Optomechanics and quantum
measurement
Aashish A. Clerk
Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8

1

Contents
1

Introduction

1

2

Basic quantum cavity optomechanics theory
2.1 Optomechanical Hamiltonian
2.2 Dissipation and noise
2.3 Driving and output field
2.4 Displacement transformation
2.5 Linearized regime of optomechanics

2
2
3
5
6
7

3

Quantum limit on continuous position detection
3.1 General problem: minimizing total detector added noise
3.2 Quantum noise spectral densities: some essential features
3.3 Heisenberg inequality on detector quantum noise
3.4 Power gain and the large gain limit
3.5 Defining the quantum limit
3.6 Derivation of the quantum limit
3.7 Simple limits and discussion
3.8 Applications to an optomechanical cavity (with and without input squeezing)

10
10
12
16
20
21
24
26

Backaction evasion and conditional squeezing
4.1 Single quadrature measurements
4.2 Two-tone QND scheme
4.3 Conditional squeezing: heuristic description
4.4 Stochastic master equation description of a conditional measurement
4.5 Conditional back-action evading measurement
4.6 Feedback to create unconditional squeezing
4.7 Extensions of back-action evasion techniques

33
33
34
38

4

Appendix A
References

Derivation of power gain expression

28

39
44
46
47
48
51

1
Introduction
These notes summarize lectures given at the 2015 Les Houches School on Optomechanics. The first part of the notes give a quick review of the basic theory of quantum
optomechanical systems, based largely on linearized Heisenberg-Langevin equations.
The notes then focus on selected topics relating to quantum measurement and quantum optomechanics. Chapter 3 gives comprehensive discussion of the quantum limiton the added noise of a continuous position detector, following the quantum linear response approach. While much of this discussion can already be found in (Clerk, 2004;
Clerk et al., 2010), I provide a greater discussion here of the role of noise correlations,and how these can be achieved in an optomechanical cavity (by using squeezed input
light, or by modifying the choice of measured output quadrature). Chapter 4 turns
to a discussion of back-action evading measurements of a mechanical quadrature, discussing how this can be achieved in a two-tone driven cavity system. I also provide
a quick introduction to the theory of conditional continuous quantum measurement,
and use it to discuss how a back-action evading measurement can be used to produce
conditional mechanical squeezed states.

2
Basic quantum cavity optomechanics
theoryThis chapter will present a “quick and dirty” introduction to the basic theoreticallanguage used to describe quantum optomechanical systems. More complete introductions to some of the topics covered here can be found in (Clerk et al., 2010; Aspelmeyer
et al., 2014; Clerk and Marquardt, 2014)

2.1

Optomechanical Hamiltonian

We start by considering a standard optomechanical system, consisting of a single modeof a resonant electromagnetic cavity whose frequency ωcav depends on the position xof a mechanical resonator. Both the mechanical mode and cavity mode are harmonic
oscillators, and the Hamiltonian takes the form (~ = 1):
ĤOM = ωcav [x̂]â† â + ωM b̂† b̂

(2.1)

where â is the annihilation operator for the cavity mode, b̂ is the annihilation operator
for the mechanical mode. Finally, as we are typically interested in small mechanicaldisplacements, we can Taylor expand the dependence of ωcav on x keeping just the firstterm. Writing the mechanical position in terms of creation and destruction operators,
the optomechanical Hamiltonian takes the form:ĤOM = ωcav â† â + ωM b̂† b̂ + gâ† â b̂ + b̂† ≡ Ĥ0 + Ĥint
(2.2)
where
dωcav
dωcav
xZPF =
g=
dx
dx

r

~
.
2mωM

(2.3)

It is worth noting a few basic features of this Hamiltonian:• The position x of the mechanical resonator sets the cavity frequency; thus, if you
drive the cavity with a monochromatic laser, the average cavity photon numberhâ† âi will also depend on the mechanical position.
• Anything that multiplies x̂ in a Hamiltonian acts as a force on the mechanicalresonator. Hence, the cavity photon number n̂ = â† â is a force on the mechanicalresonator (i.e. the radiation pressure force).
• Without loss of generality, we have defined the optomechanical interaction witha plus sign in this chapter. Hence, a positive displacement x > 0 of the mechanics

Dissipation and noise3

results in an increased cavity frequency. We also use g to denote the single-photon
optomechanical coupling strength in this chapter (the same quantity which isdenoted g0 in other works, e.g. (Aspelmeyer et al., 2014)).
A rigorous derivation of this Hamiltonian (for the specific case of a Fabry-Perot
resonator with a moveable end mirror) was given in (Law, 1995). This derivation keeps
all the resonant modes of the optical cavity, and shows how in principle one also obtains interaction terms where the mechanical resonator can mediate scattering between
different optical modes, and also terms corresponding to the dynamical Casamir effect,
where, e.g. , destruction of a phonon can result in the creation of a pair of photons.
Such additional terms are of negligible importance in the standard situation where themechanical frequency ωM is much smaller than all optical frequency scales.
Returning to the basic Hamiltonian ĤOM above, one sees immediately that thecavity photon number n̂ commutes with Ĥ and is thus a conserved quantity. Thus,in the absence of any driving or coupling to dissipation, ĤOM can easily be exactly
diagonalized. For a fixed photon number, the optomechanical interaction corresponds
to a static force on the mechanics, which simply shifts its equilibrium position an
amount ∆xn = −2(g/ωM )xZPF . The eigenstates are just tensor products of statesof fixed photon number with displaced harmonic oscillator eigenstates. This is conveniently described by making a polaron transformation, i.e. a n̂-dependent displacementtransformation of the mechanical resonator:g n̂ b̂ − b̂†(2.4)
Û = exp (−ip̂∆xn ) = exp
ωM
The transformed Hamiltonian takes the form:
g2
g2 † ††
Û ĤOM Û = ωcav −â† â + ωM b̂† b̂ −â â ââ
ωM
ωm

(2.5)

In this new frame, we see explicitly from the last term that the eigenenergies of the
Hamiltonian have a nonlinear dependence on photon number: the mechanical resonator mediates an optical Kerr-type nonlinearity (or equivalently, photon-photon
interaction). In the simplest picture, we simply combine the two features noted above:as ωcav depends on x, and x depends on n̂ (as it is a force), the cavity frequency will
depend on n̂. This yields the n̂2 term above. In a more quantum picture, note thatto leading order, the interaction Ĥint creates or destroys a mechanical excitation (aphonon) with a matrix element proportional to n̂. While such a process would notconserve energy, to second order we could have an energy conserving process that involves a virtual state with a mechanical excitation. The amplitude of such a process
would be proportional to n̂2 and inversely proportional to ωM (i.e. the energy cost of
the virtual state). This is yet another way to understand the last term in Eq. (2.5).

2.2

Dissipation and noise

Next, we need to include the coupling of both the cavity and the mechanical resonatorto their respective dissipative environments, as well any driving terms (e.g. a coherent
laser drive of the cavity). We follows the standard input-output theory route to treat

4

Basic quantum cavity optomechanics theory

these effects (see, e.g. (Walls and Milburn, 2008; Clerk et al., 2010)), where the cavity
mode (mechanical mode) acquires an energy damping rate κ (γ). To obtain theseeffects, one needs to include terms in the Hamiltonian describing the dissipative baths
and their coupling to the system:
Ĥ = ĤOM + Ĥκ + Ĥγ(2.6)

Consider first the cavity dissipation, described by Ĥκ . This just describes a linear coupling between the cavity mode and extra-cavity photon modes, which are themselves
just free bosons (lowering operators b̂q ):rXκ X ††â b̂q − h.c.
(2.7)
Ĥκ =
ωq b̂q b̂q − i2πρ qqAs is standard, we approximate the the bath density of states to be a frequencyindependent constant (which is typically an excellent approximation, as we are onlyinterested in a small range of bath frequencies centred around the cavity frequency
ωcav ):X
δ(ω − ωq ) = ρ(2.8)
qAs described in, e.g. (Clerk et al., 2010), one can now derive the effective HeisenbergLangevin equation of motion for the cavity field. This involves first solving the Heisenberg equation of motion for the bath operators b̂q , and then substituting these into the
Heisenberg equation of motion for the cavity mode lowering operator â. One obtains:
h
i κ
√
d
â = −i â, ĤOM − â − κâin
dt
2

(2.9)

The second term on the RHS describes the a simple linear damping of the cavity(resulting from photons leaking from the cavity to the bath), while the third term
describes a driving of the cavity mode by noise emanating from the bath. Note that†âin has been
√ normalized so that âin âin represents a photon number flux (i.e. âin has
units of 1/ time).
Taking the bath to be in a thermal equilibrium state, one finds that the operatorvalued input noise âin (t) is Gaussian (i.e. fully characterized by two-point correlation
functions). Further, for the typically small frequency scales of interest, it can also beapproximated as being white noise. One obtains:DEâin (t)â†in (t0 ) = δ(t − t0 ) (1 + n̄cth )(2.10)
EDâ†in (t)âin (t0 ) = δ(t − t0 )n̄cth(2.11)
Here, n̄cav
th is a Bose-Einstein occupancy factor evaluated at the cavity frequency andat the bath temperature. Note that the input noise satisfies the canonical commutation
relationhiâin (t), â†in (t0 ) = δ(t − t0 )(2.12)

Driving and output field5

One can treat the effects of mechanical dissipation in a completely analogous manner.
The Heisenberg-Langevin equation of motion for the mechanical lowering operator
takes the form
h
i γ
√
d
b̂ = −i b̂, ĤOM − b̂ − κb̂in(2.13)
dt
2
The mechanical input noise has correlation functions analogous to those in Eqs. (2.10),(2.11),
M
except that n̄cav
th is replaced by n̄th , a Bose-Einstein factor evaluated at the mechanicalfrequency and mechanical bath temperature.
Finally, we note that the approximations we have made in treating the bath(i.e. taking it to have a constant density of states, and taking its noise correlation
functions to be delta-correlated) correspond to treating it as a Markovian bath (a vanishing correlation time, absence of memory effects). The resulting Heisenberg-Langevinequations are completely local in time.

