PHYSICAL REVIEW A 93, 063809 (2016)

Quantum backaction and noise interference in asymmetric two-cavity optomechanical systems
Yariv Yanay, Jack C. Sankey, and Aashish A. Clerk
Department of Physics, McGill University, Montreal, Canada H3A 2T8(Received 11 April 2016; published 8 June 2016)
We study the effect of cavity damping asymmetries on backaction in a “membrane-in-the-middle”
optomechanical system, where a mechanical mode modulates the coupling between two photonic modes. We
show that when the energy difference between the optical modes dominates (i.e., in the adiabatic limit) thissystem generically realizes a dissipative optomechanical coupling, with an effective position-dependent photonic
damping rate. The resulting quantum noise interference can be used to ground-state cool a mechanical resonatorin the unresolved sideband regime. We explicitly demonstrate how quantum noise interference controls linear
backaction effects and show that this interference persists even outside the adiabatic limit. For a one-port cavity
in the extreme bad cavity limit, the interference allows one to cancel all linear backaction effects. This allows
continuous measurements of position-squared, with no stringent constraints on the single-photon optomechanicalcoupling strength. In contrast, such a complete cancellation is not possible in the good cavity limit. This places
strict bounds on the optomechanical coupling required for quantum nondemolition measurements of mechanicalenergy, even in a one-port device.
DOI: 10.1103/PhysRevA.93.063809

I. INTRODUCTION

The field of quantum optomechanics has largely focusedon a relatively simple system, where photons in a single modeof a resonant cavity interact with the motion of a mechanicalresonator. This kind of system has been at the heart of a numberof recent experimental breakthroughs, ranging from the nearground-state cooling of a mechanical mode [1–3] to thegeneration of squeezed light [4,5]. Potentially richer behavior
can be realized in a system where a mechanical resonator
couples to two optical modes, which are in turn tunnel-coupled
to one another [6]. By using the avoided crossing of the
optical normal modes, such systems can realize an effective
quadratic optomechanical coupling, where the adiabatic cavity
normal-mode frequencies have no linear dependence on themechanical position x, but instead depend on x 2 . Such devices
could allow quantum nondemolition (QND) measurement ofphonon and photon number [6–8] and can realize novel kindsof optomechanical cooling and squeezing [9–11]. Quadratic
couplings were first realized in “membrane-in-the-middle”
systems, where a movable membrane is placed between two
fixed mirrors of a Fabry-Perot cavity [6,12], and more recently
in several other experimental setups [13–16].
A key issue in such systems is the presence of residual linear
backaction, that is, the linear coupling of external noise sources
to the mechanics through the cavity. Such backaction hinders
QND phonon-number detection. As discussed extensively by
Miao et al. [17], linear backaction persists even though the
adiabatic normal-mode frequencies depend on x 2 , because theadiabatic wave functions depend linearly on x. When eachcavity normal mode is coupled to an independent dissipative
reservoir, the linear noise is proportional to the product of bothdissipation rates, suggesting that it could be eliminated if one
had a truly single-port cavity.
In this work, we revisit linear backaction effects inquadratic-coupling optomechanics, focusing on the case ofa realistic, asymmetric two-port cavity. In general, the two
dissipative ports each couple to both photonic normal modes
as illustrated in Fig. 1, implying that the two modes see
2469-9926/2016/93(6)/063809(9)correlated dissipation and noise. In the adiabatic limit, where
the splitting of the photonic modes is much greater than allother energy scales, we show that one obtains an effective
dissipative optomechanical coupling [18], where the damping
rate of each normal mode depends on x. The resulting quantum
noise interference implies that one could ground-state cool amechanical resonator in the unresolved sideband limit [18].
We also explain in detail how this quantum noise interference
persists even outside the adiabatic limit and show how
standard approximations, which ignore noise correlations,can give misleading results. Note that a related quantum
noise cancellation effect was discussed in Ref. [19], wherea two-mode optomechanical system couples to a single opticalreservoir.
Turning to measurement physics, we show that theinterference-based noise cancellation can only be used tocancel all linear backaction effects in the unresolved sideband
limit, where the cavity damping rate is greater than themechanical frequency. In contrast, one needs to be in the
opposite (good cavity) limit to make a QND measurementof mechanical energy [6–8]. As a result, even in an ideal
one-port device QND measurement of the mechanical energythat can resolve single-phonon jumps requires a single-photon
optomechanical coupling that is larger than the cavity damping
rate. More optimistically, the cancellation of linear backaction
in single-port system in the bad cavity limit allows one to makea measurement of x 2 ; such nonlinear measurement can lead to
nonclassical mechanical states [20,21].

II. MODEL
A. Hamiltonian and coupling to environment

The two-cavity optomechanical system of interest is depicted in Fig. 1. Its Hamiltonian takes the form

063809-1

Ĥ = Ĥopt + Ĥκ + Ĥm + Ĥγ + Ĥint .

