Quantum Noise Interference and Back-action Cooling in Cavity Nanomechanics
Florian Elste,1S. M. Girvin,2and A. A. Clerk1
1Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8
2Department of Physics, Yale University, New Haven, Connecticut 06520, USA
(Dated: March 12, 2009)
We present a theoretical analysis of a novel cavity electromechanical system where a mechanical
resonator directly modulates the damping rate κ of a driven electromagnetic cavity.
that via a destructive interference of quantum noise, the driven cavity can effectively act like a zero-
temperature bath irrespective of the ratio κ/ωM, where ωM 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 ‘good cavity’ limit κ ? ωM. The system described here could be implemented
directly using setups similar to those used in recent experiments in cavity electromechanics.
PACS numbers:42.79.Gn, 07.10.Cm, 42.50.Lc.
Introduction.— Much of the rapid progress in fabri-
cating and controlling nanomechanical devices has been
fueled by numerous promising technological applications.
However, progress has also been motivated by the real-
ization that such systems are ideally poised to allow the
exploration of several fundamental quantum mechanical
effects.Various studies have addressed such issues as
entanglement, quantum-limited measurement and Fock
state detection in systems containing nanoscale mechan-
ical resonators [1, 2, 3, 4]. The issue of quantum back-
action has also received considerable attention in these
systems [5, 6]. Understanding the back-action properties
of a detector in a quantum electromechanical or optome-
chanical system is necessary if one wishes to do quantum-
limited position detection. Further, this back-action can
in some cases be exploited to achieve significant non-
equilibrium cooling of the mechanical resonator. This is
of particular importance, since a pre-requisite to seeing
truly quantum behaviour in these systems is the ability
to cool the mechanical resonator to near its ground state.
A particularly effective back-action cooling scheme is
so-called “cavity cooling” [7, 8, 9, 10]. Here, a mechan-
ical resonator is dispersively coupled to a driven elec-
tromagnetic cavity (i.e. the cavity’s resonance frequency
depends on the mechanical displacement x). By moni-
toring the frequency of the cavity, one can make a sen-
sitive measurement of x. In addition, the cavity photon
number necessarily acts as a noisy force on the mechan-
ics; for a suitably chosen cavity drive, this force can be
used to effectively cool the mechanical resonator. This
approach has been used in a number of recent experi-
ments, both with 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 could in principle allow ground state cool-
ing of the mechanical resonator [9, 10]. Similar physics
even occurs in seemingly very different systems, e.g. a
superconducting single-electron transistor coupled to a
mechanical resonator [15, 16].
In this Letter, we present a theoretical analysis of
a generic electromechanical (or optomechanical) system
which is the dual of the dispersive coupling discussed
above. We again consider a driven cavity coupled to a
mechanical resonator: now, however, the mechanical dis-
placement x does not change the cavity frequency, but
rather changes its damping rate κ. As we will discuss,
such a coupling could be achieved in a microwave cav-
ity system, or in an optical cavity containing a moveable
‘mebrane-in-the-middle’ [12, 17]. We show that for such
a dissipative cavity-mechanical resonator coupling, inter-
ference effects are important in determining the quantum
back-action 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 ωM
to the cavity linewidth κ; as such, ground state cool-
ing is possible without requiring the ‘good cavity’ limit
ωM? κ. This is in sharp contrast to the case of a disper-
sive coupling, where ground state cooling is only possible
if ωM ? κ. From a practical perspective, being able
to deviate from the good cavity limit is advantageous,
as it allows one to use small drive detunings and hence
achieve much larger effective cavity-mechanical resonator
couplings. We show that this destructive interference ef-
fect persists in the case where one has both a dissipative
and dispersive coupling; we also show that the dissipative
coupling can also allow for a quantum-limited position
Model.— We consider a mechanical oscillator (fre-
quency ωM, displacement x) whose motion weakly mod-
ulates the damping rate κ and resonant frequency ωRof
a driven electromagnetic cavity. For small displacements,
both ωRand κ will have a linear dependence on x, and
we can describe the system via the Hamiltonian (? = 1):
ˆH = ωRˆ a†ˆ a + ωMˆ c†ˆ c +?
