PRL 102, 207209 (2009)PHYSICAL REVIEW LETTERS

week ending
22 MAY 2009

Quantum Noise Interference and Backaction Cooling in Cavity Nanomechanics
Florian Elste,1 S. M. Girvin,2 and A. A. Clerk1
1

Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8
2
Department of Physics, Yale University, New Haven, Connecticut 06520, USA(Received 17 March 2009; published 22 May 2009)

We present a theoretical analysis of a novel cavity electromechanical system where a mechanicalresonator directly modulates the damping rate  of a driven electromagnetic cavity. We show that via adestructive interference of quantum noise, the driven cavity can effectively act like a zero-temperaturebath irrespective of the ratio =!M , where !M is the mechanical frequency. This scheme thus allows one
to cool the mechanical resonator to its ground state without requiring the cavity to be in the so-called goodcavity limit   !M . The system described here could be implemented directly using setups similar to
those used in recent experiments in cavity electromechanics.
DOI: 10.1103/PhysRevLett.102.207209

PACS numbers: 85.85.+j, 42.50.Lc, 42.50.Pq

Introduction.—Much of the rapid progress in fabricating
and controlling nanomechanical devices has been fueledby numerous promising technological applications.
However, progress has also been motivated by the realization that such systems are ideally poised to allow the
exploration of several fundamental quantum mechanicaleffects. Various studies have addressed such issues as entanglement, quantum-limited measurement, and Fock state
detection in systems containing nanoscale mechanical resonators [1–4]. The issue of quantum backaction has also
received considerable attention in these systems [5,6].
Understanding the backaction properties of a detector ina quantum electromechanical or optomechanical system isnecessary if one wishes to do quantum-limited position
detection. Further, this backaction can in some cases be
exploited to achieve significant nonequilibrium cooling of
the mechanical resonator. This is of particular importance,since a prerequisite to seeing truly quantum behavior inthese systems is the ability to cool the mechanical resonator to near its ground state.
A particularly effective backaction cooling scheme is‘‘cavity cooling’’ [7–10]. Here, a mechanical resonator isdispersively coupled to a driven electromagnetic cavity(i.e., the cavity’s frequency depends on the mechanical displacement x). By monitoring the frequency of the cavity,
one can make a sensitive measurement of x. In addition, thecavity photon number necessarily acts as a noisy force on
the mechanics; for a suitably chosen cavity drive, this force
can be used to effectively cool the mechanical resonator.
This approach has been used in recent experiments, bothwith optical cavities coupled to mechanical resonators[11,12], as well as in systems using microwave cavities[13,14]. Theoretically, it has been shown that such schemes
can allow ground-state cooling of the mechanical resonator[9,10]. Similar physics even occurs in seemingly very
different systems, e.g., a superconducting single-electrontransistor coupled to a mechanical resonator [15,16].
In this Letter, we present a theoretical analysis of ageneric electromechanical (or optomechanical) system0031-9007=09=102(20)=207209(4)which is the dual of the dispersive coupling discussed
above. We again consider a driven cavity coupled to amechanical resonator: now, however, the mechanical displacement x does not change the cavity frequency, butrather changes its damping rate . In the ideal case, thisdamping will be dominated by the coupling of the cavity tothe port used to drive it. As we will discuss, such a
dissipative coupling arises naturally in systems where a
microwave cavity is coupled to a nanomechanical resonator [17]; it could also be realized in an optical cavity
containing a moveable ‘‘mebrane in the middle’’ [12,18].
We show that for such a dissipative cavity-mechanicalresonator coupling, interference effects are important indetermining the quantum backaction effects on the mechanical system; this is not the case for a purely dispersive
coupling. In particular, we show that one can use destructive interference to allow the cavity to act as an effective
zero-temperature bath, irrespective of the ratio of the mechanical frequency !M to the cavity linewidth ; as such,ground-state cooling is possible without requiring the‘‘good cavity’’ limit !M  . This is in sharp contrast to
the case of a dispersive coupling, where ground-state cooling is only possible if !M  . From a practical perspective, being able to deviate from the good cavity limit isadvantageous, as it allows one to use small drive detunings
and hence achieve much larger effective cavity-mechanicalresonator couplings. We show that this destructive interference effect persists in the case where one has both adissipative and dispersive coupling; we also show that adissipative coupling can allow for a quantum-limited position measurement.
Model.—We consider a mechanical oscillator (frequency !M , displacement x) whose motion weakly modulates the damping rate  and resonant frequency !R of adriven electromagnetic cavity. For small displacements,
both !R and  will have a linear dependence on x, andwe can describe the system via the Hamiltonian (@ ¼ 1):
PH^ ¼ !R a^ y a^ þ !M c^ y c^ þ q !q b^yq b^q þ H^ damp þ H^ int þ H^  .
The first two terms describe the cavity and mechanical207209-1

