contribution ⌳2共t兲 to the decay of ␳↑↓共t兲.
In the limit of zero temperature, there is no thermal contribution to the decay of ␳↑↓ 共i.e., there is no zero-temperaturedephasing兲, and ␴x = ␴ p = 1 for all times. This corresponds tothe limit considered in Ref. 关7兴; as we will see, the current
approach yields the same answer found in that work. In the
more general case of a nonzero temperature, the thermal
number splitting will also influence the “drive” contribution
to the number splitting, through the dynamics of ␴x共t兲
+ ␴ p共t兲.
To proceed, we make use of the small-␥ solution for s̄共t兲
obtained in the previous section 关cf. Eq. 共32兲兴. We stressagain that this result would be exact if one started by making
a rotating wave approximation in Eq. 共9兲, as is commonly
done in the quantum optics community. We also stress that
corrections to this approximation are negligible in the weakdamping, weak coupling 共␭  1, ␥  ⍀兲 limit of interest. We
thus find
1 − Me−2i␭␣t2,
1 − Me−2i␭␣t1

共60兲

1 − Me−2i␭␣t2.
1 − Me−2i␭␣t1

共61兲

g1共t1,t2兲 = e−i共1+␭␣兲共t1−t2兲
g2共t1,t2兲 = ei共1−␭␣兲共t1−t2兲

The parameters M and ␣ are temperature dependent, and are
given by Eqs. 共33兲 and 共34兲, respectively. Note that g1 and g2
are not time-translational invariant, as they are sensitive tothe initial time t = 0 at which the coupling was turned on.
Also note that in the zero temperature limit, we have M→ 0 and ␭␣ → ␭ − i␥ / 2. The propagators g1 and g2 then reduce to the expected propagators for a damped oscillator
having a frequency of ⍀ ± ␭. The propagators also have thisform at long times 共t1␥, t2␥  1兲 in the finite-temperaturecase.
We can now write a1共t兲 and a2共t兲 as
a1共t兲 = a1,se−i␻dt + ã1共t兲,

共62兲

a2共t兲 = a2,sei␻dt + ã2共t兲.

共63兲Here, ã1共t兲 and ã2共t兲 describe the transient behavior of a1 anda2. This decomposition then allows us to write the drivedependent contribution to ␳↑↓共t兲, ⌳2共t兲, ast

dt g2共t,t⬘兲关F+共t⬘兲兴* ,t⬘

dt⌳2共t兲 ⬅ exp关− i⌬drt − ⌫␸,drt兴exp关Qtrans共t兲兴,where− i⌬dr − ⌫␸,dr ⬅ − 2i␭a1,sa2,s ,with042302-7

共64兲

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMI冕 ⬘t

Qtrans共t兲 ⬅ − 2i␭

dt 关ã1共t⬘兲ã2共t⬘兲 + a1,sã2共t⬘兲e−i␻dt⬘

ã2共t兲 =0+ a2,sã1共t⬘兲e+i␻dt⬘兴.

共65兲

with

We thus see that the long time behaviour of a1 , a2 will set thedrive contribution ⌫␸,dr to the dephasing rate of the qubit
共i.e., the exponential decay of ␳↑↓ at long times兲, while thetransient behavior of these two functions will be 共presumably兲 responsible for the number splitting effect.

1关A2ei共1−␭␣兲t + B2ei共␻d−2␭␣兲t兴,
1 − Me−2i␭␣t冋

A1 = 共1 − M兲 a共0兲 −

冉

iF/2
2M
i␭␣
1−
i␦ + i␭␣
共1 − M兲 i␦ − i␭␣

共71兲

冊册

,

共72兲

冋

A2 = 共1 − M兲 关a共0兲兴* −冉

A. Results for the drive contribution to the dephasing rate

In the presence of a drive, the total dephasing rate of thequbit 关i.e., the exponential decay rate of ␳↑↓共t兲 at long times兴
will be given by⫻ 1+− iF*/2− i␦ + i␭␣