Hamiltonians, while Hγdescribes the intrinsic mechani-
cal damping by an equilibrium bath at temperature Teq.
The third term inˆH describes the bosonic bath responsi-
The first two terms describe the cavity and mechanical
arXiv:0903.2242v2 [cond-mat.mes-hall] 13 Mar 2009
SFF(ω) / SFF,max
∆ = κ/2, for different ratios of the dispersive to dissipa-
tive optomechanical couplings.
˜ A/˜B = 10, the long-dashed green curve to
˜ A/˜B = 0.75 and the short-dashed red curve to˜ A/˜B = 0.
Each curve has been scaled by its maximum value. (b) Num-
ber of oscillator quanta nosc versus drive detuning ∆, ob-
tained by solving the full equations of motion. The dotted
blue curve is the Bose-Einstein factor neff corresponding to
the back-action effective temperature Teff when κ = ωM (i.e.
neff = SFF(−ωM)/(SFF(ωM) − SFF(−ωM)). The remaining
curves correspond to˜B¯ a = 0.2, neq = 50, γ = 10−6ωM, and
κ/ωM = 0.2 (green solid curve), κ/ωM = 1.0 (purple long-
dashed curve), κ/ωM = 5.0 (red short-dashed curve).
(a) Back-action force noise spectral density for
The solid blue curve cor-
ble for the dissipation and driving of the cavity. Working
in the usual Markovian limit where κ ? ωR, and where
the bath density of states ρ can be treated as energy-
independent over relevant frequencies, the cavity-bath
interaction takes the form [18, 19]:
We stress that the Markovian limit is well justified both
for cavities used in optomechanical and microwave elec-
tromechanical systems. The coupling between the cavity
and mechanics is then described by:
The dimensionless coupling strengths ˜A,˜B above are
given by˜Bκ = (dκ/dx)xzpt,˜Aκ = (dωR/dx)xzpt, where
xzptis the zero-point motion amplitude of the mechani-
cal oscillator. Setting˜B = 0 recovers a purely dispersive
coupling, while setting˜A = 0 corresponds to a purely
dissipative optomechanical coupling.
Quantum noise.— We may now identify the quantum
back-action force operatorˆF ≡ −(d/dx)ˆHint. For a suf-
ficiently weak optomechanical coupling, linear-response
theory applies, and the unperturbed quantum noise spec-
trum ofˆF will determine both the back-action damp-
ing and back-action heating of the mechanical res-
onator by the driven cavity [15, 19].
spectrum is SFF(ω) =
that SFF(ωM) is proportional to the Fermi’s golden
rule rate for the absorption of a mechanical quan-
tum by the driven cavity, while SFF(−ωM) is pro-
portional to the corresponding emission rate.
ˆHdamp = −i
ˆHint = (ˆ x/xzpt)
˜BˆHdamp+˜Aκˆ a†ˆ a
effective temperature associated with the back-action
noise as seen by the mechanical oscillator is then
given by kBTeff ≡ ωM(log[SFF(ωM)/SFF(−ωM)])−1,
while the back-action damping is given by γBA
the total mechanical damping rate to be small enough
that the resonator is only sensitive to noise at ω = ±ωM.
We calculate SFF(ω) by linearizing the full quantum
dynamics about the solutions of the classical equations
of motion (see below), and by making use of the input-
output formalism of quantum optics [18, 19]. Letting
∆ = ωdrive−ωRbe the detuning of the cavity drive, and
¯ a = ?ˆ a?, one finds:
zpt(SFF(ωM) − SFF(−ωM)).These results presume
SFF(ω) = κ
ω + 2∆ −2˜
(ω + ∆)2+ κ2/4.
In the limit˜B → 0 of a purely dispersive coupling,
Eq. (3) reduces to a simple Lorentzian, in agreement with
Refs. [9, 10]. This form has a simple interpretation: it
corresponds to the Lorentzian density of final states rele-
vant to a Raman process where an incident drive photon
gains an energy ?ω while attempting to enter the cavity.
The optimal back-action cooling discussed in Refs. [9, 10]
involves choosing ∆ = −ωMand κ ? ωM. For these pa-
rameters, 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 is greatly
suppressed, as the drive photon is moved even further
from the cavity resonance. Thus, the˜B = 0 form of
SFF(ω) and the resulting cooling are simply explained
as a density of states effect.
