PRL 104, 213603 (2010)

PHYSICAL REVIEW LETTERS

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Quantum Measurement of Phonon Shot Noise
A. A. Clerk,1 Florian Marquardt,2 and J. G. E. Harris3
1

Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8
Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics,
Ludwig-Maximilians-Universität München Theresienstr. 37, D-80333 München, Germany,
Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany,and Max Planck Institute for the Science of Light, Günter-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
3
Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA(Received 17 February 2010; revised manuscript received 3 May 2010; published 27 May 2010)2

We provide a full quantum mechanical analysis of a weak energy measurement of a driven mechanicalresonator. We demonstrate that measurements too weak to resolve individual mechanical Fock states cannonetheless be used to detect the nonclassical energy fluctuations of the driven mechanical resonator, i.e.,‘‘phonon shot noise’’. We also show that the third moment of the oscillator’s energy fluctuations provides
a far more sensitive probe of quantum effects than the second moment, and that measuring the third
moment via the phase shift of light in an optomechanical setup directly yields the type of operator
ordering postulated in the theory of full-counting statistics.
DOI: 10.1103/PhysRevLett.104.213603

PACS numbers: 42.50.Lc, 07.10.Cm, 72.70.+m

Introduction.—There has been considerable interest recently in preparing and detecting quantum mechanicalbehavior in mechanical resonators. This general goal has
been pursued using both optomechanical systems [1](where the mechanics is coupled to an optical cavity) andelectromechanical systems [2–6] (where the mechanics is
coupled to an electrical circuit). Not only do such studies
attempt to test quantum mechanics in a new regime of large
mass, they also have the potential of furthering our understanding of quantum dissipative processes and the boundary between classical and quantum physics.
A simple example of truly quantum behavior is theenergy quantization expected of a quantum oscillator.
Detecting this quantization directly by, e.g., observing
quantum jumps between different Fock states [7], is extremely challenging [8–10]. One requires a detector which
couples directly to the energy of the mechanical oscillator,
not to its position x^ as is more common; further, thiscoupling must be strong enough to resolve a mechanicalenergy quantum within the lifetime of a Fock state. Recent
optomechanical experiments [10] have demonstrated coupling to x2 , which is equivalent to energy in the rotating
wave approximation. However, it remains a challenge tosatisfy all of the requirements for achieving a practicalquantum nondemolition (QND) measurement of individual
quantum jumps.
As a result, it would be desirable to detect energy
quantization in a mechanical oscillator using only thepresently existing resource of a detector that couples
weakly to energy. This is the goal of this Letter. Similar
to previous studies, we consider the QND measurement of
the energy fluctuations of a dissipative mechanical resonator, but now consider the case where the mechanical resonator is strongly driven. This drive will result in a large
average number of quanta in the resonator, n  1. Our
0031-9007=10=104(21)=213603(4)focus will be on the fluctuations of energy about thisaverage value; in the low-temperature limit, these fluctuations are completely quantum in nature and reflect thediscreteness of the oscillator’s energy. As the magnitudeof this ‘‘phonon shot noise’’ scales with the magnitude of
the applied drive, one can make it large enough to detecteven if the detector-oscillator coupling is too weak toresolve individual Fock states. We show that this is possible even in the presence of strong cavity cooling, which isnecessary to ensure that the phonon shot noise dominates
the thermal noise. We also analyze the fundamental backaction of the measurement. We show that if the mechanicalresonator is driven on resonance, the measurement is backaction evading, having a formal equivalence to a singlequadrature position measurement [11]. As such, there is nofundamental quantum limit on the continuous monitoringof phonon shot noise.
Our study also reveals new physics associated with theenergy fluctuations of a driven quantum resonator, namely,that higher moments of the fluctuations are far more sensitive to the difference between the quantum and classicallimit than the second moment.
Model.—We consider a generic setup in which a detector
is weakly coupled to the energy of a damped, driven
mechanical resonator. We let F^ denote the detector quantity which directly couples to the oscillator number opera^ and I^ denotes the detector quantity that is
tor n^ ¼ c^ y c,monitored. We take @ ¼ 1. The Hamiltonian is:^ (1)H^ ¼ !M c^ y c^ þ H^   fðei!D t c^ þ H:c:Þ þ H^ det þ F^  n:
Here H^  describes the damping (at a rate ) of the mechanical resonator by a thermal bath, f is the mechanicaldriving amplitude (!D ¼ !M þ ), and H^ det is the detector Hamiltonian. The drive f yields an average number of^  n ¼ 4f2 =ð42 þ 2 Þ. An exmechanical quanta hni