2Mi␭␣
共1 − M兲 i␦ + i␭␣

冊册

共73兲

,

共66兲

B1 = − 2i␭␣ M

共74兲

The purely thermal contribution ⌫␸,th is given in Eq. 共44兲.
Using the results of this section, we can calculate the drive
contribution to the dephasing rate. One first finds the stationary amplitudes a1,s and a2,s. Letting ␦ ⬅ ⍀ − ␻d, we haveiF/2,
共i␦ + i␭␣兲共i␦ − i␭␣兲

B2 = − 2i␭␣ M− iF*/2.
共i␦ + i␭␣兲共i␦ − i␭␣兲共75兲

⌫␸ = ⌫␸,th + ⌫␸,dr .

iF/2=i␦ + i␭␣

iF/2,
␥
i共␦ + ⌬th兲 + + ⌫␸,th2− iF*/2=− i␦ + i␭␣− iF*/2.
␥− i共␦ − ⌬th兲 + + ⌫␸,th2

a1,s =

a2,s =

共67兲

共68兲

1. Zero temperature limit

Here, ⌬th is the thermal frequency shift of the qubit 关cf. Eq.
共43兲兴; to lowest order in ␥, ⌬th = ␭. Note that the stationary
amplitudes are temperature-dependent.
Using these expressions, and letting ␥th ⬅ 共␥ + 2⌫␸,th兲, wefind that the drive-dependent contribution to the qubit
dephasing rate takes the form:⌫␸,dr =

兩F兩22冋␭⌬th␥th共␦ + ⌬th兲2 +冉 冊 册冋
␥th 2
2

共␦ − ⌬th兲2 +

冉 冊册
␥th 2
2

The contribution to the number splitting from the transients of a1 and a2 is given by Eq. 共65兲. In general, the result
will be a complex pattern of peaks, with no simple relation tothe original photon-number distribution of the mode. There
are however several important situations where things simplify, as we now show.

In the zero temperature limit, we have B1 = B2 = 0, and
共76兲

A2 = 关a共0兲兴* − a2,s .

共77兲

Only three terms are present in Qtrans:

冤

exp关Qtrans共t兲兴 = exp.

共69兲
Note that, at zero temperature, ⌬th = ␭ and ⌫␸,th = 0. In thislimit, Eq. 共69兲 reduces to the expression found in Ref. 关7兴.
We see that the effect of finite temperatures is to both provide an extra term in the dephasing rate of the qubit 关i.e.,⌫␸,th in Eq. 共44兲兴, and also to modify the form of the drivedependent contribution to the dephasing rate.
B. Results for the drive contribution to the dephasing
spectrum

As discussed, the transient parts of a1共t兲 and a2共t兲 willlead to oscillating contributions to ␳↑↓. Using the above results, one finds that these transients are given byã1共t兲 =A1 = a共0兲 − a1,s ,1关A1e−i共1+␭␣兲t + B1e−i共␻d+2␭␣兲t兴, 共70兲
1 − Me−2i␭␣t

A1A2 −2i␭␣t
− 1兲
共e
␥
1−i
2␭

冋
冋

冥

册
册

⫻ exp

2a1,sA2␭ i␦t −i␭␣t
− 1兲
共e e
␭␣ − ␦

⫻ exp

2a2,sA1␭ −i␦t −i␭␣t
共e e− 1兲 .
␭␣ + ␦

共78兲

Each factor above would yield a Poisson distribution ofoscillating exponentials. The combination of all three would
thus yield a complex distribution of peaks in the spectrum of␳↑↓, with weights being given as a convolution of three Poisson distributions. Reference 关7兴 considered an initial condition where the oscillator is in a state relaxed to the qubit
being down: a共0兲 = 共a2,s兲*. This is the correct initial state foran absorption experiment 关cf. Eq. 共6兲兴, where the coupling isalways on, and the qubit is initially in its ground state. In thiscase A2 = 0, and only the third factor in Eq. 共78兲 contributes.
One thus finds that Fourier transform of ␳↑↓共t兲 consists of aset of Poisson-distributed peaks whose frequency spacing is042302-8