In the more general case where the dissipative op-
tomechanical coupling˜B is also non-zero, we see that
SFF(ω) is not a simple Lorentzian; as such, the back-
action physics cannot be interpreted solely in terms of a
density of states effect. In general, SFF(ω) has an asym-
metric Fano lineshape (see Fig. 1); in particular, the cav-
ity emission noise SFF(−ωM) is equal to zero whenever
the detuning ∆ = ωM/2 + (˜A/˜B)κ ≡ ∆opt. Thus, for
this value of the detuning, one finds that the cavity acts
as an effective zero-temperature bath, irrespective of the
ratio κ/ωM. For ∆ = ∆opt, one has:
3ωM/2 + (˜A/˜B)κ
For ∆ = ∆opt, the weak-coupling, quantum noise ap-
proach yields the equilibrium number of quanta in the
oscillator to be nosc = γneq/(γBA,opt+ γ), where γ is
the intrinsic damping rate of the oscillator, and neq is
the Bose-Einstein factor associated with the bath tem-
perature Teq. One can thus cool to the ground state for
a sufficiently large coupling and/or cavity drive. This
possibility of ground state cooling for an arbitrary κ/ωM
ratio is a main result of this paper. Note that the optimal
detuning ∆optis greater than 0: ground state cooling is
possible even though the drive photons would seemingly
need to “burn off” energy to enter the cavity. This is in
stark contrast to the purely dispersive case where a posi-
tive detuning leads to heating and a negative-damping
instability; it highlights the fact that the back-action
physics here is not simply a density of states effect. Note
finally that there is also an “optical spring” effect asso-
ciated with the back-action; it can be obtained directly
from SFF(ω) .
A heuristic explanation of the Fano form of SFF(ω)
can be given. Fano lineshapes arise generically as a re-
sult of interference between resonant and non-resonant
processes; the situation is no different here. In the usual
˜B = 0 case, the only source of back-action force noise
is the number fluctuations of the cavity field ˆ a. How-
ever, when˜B ?= 0, the mechanical oscillator mediates
the coupling between the cavity and the cavity’s dissi-
pative bath. As a result, it is subject to two sources of
noise: the shot noise associated with the driving of the
cavity, as well as the fluctuations of ˆ a. The first of these
noise processes is white, whereas the second is not: it
is simply the shot noise incident on the cavity filtered
by the ω-dependent cavity susceptibility. The interfer-
ence between these two noises yields a Fano lineshape for
SFF(ω), and the destructive interference at ∆ = ∆opt
which causes Teff = 0. Note that Fano interference in
quantum electromechanical systems has recently been
discussed in Ref. , albeit in a completely different
Equations of motion.— To address whether the de-
structive noise interference effect persists beyond the sim-
plest weak-coupling regime, we now examine the full so-
lutions of the linearized Heisenberg equations of motion
for our system. For simplicity, we focus on the case˜A = 0
in what follows. Working in frame rotating at the drive
frequency, and writing ˆ a = ¯ a+ˆd, the linearized equations
take the form:
˙ˆ c = −
Here¯binis the amplitude of the coherent cavity drive,ˆξ
describes vacuum noise entering the cavity from the drive
port, and ˆ η describes thermal noise associated with the
mechanical damping γ; bothˆξ and ˆ η are operator-valued
white noise [18, 19]. The back-action force noise operator
in Eq. (6) takes the form:
ˆ c −√γˆ η + iˆf.
¯ a +
??ˆ c + ˆ c†?−√κˆξ,(5)
ˆf = i
√κ(¯ a∗ˆξ −¯b∗
inˆd) + h.c.
We see the two contributions to the back-action discussed
earlier: the first term corresponds to the direct contribu-
tion of the drive shot noise, while the second term rep-
resents the contribution from fluctuations in the cavity
field. As discussed, the interference of these terms yields
a Fano lineshape.