(1)©2016 American Physical Society

YARIV YANAY, JACK C. SANKEY, AND AASHISH A. CLERK

ωm γ

Ξ̂γ

ωc + JΞ̂L κLκR

ωc − JΞ̂R

FIG. 1. Schematic of the two-cavity optomechanical systemdescribed by Eq. (1). Two cavity normal modes, with energies ωc ± J ,couple to a moving membrane (frequency ωm ). The optical normal
modes are both coupled to two independent reservoirs, L and R, withdifferent damping rates, while the membrane is coupled to a thirdreservoir at damping rate γ .

The Hamiltonian describing the cavities in the absence of
optomechanical coupling is ( = 1 throughout)
†

†

†

†

Ĥopt = ωc (âL âL + âR âR ) − J (âL âR + âR âL )††= (ωc − J )â+ â+ + (ωc + J )â− â− ,(2)where âL and âR are the annihilation operators for a pair of
localized cavity modes L and R (e.g., to the left and to theright of a membrane at the center of an optical cavity), andJ describes a tunnel coupling between them. In the second
line, we have diagonalized this Hamiltonian in terms of the
optical normal modes â± = âL√±2âR . The mechanical resonator(frequency ωm , annihilation operator b̂, position operator x̂)is described by Ĥm = ωm b̂† b̂. The optomechanical couplingtakes the form Ĥint = −(x̂/xzpt )F̂opt , where
†

†

†

†

F̂opt = g(âL âL − âR âR ) = g(â+ â− + â− â+ )PHYSICAL REVIEW A 93, 063809 (2016)

While Eq. (4) seems innocuous enough, it implies thatunless κR = κL , the optical eigenmodes â± will not be coupled
to independent baths. This implies both that dissipation cancouple the two optical normal modes and that their fluctuations
are correlated. While it is standard in many quantum optics
contexts to ignore such noise correlations between modes thatare well-separated in frequency, we show they are significant inthis class of systems, as the effects of nonresonant fluctuations
are important. Exploring the full consequences of these
correlations for a two-cavity system when κL = κR is the main
goal of our work.
B. Heisenberg-Langevin equations

To see this noise correlation explicitly, it is useful toconsider the Heisenberg-Langevin equations for our systemin the absence of optomechanical coupling. We consider the
standard case where the cavity modes are coherently driven at
the frequency ωdr near the symmetric normal-mode frequency
ωc − J . Working in a rotating frame at the drive frequency,
the optical Hamiltonian takes the form†

ˆ L,R are bath operators and Ĥext describes the freewhere bath modes. We treat the dissipation as per standard input
output theory (see, e.g., Refs. [24,25]). For J = 0, the L mode
(R mode) has an amplitude damping rate of κL /2 (κR /2).
Taking L and R to see independent reservoirs describes the
dynamics of a general, asymmetric mebrane-in-the-middle
type setup [6], where each end mirror of the cavity has anonzero transmission. We discuss the more general situation,
where the two optical eigenmodes are generically coupled totwo dissipative reservoirs (e.g., a one port setup with internalloss) in Appendix B.

(5)where δ = ωdr − (ωc − J ) is the detuning of the driving
frequency from the â+ resonance.
By applying standard input-output theory [24,25], theHeisenberg-Langevin equations of motion for our system (atRR
g = 0) are easily found. Defining κ̄ = κL +κ
and κ = κL −κ,22
we have˙â− = − κ̄ + iδ − 2iJ â− − κ â+
2
2


 


κL in
κR in
κL
κR
+
+
α −
α
ξ̂L −
ξ̂R ,
2 L
2 R
2
2

(3)

and xzpt is the ground-state position uncertainty of the oscillator. Note that we have scaled the backaction force operator tohave units of a rate. In the case of a membrane-in-the-middle
type setup, a full derivation of the kind of few-mode description
given here is provided in Refs. [7,22]; limitations of the modelare discussed in Refs. [22,23].
The remaining terms in Eq. (1) describe dissipation. As
is standard, the mechanical dissipation (Ĥγ ) is described bya coupling to a thermal Markovian reservoir, giving rise toan amplitude damping rate of γ /2. The treatment of cavity
dissipation (described by Ĥκ ) requires slightly more care. We
take each of the L and R modes to be coupled to independent,
zero-temperature Markovian reservoirs. The dissipation of thecavities is thus described by
√ ˆ†
√ ˆ†
Ĥκ = −i( κL 
κR R âR − H.c.) + Ĥext ,(4)
L âL +†

Ĥopt = −δ â+ â+ + (2J − δ)â− â− ,

(6)


κ̄
κ
â˙ + = −+ iδ â+ −
â−
2
2


 


κL in
κR in
κL
κR
+
α +
α
+
ξ̂L +
ξ̂R .
2 L
2 R
2
2
(7)
We are considering the general case where a coherent cavity
drive is applied both at the left port and the right port, withrespective classical input field amplitudes αLin and αRin . The ξ̂i (t)operators (i = L and R) describe operator-valued Gaussian
white noise, i.e., incident vacuum fluctuations entering the left
and right ports. They have zero mean and correlation functions††
ξ̂i (t)ξ̂j (t  ) = 0 and ξ̂i (t)ξ̂j (t  ) = δij δ(t − t  ).
One sees clearly that the noise fluctuations driving â+ andâ− are in general correlated with one another. These noises areof course identical (and hence completely correlated) in thecase of a one-port cavity, κR = 0. The only case where there
is no correlation is when κR = κL . In this case, the situation
is identical to having the + and − optical modes coupled toindependent reservoirs.