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Ó 2010 The American Physical Society

PRL 104, 213603 (2010)

PHYSICAL REVIEW LETTERS

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TABLE I. Optomechanical device parameters. m, !M , Q: membrane mass, resonance frequency, and quality factor. F, L, Pin : cavity
finesse, length, and input power. T: bath temperature. With the exception of T and nth these parameters are those of the device
demonstrated in Refs. [13,18]. They allow a peak-to-noise ratio S ¼ 7:9 for the phonon shot-noise measurement.
m: 40 ng!M2 : 1 MHz

Q: 1:2  107

F: 5  104

L: 63 mmample of such a system is the ‘‘membrane-in-the-middle’’
optomechanical system discussed in Refs. [10,12,13]. The
detector here is a driven optical cavity whose frequency !Rdepends quadratically on the displacement of a dielectricmembrane placed in the cavity. Within a rotating-wave^
approximation, one obtains the desired coupling to n,with F^ ¼ An^ cav , where n^ cav is the cavity photon number
and A ¼ ðd2 !R =dx2 Þx2zpt is the quadratic optomechanicalcoupling. The output operator I^ here would correspond to
the error signal in a Pound-Drever-Hall scheme used to^ Quadratic
monitor the cavity frequency (and hence n).
optomechanical couplings could also be realized in otherways (see, e.g., Fig. 1 in Ref. [11]).
For the weak coupling situation we focus on, linear^^
response theory applies, implying hIðtÞi
¼ hnðtÞi,where is the detector response coefficient. We focus on theexperimentally relevant limit where the oscillator energy
evolves slowly enough that we can ignore the frequency
dependence of . A calculation yields the symmetrized
noise spectrum of I^ at low frequencies:
SII ½! ¼ 2 ðSnn;imp þ Snn ½! þ Snn;BA ½!Þ; þ 2nth Þð½! þ  þ ½!  Þ;Snn ½! ¼ nð1

(2a)
(2b)

where ½! ¼ ð=2Þ=ð!2 þ 2 =4Þ, and  is the total
damping of the mechanical oscillator. Given the large
mechanical drive, we retain only leading terms in n  1.
The first term in Eq. (2a), Snn;imp , describes the intrinsic
output noise floor of the detector (i.e., the imprecision
noise), while the second term corresponds to the amplified
number fluctuations Snn of the mechanical resonator. These
number fluctuations [cf. Eq. (2b)] have both a quantum
shot-noise part which is nonvanishing at zero-temperature
and a classical part proportional to the effective number ofthermal quanta in the oscillator, nth . Finally, the third termin Eq. (2a) describes a backaction contribution to the output spectrum: by virtue of the detector-oscillator couplingin Eq. (1), fluctuations of F^ will result in fluctuations of themechanical oscillator frequency, thus enhancing its number fluctuations and yielding extra noise in the output.
Resolving phonon shot noise.—We begin by ignoring theeffects of backaction. A first requirement is to have thequantum shot-noise contribution to Snn overwhelm theclassical, thermal contribution. We thus require a cold
oscillator, nth  1. In the optomechanical setup, this could
be achieved by using a second, linearly coupled, opticalmode whose linewidth is smaller than !M to laser cool themechanical mode, as discussed in Refs. [14–16]; thismethod allows ground-state cooling even in the presenceof a strong mechanical drive [17]. A simultaneous use of