Upon solving the equations of motion, the oscillator
spectrum Scc=?dt?ˆ c†(t)ˆ c(0)e−iωt? is found to be:
Scc(ω) = |˜ χM(−ω)|2[σeq(−ω) + SFF(ω)]
˜ χM(ω) = χM(ω)/[1 + iχM(ω)Σ(ω)],
Σ(ω) = −i˜B2|¯ a|2χ∗
σeq(ω) = γneq
(1 + 2neq)
Here, we denote the bare mechanical and cavity sus-
ceptibilties by χM(ω) = (−i(ω − ωM) + γ/2)−1and
χR(ω) = (−i(ω + ∆) + κ/2)−1. Eq. (8) has a simple
interpretation: the oscillator responds with a modified
susceptibility ˜ χM to two independent fluctuating forces,
corresponding to the two terms in the equation. The
first, described by σeq, is the fluctuating thermal force
associated with the intrinsic oscillator damping γ. The
second is the back-action from the driven cavity. We see
that its form is not affected by the coupling strength:
the same spectral density SFF(ω) (evaluated at˜A = 0)
found earlier in the weak-coupling regime (c.f. Eq. (3))
appears here. Thus, the quantum noise interference dis-
cussed above continues to play a role even for moderate
coupling strengths. A strong cavity-mechanical resonator
coupling will nonetheless modify the physics, as the de-
structive interference occurring when ∆ = ∆optonly oc-
curs at the single frequency ωM. For a sufficiently strong
coupling, the oscillator’s total damping will become large
enough that the oscillator will be sensitive to noise at fre-
quencies away from ωM, frequencies where the destruc-
tive interference is not complete. As a result, the cavity
will no longer appear to the oscillator as an effective zero-
temperature bath. Note that when˜A ?= 0, the spectrum
Scc(ω) still has the general form given by Eqs. (8, 9, 11):
one now simply uses the full form of SFF(ω), as well as
a self-energy that contains extra terms ∝˜A.
To see whether this resonance-broadening effect as well
as other strong coupling effects preclude ground state
cooling at the optimal detuning ∆ = ∆opt, we calcu-
late the average number of oscillator quanta noscdirectly
by integrating Scc(ω) in Eq. (8). In Fig. 2 we show the
expected cooling for realistic parameter values, as a func-
tion of the cavity drive strength. Strong-coupling effects
lead to an optimal drive strength, beyond which nosc
γ/ωM = 10-5
γ/ωM = 10-6
FIG. 2: (a) Number of mechanical quanta nosc versus cou-
pling strength˜B|¯ a|, for˜ A = 0, neq = 50, κ/ωM = 1 and
for an optimal detuning ∆ = ωM/2; the mechanical damping
γ is as marked. Dashed curves are results from the linear-
response, 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) cou-
pled to a transmission line (impedance Z0) via an x-dependent
starts to increase. Nonetheless, the minimum value of
nosccan still be significantly less than 1. Again, we stress
that the dissipative coupling discussed here is in many
ways advantageous to a system with a purely dispersive
optomechanical coupling, as one does not need to use a
large detuning to achieve a low effective temperature.
Physical realization.— The dissipative optomechani-
cal coupling analyzed here could be realized in a mi-
crowave electromechanical system similar to those stud-
ied in Refs. [13, 14]. In such systems, a capacitor C1
couples the cavity to a transmission line (impedance Z0)
which both drives and damps the cavity. One would now
need to make C1 mechanically compliant (see Fig. 2b).
In general, such a setup will have both dissipative and
dispersive optomechanical couplings (i.e.˜A ?= 0,˜B ?= 0).
A careful analysis shows that in the physically relevant
regime C1 ? C, one can have ˜A ? ˜B if the cavity
impedance ZR =
Z0? 50Ω. For example, taking experimentally achiev-
able parameters ωR = 2π × 10 GHz, C = 3.2 pF and
C1 = 0.01C results in κ/ωR ? 10−3and |˜A/˜B| ? 2.5.
Eq. (3) then implies that ground-state cooling via de-
structive noise interference is possible if one uses a drive
detuning ∆ = ∆opt= ωM/2 + 2.5κ. We stress this con-
clusion is independent of the magnitude of ωM/κ; in con-
trast, if one had a purely dispersive cavity - mechanical
resonator coupling, ground state cooling is only possi-
ble if ωM ? κ. A second setup where the dissipative
optomechanical coupling could be realized is an optical
Fabry-Perot cavity containing a thin, moveable dielectric
slab. If the cavity is constructed with mirrors having dif-
ferent reflectivities, the motion of the membrane will nat-
urally modulate the damping rate of the optical modes.