063809-2

QUANTUM BACKACTION AND NOISE INTERFERENCE IN . . .
C. Effective dissipative optomechanical coupling

For more intuition into the origin of linear backaction
effects in this system, it is useful to consider the structureof the “adiabatic” cavity eigenmodes: if x̂ is treated as astatic, classical parameter, then what are the cavity eigenmodes
for a given value of x? One finds the lower-energy adiabatic
eigenmode is described byâ+ [x] = cos[θ (x)]âL + sin[θ (x)]âR ,(8)where cot 2θ (x) = gx/J xzpt . Its frequency is given by

ω+ [x] = ωc − J 2 + (gx/xzpt )2 and depends quadraticallyon x.
Despite the x 2 dependence of the adiabatic mode frequencies, the dissipation of the adiabatic optical modes can leadto linear backaction. As it couples to the environment throughthe two leaky mirrors at L and R, the dissipation rate of the +adiabatic mode is
κ+ [x] = cos2 [θ (x)]κL + sin2 [θ (x)]κR
gx/xzpt= κ̄ + κ.
J 2 + (gx/xzpt )2(9)

Unless κL = κR , the effective damping rate of the adiabatic
mode depends on whether the mode is localized more on theleft or on the right. As the “wave function” of this adiabatic
mode depends on x, one obtains a so-called dissipative
optomechanical coupling [18], where the damping rate ofan optical mode depends on x. Note that this represents a
potentially simpler method for realizing a dissipative optomechanical coupling than the Michelson-Sagnac interferometer
proposed in Ref. [26] and realized experimentally in Ref. [27].
We note that recent experiments using trampoline-style resonators in a Fabry-Perot cavity with κL  κR observed large
position-dependent photonic damping [28].
Crucially, to leading order in 1/J , we see that even though
the adiabatic mode frequency has a quadratic dependence on x,
κ+ [x] depends linearly on x for small displacements. This implies that there will be information on x available in the cavity
output field, opening the door to unwanted linear backaction
effects. It also suggests that the unusual quantum noise physics
of a dissipative optomechanical coupling will be relevant here,
namely, the possibility of Fano-style interference [17,18].
Note that throughout this discussion we have focused on
the lower-energy adiabatic mode, â+ . A full discussion must
include the higher-energy adiabatic mode as well, as the two
are in general coupled by dissipation. Such coupling effects
only contribute at higher orders in 1/J .

PHYSICAL REVIEW A 93, 063809 (2016)

only needs to consider the drive-enhanced optomechanicalcoupling, i.e., one can linearize the F̂opt operator in thefluctuations of â± .
A. Noise spectrum and noise amplitudes

The (unsymmetrized) quantum noise spectral density ofF̂opt is defined as [25,29,30]
SF F [ω] = dteiωt F̂opt (t)F̂opt (0).
(10)
As long as the features of SF F [ω], such as peaks and dips,
are wider than the mechanical linewidth, the cavity can beunderstood as an effective thermal bath for the mechanics, with
an optomechanical damping rate and an effective thermal
occupation n̄eff given by= SF F [ωm ] − SF F [−ωm ],(11)

Here, SF F [+ωm ] describes the emission of energy fromthe mechanics into the cavity, or cooling processes, while
SF F [−ωm ] describes the absorption of energy by the mechanics, or heating processes.
In our system, the leading terms in F̂opt (which are enhancedby the classical cavity drive) will be linear in the input noise
operators. We can thus write it in terms of “noise amplitudes”
AL/R [ω] as
†F̂opt [ω] =Ai [ω]ξ̂i [ω] + A∗i [−ω]ξ̂i [ω].
(12)
i=L,R

We have defined X̂[ω] ≡
†

∞

−∞ e
†

iωt

X̂(t)dt for any operator X̂

(implying X̂ [ω] = [X̂[−ω]] ). As the input noise operators
describe vacuum noise, one immediately finds
SF F [ω] = |AL [ω]|2 + |AR [ω]|2 .