2.3

Driving and output field

To include a coherent driving of the cavity (e.g. by a laser), we simply allow the input
field âin to have an average value āin (t). One can easily confirm that this is completely
equivalent to having added an explicit linear driving termĤdrive = −i āin (t)â† − h.c.
(2.14)
to the Hamiltonian. In many cases, the cavity bath corresponds to modes in the waveguide (or transmission line) used to drive and measure the cavity. In this case, we arealso interested in knowing what the field emitted by the cavity into the waveguide
is. The amplitude of this outgoing field is described by the operator âout (t). In thesimple Markovian limit we are focusing on, this field is completely determined by the
input-output relation:√(2.15)âout (t) = âin (t) + κâ(t)where the intra-cavity field â is determined by the Heisenberg-Langevin equation
Eq. (2.9). Note that â(t) is driven by the input field âin (t), and hence the two terms on
the RHS here are not independent. When κ = 0 (i.e. no coupling between the wavegudie and the cavity), the input-output equation just describes a perfect reflection ofwaves off the end of the waveguide. For non-zero κ, the first term describes incident
waves that are immediately reflected from the cavity-waveguide boundary, whereas
the second term describes wave emitted from the waveguide.
Consider the simple case where we have a monochromatic coherent driving of
the cavity: āin (t) = αin e−iωL t . It is convenient to work in a rotating frame where
this driving looks time-independent. This is achieved by using the unitary Û (t) =exp iωL â† ât to transform to a new frame. In this new frame, the Hamiltonian isgiven by 1 :1 One also needs in principle to shift the frequency of the bath oscillators b̂ to make sure that inq
the new frame, the cavity-bath coupling remains time-independent. While this is normally innocuous,
it does imply the presence of negative-frequency bath modes in the new frame. The implications of
this for a driven optomechanical system are discussed in detail in (Lemonde and Clerk, 2015).

6

Basic quantum cavity optomechanics theoryĤ 0 = Û (t)Ĥ Û † (t) + idÛ (t) Û † (t)dt(2.16)and the Heisenberg Langevin equation takes the form:
h
i κ
√
√
d
â = −i â, −∆â† â + ωM b̂† b̂ + Ĥint − â − καin − κâindt
2(2.17)

where the drive detuning ∆ ≡ ωL − ωcav , and we have explicitly separated out the
average value of âin . Note that the RHS has no explicit time dependence.
In the case where there is no optomechanical coupling, one can easily solve theabove equation for the stationary value of â; one finds:
√
καin
≡α
(2.18)
hâi = − κ
2 − i∆
It thus follows from Eq. (2.15) that the average output field is given by:κ+ i∆
αin ≡ eiθ αin−i∆2hâout i = − κ2

(2.19)We thus recover the expected expression for the reflection phase θ.

2.4

Displacement transformation

Let’s now include the optomechanical interaction in Eq. (2.17). We expect again thatthe cavity drive will induce an average value for the cavity field â, and will also induce
an average value for the mechanical lowering operator b̂ (as the average photon numberof the cavity is a static force on the mechanical resonator, which will displace its
equilibrium position). It is useful to make displacement transformations to separate out
these classical mean values from the additional dynamics that arises due to the noise
operator âin and b̂in (operators which encode both classical and quantum noise driving
the system). We thus introduce displaced cavity and mechanical lowering operators dˆ
and b̂new defined via:â = ācl + dˆb̂ = b̄cl + b̂new(2.20)
where ācl , b̄cl are the classical average values for the cavity and mechanical mode operators. These are found by solving the classical, noise-free version of the HeisenbergLangevin equations Eq. (2.9) and (2.13) (i.e. one replaces â → ācl , b̂ → b̄cl in theequations and drops all noise terms). These classical equations are nonlinear, andregimes exist where multiple classical solutions can be found (the well-known optomechanical bistability, see Sec. V.A of (Aspelmeyer et al., 2014) for a discussion). We
assume however that we work in a regime where there is a unique solution to theclassical equations. As dˆ and b̂new encode all quantum effects in our system, it is common to refer to them as the quantum parts of the cavity and mechanical annihilation
operators. We stress that they are standard canonical bosonic annihilation operators.
Once the classical amplitudes have been found, one returns to the full HeisenbergLangevin equations, expressed now in terms of the operators dˆ and b̂new . One finds thatLinearized regime of optomechanics

7

there are no purely constant terms on the RHS of these equations (i.e. linear driving
terms). In particular, the coherent cavity drive αin enters only through the classicaldisplacements ācl , b̄cl . The resulting equations are equivalent to having started with acoherent Hamiltonian
ĤOM = −∆0 dˆ† dˆ + ωM b̂†new b̂new + Ĥint
Ĥint = g ā∗cl dˆ + ācl dˆ† + dˆ† dˆ b̂†new + b̂new(2.21)(2.22)

where the modified detuning ∆0 = ∆ − g(b̄cl + b̄∗cl ). We see than in this new displacedframe, the mechanical mode only interacts with the fluctuating parts of the cavityˆ dˆ† ). Note that the static, “classical” part ofphoton number n̂ (i.e. terms involving d,
the photon number |ācl |2 determines the classical mechanical displacement b̂cl , butdoes not appear explicitly in ĤOM . In what follows, we drop the subscript ”new” onb̂new to keep things clear, and also replace ∆0 by ∆.

2.5

Linearized regime of optomechanics

By strongly driving the cavity (large αin ), ācl increases in magnitude. This in turnincreases the fluctuations in the intracavity photon number, namely the term that islinear in the dˆ operators. Correspondingly (as per Eq. (2.22)), the large drive enhances
the quadratic terms in the optomechanical interaction Hamiltonian. It is thus common to introduce a drive enhanced many-photon optomechanical interaction strength,
defined as 2√
(2.23)G ≡ g |ācl | = g n̄cav
In almost all current experiments, the single photon strength g is too weak to directly
play a role (i.e. g  κ, ωM ), and appreciable optomechanical effects are only obtained
when the cavity is strongly driven and G made large. In this regime where G  g,it a good approximation to only retain the drive-enhanced terms in Eq. (2.22). The
physics is then described by the approximate quadratic Hamiltonian:Ĥlin = −∆dˆ† dˆ + ωM b̂† b̂ + G(b̂ + b̂† )(dˆ + dˆ† )(2.24)

with corresponding Heisenberg-Langevin equations:

 √
d ˆ
d = (i∆ − κ/2)dˆ − iG b̂ + b̂† − κdˆin
dt

 √
d
b̂ = (−iωM − γ/2)b̂ − iG dˆ + dˆ† − γ b̂indt

(2.25)(2.26)

Note that we have made a gauge transformation on the cavity field to absorb thephase of the classical cavity field ācl . In this regime, the equations of motion for thecavity and mechanical lowering operators are purely linear, and thus it is often termed
2 It is also common notation to use g (instead of g) to denote the single-photon optomechanical0
coupling strength, and use g (instead of G) to denote the many-photon coupling strength. We prefer
the notation used here, both as it avoids having to use yet another subscript, and because the use ofa capital letter more dramatically emphasizes the coupling enhancement by the drive.

8

Basic quantum cavity optomechanics theory

the linearized regime of optomechanics. The physics is just that of a system of two
linearly coupled harmonic oscillators, albeit one with a great deal of tunability, andone where the dissipative rates and effective temperatures of the two oscillators canbe very different. The effective frequency of the photonic oscillator can be tuned bychanging the frequency of the cavity drive, and the strength of the interaction G canbe tuned by changing the amplitude of the cavity drive.
To get a feel for the various interesting things that can be done with linearized
optomechanics, we quickly sketch different interesting possibilities below.
2.5.1

Beam-splitter Hamiltonian

Consider Eq. (2.27) in the regime where ∆ = −ωM , implying that the cavity drive
frequency is detuned to the red of the cavity resonance an amount exactly equal to themechanical frequency (a so-called red-sideband drive). The two effective oscillators in
Eq. (2.27) are then resonant. If we further assume that ωM  κ and that G  ωM ,the interaction terms that create or destroy a photon-phonon pair are sufficiently nonresonant to not play a large role in the dynamics. We can thus make a rotating wave
approximation, keeping only the energy conserving interaction terms. The Hamiltonian
then takes the simple form:Ĥlin = −∆dˆ† dˆ + ωM b̂† b̂ + G(b̂† dˆ + dˆ† b̂)(2.27)

This interaction converts photons to phonons and vice-versa, and is known as a beamsplitter Hamiltonian or “swap” Hamiltonian. It is at the heart of quantum statetransfer applications of optomechanics (see, e.g., (Hill et al., 2012; Dong et al., 2012;
Andrews et al., 2014)). It also is at the heart of optimal cavity-cooling schemes, where
one uses the driven cavity to cool a thermal mechanical resonator to close to the groundstate (Marquardt et al., 2007; Wilson-Rae et al., 2007). The above swap-Hamiltonian
can be used to transfer mechanical excitations to the cavity mode, where they arequickly emitted (at a rate κ) to the cavity bath.
2.5.2

Entangling Hamiltonian

Consider Eq. (2.27) for the opposite detuning choice, where ∆ = ωM : the cavity drive
frequency is now detuned to the blue of the cavity resonance an amount ωM . If we again
assume that the good-cavity condition ωM  κ is fulfilled and that G  ωM , we canagain make a rotating-wave approximation where we only retain energy conserving
terms. As now the cavity photons have an effective negative frequency, the energy
conserving terms correspond to creating or destroying pairs of excitations, and the
Hamiltonian becomes:Ĥlin = −ωM dˆ† dˆ + ωM b̂† b̂ + G(b̂† dˆ† + dˆb̂)(2.28)The dynamics of this Hamiltonian creates photons and phonons in pairs, and leads
to states with a high degree of correlations between the mechanics and light (i.e. thephoton number and phonon number are almost perfectly correlated). Such states are
known as “two-mode squeezed states”, and their correlations correspond to quantum
entanglement. Such optomechanical entanglement has recently been measured. The

Linearized regime of optomechanics

9

above Hamiltonian also corresponds to the Hamiltonian of a non-degenerate parametric amplifier. It can be used for near quantum-limited amplification. It also can exhibit
dynamical instabilities for sufficiently large values of G.