Ó 2009 The American Physical Society

PRL 102, 207209 (2009)PHYSICAL REVIEW LETTERS

Hamiltonians, while H describes the intrinsic mechanicaldamping by an equilibrium bath at temperature Teq . The
third term in H^ describes the bosonic bath responsible forthe dissipation and driving of the cavity. Working in theusual Markovian limit where   !R , and where the bathdensity of states  can be treated as energy independent
over relevant frequencies, the cavity-bath interaction takes
the form [19,20]
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 X y^^^(1)¼
iða^ bq  b^yq aÞ:
H damp2 q
The cavity-mechanical coupling is then described by~ a^ y aÞ:^^ zpt Þð12B~H^ damp þ AH^ int ¼ ðx=x(2)

~ B~ above are given
The dimensionless coupling strengths A,~ ¼ ðd!R =dxÞxzpt , where xzpt is~ ¼ ðd=dxÞxzpt , A
by B
the zero-point motion amplitude of the mechanicaloscillator.
To proceed, we linearize the full quantum dynamics
about the solutions of the classical equations of motion
for the uncoupled cavity, making use of the input-output
formalism of quantum optics [19,20]. Working in frame
rotating at the drive frequency, and writing a^ ¼ a þ d^
^ the linearized Heisenberg equations of(where a ¼ hai),motion take the form
 ~




pﬃﬃﬃﬃ

b
B
d^_ ¼ i  d^  ^  a þ pinﬃﬃﬃﬃ þ iA~ a ðc^ þ c^ y Þ;

2
2
(3)



pﬃﬃﬃﬃ
^
c^_ ¼  i!M þ c^  ^ þ ixzpt F:(4)2Here bin is the amplitude of the coherent cavity drive,  ¼!drive  !R is the detuning of the cavity drive, ^ describes
vacuum noise entering the cavity from the drive port, and ^
describes thermal noise associated with the mechanicaldamping  [19,20]. The backaction force operator F^ ¼
ðd=dxÞH^ int in Eq. (4) takes the form
pﬃﬃﬃﬃ

~
A  ^ iB~   ^^ þ H:c: (5)a d ða   bin dÞF^ ¼ 
2xzpt
xzpt
Quantum noise.—For a sufficiently weak optomechanical coupling, linear-response theory applies, and the unperturbed quantum noise spectrum of F^ (i.e., calculated at
A~ ¼ B~ ¼ 0) determines both the backaction damping andheating of the mechanical resonator by the driven cavity [15,20]. The relevant spectrum is SFF ð!Þ ¼
R^ Fð0Þi
^dei! hFðÞ
0 . Recall that SFF ð!M Þ is proportional
to the Fermi’s golden rule rate for the absorption of amechanical quantum by the driven cavity, while
SFF ð!M Þ is proportional to the corresponding emission
rate. The effective temperature associated with the backaction noise as seen by the mechanical oscillator is thengiven by kB Teff  !M ðlog½SFF ð!M Þ=SFF ð!M ÞÞ1 ,while the backaction damping is given by BA ¼

week ending
22 MAY 2009x2zpt ½SFF ð!M Þ  SFF ð!M Þ. These linear-response results presume the total mechanical damping rate to besmall enough that the resonator is only sensitive to noiseat ! ¼ !M ; as we will see, deviations will occur whenthe effective optomechanical coupling becomes large
enough to make BA comparable to !M .
One finds from Eqs. (3) and (4) that the uncoupled cavity’s backaction force noise spectrum SFF ð!Þ is given by ~ 2 ½! þ 2  2A~ 2