(13)

In addition to controlling the backaction noise spectral density,
the noise amplitudes Ai [ω] also directly determine how well
one can make a linear measurement of position x from theoutput light leaving the cavities; large linear backaction effects
come hand in hand with large amounts of information on x in
the output field. This is shown explicitly in Appendix A.
The amplitudes Ai [ω] are found by solving the HeisenbergLangevin Eqs. (6) and (7) in the Fourier domain. They are eacha sum of two terms, corresponding to the two optical normal
modes â+ and â− :iAL/R [ω] = √
×

2

G√
κL/R
2J˜
εm
εm + i κ∓ J 1 ± 2J2

ω + δ − J + i κ̄2III. BACKACTION QUANTUM NOISE SPECTRUM

We now turn to the question of how noise correlations
influence the linear optomechanical backaction on the mechanical resonator. We focus on the standard case where
the optomechanical coupling is sufficiently weak that opticalbackaction effects on the mechanical resonator can be fully
understood using linear response theory. This is equivalent to
extracting backaction effects from the quantum noise spectrumof the optical force operator F̂opt , evaluated to leading order ing [25,29,30]. We further assume that g is so weak that onen̄eff = SF F [−ωm ].

−εm± J 1 ± 2J
∓2J + i κ2
ω + δ − J − 2J˜ + i κ̄

,

(14)

2

where

063809-3

G = g|â+ |,
J˜ =

J2 −

∗
â− 
εm=,2J
â+ κ 2, J = J − J˜.
2(15)

YARIV YANAY, JACK C. SANKEY, AND AASHISH A. CLERKPHYSICAL REVIEW A 93, 063809 (2016)

Here, G is the many-photon optomechanical coupling, εm isthe ratio of the average amplitudes of the optical eigenmodes,and J is the correction to the eigenmode splitting frequency
due to dissipation.
The form of Ai [ω] has a simple interpretation: the vacuum
noise entering each port of the cavity can contribute to the force
noise in two ways, either by exciting the symmetric mode [thefirst term in the brackets of Eq. (14)], or the anti-symmetricmode [the second term in Eq. (14)]. For small frequencies,
the first process is near-resonant, while the second far fromresonant. From Eqs. (14) and (13), we see that interference
between these two “paths” will in general be important todetermining the final value of the noise spectrum. In particular,
we have the possibility of using destructive interference tostrongly suppress the noise at a given frequency. The fact thatinterference could play a role in the backaction noise in thissystem when κR = 0 was briefly mentioned by Miao et al. [17].
The interference here is also reminiscent of the backaction
cancellation approach used by Caniard et al. [31], where two
mechanical modes responded to the same fluctuating radiationpressure force.
B. Large- J “adiabatic” limit

We now specialize to the usual situation where thenormal-mode splitting J is large, taking |ω|,|δ|,κ̄
2J .
The amplitudes determining the force noise then simplify, toleading order in 1/J , toi
AL/R [ω] = √
√

2

G√
κL/R
2J

√
κ α in − κ α in−δ + i κ̄2
ω + δ + i κ̄2

∗

∓1 ,

(16)

where = √κLL αLin +√κRR αRin .
L
R
In this regime, the nonresonant term in each amplitude
reduces to a frequency-independent constant. The resulting
interference implies that the quantum noise spectral density
as a function of ω [as given by Eq. (13)] is the sum of two
Fano line shapes [32,33]. In general, Fano line shapes can
exhibit perfect destructive interference with a vanishing net
amplitude. In our case, this will not be possible for both AL
and AR simultaneously, because of the sign difference in thesecond term in Eq. (16). Hence, at best interference can beused to cancel the noise coming from one port, at one particularfrequency.
The above destructive interference becomes even more
explicit when one looks at the spectral density, which to leading
order in 1/J is
SF F [ω] =

G2
κ̄
4J 2

 κ
 (ω + 2δ) + δ + i κ̄κL κ R
κ̄
2
×
+


κ̄ 2
ω + δ + i κ̄ 2

2
− κ
)
κ̄

.

2

(17)
By setting the drive detuning δ (as well as the relative amplitudeof the drives applied to each port) one can suppress the second
term at a particular frequency, minimizing the noise at thisfrequency. This is a direct manifestation of the destructive
interference discussed above. In particular, for backaction
cooling applications, one could chose to minimize the noise

1.2
1.0
0.8
0.6
0.4
102

0.2

1
0.0

−2

−1

0

1

2

FIG. 2. The spectral noise density SF F [ω] versus frequency ω,
for a two-mode optomechanical cavity driven from the left at the +mode resonance (i.e., δ = 0). We set J = 10κ̄ to be in the adiabatic
regime; curves correspond to different values of κR . At ω = 0, thenoise density is suppressed as κR /κ̄. For ω  κ̄, the noise density
returns to a near-constant value. Inset: SF F [ω], but now plotted over a
wider range of frequencies. One clearly sees two resonances (one near
ω = 0, one near ω ∼ 2J ), corresponding to the two optical normal
modes. At this scale all three curves overlap.