USING A QUBIT TO MEASURE PHOTON-NUMBER …

PHYSICAL REVIEW A 75, 042302 共2007兲

approximately 共in the small-␥ limit兲 ⌬␻ = ␦ + ␭. Note that if
we assumed the spectrum of ␳↑↓ directly reflected the number
statistics of the oscillator 共cf. Eq. 共7兲兴, we would expect a
spacing between peaks of ⌬␻ = 2␭. Also note that though one
does obtain a Poisson distribution of peaks, the mean of this
distribution does not correspond to 兩a共0兲兩2 共i.e., the mean of
the original number distribution before the measurement was
turned on兲. It is unclear to what extent we can say we have“measured” the initial number distribution of the oscillator
in this case. Note that if we specialize further, and chose thedetuning of the driving force to be ␦ = ␭, then the simple
expectation of Eq. 共7兲 does hold 关7兴: One obtains peaks withthe expected spacing 2␭, and whose weights correspond to a
Poisson distribution with mean 兩a共0兲兩2.
Let us now consider a different approach. Returning to thefull expression, we see that only the first factor in Eq. 共78兲yields a set of peaks with the expected spacing of ⌬␻ = 2␭,regardless of the detuning ␦. Consider the “strong measurement regime,” where ␭ is much larger than both ␦ and ␥ / 2.
Further, take the initial state of the oscillator to correspond to␭ = 0 共i.e., a driven damped oscillator, but no coupling to thequbit兲. One then has 兩a共0兲 兩  兩a1,s 兩 , 兩a2,s兩. In this limit, one
can approximate A1 ⯝ a共0兲, A2 ⯝ a共0兲*. Only the first factor
in Eq. 共78兲 will contribute to the peak spacing; the second
and third terms will be suppressed by the small factors
兩a1,s / a共0兲兩 and 兩a2,s / a共0兲兩, respectively. Thus

冤

exp关Qtrans共t兲兴 ⯝ exp

冥兩a共0兲兩2 −2i␭t −␥te − 1兲 .
共e
i␥
1−
2␭

共79兲

The Fourier transform of ␳↑↓共t兲 will thus consist of a
Poisson-distributed set of peaks with the correct spacing of⌬␻ = 2␭; moreover, in the small damping limit ␥  ␭, themean of this distribution agrees with the mean of the originalstate, 兩a0兩2. We have thus shown that at zero temperature, one
can reliably extract the initial number distribution of thedriven oscillator 共arbitrary detuning ␦兲 from the time dependence of ␳↑↓ using the “number splitting” interpretation of
Eq. 共7兲. To do so, one must keep the qubit-oscillator coupling␭ off before t = 0; further, for t ⬎ 0, the coupling must be
suitably large: ␭  ␦ , ␥. Note that one may still have ␭  1.
It is worth emphasizing that the initial condition we consider
here 共i.e., Hint = 0 before t = 0兲 is not appropriate for the absorption experiment of Ref. 关5兴. However, it could conceivably be realized in experiments involving a nanomechanicaloscillator capacitively coupled to a qubit, as in such systems
the qubit-oscillator coupling is proportional to a tunable voltage.

dt

0

=

e−2i␭␣t⬘共80兲

共1 − Me−2i␭␣t⬘兲2冋册

兩a共0兲兩2 −2i␭␣te−2i␭␣t − 1
共e− 1兲 1 + M.
␣
1 − Me−2i␭␣t⬘

共81兲

If we now combine the above results for the drive contribution to ␳↑↓共t兲 with the thermal contribution ⌳1共t兲 关cf. Eq.
共36兲兴 we find␳↑↓共t兲 = e−i␭␣te−i⌬drte−⌫␸,drt P̃共k = − 2␭␣ ;Teff,Feff, ␦兲 共82兲=e−i⌬drte−⌫␸,drt 兺 P共n;Teff,Feff, ␦兲e−2i␭␣共n+1/2兲t .
n

共83兲
Here, P共n ; T , F , ␦兲 is the photon number distribution of an
oscillator at temperature T which is driven by a force ofmagnitude F and detuning ␦; P̃共k兲 = 兺neik P共n兲 is the corresponding generating function. The explicit form of this distribution is given in the Appendixes. To see the correspondence most directly, make the substitutions e−␤⍀ → M and k→ −2␭␣t in the generating function given in Eqs.
共B14兲–共B16兲. We find that in the small damping limit, Teffand Feff appearing in Eq. 共83兲 are almost identical to theinitial temperature and driving force on the oscillator: Teff isagain given by Eq. 共40兲 andFeff =