3
Quantum limit on continuous
position detection
One of the key motivations for studying optomechanics is the possibility to use lightto measure mechanical motion with a precision limited by the fundamental constraints
imposed by quantum mechanics. In some cases, one can even devise schemes that transcend the quantum constraints that limit more conventional measurement strategies.
In this chapter, we will review the most general and rigorous formulation of the quantum limit on continuous position detection for a generic linear-response detector. The
general derivation will make the origin of this quantum limit clear, as well establish aset of requirements which must be met to achieve it. We will then apply this generalformalism to the basic optomechanical cavity as introduced in the previous chapter.
The presentation here is closely related to that developed in (Clerk, 2004) and (Clerk
et al., 2010). In contrast to those works, we provide a discussion of how squeezing canbe used to generate the noise correlations needed to reach the quantum limit in thefree mass limit.

3.1

General problem: minimizing total detector added noise

To motivate things, let’s return to Eq. (2.19) for the reflection phase shift for a driven
single-sided cavity, and give a classical discussion of how our cavity could function as
a position detector. For a fixed cavity drive frequency, the reflection phase shift θ isa function of the cavity frequency ωcav . With the optomechanical coupling, the instantaneous cavity frequency becomes a function of the mechanical position. Consider
first the simple case where x is fixed to some unknown position x0 . The mechanicaldisplacement shifts the cavity frequency by an amount gx0 /xZPF . For small displacements, the change in phase will be linear in x0 . In the optimal case where ∆ = 0, we
have
4g x0
θ=π+
≡ θ0 + ∆θ
(3.1)
κ xZPF(with θ0 = π) and the output field from the cavity will have the form
aout = ei(θ0 +∆θ) ain ' eiθ0 (1 + i∆θ) ain = − (1 + i∆θ) ain(3.2)

where ain is the amplitude of the incident cavity driving field. Information on themechanical displacement x0 is encoded in the change in reflection phase ∆θ, andGeneral problem: minimizing total detector added noise

11

will thus be optimally contained in the phase quadrature of the output field. Taking(without loss of generality) ain to be real, this quadrature is defined as:
Xφ (t) = −i (aout (t) − a∗out (t))(3.3)

Using the standard technique of homodyne interferometry, this output-field quadrature
can be directly converted into a photocurrent I(t) ∝ Xφ (see e.g. (Walls and Milburn,2008; Clerk et al., 2010)). This photocurrent will have intrinsic fluctuations (due toshot noise in the light), and hence will have the form:
I(t) = λx0 + δI0 (t)(3.4)

The first term is the “signal” associated with the measurement, where λ ∝ g parameterizes the response of the photocurrent to changes in position. The second term
represents the imprecision noise in the measurement, i.e. the intrinsic fluctuations in
the photocurrent. At each instant t, δI0 (t) is a random variable with zero mean. To
determine x0 , one would need to integrate I(t) over some finite time interval to resolve
the signal above these intrinsic fluctuations. It is common to scale these intrinsic output fluctuations by the response coefficient λ, and think of this noise as an equivalentmechanical position noise δximp (t) ≡ δI0 (t)λ.
If the mechanical resonator is now in motion, and ωM is sufficiently small (i.e. muchsmaller than κ), then the cavity will be able to adiabatically follow the mechanicalmotion. Eq. (3.4) will then still hold, with the replacement x0 → x0 (t). Our goal here
will not be to be measure the instantaneous position of the mechanical resonator: weconsider the standard situation where the mechanical resonator is in motion (oscillating!), and the optomechanical coupling is far too weak to measure position in a timeshort compared to the mechanical period. Instead, our goal will be to determine thequadrature amplitudes describing the oscillating motion, i.e. the amplitudes of the sineand cosine components of the motion:x(t) = X(t) cos(Ωt) + Y (t) sin(Ωt).

(3.5)

These amplitudes typically evolve on a time scale much longer than the mechanicalperiod.
One might think that in principle, one could increase the signal-to-noise ratio
indefinitely by increasing λ (e.g. by increasing g or the power associated with the cavity
drive). From a quantum point of view, we know the situation cannot be that simple, as
the quadrature amplitudes X and Y become non-commuting, conjugate observables,
and cannot be known simultaneously with arbitrary precision. More concretely, in thequantum case, we absolutely need to think about an additional kind of measurement
noise: the disturbance of the mechanical position by the detector, otherwise known asback-action. Returning to the optomechanical Hamiltonian of Eq. (2.2), we see thatfluctuations of cavity photon number act like a noisy force on the mechanical resonator.
This will give rise to extra fluctuations in its position, δxBA (t). Eq. (3.4) thus needs
to be updated to have the form:
I(t) = λ [x0 (t) + δxBA (t)] + δI0 (t) ≡ λ [x0 (t) + δxadd (t)](3.6)

We have introduced here the total added noise of the measurement δxadd (t) = δxBA (t)+δI0 (t)/λ.

12Quantum limit on continuous position detection

The goal is now to make this total added noise as small as possible, and to understand if there are any fundamental limits to its size. Simply increasing the couplingstrength or optical power no longer is a good strategy: in this limit, the second “imprecision” noise term in δxadd (t) will become negligible, but the first back-action term will
be huge. Similarly, in the opposite limit where the coupling (or optical drive power) is
extremely weak, back-action noise will be negligible, but the imprecision noise noise
will be huge. Clearly, some trade-off between these two limits will be optimal. Further,
one needs to consider whether these two kinds of noises (imprecision and back-action)
can be correlated, and whether this is desirable.
Our goal in what follows will be to first review how one characterizes the magnitude
of noise via a spectral density, and then to establish the rigorous quantum limit on
how small one can make the added noise δxadd (t).

3.2

Quantum noise spectral densities: some essential features

In this section, we give a compact (and no doubt highly incomplete) review of somebasic properties of quantum noise spectral densities. We start however with the simpler
case of spectral densities describing classical noise.
3.2.1

Classical noise basics

Consider a classical random signal I(t). The signal is characterized by zero meanhI(t)i = 0, and autocorrelation function
GII (t, t0 ) = hI(t)I(t0 )i.

(3.7)

The autocorrelation function is analogous to a covariance matrix: for t = t0 , it tells
us the variance of the fluctuations of I(t), where as for t 6= t0 , it tells us if and how
fluctuations of I(t) are correlated with those at I(t0 ). Some crucial concepts regarding
noise are:• Stationary noise. The statistical properties of the fluctuations are time-translation
invariant, and hence GII (t, t0 ) = GII (t − t0 ).
• Gaussian fluctuations. The noise is fully characterized by its autocorrelation function; there are no higher-order cumulants.
• Correlation time. This time-scale τc governs the decay of GII (t): I(t) and I(t0 )are uncorrelated (i.e. GII (t − t0 ) → 0) when |t − t0 |  τc .