at ω = −ωm , as this minimizes n̄eff [cf. Eq. (11)]. By using adrive detuning δ = ωm /2 and assuming that we only drive thecavity from the L port (i.e., αRin = 0), we find, to leading orderin 1/J ,
G2
κR .
(18)
2J 2
Equation (18) implies that the “heating” backaction noise
vanishes completely in the limit of a one port cavity (i.e.,κR → 0), so that the cavity backaction acts like a zerotemperature reservoir for the mechanics. This opens the door
to ground-state cooling of mechanical resonators that are notin the good cavity limit, something that cannot be done ina standard, coherently driven, single-cavity optomechanicalsystem. We stress that in the large-J limit, this quantum noise
cancellation and cooling can be completely understood interms of the effective dissipative optomechanical coupling in
our system (as introduced in Sec. II C).
Shown in Fig. 2 are representative plots of SF F [ω] which
illustrate the noise cancellation effect. The width in frequencyof the noise suppression is ω ∼ κ̄. This implies that in theextreme bad cavity limit ωmκ̄, one can use this interference
to suppress all linear backaction effects, i.e., both heating andcooling. In contrast, in the good cavity limit ωm  κ̄, it is
not possible to use interference to suppress both SF F [ωm ] andSF F [−ωm ]. As a result, linear backaction effects persist evenin a pure single-port system where κR = 0. As we discuss
below, this means QND measurement of the phonon number
in a one-port cavity of this type is subject to the same toughconstraints on the single-photon optomechanical coupling g
as in a two-port cavity.
SF F [−ωM ] =C. Noise interference away from the adiabatic, large- J limit

While the noise spectrum is easiest to understand in thelarge-J limit, we find that most of the features described

063809-4

QUANTUM BACKACTION AND NOISE INTERFERENCE IN . . .

3

âR  = J J−δ âL . As long as there is no loss through the R
port, the amplitude in the right cavity can grow arbitrarily
large, allowing for perfect destructive interference. The same
intuition holds as long as κR remains small compared withωm and J ; see Sec. V for a more quantitative analysis of thisrestriction.
For κR > 0, the force noise cannot be made to vanish
at any frequency, but the effective thermal occupancy n̄eff
associated with the backaction can be minimized by choosing
an appropriate drive detuning δ. In general, to achieve a smalln̄eff , one would like to both minimize SF F [−ωm ] (i.e., the“heating” noise), while simultaneously maximizing SF F [ωm ](the “cooling” noise). The resonant structure of the noise
spectrum has its minimum and maximum near its two poles,
which are roughly 2J apart [see Eq. (14) and the inset of
Fig. 2]. This means that n̄eff is minimized for J ∼ ωm .

10

IV. CONSEQUENCES FOR QND PHONONMEASUREMENT

50

10

1
−3

−2

−1

0.5

1

0

1

2

PHYSICAL REVIEW A 93, 063809 (2016)

0.5
0.4
0.3
0.2
0.1
0.0

5

FIG. 3. Behavior of the backaction noise spectral density SF F [ω]away from the adiabatic, large-J limit. All results correspond to
its optimal value forκR = κL /20 and having set the drive detuning δ to

minimizing SF F [−ωm ]: δ = δcold = ω2m + J − J 2 + ( ω2m )2 . Top:SF F [ω] versus ω, for different values of J . All curves correspondto ωm = κ4L . We observe the crossover behavior from a Fano line
shape to a single peak as J is reduced. Bottom: The effective thermal
occupation n̄eff associated with the cavity backaction as a functionof the splitting J of the optical normal modes. It is minimized at thecrossover regime J ∼ ωm .

above appear for any value of the mode splitting. Away fromthe large-J limit, SF F [ω] does not exhibit a simple Fano
resonance, but reflects the interference of the two resonant
amplitudes written in Eq. (14). As shown in Fig. 3, one finds
a qualitative crossover in the form of the spectrum as J is
reduced below ∼κ (i.e., when the cavity normal modes are nolonger resolved).
Despite the changes in the shape of SF F [ω], we find thata perfect destructive interference of the noise is possible inthe single-port limit for any value of J . We stress that, forsmall J , one is not in the adiabatic limit, and the system is not
equivalent to having a dissipative optomechanical coupling.
For κR = 0, the noise spectral density is

A key motivation for the study of membrane-in-the-middle
style optomechanical setups is the possibility of QND measurement of mechanical energy eigenstates and the possibility to observe “quantum jumps” in the mechanical energy[6–8,13]. As sketched in Sec. II C (and derived more rigorously
in Refs. [6,7]), in the large-J limit one can adiabaticallyeliminate the off-resonant mode to obtain an effective optomechanical coupling:Ĥquad = −g2 †2â â+ (b̂ + b̂† ) .
2J +If one further assumes the good cavity limit, the b̂b̂ andb̂† b̂† terms have negligible influence, leaving only the desired
coupling: the frequency of the cavity + mode is controlled
by the number of phonons in the mechanical resonator. By
driving the + optical mode resonantly (i.e., δ = 0) and making
a homodyne measurement of the optical phase quadrature, one
can thus monitor the mechanical phonon number.
A. Measurement and backaction time scales