Bjaj
B~
:
(6)
SFF ð!Þ ¼ 
2xzpt ð! þ Þ2 þ 2 =4
In the limit B~ ! 0 of a purely dispersive coupling, Eq. (6)reduces to a simple Lorentzian [9,10]. This form has asimple interpretation: it corresponds to the Lorentziandensity of final states relevant to a Raman process where
an incident drive photon gains an energy @! while attempting to enter the cavity. The optimal backaction cooling
discussed in Refs. [9,10] requires  ¼ !M and  !M . For these parameters, the drive photons are initially
far from being on resonance with the cavity. The absorption of energy from the oscillator is resonantly enhanced,
as it moves an incident drive photon onto the cavity resonance. In contrast, emission of energy to the oscillator isgreatly suppressed, as the drive photon is moved evenfurther from resonance. Thus, the B~ ¼ 0 form of SFF ð!Þ
and the resulting cooling are simply explained as a densityof states effect.
In the more general case where the dissipative optomechanical coupling B~ is also nonzero, SFF ð!Þ is not a simple
Lorentzian; as such, the backaction physics cannot beinterpreted solely as a density of states effect. In general,
SFF ð!Þ has an asymmetric Fano line shape (see Fig. 1),
with the cavity emission noise SFF ð!M Þ vanishing when~ BÞ~  opt . Thus, forever the detuning  ¼ !M =2 þ ðA=
this optimal detuning, one finds that the cavity acts as aneffective zero-temperature bath, irrespective of the ratio
=!M . For  ¼ opt , one has
!2M
: (7)
~ BÞ
~ 2 þ 2 =4½3!M =2 þ ðA=
For  ¼ opt , the weak-coupling, quantum noise approach yields the equilibrium number of quanta in theoscillator to be nosc ¼ neq =ðBA;opt þ Þ, where  is theintrinsic damping rate of the oscillator, and neq is the BoseEinstein factor associated with the bath temperature Teq .
One can thus cool to the ground state for a sufficiently large
coupling and/or cavity drive. This possibility of groundstate cooling for an arbitrary =!M ratio is a main result of
this Letter. Note that the optimal detuning opt is greaterthan 0: ground-state cooling is possible even though thedrive photons would seemingly need to ‘‘burn off’’ energy
to enter the cavity. This is in stark contrast to the purely
dispersive case where a positive detuning leads to heating
and a negative-damping instability. Note finally that there
is also an ‘‘optical spring’’ effect associated with the backaction; it can be obtained directly from SFF ð!Þ [9].

207209-2 2BA;opt ¼ B~2 jaj

1.0

2

 

 


Reð!Þ
ð!Þ 




ð!Þ
¼

n
ð1
þ
2n
Þ
1
þ
þ


eq
eq
eq :


 2!M 
!M

1

0.8
0.6

nosc

0.1

(11)

0.4
0.2
0.0

(a)

0.01

-3

-2

-1

0

1

2

3

ω/κ

0

0.2

(b)

0.4

0.6

∆/ωM

FIG. 1 (color online). (a) Backaction force noise spectral density for  ¼ =2, for different ratios of the dispersive to dissipative optomechanical couplings. The solid blue curve~ B~ ¼ 10, the long-dashed green curve tocorresponds to A=~ B~ ¼ 0:75, and the short-dashed red curve to A=~ B~ ¼ 0. EachA=
curve has been scaled by its maximum value. (b) Number ofoscillator quanta nosc versus drive detuning , obtained bysolving the full equations of motion for a purely dissipative
optomechanical coupling (i.e., A~ ¼ 0). The dotted blue curve isthe Bose-Einstein factor neff corresponding to the backaction
effective temperature Teff when  ¼ !M [i.e., neff ¼
SFF ð!M Þ=½SFF ð!M Þ  SFF ð!M Þ]. The remaining curves
correspond to B~ a ¼ 0:2, neq ¼ 50,  ¼ 106 !M , and =!M ¼
0:2 (green solid curve), =!M ¼ 1:0 (purple long-dashed curve),
=!M ¼ 5:0 (red short-dashed curve).