Pin : 5 W

T: 300 mK@ 2 !R: 0:9 MHz=nm2@x2

nth : 0.2

different optical modes for cooling and measurement canbe achieved in the device of Ref. [13]. The use of lasercooling comes at a price: it increases the total mechanicaloscillator damping compared to its intrinsic value 0 . Thisin turn reduces the resolvability of the oscillator peak in the
output spectrum. One has the simple relation  ¼
0 ðnbath =nth Þ, where nbath is the bath temperature (expressed as a number of quanta).
Resolving the peak associated with Snn ½! in the output
spectrum corresponds to continuously measuring the phonon shot noise. Even without backaction effects, it is notclear that this is feasible, given both that strong cooling isrequired, and that the intrinsic detector-oscillator couplingis weak. As a measure of the resolvability of the phononshot noise, we consider the peak-to-noise ratio S Snn ½0=Snn;imp . In the interesting limit nth ! 0, n  1,
and taking  ¼ 0 for a maximal S, we find

4n=
4n
0
 th ð0Þ :
¼ 8nn
S ¼ 
¼ 
Snn;imp Snn;imp 0 

(3)Here, ð0Þ ¼ 1=ð20 nbath Snn;imp Þ is the signal-to-noise ratio introduced in Ref. [10] associated with resolving asingle mechanical quantum jump. The fact that S / nthreflects the increase in  associated with cavity cooling.
We see that with a suitably strong mechanical drive, S > 1even though the measurement strength is weak (i.e., ð0Þ <1) and though a significant amount of backaction cooling isrequired (i.e., nth  1  nbath ).
We can apply our general analysis to the device ofRef. [13] (see Table I for parameters). This device achieves
values of @2 !R =@x2 up to 103 greater than in Ref. [10]while maintaining negligible optical absorption [13]. We
assume the device is precooled inside a cryostat to a bathtemperature T ¼ 300 mK, making laser cooling to nth < 1feasible [15,16]. We also assume the membrane is driven toan amplitude of 2 nm. This is below the onset of dynamical
bistability in these devices [18], and corresponds to meanphonon number n ¼ 4:76  1012 . The resulting peak-tonoise ratio is S ¼ 7:9. Thus, by combining the measurement scheme presented here with the enhanced @2 !R =@x2
demonstrated in Ref. [13], it should be possible to observe
energy quantization. This is in sharp contrast to the proposal for detecting individual phonons in an optomechanical device, which would require substantial improvements
to the membrane, the cavity, and their coupling [10].
Backaction.—We now consider the effects of the fundamental measurement backaction. These result from fluctu^ which lead to frequency
ations of the input operator F,
noise of the mechanical oscillator and enhanced mechani-

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cal energy fluctuations. One might expect that to prevent
this backaction contribution from obscuring the phononshot-noise signature, there will be a limit to how large one and hence a limit on the maximum signal-tocan make n,noise ratio S in Eq. (3). A similar situation arises in thecontinuous monitoring of zero-point position fluctuations,where the peak-to-noise ratio S 3 [19].
To analyze the effects of backaction in the large n limit,
we let d^ denote the fluctuations of the mechanical oscil^ ¼
lator annihilation
operator from its average value: cðtÞpﬃﬃﬃ^
ei!D t ½ n ei þ dðtÞ.
In the rotating frame, the mechani^
cal oscillator Hamiltonian takes the form H^ M ¼ d^y d.
ﬃﬃﬃ the oscillator number
Further, to leading order in pn,^^ pﬃﬃﬃ’ n þ n XðtÞand the coupling
operator becomes nðtÞ
^ þ
^
^
^
^
Hamiltonian H int ¼ n F X , where XðtÞ ¼ ½ei dðtÞi ^ ye d ðtÞ. Measuring the oscillator energy thus corre^
sponds to a measurement of the ‘‘position’’ operator X,
and a continuous measurement of the phonon shot noise to^ As witha measurement of the zero-point fluctuations of X.
standard continuous position detection, there will in general be a backaction associated with this measurement, asX^ does not commute with H^ M . However, in the special caseof a resonant oscillator drive (i.e.,  ¼ 0), H^ M vanishes. X^
is thus trivially a QND observable that can be measured
without a backaction limit. Thus, for a resonant drive, and there is no backaction associated withto leading order in n,
the phonon shot-noise measurement; we thus anticipate
that there is no backaction-induced quantum limit on S.
Formally, the measurement of phonon shot noise in thislarge-n limit corresponds to a ‘‘QND in time’’ measurement of a single motional quadrature [11].
To verify the lack of backaction limit, we need todescribe the leading nonvanishing backaction contribution
when  ¼ 0. This arises from the term d^y d^ in n^ that wasneglected above. Treating the fluctuations of F^ as being
Gaussian (as is appropriate for the weak couplings we
consider), we find that the leading-order backaction-driven
2 
number fluctuations are given by Snn;BA ½0 ¼ 4n ðSFF Þ2 .
The total added noise of the measurement includes thiscontribution, plus the imprecision noise of the detector [first term in Eq. (2a)]. The total added noise
can be represented as an effective number of thermal
oscillator quanta nadd via nadd ¼ ðSnn;imp þ Snn;BA ½0Þ=ð2Snn ½0; nth ¼ 0Þ. Assuming our detector has quantumlimited noise properties and thus optimizes the
Heisenberg noise inequality Snn;imp SFF 1=4 [19], we the minimal possible value
find that for a fixed value of n,
of nadd is given bynadd jmin ¼