Because of the intrinsic noise in the measured homodyne
current, it will take a finite amount of time to resolve themechanical phonon number. A standard argument [17] showsthat the time needed to resolve the mechanical energy to better
than one quanta is

SF F [ω]2J (ω + 2δ) − δ(ω + δ)
2G2 κL =κLκL  .
2
(2J − δ) 2J ω + δ + i 4 − (ω + δ) ω + δ + i 2(19)

It can always be made tovanish at SF F [−ωm ] by setting thedetuning δ = ω2m + J − J 2 + ( ω2m )2 ≡ δcold . It might seem
surprising that the interference persists even for very smallvalues of J , as one would expect to recover the physics
of a standard, single-cavity optomechanical system. This is
not the case. On the dissipation-free side of the cavity, theaverage amplitude is inversely proportional J : for κR = 0,(20)τmeas ∼

J 2 κL.
G2 g 2(21)

As discussed previously [6,7,17], a successful QND measurement requires that linear backaction effects do not cause atransition of the mechanical state before the measurement canresolve it. For a situation where the mechanical resonator isprepared near its ground state, a minimal requirement is thatthe measurement time τmeas be shorter than the lifetime of boththe mechanical ground state and the n = 1 Fock state due tobackaction. This is illustrated in Fig. 4.
Fermi’s Golden rule lets us directly relate the lifetime of
the nth mechanical Fock state to the backaction quantum noise

063809-5

YARIV YANAY, JACK C. SANKEY, AND AASHISH A. CLERKPHYSICAL REVIEW A 93, 063809 (2016)

optomechanical coupling rate must be larger than the cavity
damping rate. This is in agreement with previous work [17].
For an asymmetric cavity, we must consider separately the
resolved and unresolved sideband limits.
B. Resolved sideband limit
FIG. 4. Illustration of a QND measurement of the phonon
number. Plotted as a function of time, the gray lines indicate thebehavior of the actual mechanical phonon number, while the dashedblue lines indicate the output signal, averaged over τmeas . At times
t = 4τmeas and 8τmeas the mechanical system experiences an upward
quantum jump due to thermal noise unrelated to the cavity. In the−1= 0), or when the the typical backaction
absence of backaction (τBA,1
is longer than the measurement time (τBA,0 = 2τmeas ), these jumps
and the discreteness of the phonon numbers can be observed. Whenthe backaction dominates (τBA,1 = τmeas /2) the mechanical oscillator
appears to remain in the ground state.

spectrum,−1
τBA,n≡ (1 + n)SFF [−ωm ] + nSFF [+ωm ].

(22)

For driving through the L port at δ = 0 we find
G2
−1
= 2
τBA,n
J

 
2
κ̄ 2
ω + κR κ̄2
1
2 m
+O
κ̄ 2
2
J
ωm + 2



1
n+
.
2

(23)

As the mean backaction time decreases with n, the requirement
for effective QND measurements is τmeas /τBA,1 < 1. This ratio
is plotted, for a one-port cavity, in Fig. 5.
For a completely symmetric two-port cavity (κL = κR = κ̄),
2
−1
Eq. (23) yields τBA,1∼Gκ̄ (with a prefactor ranging from 12 toJ21). Combined with Eq. (21), the requirement τBA,1 > τmeas for
QND measurement then reduces to g  κ̄: the single-photon

C. Unresolved sideband limit and possibility of x 2 measurement

In the ωmκ̄ limit, we have already shown that noiseinterference can be used to completely cancel linear backaction
effects (see Sec. III B). Thus, not surprisingly, in this limit wefind that τBA,1 diverges as κR → 0. In the perfect one-port case,κR = 0, there is no linear backaction at all.
In this limit Eq. (20) does not allow the cavity to measure
the phonon number. Instead, the cavity will measure x̂ 2 ofthe mechanics. While such a measurement does not allow
one to detect quantum jumps in mechanical energy, its
nonlinear nature can allow the conditional generation ofhighly nonclassical mechanical states, i.e., states which exhibit
negativity in their Wigner functions [20,21]. We will explore
this physics in detail in a later work.

0.4

0.3

0.2

0.1

0.0
0.01

To see the discreteness of the mechanical energy, one
needs to be in the resolved sideband (i.e., good cavity) limit,ωm  κ̄. In this regime x̂ 2 is approximately proportional tothe phonon number operator, as discussed following Eq. (20).
−1
In the resolved sideband limit, Eq. (23) indicates τBA,1
scalesG2
as J 2 κ̄, i.e., in the same way as in the symmetric two-portcase. The requirement that τBA,1 > τmeas thus reduces again torequiring g  κ̄ for QND measurement. This means that thescale of the optomechanical coupling must be larger than both
κL and κR (and not just their product). A similar conclusion
holds in general for a single-sided cavity having internal loss:
g must be larger than the coupling κ, not just the internal-lossκ (see Appendix B).
The above conclusion differs from previous works, whichsuggested that in a perfect one-port cavity, linear backaction
effects do not present any limit to performing QND measurement. While it is true that in a single-port cavity, one
can perfectly cancel the backaction noise at ω = −ωm via
interference [cf. Eq. (17)], the noise at positive frequency
ω = +ωm remains. This noise will kill the lifetime of then = 1 Fock state, making it impossible to resolve a quantum
jump (see Fig. 4).
Finally, we note that if one manages to achieve a very
different situation than that described here, where each opticalnormal mode couples independently to a separate dissipative
reservoir, then one recovers the result of Ref. [17]: thecondition τBA,1 > τmeas reduces to g 2  κ+ κ− .