F

冑兩␣兩

冉

⯝F 1−

冊

␥2
2
共1 + 8n̄eq + 8n̄eq
兲 .
8␭2

共84兲

Note the close connection between Eq. 共83兲 and the naive“number splitting” expectation for ␳↑↓共t兲 given in Eq. 共7兲. We
have thus shown that in the “strong measurement limit”␭  ␥ , ␦, the spectrum of ␳↑↓共t兲 accurately reflects the number statistics of the mode: In this limit, each peak in the
spectrum of ␳↑↓ corresponds to a number state in the mode,and the corresponding peak weight to P共n兲. The width ␥n ofthe peak corresponding the photon number n is easily found
to be

冉

␥n = ⌫␸,dr + 共2n + 1兲 ⌫␸,th +

冊

␥
␥
− .
2
2

共85兲

In Fig. 1, we show results for the dephasing spectrum
S共␻兲 for various values of the qubit-oscillator couplingstrength ␭, where

2. Drive number splitting, finite temperature

We now consider the transient drive contribution in thepresence of a finite temperature. We again consider the“strong measurement limit” ␭  ␦ , ␥. In this limit, the initial
amplitude of the oscillator 兩a共0兲兩 is much larger than 兩a1,s兩 or兩a2,s兩. Thus the only significant term in Qtrans共t兲 is the one
proportional to A1A2, as this is the only term which scales as兩a共0兲兩2. Using A1A2 ⯝ 共1 − M兲2 兩 a共0兲兩2 关cf. Eqs. 共72兲 and 共73兲兴,
we find冕 ⬘t

Qtrans = − 2i␭A1A2

S共␻兲 =

冏冕⬁0

dt ei␻t冉␳↑↓共t兲 i␭te␳↑↓共0兲

冊冏

2

.

共86兲

Note we have shifted the spectrum so that the peaks associated with number states should appear at frequencies ␻ equal
to even multiples of 2␭. One also expects from Eq. 共7兲 thatthe weights of the different peaks in the spectrum should
correspond to the oscillator number distribution. These
weights may be approximately obtained from S共␻兲 via

042302-9

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMI
0.08

4000

S(ω)

(a)λ=0.003
3000

0.07

2000

0.06

1000

λ=0.1
λ=0.01

0.05
0

1

2

3

4

5

6

7

8

9

P(m)

0
−1

true

10

0.04
0.03

S(ω)

1000

0.01

500

0

−1

0

1

2

3

4

5

6

7

1500

S(ω)

0.02

(b) λ=0.01

1500

8

9

500

1

2

3

4

5

ω/(2λ)

6

7

8

9

冑 冕
1

␲ hm

共2m+1兲␭

共2m−1兲␭d␻ S共␻兲

15

20

10

FIG. 2. Fourier transform of the dephasing spectrum S共␻兲 关cf.
Eq. 共86兲兴 for different values of ␭; as in the main text, we have set⍀ = 1. For each plot, we have taken the oscillator damping ␥= 10−4, a driving force of magnitude 兩F 兩 = 3␥ and detuning ␦ = 0, a
temperature which corresponds to n̄eq = 1, and an initial condition
which corresponds to the driven thermal oscillator at zero qubitoscillator coupling. Based on the number splitting interpretation of
Eq. 共7兲, one expects to see peaks in S共␻兲 whenever ␻ = 2␭. For
small values of ␭, S共␻兲 is in poor correspondence to the number
distribution; note that there are additional peaks occurring roughlyat ␻ equal to odd multiples of ␭. These unwanted peaks become
increasingly insignificant as ␭ is increased. The weights of thepeaks in plot 共c兲 are in good agreement with the expected number
state distribution for the initial oscillator state 共see Fig. 1兲.