For stationary noise, it is often most useful to think about the fluctuations in thefrequency domain. In the same way that I(t) is a Gaussian random variable with zeromean, so is its Fourier transform, which we define as:1
IT [ω] = √T

Z +T /2

dt eiωt I(t),

(3.8)

−T /2

where T is the sampling time. In the limit T  τc the integral is a sum of a large
number N ≈ τTc of random uncorrelated terms. We can think of the value of the integral
as the end point of a random walk in the complex
plane which starts at the origin.
√
Because the distance traveled will scale with T , our choice of normalization makes

Quantum noise spectral densities: some essential features13

the statistical properties of I[ω] independent of the samplingtime T (for sufficiently
√
large
T
).
Notice
that
I
[ω]
has
the
peculiar
units
of
[I]
secs
which
is usually denotedT√[I]/ Hz.
The spectral density of the noise (or power spectrum) SII [ω] answers the question“how big is the noise at frequency ω?”. It is simply the variance of IT (ω) in thelarge-time limit:
SII [ω] ≡ lim h|IT [ω]|2 i = lim hIT [ω]IT [−ω]i.
T →∞

T →∞

(3.9)

A reasonably straightforward manipulation (known as the Wiener-Khinchin theorem)
tells us that the spectral density is equal to the Fourier transform of the autocorrelation
functionZ +∞SII [ω] =dt eiωt GII (t).
(3.10)−∞

We stress that Eq. (3.9) provides a simple intuitive understanding of what a spectral
density represents, whereas in theoretical calculations, one almost always starts withthe expression in Eq. (3.10). We also stress that since the autocorrelation function
GII (t) is real, SII [ω] = SII [−ω]. This is of course in keeping with Eq. (3.8), whichtells us that negative and positive frequency components of the noise are related bycomplex conjugation, and hence necessarily have the same magnitude.
3.2.2

Definition of quantum noise spectral densities

In formulating quantum noise, one turns from a noisy classical signal I(t) to a Heisenbergˆ
picture Hermitian operator I(t).
Similar to our noisy classical signal, one needs toˆ statistically. One can thus introduce a quantumthink about measurements of I(t)noise spectral density which completely mimics the classical definition, e.g.:Z +∞Sxx [ω] =dt eiωt hx̂(t)x̂(0)i.
(3.11)−∞We have simply inserted the quantum autocorrelation function in the classical definition. The expectation value is the quantum statistical average with respect to thenoisy system’s density matrix; we assume that this is time-independent, which thenalso gives us an autocorrelation function which is time-translational invariant.
What makes quantum noise so quantum? There are at least three answers to thisquestion:• Zero-point motion. While a classical system at zero-temperature has no noise,quantum mechanically there are still fluctuations, i.e. Sxx [ω] need not be zero.
• Frequency asymmetry. Quantum mechanically, x̂(t) and x̂(t0 ) need not commute
when t 6= t0 . As a result the autocorrelation function hx̂(t)x̂(t0 )i can be complex,
and Sxx [ω] need not equal Sxx [−ω]. This of course can never happen for a classicalnoise spectral density.
• Heisenberg constraints. For any system that can act as a detector or amplifier,there are fundamental quantum constraints that bound its noise. These constraints have their origin in the uncertainty principle, and have no classical counterpart.

14

Quantum limit on continuous position detection

3.2.3

Noise asymmetry and fluctuation dissipation theorem

The asymmetry in frequency of quantum noise spectral densities is a topic that is
discussed in great detail in (Clerk et al., 2010). In short, this asymmetry directlyreflects the asymmetry in the noisy system’s ability to absorb versus emit energy.
This aspect of quantum noise spectral densities provides an extremely useful routeto understanding optomechanical damping effects. While this asymmetry will not be
the main focus of our discussion here, it is useful to consider a simple but instructive
example which helps demystify two-sided quantum noise spectral densities. Consider
a harmonic oscillator that is coupled weakly to a noise force produced by a second
quantum system. This force is described by an operator F̂ , and the coupling to theharmonic oscillator isĤint = −x̂F̂ .
(3.12)

Of course, the basic optomechanical Hamiltonian is an example of such a coupling,where F̂ ∝ n̂, the intracavity photon number.
Classically, including this force in Newton’s equation yields a Langevin equation:2
M ẍ = −M ωM
x − M γcl ẋ + Fcl (t).

(3.13)In addition to the noisy force, we have included a damping term (rate γcl ). This will
prevent the oscillator from being infinitely heated by the noise source; we can thinkof it as describing the average value of the force exerted on the oscillator by the noise
source, which is now playing the role of a dissipative bath. If this bath is in thermal
equilibrium at temperature T , we also expect the oscillator to equilibrate to the same
temperature. This implies that the heating effect of Fcl (t) must be precisely balancedby the energy-loss effect of the damping force. More explicitly, one can use Eq. (3.13)
to derive an equation for the average energy of the oscillator hEi. As we are assuming
a weak coupling between the bath and the oscillator, we can take γcl  ωM , and hence
find
SF F [ωM ]dhEi = −γcl hEi +.
(3.14)
dt
2M
where SF F [ω] is the classical noise spectral density associated with Fcl (t) (c.f. Eq. (3.9)).
Insisting that the stationary value of hEi obey equipartition then leads directly tothe classical fluctuation dissipation relation:
SF F [ωM ] = 2M γcl kB T.

(3.15)

Let’s now look at our problem quantum mechanically. Writing x̂ in terms of ladder operators, we see that Ĥint will cause transitions between different oscillator Fockstates. Treating Ĥint in perturbation theory, we thus derive Fermi Golden rule transition rates Γn±1,n for transitions from the n to the n±1 Fock state. As shown explicitlyin Appendix B of (Clerk et al., 2010), these rates can be directly tied to the quantum
noise spectral density associated with F̂ . One finds:x2
Γn+1,n = (n + 1) ZPF
SF F [−ωM ] ≡ (n + 1)Γ↑ ,~2x2
Γn−1,n = (n) ZPF
SF F [ωM ] ≡ nΓ↓ .
~2(3.16)(3.17)

Quantum noise spectral densities: some essential features

15

Transitions where the noise source absorbs energy are set by the negative frequency
part of the noise spectral density, while emission is set by the positive frequency part.
We can now write a simple master equation for the probability pn (t) that theoscillator is in the nth Fock state:dpn = [nΓ↑ pn−1 + (n + 1)Γ↓ pn+1 ] − [nΓ↓ + (n + 1)Γ↑ ] pn .
dt

(3.18)

We can then connect this quantum picture to our classical Langevin equation byusing Eq. (3.18) to derive an equation for the average oscillator energy hEi. One
obtains
S̄F F [ωM ]dhEi = −γhEi +,(3.19)
dt
2M
where:x2ZPF(SF F [ωM ] − SF F [−ωM ]) ,~2
SF F [ωM ] + SF F [−ωM ]S̄F F [Ω] =.
2γ=(3.20)(3.21)We see that the quantum equation for the average energy, Eq. (3.19), has an identicalform to the classical equation (Eq. (3.14)), which gives us a simple way to connect ourquantum noise spectral density to quantities in the classical theory:• The symmetrized quantum noise spectral density S̄F F [Ω] defined in Eq. (3.21)
plays the same role as the classical noise spectral density SF F [Ω]: it heats theoscillator the same way a classical stochastic force would.
• The asymmetric-in-frequency part of the quantum noise spectral density SF F [Ω]is directly related to the damping rate γ in the classical theory. The asymmetry
between absorption and emission events leads to a net energy flow between theoscillator and the noise source, analogous to what one obtains from a classicalviscous damping force.
We thus see that there is a direct connection to a classical noise spectral density,and moreover the “extra information” in the asymmetry of a quantum noise spectral
density also corresponds to a seemingly distinct classical quantity, a damping rate. Thislatter connection is not so surprising. The asymmetry of the quantum noise is a directconsequence of (here) [F̂ (t), F̂ (t0 )] 6= 0. However, this same non-commutation causes
the average value of hF̂ i to change in response to x̂(t) via the interaction Hamiltonianof Eq. (3.12). Using standard quantum linear response (i.e. first-order time-dependent
perturbation theory, see e.g. Ch. 6 of (Bruus and Flensberg, 2004)), one finds
Z ∞δhF̂ (t)i =dt0 χF F (t − t0 )hx̂(t0 )i,(3.22)−∞

where the force-force susceptibility is given by the Kubo formula:χF F (t) ≡

Dh
iE
−i
θ(t) F̂ (t), F̂ (0) .
~(3.23)

16

Quantum limit on continuous position detection

From the classical Langevin equation Eq. (3.13), we see that part of hF̂ (t)i which isin phase with ẋ is the damping force. This leads to the definitionγ=

1(−Im χF F [Ω]) .
MΩ(3.24)An explicit calculation shows that the above definition is identical to Eq. (3.20), which
expresses γ in terms of the noise asymmetry. Note that in the language of many-body
Green functions, −Im χF F is referred to as a spectral function, whereas the symmetrized noise S̄F F [ω] is known (up to a constant) as the “Keldysh” Green function.
Quantum fluctuation-dissipation theorem and notion of effective temperature. Consider the case where the quantum system producing the noise F̂ is in thermal equilibrium at temperature T . For weak coupling, we expect that the stationary value of hEi
as given by Eq. (3.19) should match the thermal equilibrium value ~ωM (1/2 + nB [ωM ]).
Insisting that this be the case forces a relation between the damping γ (which is set
by the asymmetry of the noise, c.f. Eq. (3.20)) and symmetrized noise S̄F F [ωM ] which
is nothing more than the quantum version of the fluctuation dissipation theorem:~ωM= M γ[ωM ] ~ωM (1 + 2nB [ωM ]) . (3.25)
S̄F F [ωM ] = M γ[ωM ] ~ωM coth
2kB T
For kB T  ~ωM this reproduces the classical result of Eq. (3.15), whereas in the
opposite limit, it describes zero-point noise. We stress that Eq. (3.25) can also be
proved directly using nothing more than the fact that the system producing the noise
has a thermal-equilibrium density matrix.
What happens if our noise source is not in thermal equilibrium? In that case, it
is useful to use Eq. (3.25) to define an effective temperature Teff [Ω] from the ratio of
the symmetrized noise and damping. Re-writing things in terms of the quantum noise
spectral density, one finds 
−1
SF F [Ω]kB Teff [Ω] ≡ ~Ω ln.
(3.26)SF F [−Ω]The effective temperature at a given frequency Ω characterizes the asymmetry between
absorption and emission rates of energy ~Ω; a large temperature indicates that these
rates are almost equal, whereas a small temperature indicates that emission by thenoise source is greatly suppressed compared to absorption by the source. Away fromthermal equilibrium, there is no guarantee that the ratio on the RHS will be frequencyindependent, and hence Teff will generally have a frequency dependence.