0.05

0.10

0.50

1

5

V. BACKACTION COOLING
FIG. 5. The relative ratio between typical times for quadratic
measurement and for linear backaction, τmeas /τBA,1 [see Eqs. (21)–(23)]. As discussed in the text, to perform QND measurements, thisratio must be less than 1. Shown here for a one-port cavity, κR = 0,
κL , linearwith J = 10κL and δ = 0, driving from the left. At ωmnoise is strongly suppressed, while in the good cavity limit its value
is similar to the two-port case.

In this final section, we return to quantum noise interference
that is possible in our system and discuss further the possibilities for mechanical cooling. In the large-J , adiabatic limit,
and for κR = 0, the fact that our system allows ground-statecooling of a mechanical resonator in the bad cavity limit is not
surprising, as it directly realizes the dissipative κ(x) coupling
discussed in Ref. [18] (see Sec. II C). More surprising is the063809-6

QUANTUM BACKACTION AND NOISE INTERFERENCE IN . . .

PHYSICAL REVIEW A 93, 063809 (2016)

fact that ground-state cooling is possible even away fromthe large-J limit, where the system is not identical to thedissipative-coupling optomechanical system. As we have seenin Eq. (19) and the surrounding discussion, an effective zerotemperature can be achieved for any J given κR = 0.
Driving through the L port, setting δ = δcold [see Eq. (19)and the following discussion], we find, to leading order in κR ,Note added. Recently, we became aware of a preprint by
Khalili et al. [36] which also discusses connections between
two-mode optomechanical systems and effective dissipative
optomechanical couplings.

δ2
2G2
,
= 2 2 coldJ ωm (n̄eff /κR )

2
 2
ωm
9
5 ωm κRn̄eff =
1+
−
4
2J
3 2J
κL

+


1+

ωm
2J

2

ωm
−3
2J

2

κL κR
.
2
16ωm

(24)

(25)

The system can be effectively cooled as long as the internal
2ωm. This range of
damping is small enough that κL κR
parameters is experimentally realistic [28]. We stress that thissmall κR expansion does not assume anything about the valueof J . We also remind the reader that this result is derived within
the perturbative, quantum-noise approach. For strong drives,
where ∼ κ̄, the broadening of the mechanical resonance byleads to an additional nonzero term in n̄eff [34,35].
Finally, note that in the good cavity limit, ωm  κL , more
effective cooling is possible regardless of κR by setting thedetuning to δ ∼ −ωm , using the same physics seen in a singlemode cavity [30].
VI. CONCLUSIONS AND OUTLOOKWe have presented a thorough analysis of the residual linear
backaction noise in an asymmetric two-mode optomechanicalsystem having the form of the canonical “membrane-in-themiddle” system. Our analysis shows that in the adiabatic,large-J limit, the system has an effective dissipative optomechanical coupling, with a corresponding Fano interferencein its quantum backaction noise. Our analysis also showsthat this interference (and potential for perfect cancellation)
persists even to the nonadiabatic regime for arbitrary J .
We demonstrated that in a perfect one-port device in theunresolved sideband regime, all linear backaction effects could
be suppressed, allowing an ideal continuous measurement ofx 2 . In contrast, if one works in the good cavity limit andattempts to measure quantum jumps in the mechanical phonon
number, linear backaction cannot be completely suppressed
even in a single-port system, and one requires g  κ̄ for themeasurement to be stronger than the unwanted backaction.
We have focused here primarily on a literal membrane in the
middle style cavity, but our results hold, qualitatively, for otheroptomechanical setups where two optical modes are coupled
to the mechanics (see Appendix B). The only exception isa system where the two relevant optical normal modes see
completely independent dissipation; such systems may be themost promising avenue for the observation of mechanicalquantum jumps. An interesting future direction would be tounderstand how quantum noise interference effects of the sortdiscussed here manifest themselves in even more complex
multimode optomechanical systems.