P共m兲approx =mFIG. 3. Number distributions derived from the dephasing spectrum, using the same oscillator parameters as in Fig. 1. The blackbars correspond to the actual photon number distribution associatedwith the initial oscillator state. The remaining bars correspond to
distributions obtained from the dephasing spectrum via Eq. 共87兲.
For ␭ = 0.01, there is a good correspondence. Note that the deviations from the true distribution in this case are mainly due to theinaccuracy of using Eq. 共87兲 to obtain the weight of peaks.

(c) λ=0.1

0

10

10

1000

−1

5

共87兲

where hm is the maximum of S共␻兲 in the interval 关共2m
− 1兲␭ , 共2m + 1兲␭兴. Number distributions obtained from S共␻兲
using this procedure are shown in Fig. 2. Note that one could
be more sophisticated in extracting the weights of peaks
from S共␻兲; we use Eq. 共87兲 only for simplicity. 共See Fig. 3.兲
VI. CONCLUSIONSIn this paper, we have used a conceptually simple Wigner
function approach to study the physics of a qubit dispersively
coupled to a driven, thermal harmonic mode. In general,back-action effects prevent the qubit’s dephasing spectrum
关i.e., the Fourier spectrum of ␳↑↓共t兲兴 from being a perfectmirror of the number statistics of the initial oscillator state.
We have shown however that for a sufficiently strong qubit-

oscillator coupling, one can indeed reliably extract the photon distribution of the driven, thermal mode from the dephasing spectrum. We have also calculated the oscillator-induceddephasing of the qubit in the case of a thermal, driven mode.
ACKNOWLEDGMENTS

We thank Alexandre Blais and Mark Dykman for usefulconversations. This work was supported by NSERC, FQRNTand the Canadian Institute for Advanced Research.
APPENDIX A: CORRECTIONS TO THE THERMALDEPHASING RATEIn the main text, the thermal dephasing rate is calculated
starting from the master equation Eq. 共9兲, using a weakdamping approximation. As discussed, this approximation isequivalent to having dropped terms proportional to a2 and共a†兲2 in Eq. 共9兲, as is often done in the quantum optics literature; this approximation was also used in Refs. 关6,7兴. In
this section, we discuss corrections arising when one goesbeyond this approximation.
1. Results from the full, non-Lindblad master equation

If one retains the counter-rotating terms in Eq. 共9兲, one
finds corrections to the result Eq. 共44兲. These corrections areirrelevant in the small damping, small coupling 共␥ , ␭  1兲limit of interest. Explicitly, expanding ⌫␸,th in ␥ for a fixed ␭,
one finds that corrections to Eq. 共44兲 only arise at order ␥3:关⌫␸,th兴full = ␥n̄eq −␥3 1 − 3␭2 + 4␭4n̄eq共1 + n̄eq兲共1 + 2n̄eq兲
4␭2 共1 − ␭2兲2

+ O共␥5兲.

共A1兲

Comparing against the corresponding expansion of the ap-

042302-10

USING A QUBIT TO MEASURE PHOTON-NUMBER …

PHYSICAL REVIEW A 75, 042302 共2007兲

proximate answer, Eq. 共46兲, we see that the correction terms
arising from using the full equation are suppressed by a factor ␭2 compared to the order-␥3 terms in approximate answer.
Similarly, if we expand the full answer to lowest order ␭,
we find that corrections to Eq. 共48兲 are suppressed by ␥2:D̂ = exp共␣0â† − ␣*0â兲.

冉 冊

4n̄eq共1 + n̄eq兲
␥
1+
4
␥

Finally, ␣0 is the coherent state amplitude corresponding to
the driven, damped motion of the oscillator; only its magnitude will be important for the number statistics. We have

+ O共␭4兲.

共A2兲

P 0共 ␣ 兲 =册␳=P共n兲 = 具n兩␳兩n典

共A4兲

=

冕

Note that this expression is identical to the previous Eq. 共44兲
except for terms which are small as ␥ or ␭. To see thisexplicitly, we can expand this result for small ␭ to obtain=e ␤⍀ − 1␲where冑2 − ␥2 + 2冑1 + 4i␥␭共2n̄eq + 1兲 − 4␭2 .

共B5兲

冕

d␣ P0共␣兲兩␣ + ␣0典具␣ + ␣0兩.