The resulting value of˜B can be easily derived using the
approach of Ref. .
Position detection.— A straightforward calculation
shows that the power spectrum ofˆbout, the output field
?L/C is made slightly smaller than
from the cavity, is modified in a simple way by the
coupling to the oscillator: the oscillator adds a term
Sxx(−ω)SFF(ω) (where ω is measured from the drive fre-
quency); this holds for both˜A,˜B ?= 0. Here, Sxx(ω) =
?dτeiωτ?ˆ x(τ)ˆ x(0)?.
ture of the oscillator. Note that for an optimal detuning
∆ = ∆opt, the destructive interference effect means that
the peak at ω = ωMin the output spectrum will vanish.
This could serve as an interesting test of our predictions.
Finally, specializing to the case where ωM ? κ, one
can also use the dissipative optomechanical coupling to
make a quantum-limited, weak continuous position mea-
surement of the oscillator. One drives the cavity off res-
onance, and performs a homodyne detection of the ap-
propriate quadrature of the output field from the cavity.
One finds from Eqs. (5),(6) that for any non-zero detun-
ing ∆, the corresponding back-action noise is as small
as is allowed by the uncertainty principle, meaning that
one can reach the quantum limit on the total added noise
Conclusions.— We have considered a generic system
where a mechanical resonator modulates the damping of
a driven quantum electromagnetic resonator, and demon-
strated how Fano interference is important to the back-
action physics. In particular, one can have a perfect
destructive interference which allows the back-action to
mimic a zero-temperature environment; as such, such a
system should allow near ground-state cooling of the me-
We thank J. Harris for useful comments. This work
was supported by NSERC, CIFAR and the NSF under
grants DMR-0653377 and DMR-0603369.
One can thus use the peaks at
ω = ±ωM in the output spectrum to infer the tempera-
 A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys.
Rev. Lett. 88, 148301 (2002).
 D. H. Santamore, A. C. Doherty, and M. C. Cross, Phys.
Rev. B 70, 144301 (2004).
 M. P. Blencowe, Phys. Rep. 359, 159 (2004).
 A. A. Clerk, Phys. Rev. B 70, 245306 (2004).
 T. J. Kippenberg and K. J. Vahala, Science 321, 1172
 A. Naik, O. Buu, M. D. LaHaye, A. D. Armour, A. A.
Clerk, M. P. Blencowe, and K. C. Schwab, Nature (Lon-
don) 443, 193 (2006).
 M. I. Dykman, Sov. Phys. Solid State 20, 1306 (1978).
 V. B. Braginsky and S. P. Vyatchanin, Phys. Lett. A 293
 F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,
Phys. Rev. Let. 99, 093902 (2007).
 I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kip-
penberg, Phys. Rev. Lett. 99, 093901 (2007).
 A. Schliesser, R. Rivi` ere, G. Anetsberger, O. Arcizet, and
T. J. Kippenberg, Nature Phys. 4, 415 (2008).
 J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Mar-
quardt, S. M. Girvin, and J. G. E. Harris, Nature (Lon-
don) 452, 06715 (2008).
 C. A. Regal, J. D. Teufel, and K. W. Lehnert, Nature
Phys. 4, 555 (2008).
 J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W.
Lehnert, Phys. Rev. Lett. 101, 197203 (2008).
 A. A. Clerk and S. D. Bennett, New J. Phys. 7, 238
 M. P. Blencowe, J. Imbers, and A. D. Armour, New J.
Phys. 7, 236 (2005).
 A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D.
Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, and
J. G. E. Harris, New J. Phys. 10, 095008 (2008).
 D. F. Walls and G. J. Milburn, Quantum Optics
(Springer, Berlin, 1994).
 A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt,
and R. J. Schoelkopf, arXiv:08104729 (2008).
 D. A. Rodrigues, Phys. Rev. Lett. 102, 067202 (2009).