Fano line shapes arise generically as a result of interference between resonant and nonresonant processes; thesituation is no different here. In the usual B~ ¼ 0 case, theonly source of backaction force noise is the number fluc^ However, when B~ Þ 0, thetuations of the cavity field a.
mechanical oscillator mediates the coupling between thecavity and the cavity’s dissipative bath. As a result, it issubject to two sources of noise, corresponding to the twoterms in Eq. (5): the shot noise associated with the driving^ The first ofof the cavity, as well as the fluctuations of a.
these noise processes is white, whereas the second is not: it
is simply the shot noise incident on the cavity filtered by
the !-dependent cavity susceptibility. The interference
between these two noises yields a Fano line shape for
SFF ð!Þ, and the destructive interference at  ¼ opt whichcauses Teff ¼ 0. Note that Fano interference in electromechanical systems has recently been discussed in Ref. [21],albeit in a very different context.
Full solution.—To address whether the destructive noiseinterference effect persists beyond the simplest weakcoupling regime, we now examine the full solution of
Eqs. (3) and (4); for simplicity, we focus
A~ ¼ 0. One
R on
y
i!t
^
finds for the oscillator spectrum Scc ¼ dthc^ ðtÞcð0Þie,~ M ð!Þj2 ½
Scc ð!Þ ¼ j

eq ð!Þ þ SFF ð!Þ;(9)

 2 M ð!ÞR ð!ÞR ð!Þ
ð!Þ ¼ iB~2 jaj !M ½2  342 þ i!;(10)γ/ωM = 10-5

10γ/ωM = 10-6

(a)

(b)
Z0
C1(x)

L

0.1

C

0.001
0.

(8)

where
~ M ð!Þ ¼ M ð!Þ=½1 þ iM ð!Þð!Þ;

Here, we denote the bare mechanical and cavity susceptibilities by M ð!Þ ¼ ½ið!  !M Þ þ =21 andR ð!Þ ¼ ½ið! þ Þ þ =21 . Equation (8) has a simple interpretation: the oscillator responds with a modified~ M to two independent fluctuating forces,
susceptibility 
corresponding to the two terms in the equation. The first,
described by eq , is the fluctuating thermal force associated with the intrinsic oscillator damping . The second is
the backaction from the driven cavity. We see that its form
is not affected by the coupling strength: the same spectral
density SFF ð!Þ (evaluated at A~ ¼ 0) found earlier in the
weak-coupling regime [c.f. Eq. (6)] appears here. Thus, thequantum noise interference discussed above continues toplay a role even for moderate coupling strengths. A strong
cavity-mechanical resonator coupling will nonethelessmodify the physics, as the destructive interference occurring when  ¼ opt only occurs at the single frequency!M . For a sufficiently strong coupling, the oscillator’s total
damping will become large enough that the oscillator will
be sensitive to noise at frequencies away from !M , frequencies where the destructive interference is not complete. As a result, the cavity will no longer appear to theoscillator as an effective zero-temperature bath. Note thatwhen A~ Þ 0, the spectrum Scc ð!Þ still has the general form
given by Eqs. (8), (9), and (11): one now simply uses thefull form of SFF ð!Þ, as well as a self-energy that contains~
extra terms / A.
To see whether this resonance-broadening effect as wellas other strong-coupling effects preclude ground-statecooling at the optimal detuning  ¼ opt , we calculate theaverage number of oscillator quanta nosc directly by integrating Scc ð!Þ in Eq. (8). In Fig. 2 we show the expected
cooling for realistic parameter values, as a function of thecavity drive strength. Strong-coupling effects lead to an

nosc

SFF(ω) / SFF,max

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22 MAY 2009PHYSICAL REVIEW LETTERS

PRL 102, 207209 (2009)

0.2
~
B|a|

0.4

FIG. 2 (color online). (a) Number of mechanical quanta nosc
~ aj,
 for A~ ¼ 0, neq ¼ 50, =!M ¼ 1
versus coupling strength Bj
and for an optimal detuning  ¼ !M =2; the mechanical damping  is as marked. Dashed curves are results from the linearresponse, quantum noise approach, whereas solid curves are
obtained from the full solutions of the equations of motion.
(b) Schematic of a cavity (modeled as an LC resonator) coupled
to a transmission line (impedance Z0 ) via an x-dependent capacitance C1 ðxÞ.