3.3

Heisenberg inequality on detector quantum noise

3.3.1

Generic two-port linear response detector

Having discussed two of the ways quantum noise spectral densities differ from theirclassical counterparts (zero-point noise, frequency asymmetry), we now turn to thethird distinguishing feature: there are purely quantum constraints on the noise properties of any system capable of acting as a detector or amplifier. We will be interested

Heisenberg inequality on detector quantum noise

signalsourceinput output
operator

17"load"

detector
Fig. 3.1 (Color online) Schematic of a generic linear response detector.

in the generic two-port detector sketched in Fig. 3.1. The detector has an input port
characterized by an operator F̂ : this is the detector quantity which couples to thesystem we wish to measure. Similarly, the output port is characterized by an operator
ˆ this is the detector quantity that we will readout to learn about the system coupledI:
to the input. In our optomechanical system, F̂ would be the cavity photon numbern̂, while Iˆ would be proportional to the phase quadrature of the output field of thecavity (c.f. Eq. (3.3)).
We will be interested almost exclusively in detector-signal couplings weak enoughthat one can use linear-response to describe how Iˆ changes in response to the signal. For
example, if we couple an input signal x̂ to our detector via an interaction HamiltonianĤint = x̂ · F̂ ,(3.27)

linear response tells us that the change in the detector output will be given by:ˆδhI(t)i=Z ∞−∞dt0 χIF (t − t0 )hx̂(t0 )i,

DEi
ˆ F̂ (0)] .
χIF (t) = − θ(t) [I(t),~(3.28)(3.29)

This is completely analogous to the way we discussed damping, c.f. Eq. (3.24). As isstandard in linear-response, the expectation value in Eq. (3.29) is with respect to thestate of the system (signal plus detector) at zero coupling (i.e. Ĥint = 0). Also, withoutˆ and hF̂ i are zero in the absence of any
loss of generality, we will assume that both hIi
coupling to the input signal.
Even on a classical level, any noise in the input and output ports will limit ourability to make measurements with the detector. Quantum mechanically, we have seenthat it is the symmetrized quantum spectral densities that play a role analogous toclassical noise spectral densities. We will thus be interested in the quantities S̄II [ω]and S̄F F [ω]. Given our interest in weak detector-signal couplings, it will be sufficient
to characterize the detector noise at zero-coupling to the detector.
In addition to S̄II , S̄F F , we will also have to contend with the fact that the noisein Iˆ and F̂ may be correlated. Classically, we would describe such correlations via a
correlation spectral density SIF [ω]:18Quantum limit on continuous position detection∗

SIF [ω] ≡ lim hIT [ω] (FT [ω]) i =
T →∞

Z ∞
−∞

dt hI(t)F (0)ieiωt ,(3.30)

where the Fourier transforms IT [ω] and FT [ω] are defined analogously to Eq. (3.8). Not
surprisingly, such classical correlations correspond to a symmetrized quantum noise
spectral densityZ
1 ∞ˆ F̂ (0)}ieiωt .
dt h{I(t),(3.31)S̄IF [ω] ≡2 −∞Note that the classical correlation density SIF [ω] is generally complex, and is only
guaranteed to be real at ω = 0; the same is true of S̄IF [ω].
Finally, we normally are only concerned about how large the output noise is compared to the magnitude of the “amplified” input signal at the output (i.e. Eq. (3.28)).
It is thus common to think of the output noise at a given frequency δIT [ω] as an equivalent fluctuation of the signal δzimp [ω] ≡ δIT [ω]/χIF [ω]. We thus define the imprecision
noise spectral density and imprecision-back-action correlation density as:S̄zz [ω] ≡3.3.2

S̄II [ω],|χIF [ω]|2S̄zF [ω] ≡S̄IF [ω].
χIF [ω](3.32)

Motivation and derivation of noise constraint

We can now ask what sort of constraints exist on the detector noise. In almost all
relevant cases, our detector will be some sort of driven quantum system, and hence
will not be in thermal equilibrium. As a result, any meaningful constraint should not
rely on having a thermal equilibrium state. Classically, all we can say is that thecorrelations in the noise cannot be bigger than the noise itself. This constraint takes
the form of a a Schwartz inequality, yielding
2

Szz [ω]SF F [ω] ≥ |SzF [ω]| .

(3.33)

Equality here implies a perfect correlation, i.e. IT [ω] ∝ FT [ω].
Quantum mechanically, additional constraints will emerge. Heuristically, this canbe expected by making an analogy to the example of the Heisenberg microscope.
In that example, one finds that there is a tradeoff between the imprecision of themeasurement (i.e. the position resolution) and the back-action of the measurement(i.e. the momentum kick delivered to the particle). In our detector, noise in Iˆ willcorrespond to the imprecision of the measurement (i.e. the bigger this noise, the harder
it will be to resolve the signal described by Eq. (3.28)). Similarly, noise in F̂ is the
back-action: as we already saw, by virtue of the detector-signal coupling, F̂ acts as anoisy force on the measured quantity ẑ. We thus might naturally expect a bound on
the product of S̄zz S̄F F .
Alternatively, we see from Eq. (3.29) that for our detector to have any responseˆ and F̂ (t0 ) cannot commute for all times. Quantum mechanically, we know
at all, I(t)that uncertainty relations apply any time we have non-commuting observables; here
things are somewhat different, as the non-commutation is between Heisenberg-picture
operators at different times. Nonetheless, we can still use the standard derivation of an

Heisenberg inequality on detector quantum noise

19

uncertainty relation to obtain a useful constraint. Recall that for two non-commuting
observables Â and B̂, the full Heisenberg inequality is (see, e.g. (Gottfried, 1966))(∆A)2 (∆B)2 ≥iE 2
oE2 1 Dh
1 Dn
Â, B̂
.
Â, B̂
+4
4(3.34)

Here we have assumed hÂi = hB̂i = 0. We now take Â and B̂ to be cosine-transformsof Iˆ and F̂ , respectively, over a finite time-interval T :r Z T /2
r Z T /222
ˆÂ ≡dt cos(ωt + δ) I(t), B̂ ≡dt cos(ωt) F̂ (t). (3.35)
T −T /2
T −T /2
Note that we have phase shifted the transform of Iˆ relative to that of F̂ by a phase δ.
In the limit T → ∞ we find


∗ 2

2 ~2(χF I [ω])iδiδS̄zz [ω]S̄F F [ω] ≥ Re e S̄zF [ω]+Re e. (3.36)
1−
4
χIF [ω]We have introduced here a new susceptibility χF I [ω], which describes the reverse
response coefficient or reverse gain of our detector. This is the response coefficientˆ and seerelevant if we used our detector in reverse: couple the input signal ẑ to I,
how hF̂ i changes. A linear response relation analogous to Eq. (3.28) would then apply,with F ↔ I everywhere. For the optomechanical system we are most interested in,this reverse response coefficient vanishes: coupling to the output field from the cavity
cannot change the intracavity photon number (and hence F̂ ). We thus take χF I [ω] = 0
in what follows (see (Clerk, Devoret, Girvin, Marquardt and Schoelkopf, 2010) for
further discussion on the role of a non-zero χF I [ω]).
If we now maximize the RHS of Eq. (3.36) over all values of δ, we are left with the
optimal bound



~2
2S̄zF [ω]2S̄zz [ω]S̄F F [ω] − S̄zF [ω] ≥1+∆,(3.37)
4~where
1 + y 2 − 1 + |y|2∆[y] =,
2(3.38)

Note that for any complex number y, 1 + ∆[y] > 0. Related noise constraints onlinear-response detectors are presented in (Braginsky and Khalili, 1996) and (Averin,2003).
We see that applying the uncertainty principle to our detector has given us arigorous constraint on the detector’s noise which is stronger than the simple classicalbound of Eq. (3.33) on its correlations. For simplicity, consider first the ω → 0 limit,
where all noise spectral densities and susceptibilities are real, and hence the term
involving ∆[y] vanishes. The extra quantum term on the RHS of Eq. (3.36) thenimplies:20

Quantum limit on continuous position detection• The product of the imprecision noise S̄zz and back-action noise S̄F F cannot be
zero. The magnitude of both kinds of fluctuations must be non-zero.
• Moreover, these fluctuations cannot be perfectly correlated with one another: we2cannot have S̄zF = S̄zz S̄F F .
The presence of these extra quantum constraints on noise will lead directly (andrigorously!) to the fundamental quantum limits on continuous position detection. As wewill see, reaching this quantum limit requires one to use a detector which has “ideal”
quantum noise (i.e. noise spectral densities for which the inequality of Eq. (3.37)
becomes an equality).