共B6兲

⫻

0

共B8兲

2 ␤⍀−2r兩␣ 兩cos ␪−兩␣ 兩200d␪ e−r e

0

共r + 兩␣0兩 + 2r␣0 cos ␪兲n
.
n!
2

n

⫻

共B9兲

冉

e ␤␻ − 1
␲

冊冕 冕
⬁

2␲

r dr

d␪ exp关− r2共e␤␻ − eik兲兴00⫻exp关− 2r兩␣0兩cos ␪共1 − eik兲兴冋= exp关− 兩␣0兩2共1 − eik兲兴 2共e␤␻ − 1兲APPENDIX B: NUMBER STATISTICS FOR A DRIVENTHERMAL MODE

where ␳0 is the equilibrium density matrix of the oscillator,and D̂ is the displacement operator:

2␲

r dr

兩␣ + ␣0兩2nn!

P共k兲 ⬅ 兺 P共n兲eikn = exp关− 兩␣0兩2共1 − eik兲兴

The second term above corresponds to the correction comingfrom using the nonrotating wave interaction Hamiltonian; it
is suppressed by a factor ␥2. Thus, as claimed, the nonrotating terms in the qubit-oscillator interaction Hamiltonian Hint
play no role in the weak damping, weak coupling共␭  1 , ␥  ⍀兲 limit of interest. Note that ⌫␸,th for an interaction Hamiltonian Hint,2 was also calculated in Ref. 关15兴;
they found a result similar to Eq. 共A5兲 above.

共B1兲

冕 冕
⬁

2

The corresponding generating function is thus

共A5兲In this appendix, we calculate the generating function for
the number distribution for a driven harmonic oscillator
which is also coupled to an equilibrium dissipative bath. The
Wigner function for such a state will be a Gaussian, with thevariances determined by equipartition. The means of theGaussian will follow the equations of motion for a damped,
driven oscillator. In general, we can represent this state in the
form

共B7兲

d␣ P0共␣兲e−兩␣ + ␣0兩

2

␭2关⌫␸,th兴Hint,2 = 关4n̄eq共1 + n̄eq兲 + ␥2共1 + 2n̄eq兲2兴 + O共␭4兲.
␥␳driven = D̂␳0D̂−1 ,

1 ␤ប⍀
共e
− 1兲exp关− 兩␣兩2共e␤⍀ − 1兲兴.
␲The number distribution is then given by

共A3兲

A=

共B4兲

It thus follows that2i␭ 2 8i␭
␥2
1+
−1 ,
+
n̄eq −
4␥␥2

冕

d␣ P0共␣兲兩␣典具␣兩,␳0 =One can also calculate the thermal dephasing rate of thequbit using an qubit-oscillator interaction Hamiltonian which
retains counter-rotating terms, cf. Eq. 共50兲. It was recently
suggested that such terms could play a significant role 关15兴.
Repeating the treatment which led to the full dephasing rate
above 共i.e., no small damping approximation兲, we obtain冋 冑冉 冊

共B3兲

To proceed, it is useful to represent ␳0 using a P function:

2. Results for a counter-rotating
qubit-oscillator interaction␥关⌫␸,th兴Hint,2 = Re A
2

兩F兩2/4.
共␻d − ⍀兲2 + ␥2/4

兩 ␣ 0兩 2 =2关⌫␸,th兴full = ␭2

共B2兲

冕

⬁

0

r dr

册

⫻exp关− r2共e␤␻ − eik兲兴I0关2r兩␣0兩共1 − eik兲兴 . 共B10兲
The first exponential factor above corresponds a generating
function for a Poisson distribution; this would correspond to
the number statistics of a driven, zero temperature oscillator
共i.e., coherent state statistics兲. The second term corresponds
to the effect of a nonzero temperature. In the absence of theBessel-function factor, it would yield the generating function
for a thermal distribution. In the presence of a drive, theBessel-function factors modifies the form of this term. Thus,
not surprisingly, the number statistics of a driven thermal
oscillator is not simply a convolution of a Bose-Einstein distribution with a Poisson distribution.