ACKNOWLEDGMENT

This work was supported by NSERC.
APPENDIX A: CONNECTION TO EFFECTIVE LINEAR
MEASUREMENT

We show briefly here that the noise amplitudes Ai [ω]introduced in Eq. (12) control how well one could make ameasurement of linear position from the output light leaving
the cavity. From standard input-output theory, the output field
for port i = L and R is given by [25]√(A1)âiout = âiin − κi âi .
Using our Heisenberg-Langevin equations, one straightforwardly finds that the fluctuating part of the output field is givenby
Bij [ω]ξ̂j [ω] − iAi [ω]x̂[ω].
(A2)ξ̂iout [ω] =j =L,R

Here, the Bij [ω] are complex functions of frequency which are
independent of the optomechanical coupling g; they determine
the output fluctuations in the absence of any coupling. The
second term describes how the output fields depend linearlyon x̂; note that the Heisenberg operator x̂ here includes theeffects of backaction. We thus see that the linear response
kernel linking the output field to x̂[ω] is identical to thenoise amplitudes Ai [ω]. This result also follows from standard
quantum linear response theory (i.e., the Kubo formula), whichsays that the relevant linear response kernel is given by thecommutator of ξiout (t) and the backaction force operator F̂ (t)(see, e.g., Ref. [25]).
It is illustrative to examine this output in the time domain
where we can write tξ̂L (t) =G(t − t  )x̂(t  )dt (A3)−∞for ξ̂L (t), the portion of the output due to coupling to themechanics. The response function is given by, to first orderin 1/J ,
G κL κ̄ − κ̄ τ
e 2 − δ(τ − η) .
(A4)G(τ ) =2J 2 2having taken = 1 and δ = 0. Here, the first terms result fromthe resonant piece of AL [ω], while the second term comes fromthe off-resonant piece; η is a positive infinitesimal.
We can see in Eq. (A4) the two regimes described inSec. IV, caused by the different response rates of the twocavity modes. The response of the rapidly oscillating − modeis near immediate. In the bad cavity regime, when ωmκ̄, theresponse of the + mode is faster than the rate of change in x̂,leading to a perfect destructive interference. In the good cavity
regime, ωm  κ̄, the slow response of the + mode meansx̂ information is averaged out. However, this still leaves the063809-7

YARIV YANAY, JACK C. SANKEY, AND AASHISH A. CLERK

information leaking through the − mode, which is no longer
canceled.
APPENDIX B: GENERIC SETUP

For completeness, we now discuss the generic case, where
the cavity is coupled to its environment by some general set of
dissipation channels. The damping Hamiltonian of any system
coupled to a single driven port and any number of additional++
κ+ = κdr+ κint,PHYSICAL REVIEW A 93, 063809 (2016)

internal loss channels can be written in the form+−
ˆ †dr κdrĤdamp = −i â+ + κdr
â−


+
−
ˆ †int κintâ+ − κintâ− + H.c.,
− i

(B1)

ˆ dr and int are two independent linear combinations
where of the various dissipation modes, defined so that there is noˆ int . We define here
driving through 

−
+
−
−
+
−
κ− = κdr+ κint, κdr = κdr+ κdr,κint = κint
+ κint
,


+ −
+ −
κ = κdrκdr − κint
κint .

κ̄ =κdr + κint
,
2

δκ =

κ+ − κ−,2(B2)

At g = 0, the equations of motion are



κ−
κ
− in
−
â˙ − = −+ i(2J − δ) â− −â+ + κdr(α + ξ̂dr ) − κintξ̂int ,
2
2




+
˙â+ = − κ+ − iδ â+ − κ â− + κ + (α in + ξ̂dr ) + κintξ̂int ,
dr
2
2

(B3)

and we find∗/td
δ + i κ2+ − i κεmâ− 
2
td
=
=
κ−
2J
â+ 
−2J + δ + i 2 − i κ
t
2 d

(B4)

and
Adr/int [ω] =εmεm
td/i ∓ J td/i ± 2J
+ i κ
1 ± 2Jεm 1 + iδκ
iG  +
2J2
κdr/int ×
ω + δ − J + i κ2+2J˜−
where td/i =−+κdr/int/κdr/int
and here
J˜ =

εm
εm
td/i ∓ J td/i ± 2J
∓2J td/i 1 + iδκ+ i κ
1 ± 2J2J2κω + δ − J − 2J˜ + i −

,

(B5)

2





δκ 2
κ 2
J +i
−
2
2

J = J + i

δκ
− J˜.
2

For a small internal loss, the noise spectral density becomes, to leading order in 1/J ,






td
td 2
κ+
1
G2
−
2
−
SFF =
1+
+i
.

 κ |ω + 2δ| + κint ω + δ 1 −4J 2 ω + δ + i κ+ 2 dr
ti
2
ti 

(B6)

(B7)

2

We see that, while the factors vary, the structure of the force noise and the noise density spectrum are similar to those discussed
in the main text and seen in Eqs. (14) and (17).
2 −In particular, for the purpose of QND measurement, a single-port system in the good cavity limit has τBA,1 ∼ G
κ , leadingJ 2 dr−. Thus, QND measurements are possible when the off-resonant channel is not coupled to the driving
to the requirement g  κdrport.

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