207209-3

PRL 102, 207209 (2009)PHYSICAL REVIEW LETTERS

optimal drive strength, beyond which nosc starts to increase; this additional heating effect scales asðBA =!M Þ2 . Nonetheless, the minimum value of nosc canstill be significantly less than 1.
Physical realization.—The dissipative optomechanicalcoupling analyzed here could be realized in a microwave
electromechanical system similar to those studied inRefs. [13,14]. In such systems, a capacitor C1 couples thecavity to a transmission line (impedance Z0 ) which bothdrives and damps the cavity. One would now need to makeC1 mechanically compliant [see Fig. 2(b)]. In general, sucha setup will have both dissipative and dispersive optomechanical couplings (i.e., A~ Þ 0, B~ Þ 0). A careful analysis
shows that in the physically relevant regime C1  C, one
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
can have A~ ’ B~ if the cavity impedance ZR ¼ L=C ismade slightly smaller than Z0 ’ 50. For example, taking
experimentally achievable parameters !R ¼ 2  10 GHz,
C ¼ 3:2 pF and C1 ¼ 0:01C results in =!R ’ 103 and~ Bj~ ’ 2:5. Equation (6) then implies that ground-statejA=
cooling via destructive noise interference is possible if one
uses a drive detuning  ¼ opt ¼ !M =2 þ 2:5. We stress
this conclusion is independent of the magnitude of !M =;in contrast, if one had a purely dispersive cavitymechanical resonator coupling, ground-state cooling isonly possible if !M  . A second setup where the dissipative optomechanical coupling could be realized is an
optical Fabry-Pérot cavity containing a thin, moveable
dielectric slab. If the cavity is constructed with mirrors
having different reflectivities L and R , the motion of themembrane will naturally modulate the damping rate of the
optical modes. The resulting value of B~ can be easily derived using the approach of Ref. [18]. Note that in such asystem, one can position the membrane such that the optical mode frequency is insensitive to linear changes in x[12]: one thus naturally achieves A~ ¼ 0 and B~ /ðR  L Þ Þ 0.
Position detection.—A straightforward calculationshows that the power spectrum of b^out , the output fieldfrom the cavity, is modified in a simple way by the coupling to the oscillator: the oscillator adds a term
Sxx ð!ÞSFF ð!Þ (where ! is measured from the drive~ B~ Þ 0. Here, Sxx ð!Þ ¼
frequency); this holds for both A,
R
i! ^
^
de hxðÞxð0Þi.
One can thus use the peaks at ! ¼
!M in the output spectrum to infer the temperature of theoscillator. Note that for an optimal detuning  ¼ opt , thedestructive interference effect means that the peak at ! ¼!M in the output spectrum will vanish. This could serve as
an interesting test of our predictions.
Finally, specializing to the case where !M  , one canalso use the dissipative optomechanical coupling to make aquantum-limited, weak continuous position measurementof the oscillator. One drives the cavity off resonance, andperforms a homodyne detection of the appropriate quadrature of the output field from the cavity. One finds from
Eqs. (3) and (4) that for any nonzero detuning , thecorresponding backaction noise is as small as is allowed

week ending
22 MAY 2009by the uncertainty principle, meaning that one can reachthe quantum limit on the total added noise [20].
Conclusions.—We have considered a generic system
where a mechanical resonator modulates the damping ofa driven quantum cavity, and demonstrated how Fanointerference is important to the backaction physics. In
particular, one can have a perfect destructive interference
which allows the backaction to mimic a zero-temperatureenvironment; as such, such a system should allow nearground-state cooling of the mechanics.
We thank J. Harris for useful comments. This work was
supported by NSERC, CIFAR, and the NSF under Grants
No. DMR-0653377 and No. DMR-0603369.