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PHYSICAL REVIEW LETTERS

3
;
 1=3
16ð2nÞ

(4)

where the minimum is achieved for an optimal measurement strength satisfying Snn;imp  ¼ ð32n 2 Þ1=3 . We thus seethat even with the inclusion of backaction effects, theadded noise of the measurement (referred back to theoscillator) can be made arbitrarily small by driving theoscillator on resonance with a sufficient strength. Notethat this is not true if one drives the oscillator off resonance. In this case, even in the limit n ! 1, the added
noise nadd cannot be made smaller than j=j. The device
described in Table I is not optimally coupled, but backaction effects nonetheless only yield nadd ’ 6  108 .
Finally, in the optomechanical realization of our scheme,
one must also consider the backaction mechanism analyzed in Ref. [20]. This mechanism is absent in the ideal
case of a one-port cavity; in the more general case of a twoport cavity, the constraint set by this mechanism on thecavity damping is much weaker for our scheme than for asingle-phonon measurement, as our scheme utilizes muchsmaller optomechanical couplings.
Nonclassical higher moments.—While the phonon shotnoise described by the zero-temperature limit of Snn ½! is
a completely quantum phenomenon, its form is not so
different from the classical, high-temperature prediction
for the energy fluctuations of a driven oscillator. The
quantum, zero-temperature limit of Snn ½! is simply obtained by taking the corresponding high-temperature classical expression [second term in Eq. (2b)], and replacing
nth with 0.5. In other words, the quantum shot noise isidentical to the classical expression evaluated at a temperature T ¼ @!=2kB . The energy fluctuations of a driven
oscillator are however not Gaussian. We now show that
this simple correspondence between classical and quantum
distributions fails starting with the third moment.
For simplicity, we consider theof the timeR fluctuations^ 0 Þ in the long-time
integrated phonon number m^ ¼ t0 dt0 nðt
limit. Experimentally, one would attempt to measure thisquantity by time-integrating the detector output IðtÞ; its^ ¼ hnit.
^ In considering themean is simply given by hmi
second and higher moments of m^ quantum mechanically,^ operacare must be taken to account for the fact that nðtÞtors at different times do not commute. One must treat theentire measurement quantum mechanically (not just theoscillator) to extract the proper definition of the higher
moments as measured in the experiment.
The above ordering problem has been addressed extensively in the study of the counting statistics of electrontransport through phase coherent conductors. Several different models of an ideal measurement yield the samedefinition for the higher moments, the so-called Keldysh
ordering [21,22]. For the second central moment, one
obtains in the long-time limit hhm^ 2 ii ¼ Snn ½0t, as would^ However, forbe obtained directly from the definition of m.
the third central moment, a far less obvious answer is given^(n^  n^  hni)hhm^ 3 iiK ¼