3.4

Power gain and the large gain limit

Before finally turning to defining and deriving the quantum limit on the added noise,
there is one more crucial aspect of the detector to address: the notion of “power gain”.
We are interested in detectors that turn the motion x(t) of the mechanical resonator
into a “large” signal in the output of the detector, a signal so large that we do notneed to worry about how this detector output is then read out. To be able to say thatour detector truly amplifies the motion of the oscillator, it is not sufficient to simply
say the response function χIF must be large (note that χIF is not dimensionless!).
Instead, true amplification requires that the power delivered by the detector to afollowing amplifier be much larger than the power drawn by the detector at its input–
i.e., the detector must have a dimensionless power gain GP [ω] much larger than one.
If the power gain was not large, we would need to worry about the next stage in theamplification of our signal, and how much noise is added in that process. Having alarge power gain means that by the time our signal reaches the following amplifier, it
is so large that the added noise of this following amplifier is unimportant.
The power gain of our position detector can be defined by imagining a situation
where one couples a second auxiliary oscillator to the output of the detector, suchthat I(t) acts as a driving force on this oscillator. The power gain is then definedas the power delivered to this auxiliary oscillator, divided by the power drawn fromthe measured mechanical resonator coupled to the detector output (optimized overproperties of the auxiliary oscillator). The calculation is presented in Appendix A,
and the result is the simple expression:|χIF [ω]|2Pout=(3.39)
GP [ω] ≡ max
Pin
4 (Im χF F [ω]) (Im χII [ω])The susceptibility χII [ω] is defined analogously to Eq. (3.23). Note that if there isno additional back-action damping of the measured oscillator by the detector, then
Im χF F vanishes, and the power gain is strictly infinite. As we will see, this the casefor an optomechanical cavity driven on resonance.
Having a large power gain also implies that we can treat the detector quantitiesˆ and F̂ (t) as being effectively “classical”. A large power gain over some relevantI(t)frequency range implies that the imaginary parts of χII and χF F are negligible overthis range. From Eqs. (3.20) and (3.24), this implies that the quantum noise spectral
densities SII [ω], SF F [ω] are to a good approximation symmetric over these frequencies,

Defining the quantum limit

21

just like a classical noise spectral density. This in turn implies that one can effectively
ˆ I(tˆ 0 )] = [F̂ (t), F̂ (t0 )] = 0 (i.e. these operators commute at differnent times,
treat [I(t),just like a classical noisy function of time). We note that in many discussions of linear quantum measurements, the fact that the detector input and output quantities
commute with themselves at different times is taken as a starting assumption in calculations (see, e.g. (Braginsky and Khalili, 1992; Khalili et al., 2012), as well as thelectures by Y. Chen).
Requiring both the quantum noise inequality in Eq. (3.37) to be saturated at
frequency ω as well as a large power gain (i.e. GP [ω]  1) leads to some important
additional constraints on the detector, as derived in Appendix I of (Clerk et al., 2010):p• (2/~)Im S̄zF [ω] is small like 1/ GP [ω].
• The detector’s effective temperature must be much larger than ~ω; one findspkB Teff [ω] ∼ GP [ω]~ω.
(3.40)
Conversely, it is the largeness of the detector’s effective temperature that allows
it to have a large power gain.

3.5

Defining the quantum limit

Having now understood the proper way to discuss the “size” of noise, as well as the existence of quantum constraints on noise, we can now return to the question posed at thestart of this chapter: how small can we make the added noise δxadd (t) (c.f. Eq. (3.6))of a generic linear-response position detector? We assume that, like in our optomechanical setup, the detector couples to mechanical position via a Hamiltonian:Ĥint = x̂ · AF̂
(3.41)
Note that unlike the generic system-detector interaction in Eq.(3.27), we have included
a coupling strength in the definition of the system operator F̂ . We assume that A is
weak enough that the detector output responds linearly to changes in position.
3.5.1

Added noise spectral density

We start by returning to the heuristic classical expression in Eq. (3.6) for the detector
output current I(t) and added noise δxadd (t), and Fourier transform these expression.
From linear response theory, we know that the response coefficient λ in that expression
should be replaced by λ → AχIF [ω], where the frequency dependence parameterizes
that the detector output will not respond instantaneously to changes at the input.
I[ω] = AχIF [ω] (x[ω] + δxadd [ω])
δxadd [ω] = δxBA [ω] +δI0 [ω]AχIF [ω](3.42)(3.43)

The first back-action term is just the mechanical response to the back-action force
fluctuations:22

Quantum limit on continuous position detection

δxBA [ω] = Aχxx [ω]δF [ω],(3.44)

1

where χxx [ω] is the oscillator’s force susceptibility , and is given by−12.
M χxx [ω] = ω 2 − ωM
+ iωγ0(3.45)To state the quantum limit on position detection, we first define the total measured
position fluctuations xmeas [ω] as simply the total detector output I[ω] referred backto the oscillator:
xmeas [ω] = Itot [ω]/(AχIF [ω]).
(3.46)
If there was no added noise, and further, if the oscillator was in thermal equilibrium attemperature T , the spectral density describing the fluctuations δxmeas (t) would simply
be the equilibrium fluctuations of the oscillator, as given by the fluctuation-dissipation
theorem:


~ω
meas
eq
S̄xx
[ω] = S̄xx
[ω, T ] = ~ coth[−Im χxx [ω]](3.47)2kB T
x2 (1 + 2nB ) X
γ0
= ZPF
.
(3.48)
2(ω − σΩ)2 + (γ0 /2)2σ=±Here, γ0 is the intrinsic damping rate of the oscillator, which we have assumed to be Ω.
Including the added noise, and for the moment ignoring the possibility of any
additional oscillator damping due to the coupling to the detector, the above result
becomes
meas
eq
add
S̄xx
[ω] = S̄xx
[ω, T ] + S̄xx[ω](3.49)

where the last term is the spectral density of the added noise (both back-action andimprecision noise).
We can now, finally, state the quantum limit on continuous position detection: ateach frequency ω, we must have
add
eq
S̄xx
[ω] ≥ S̄xx
[ω, T = 0].

(3.50)The spectral density of the added noise cannot be made arbitrarily small: at eachfrequency, it must be at least as large as the corresponding zero-point noise. Note that
we do not call this the “standard” quantum limit. As we will discuss later, what is
usually termed the standard quantum limit (e.g. in the gravitational wave detection
community) only coincides with Eq. (3.50) exactly at resonance (ω = ωM ), and forother frequencies does not represent any kind of true quantum bound. The various
contributions to the added noise spectral density are shown in Fig. 3.2.
Finally, the above result can be refined to include situations where the coupling to
the detector also changes the mechanical damping (in addition to driving it with extra
1 Strictly speaking, with our definitions the force susceptibility is −χ [ω], and the force drivingxx
the mechanics is -F̂ . This is because we took the interaction Hamiltonian to be Ĥint = +x̂F̂ insteadof the more physical Ĥint = −x̂F̂ .

Defining the quantum limit

Sxx,tot [ω] / S0xx[Ω]

2.0

23

total noise

1.5

eq. + backaction noise

1.0

equilibrium
noise (T=0)

0.5
0.0
0.0

0.4

0.8

Frequency ω/Ω

1.2

Fig. 3.2 Spectral density of a mechanical resonator’s position fluctuations as measured by aquantum limited position detector. The mechanical resonator’s intrinsic fluctuations (for thecase of a zero-temperature oscillator) are shown in blue. The back-action of the position detector will effectively heat the mechanical resonator, increasing the area under the Lorentzian;this is shown as the orange area. Finally, there is also imprecision noise, the fluctuations in
the detector output that would be present even without any coupling to the mechanical resonator. These typically give frequency independent noise, giving a flat background. The ideal
case where the quantum-limit is reached on the added noise at mechanical resonance is theyellow area.

fluctuations). Letting γBA denote this extra damping, the added noise is now defined
via
meas
S̄xx
[ω] =

γ0
add
S̄ eq [ω, T ] + S̄xx[ω],γBA + γ0 xx(3.51)

where the susceptibility χxx now involves the total damping of the oscillator, i.e.:2
M χxx [ω] = ω 2 − ωM+ iω(γ + γBA )−1.

(3.52)

With this definition, the quantum limit on the added noise is unchanged from the limit
stated in Eq. (3.50).
3.5.2

A possible correlation-based loophole?

Our heuristic formulation of the quantum limit naturally leads to a possible concern.
Even though quantum mechanics may require a position measurement to have a backaction (as position and momentum are conjugate quantities), couldn’t this back-action

24

Quantum limit on continuous position detection

noise be perfectly anti-correlated with the imprecision noise? If this were the case, theadded noise δx(t) (which is the sum of the two contributions, c.f. Eq. (3.43)) could bemade to vanish.
One might hope that this sort of loophole would be explicitly forbidden by thequantum noise inequality of Eq. (3.37). However, this is not the case. Even in the ideal
case of zero reverse gain, one can achieve a situation where back-action and imprecision
are perfectly correlated at a given non-zero frequency ω. One needs:• The correlator S̄IF [ω] should purely imaginary; this implies that the part of F (t)that is correlated with I(t) is 90 degrees out of phase. Note that S̄IF [ω] can only
be imaginary at non-zero frequencies.
• The magnitude of S̄IF [ω] should be larger than ~/2

Under these circumstances, one can verify that there is no additional quantum constraint on the noise beyond what exists classically, and hence the perfect correlation
condition of S̄F F [ω]S̄II [ω] = |S̄IF [ω]|2 is allowable. The π/2 phase of the back-actionimprecision correlations are precisely what is needed to make δxadd [ω] vanish at theoscillator resonance, ω = Ω.
As might be expected, this seeming loophole is not a route to ideal, noise-free
position detection free from quantum constraints. As already discussed, we need tobe more careful in specifying what we want our detector to do. We aren’t interested
in just having the mechanical motion show up in the detector output I(t), we want
there to be amplification associated with this process– the mechanical signal should be“bigger” at the output than it is at the input. It is only when we insist on amplificationthat there are quantum constraints on added noise; a passive transducer need not add
any noise. On a heuristic level, one could view amplification as an effective expansionof the phase space of the oscillator. Such a pure expansion is of course forbidden by
Liouville’s theorem, which tells us that volume in phase space in conserved. The way
out is to introduce additional degrees of freedom, such that for these degrees of freedom
phase space contracts. Quantum mechanically such degrees of freedom necessarily havenoise associated with them (at the very least, zero-point noise); this then is the sourceof the limit on added noise.
The requirement that our detector produces a large signal is that the power gain(as defined in Sec. 3.4) should be much larger than one. In that case, back-actionimprecision correlations must be purely real, and the possibility of zero added noise(due to perfect correlations) is excluded.