[1] A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys.
Rev. Lett. 88, 148301 (2002).
[2] D. H. Santamore, A. C. Doherty, and M. C. Cross, Phys.
Rev. B 70, 144301 (2004).
[3] M. P. Blencowe, Phys. Rep. 395, 159 (2004).
[4] A. A. Clerk, Phys. Rev. B 70, 245306 (2004).
[5] T. J. Kippenberg and K. J. Vahala, Science 321, 1172
(2008).
[6] A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.
Clerk, M. P. Blencowe, and K. C. Schwab, Nature(London) 443, 193 (2006).
[7] M. I. Dykman, Sov. Phys. Solid State 20, 1306 (1978).
[8] V. B. Braginsky and S. P. Vyatchanin, Phys. Lett. A 293,
228 (2002).
[9] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,
Phys. Rev. Lett. 99, 093902 (2007).
[10] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J.
Kippenberg, Phys. Rev. Lett. 99, 093901 (2007).
[11] A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, andT. J. Kippenberg, Nature Phys. 4, 415 (2008).
[12] J. D. Thompson, B. M. Zwickl, A. M. Jayich, F.
Marquardt, S. M. Girvin, and J. G. E. Harris, Nature(London) 452, 72 (2008).
[13] C. A. Regal, J. D. Teufel, and K. W. Lehnert, Nature Phys.
4, 555 (2008).
[14] J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert,
Phys. Rev. Lett. 101, 197203 (2008).
[15] A. A. Clerk and S. D. Bennett, New J. Phys. 7, 238(2005).
[16] M. P. Blencowe, J. Imbers, and A. D. Armour, New J.
Phys. 7, 236 (2005).
[17] In recent experiments with microwave cavity electromechanical systems (e.g., Refs. [13,14]), the system has been
engineered to have a minimal dissipative coupling, i.e.
~ the possible utility of B~ Þ 0 has not been realized,B~  A;
before.
[18] A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D.
Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, andJ. G. E. Harris, New J. Phys. 10, 095 008 (2008).
[19] D. F. Walls and G. J. Milburn, Quantum Optics (Springer,
Berlin, 1994).
[20] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt,
and R. J. Schoelkopf, arXiv:08104729.
[21] D. A. Rodrigues, Phys. Rev. Lett. 102, 067202 (2009).

207209-4

PHYSICAL REVIEW LETTERS

PRL 103, 149902 (2009)

week ending
2 OCTOBER 2009

Erratum: Quantum Noise Interference and Backaction Cooling in Cavity Nanomechanics[Phys. Rev. Lett. 102, 207209 (2009)]
Florian Elste, A. A. Clerk, and S. M. Girvin(Received 14 September 2009; published 30 September 2009)
DOI: 10.1103/PhysRevLett.103.149902

PACS numbers: 85.85.+j, 42.50.Lc, 42.50.Pq, 99.10.Cd

In Ref. [1], the dimensionless dispersive cavity-mechanical resonator coupling parameter A~ should have been definedwith an overall minus sign. The proper definition for A~ isd! xzpt
A~   R
:
dx 

(1)

Further, the sign preceding A~ in Eq. (2) for the interaction Hamiltonian, Eq. (3) for the cavity field equation of motion, and~ The
Eq. (5) for the backaction force operator is incorrect. In each of these equations, one should simply replace A~ by (A).
~sign of A in the remaining equations and discussion of the Letter is correct as given; in particular, the optimal detuning~ BÞ.
~ Note that when A~ > 0
needed for the backaction to mimic a zero-temperature bath is given by opt ¼ ð!M =2Þ þ 2ðA=(i.e., d!R =dx < 0), opt is necessarily positive.
In addition, Ref. [1] treats (as stated) only the ideal case of an overcoupled cavity, where the damping rate  of the cavity
is completely dominated by the port used to drive it. In the more general case, the total dissipation  will also have acontribution from internal losses and/or other ports besides the drive port. In this more general case, one can write  ¼
D þ int , where D is the damping resulting from the port used to drive the cavity, and int is the damping resulting frominternal losses and other ports. Assuming that these additional sources of dissipation are at an effective temperature much
lower than @!c =kB , and that the x dependence of  occurs solely through the modulation of D , the backaction noise
spectral density SFF ð!Þ given in Eq. (6) of Ref. [1] takes the form
2A~
2A~
2
2
2
~

 2 D ½! þ 2  B~D  þ int ½ð  B~D Þ þ 4 
BD jaj;
SFF ð!Þ ¼
2xzpt
ð! þ Þ2 þ 2 =4(2)d xzpt
B~ D  D
:
dx D

(3)

where we have definedFor int Þ 0, perfect destructive interference of the backaction noise at ! ¼ !M is not possible, and the cavity is nolonger able to mimic an effective zero-temperature bath at the optimal detuning opt . Instead, one finds that in the limitint =  1, the effective temperature of the backaction (measured as a number of oscillator quanta) at the optimal
detuning is given by~n BA ¼

2 þ ð3! þ 2 B~A Þ2 int
D

16!2



þO

 2 
int
:


(4)

One thus sees that having a sufficiently small int is an important condition for achieving significant cooling using the noiseinterference effect described in Ref. [1].
[1] F. Elste, S. M. Girvin, and A. A. Clerk, Phys. Rev. Lett. 102, 207209 (2009).

0031-9007=09=103(14)=149902(1)149902-1

Ó 2009 The American Physical Society