Zt
0

dt1

Zt
0

dt2

Zt
0

^ 1 Þnðt^ 2 Þnðt^ 3 Þi;dt3 ghnðt3
g ¼ ½1  ðt1  t2 Þðt3  t2 Þ:
2(5)

The classical definition would simply have g ¼ 1; instead,

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PHYSICAL REVIEW LETTERS

40

0.1

30

0.05

20
0
10
-0.05
0
-2

-1

0

1

2

-0.1
-3

-2

-1

0

1

2

3

FIG. 1 (color online). (a) Third moment hhm3 ii of integrated
energy fluctuations resulting from phonon shot noise, as afunction of the drive detuning , for different nth . (b) Zoom ofplot in (a), showing that at low temperatures, hhm3 ii can become
negative in the quantum limit.

the quantum answer involves neglecting contributions to
the integral where the middle n^ operator appears at theearliest time. This ordering follows from a consideration of
how the measured operator n^ influences the evolution of
the density matrix of the detector [21,22]. Importantly, one
can derive Eq. (5) explicitly for the single-port version of
the optomechanical system discussed above. To see this,
we assume that the output field from the cavity is subjected
to homodyne detection, and that the mechanical damping
is much smaller than the cavity damping. Using standard
input-output theory [19], the output field from the mixer^ ¼þ
used in the homodyne setup will have the form bðtÞ^BnðtÞ,
where  parameterizes the large magnitude of theclassical reference beam used, B is a constant proportional
to the optomechanical coupling, and we omit terms describing shot noise. The intensity I^ ¼ b^y b^ is then measured using a photodetector; I^ ¼ b^y b^  2 is the outputof the measurement.
^ the third moment
As I^ / n,of m^ will be proportionalR^
to the third moment of dtIðtÞ.
Experimentally, thiswould be directly obtained from the observed third moment of the number of photons detected in the output portof the homodyne setup within a given time interval. To
calculate this quantity, one has to use Glauber photodetection theory [23]. In particular, the third-order intensity correlation will be given by h:Iðt1 ÞIðt2 ÞIðt3 Þ:i ¼^ c Þbðt
^ b Þbðt
^ a Þi, where ta < tb < tc dehb^y ðta Þb^y ðtb Þb^y ðtc Þbðtnotes the time-ordered listing of t1 , t2 and t3 . By thus
calculating the leading contribution (in ) to the third
moment of the number of detected photons in the output
port of the homodyne setup, one directly recovers the
Keldysh ordering described by Eq. (5). To our knowledge,
this is the first demonstration of how the Keldysh ordering
arises in a realistic measurement setup.
Applying this definition to our system, we find
 3  4ð=Þ2 þ 16nth ðnth þ 1Þ6nt
hhm3 ii ¼ 2:(6)ð1 þ 42 =2 Þ2
We have neglected a purely thermal contribution to Eq. (6) In the high-temperature classiwhich is independent of n.

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cal limit (i.e., nth  1), the skewness of m^ is always
positive, and the dependence on the detuning  of themechanical drive enters only through the oscillator’s susceptibility. In contrast, in the quantum limit nth ! 0, theskewness can be positive or negative depending on 
(Fig. 1). Thus, the third moment of energy fluctuations
has a much greater sensitivity to whether one is in thequantum or classical limit: the quantum expression does
not simply correspond to evaluating the classical expression at an effective temperature T ¼ @!M =2kB . Note thatthe use of the proper Keldysh ordering is crucial to obtaining Eq. (6). Note also that the full distribution of the driven
oscillator’s energy fluctuations can be directly obtainedusing the approach of Ref. [24].
We thank J. C. Sankey for discussions. A. C. acknowledges funding from CIFAR, NSERC and the Alfred Sloan
Foundation; F. M. from the DFG (Emmy-Noether, SFB
631, NIM), GIF and DIP; J. G. E. H. from NSF and
AFOSR. This material is based upon work supported by
DARPA under Grant No. N66001-09-1-2100.

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