3.6

Derivation of the quantum limit

We now turn to a rigorous proof of the quantum limit on the added noise given in
Eq. (3.50). From the classical-looking Eq. (3.43), we expect that the symmetrized
quantum noise spectral density describing the added noise will be given by∗
2Re χ∗IF (χxx ) S̄IF
S̄II
2
2
+
A
|χ
|
S̄
+
(3.53)
S̄xx,add [ω] =xxFF|χIF |2 A2|χIF |2


S̄zz
2
∗
= 2 + A2 |χxx | S̄F F + 2Re (χxx ) S̄zF .
(3.54)A

Derivation of the quantum limit

25

In the second line, we have introduced the imprecision noise S̄zz and imprecision
back-action correlation S̄zF as in Eq. (3.32). We have also omitted writing the explicit
frequency dependence of the gain χIF , susceptibility χxx , and noise correlators; theyshould all be evaluated at the frequency ω. Finally, the oscillator susceptibility χxx
here is given by Eq. (3.52), and includes the effects of back-action damping. While
we have motivated this equation from a seemingly classical noise description, the fullquantum theory also yields the same result: one simply calculates the detector output
noise perturbatively in the coupling to the oscillator (Clerk, 2004).
The first step in determining the limit on the added noise is to consider its dependence on the coupling strength strength A. If we ignore for a moment the detectordependent damping of the oscillator, there will be an optimal value of the couplingstrength A which corresponds to a trade-off between imprecision noise and back-action(i.e. first and second terms in Eq. (3.53)). We would thus expect S̄xx,add [ω] to attaina minimum value at an optimal choice of coupling A = Aopt where both these terms
make equal contributions. Defining φ[ω] = arg χxx [ω], we thus have the boundhipS̄zz S̄F F + Re e−iφ[ω] S̄zF ,(3.55)
S̄xx,add [ω] ≥ 2|χxx [ω]|where the minimum value at frequency ω is achieved whensS̄zz [ω]2.
Aopt =|χxx [ω]|2 S̄F F [ω](3.56)

Using the inequality X 2 + Y 2 ≥ 2|XY | we see that this value serves as a lower boundon S̄xx,add even in the presence of detector-dependent damping. In the case where
the detector-dependent damping is negligible, the RHS of Eq. (3.55) is independent ofA, and thus Eq. (3.56) can be satisfied by simply tuning the coupling strength A; inthe more general case where there is detector-dependent damping, the RHS is also afunction of A (through the response function χxx [ω]), and it may no longer be possible
to achieve Eq. (3.56) by simply tuning A.
While Eq. (3.55) is certainly a bound on the added displacement noise S̄xx,add [ω],it does not in itself represent the quantum limit. Reaching the quantum limit requires more than simply balancing the detector back-action and intrinsic output noises(i.e. the first two terms in Eq. (3.53)); one also needs a detector with “quantum-ideal”
noise properties, that is a detector which optimizes Eq. (3.37). Using the quantum
noise constraint of Eq. (3.37) to further bound S̄xx,add [ω], we obtain
"s

#


2
Re e−iφ[ω] S̄zF
S̄zF
S̄zF
S̄xx,add [ω] ≥ ~ |χxx [ω]|1+∆+,(3.57)
+
~/2
~/2
~/2
where the function ∆[z] is defined in Eq. (3.38). The minimum value of S̄xx,add [ω] in
Eq. (3.57) is now achieved when one has both an optimal coupling (i.e. Eq. (3.56))
and a quantum limited detector, that is one which satisfies Eq. (3.37) as an equality.
Next, we further specialize to the relevant case where the detector acts as a good
amplifier, and has a power gain GP [ω]  1 over all frequencies of interest. As discussed, this implies that the ratio S̄zF is purely real up to small 1/GP corrections (see26

Quantum limit on continuous position detection

Appendix I of Ref. (Clerk et al., 2010) for more details). This in turn implies that∆[2S̄zF /~] = 0; we thus havesS̄xx,add [ω] ≥ ~|χxx [ω]|  1 +



S̄zF
~/2

2


S̄zF 
.
+ cos (φ[ω])~/2(3.58)

Finally, as there is no further constraint on S̄zF (beyond the fact that it is real),
we can minimize the expression over its value. The minimum S̄xx,add [ω] is achieved fora detector whose cross-correlator satisfies
S̄zF [ω]optimal=2~~ ω 2 − ωM
cot φ[ω] = −
,
2
2
ωγ

(3.59)

with the minimum value of the added noise being given precisely byS̄xx,add [ω]min= ~|Im χxx [ω]| = lim S̄xx,eq [ω, T ],T →0

(3.60)

in agreement with Eq. (3.50). Thus, in the limit of a large power gain, we have thatat each frequency, the minimum displacement noise added by the detector is precisely
equal to the noise arising from a zero temperature bath.

3.7

Simple limits and discussion

We have thus provided a rigorous derivation of the quantum limit on the added noiseof a continuous position detector which possesses a large gain. The derivation shows
explicitly what is needed to reach the quantum-limit, namely:
1. A detector with quantum limited noise properties, that is one which optimizes
the inequality of Eq. (3.37).
2. A coupling A which satisfies Eq. (3.56).
3. A detector cross-correlator S̄IF which satisfies Eq. (3.59).
It is worth stressing that Eq. (3.59) implies that it will not in general be possible
to achieve the quantum limit simultaneously at all frequencies, as the needed amountof back-action-imprecision correlation varies strongly with frequency. We consider afew important limits below.
3.7.1

Quantum limit on added noise at resonance ω = ωM

To reach the quantum limit on the added noise at the mechanical resonance, Eq. (3.59)
tells us that S̄zF [ω] must be zero: back-action and imprecision noises should be completely uncorrelated. If this is the case, reaching the quantum limit simply involves
tuning the coupling A to balance the contributions from back-action and imprecision
to the added noise. As we will see, in our optomechanical cavity, this is equivalent to
optimizing the choice of the driving power.

Simple limits and discussion

27

This remaining condition on the coupling (again, in the limit of a large power gain)
may be written asγBA [Aopt ]~Ω=.
(3.61)
γ0 + γBA [Aopt ]4kB Teff
As γBA [A] ∝ A2 is the back-action-induced damping of the oscillator (c.f. Eq. (3.20)),
we thus have that to achieve the quantum-limited value of S̄xx,add [Ω] with a large power
gain, one needs the intrinsic damping of the oscillator to be much larger than the backaction damping. The back-action damping must be small enough to compensate thelarge effective temperature of the detector; if the bath temperature satisfies ~Ω/kB 
Tbath  Teff , Eq. (3.61) implies that at the quantum limit, the temperature of theoscillator will be given by
Tosc ≡~Ω
γBA · Teff + γ0 · Tbath→+ Tbath .
γBA + γ04kB

(3.62)

Thus, at the quantum limit and for large Teff , the detector raises the oscillator’s
temperature by ~Ω/4kB .2 As expected, this additional heating is only half the zeropoint energy; in contrast, the quantum-limited value of S̄xx,add [ω] corresponds to thefull zero-point result, as it also includes the contribution of the intrinsic output noiseof the detector.
3.7.2

Quantum limit on added noise in the free-mass limit ω  ωMIn gravitational wave detection, one is usually interested in the added noise at frequencies far above resonance, where the mechanical dynamics are effectively like thoseof a free mass. In this case, Eq. (3.59) tells us that reaching the quantum limit on theadded noise requires back-action imprecision correlations satisfying:S̄zF [ω]optimal→−~ω
2γ(3.63)In the limit where ω  γ, the correlations are huge, implying that the optimal situation
is to have back-action and imprecision noises be almost perfectly correlated. If one
could achieve this, the quantum limited value of the added noise is given by:S̄xx,add [ω] →

~γ
mω 3

(3.64)

In contrast to the full quantum limit above, one often discusses the “standard
quantum limit” in the gravitational wave detection community. This is the minimum
added noise possible (in the free mass limit) if you use a detector with quantum-ideal
noise (i.e. saturates the inequality of Eq. (3.37)) but which has S̄zF [ω] = 0. In this2 If in contrast our oscillator was initially at zero temperature (i.e. Tbath = 0), one finds thatthe effect of the back-action (at the quantum limit and for GP  1) is to heat the oscillator to atemperature ~Ω/(kB ln 5) ' 0.62~Ω/kB .

28

Quantum limit on continuous position detection

case, the only optimization involves tuning the coupling to balance back-action andimprecision noises, and one finds:
S̄xx,add [ω]SQL= ~|χxx [ω]| →~mω 2(3.65)One sees that this larger than the true quantum limit by a large factor ω/γ.
The take-home message here is that while reaching the true quantum limit for ω ωM might be challenging, one can do much better than the “standard” quantum limit
by using a detector having back-action-imprecision correlations. In the next chapter,
we will review how injecting squeezed light into an optomechanical cavity can achieve
this goal.