042302-11

PHYSICAL REVIEW A 75, 042302 共2007兲

A. A. CLERK AND D. WAHYU UTAMI

We can evaluate the expression for P共k兲 further by making use of

P共k兲 = Ptherm共k兲Pdrive共k兲Pcorr共k兲,
with

⬁

z2l
I0共z兲 = 兺 l2
l=0 4 共l ! 兲

共B13兲

Ptherm共k兲 =共B11兲

1 − e −␤⍀,1 − e−␤⍀eik

共B14兲

Pdrive共k兲 = exp关− 兩␣0兩2共1 − eik兲兴,
and

共B15兲

⬁

冕

⬁

2

dr r1+2me−Ar =0m! −m−1A.
2兩␣0兩2le−l␤⍀ik 2l−␤⍀ ik l 共1 − e 兲
l
!
共1
−
e
e
兲
l=0

Pcorr共k兲 = 兺

共B12兲

冋

= exp 兩␣0兩2e−␤⍀Some algebra thus yields

关1兴 K. C. Schwab and M. L. Roukes, Phys. Today 58 共7兲, 36
共2005兲.
关2兴 A. Blais, R.-S. Huang, A. Wallraff, S. M. Girvin, and R. J.
Schoelkopf, Phys. Rev. A 69, 062320 共2004兲.
关3兴 A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R.-S. Huang,J. Majer, S. Kumar, S. M. Girvin, and R. J. Schoelkopf, Nature
共London兲 431, 162 共2004兲.
关4兴 I. Chiorescu, P. Bertet, K. Semba, Y. Nakamura, C. J. P. M.
Harmans, and J. E. Mooij, Nature 共London兲 431, 159 共2004兲.
关5兴 D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M.
Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H.
Devoret, S. M. Girvin, and R. J. Schoelkopf, Nature 共London兲
445, 515 共2007兲.
关6兴 M. I. Dykman and M. A. Krivoglaz, Sov. Phys. Solid State 29,
210 共1987兲.
关7兴 J. Gambetta, A. Blais, D. I. Schuster, A. Wallraff, L. Frunzio,
J. Majer, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf,
Phys. Rev. A 74, 042318 共2006兲.
关8兴 D. H. Santamore, A. C. Doherty, and M. C. Cross, Phys. Rev.
B 70, 144301 共2004兲.
关9兴 D. H. Santamore, H.-S. Goan, G. J. Milburn, and M. L.
Roukes, Phys. Rev. A 70, 052105 共2004兲.
关10兴 K. Jacobs, P. Lougovski, and M. P. Blencowe, e-print condmat/0607102.

册

共1 − eik兲2.
1 − e−␤⍀eik

共B16兲

关11兴 A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys. Rev.
Lett. 88, 148301 共2002兲.
关12兴 E. K. Irish and K. C. Schwab, Phys. Rev. B 68, 155311
共2003兲.
关13兴 E. K. Irish, J. Gea-Banacloche, I. Martin, and K. C. Schwab,
Phys. Rev. B 72, 195410 共2005兲.
关14兴 C. P. Sun, L. F. Wei, Y. X. Liu, and F. Nori, Phys. Rev. A 73,
022318 共2006兲.
关15兴 I. Serban, E. Solano, and F. K. Wilhelm, e-print cond-mat/
0606734.
关16兴 M. Steffen, M. Ansmann, R. McDermott, N. Katz, R. C. Bialczak, E. Lucero, M. Neeley, E. M. Weig, A. N. Cleland, and J.
M. Martinis, Phys. Rev. Lett. 97, 050502 共2006兲.
关17兴 M. O. Scully and M. S. Zubairy, Quantum Optics 共Cambridge
University Press, Cambridge, UK, 1997兲.
关18兴 L. S. Levitov, in Quantum Noise in Mesoscopic Systems, edited by Y. Nazarov 共Kluwer, Amsterdam, 2003兲, pp. 373–396.
关19兴 A. O. Caldeira, H. A. Cerdeira, and R. Ramaswamy, Phys.
Rev. A 40, 3438 共1989兲.
关20兴 A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A. Clerk,
M. P. Blencowe, and K. C. Schwab, Nature 共London兲 443,
05027 共2006兲.
关21兴 A. A. Clerk and D. Wahyu Utami 共unpublished兲.

042302-12