3.8

Applications to an optomechanical cavity (with and without
input squeezing)

We now apply our general approach for formulating the quantum limit to the specificcase of an optomechanical cavity detector, as introduced in the first chapter. We
consider a cavity which is strongly driven on resonance (implying that the detuning∆ = 0), and which can be treated using the linearized equations of motion introducedin chapter 1. Our starting point is thus Eq. (2.25) for the displaced cavity field, with
∆ = 0. We will further work in the regime where the mechanical frequency ωM  κ,ˆ at frequencies ω  κ. To capture the behaviour at theseand we are interested in d[ω]frequencies, we can make the adiabatic approximation, and ignore the (d/dt)dˆ termon the LHS of Eq. (2.25). Defining A = G/xZPF = gā/xZPF , we have:ˆ ' − 2iA x̂(t) − √2 dˆin (t)
d(t)
κ
κ

(3.66)

where the effective coupling strength A is given by:
A=

gācl
G
=
xZPF
xZPF

(3.67)We see that, as anticipated, the effective coupling strength is indeed dependent on thestrength of the cavity driving field. Using the input-output relation of Eq. (2.15), we
also find:√ˆ ' − 2iA
√ x̂(t) − dˆin (t)dˆout (t) = dˆin (t) + κd(t)(3.68)
κWe can use these results to calculate the needed noise properties: the fluctuationsin the back-action force acting on the mechanics, and the fluctuations in the detector output quantity, the phase quadrature of the output light defined in Eq. (3.3).
The back-action force operator corresponds to the fluctuating part of the intra-cavity
photon number. Keeping only the drive-enhanced term in this operator, we have that:
A · F̂ ≡ ~A dˆ + dˆ†(3.69)

Applications to an optomechanical cavity (with and without input squeezing)29

We need to understand the fluctuations of F̂ in the absence of any optomechanicalcoupling; we can thus substitute in Eq. (3.66) at A = 0 to find:2~ F̂ = − √ dˆin (t) + dˆ†in (t)κ(3.70)

Similarly, the output quantity is given by the phase quadrature of the cavity output
field, Eq. (3.3) 3 :Iˆ = i dout − dˆ†out4A= √ x̂(t) − i dˆin − dˆ†in ≡ AχIF x̂(t) + Iˆ0κ

(3.71)(3.72)√We thus directly can read-off both the response coefficient χIF = 4/ κ of the detector(which is frequency independent as we focus on ω  κ), and the intrinsic imprecision
noise in the output Iˆ0 .
We see that two orthogonal (and hence canonically conjugate) quadratures of the
input noise dˆin entering the cavity determine the two kinds of relevant noise (backaction and imprecision). This is to be expected: it is fluctuations in the amplitude
quadrature of the incident drive that cause photon number fluctuations and henceback-action, while it is the phase quadrature fluctuations which give imprecision noise.
3.8.1

Vacuum noise input

Taking the input noise to be vacuum noise and using Eqs. (2.10),(2.11), it is straightforward to calculate the needed noise correlators:
S̄F F [ω] =

4~2
,
κ

S̄II [ω] = 1

S̄IF [ω] = 0(3.73)

Note crucially that there are no correlations between back-action and imprecision
noise, as they correspond to conjugate quadratures of the input vacuum noise. The
vanishing of correlations can ultimately be traced back to the fact that averages likehdˆin (t)dˆin (t0 )i are always zero in the vacuum state.
It thus follows that:S̄zz S̄F F ≡S̄II S̄F F
4~2 /κ
~2
=
=
2
χIF
16/κ
4

(3.74)

Our driven cavity thus optimizes the quantum noise inequality of Eq. (3.37), but has
no back-action-imprecision correlations. From our general discussion, this implies that
it is able to reach the quantum limit on the added noise exactly at the mechanicalresonance, but away from resonance, misses the true quantum limit by a large amount.
3 There is in principle a proportionality constant between Iˆ and the phase quadrature of the output
field; however, this constant plays no role in determining the detector added noise, so we set it tounity.

30

Quantum limit on continuous position detection

In the case where one wishes to reach the quantum limit on resonance, it is interesting to ask what the optimal coupling strength is using Eq. (3.56). It is convenientto parameterize the coupling in terms of the cooperativity C, defined as:
C≡

4G2
κγ

(3.75)The optimal coupling required to reach the quantum limit on resonance then becomes:
14Noise correlations via “variational” readout
Copt =

3.8.2(3.76)

As we have seen, reaching the quantum limit in the free mass limit (i.e. at frequencies
much larger than ωM ) requires strong correlations between the back-action noise in F̂
and the imprecision noise in Iˆ0 , c.f. Eq. (3.63). These correlations are absent in thesimplest scheme described above, as the back-action and imprecision noise operators
correspond to conjugate quadratures of the input vacuum noise.
An extremely simple way to induce correlations between back-action and imprecision is to alter the choice of which quadrature of the cavity output field to measure.
Suppose instead of the choice in Eq. (3.71), we chose to measure the output quadrature:
!


F̂
†
iϕ
−iϕ
√
(3.77)
Iˆnew ≡ i e dˆout − e dˆout = cos ϕIˆold − sin ϕ2~/ κwhere the angle ϕ determines the particular choice of quadrature. ϕ = 0 corresponds
to the measuring the phase quadrature as before, and ensures that Iˆ has a maximal
sensitivity to x̂ (as x̂ only appears in the imaginary part of dˆout , c.f. Eq. (3.68)). By
taking ϕ 6= 0, we reduce the response coefficient χIF , but trivially induce back-action- imprecision correlations, as now the measured detector output includes explicitly the
back-action fluctuations F̂ . For an arbitrary choice of ϕ, one finds that S̄F F and S̄II
are unchanged from Eq. (3.73), but:χIF =4 cos ϕ√κ

2~
S̄IF [ω] = − sin ϕ √κ

(3.78)The detector still saturates the Heisenberg bound on its quantum noise:2
S̄zz S̄F F − S̄zF≡

2S̄II S̄F F − S̄IF~2
(4~2 /κ)(1 − sin2 ϕ)
=
=
2
2
χIF
16 cos ϕ/κ
4(3.79)

Thus, this strategy in principle allows one to reach the quantum limit on the added
noise for frequencies away from mechanical resonance, where one requires a non-zerocross-correlator. In the free-mass limit ω  ωM , the optimal value of S̄zF is given by
Eq. (3.63). Achieving this value requires:ω~tan ϕ =
2γ(3.80)One sees that the required choice of ϕ is frequency dependent. To achieve the optimal correlations over a finite range of frequencies, one could first apply a frequencydependent rotation to the output field from the optomechanical cavity (using, e.g. aApplications to an optomechanical cavity (with and without input squeezing)

31

second detuned cavity), and then measure a fixed quadrature. Perhaps more troublingis the fact that the RHS of the above equation is typically extremely large, implyingthat ϕ → π/2. In this limit, the measured output quantity is almost equivalent tothe amplitude quadrature of the input noise incident on the cavity, and has almost no
information on the state of the mechanics.
The above approach is known as the “variational” readout strategy in the gravitational wave detection community (Vyatchanin and Zubova, 1995; Vyatchanin andMatsko, 1996), and is also discussed (in a slightly different way) in the lectures of Yanbei Chen. While this seems like a simple strategy for reaching the quantum limit awayfrom mechanical resonance, in practice it is not a good strategy: the strong reductionin the size of the signal means that even though the intrinsic detector noise may be as
small as required by quantum mechanics, other non-intrinsic sources of added noise
will start to dominate. In short, reaching the quantum limit by throwing away signal
strength is almost never a good strategy.
3.8.3

Noise correlations via squeezing

We would like to find a way to induce the needed noise correlations for reaching thequantum limit away from mechanical resonance, while at the same time not modifying
the size of the measured signal in the output of our detector. The trick will be to modify
the input noise driving the cavity. It is convenient to introduce canonical (Hermitian)quadratures of the input noise din (t) via:
1 
(3.81)dˆin = √ X̂in + iŶin
2
As we have seen, back-action is controlled by X̂in and imprecision noise by Ŷin . A
straightforward calculation shows thatS̄zF = ~S̄Yin ,Xin

(3.82)

Hence, back-action imprecision noise correlations require a state where the fluctuationsin the two quadratures of the input noise are strongly correlated. Further, we stillrequire that the quantum noise inequality of Eq. (3.37) be saturated, even in thepresence of strong correlations. The required input noise corresponds to a quantum
squeezed state (see, e.g., (Gerry and Knight, 2005) for an extensive discussion). In
phase space, the Wigner function of such a state has elliptical iso-probability contours(as opposed to the circular contours of a vacuum state or thermal state). The elongateddirection of the ellipse should be aligned so as to yield the desired large correlations.
As the needed correlations should be negative, the “squeezed” direction of the ellipse(direction with minimal fluctuations) corresponds to the quadrature:
1 (3.83)
X̃in = √ X̂in + Ŷin2
For a squeezed state where X̃ is the squeezed quadrature, we have:S̄X̃in X̃in =

1 −2r
e
2

(3.84)

32

Quantum limit on continuous position detection

where r ≥ 0 is the squeeze parameter. Reaching the quantum limit at a frequencyω  ωM with our cavity detector thus requires one to use an input squeezing with asqueezing magnitudeγ
e−2r =.
(3.85)
2ω
The idea of using squeezing to generate strong back-action-imprecision correlations
was discussed extensively in (Pace, Collett and Walls, 1993) (albeit using a somewhat
different formulation), and is also discussed in the lectures of Y. Chen. We stress thatthe use of squeezing is to generate correlations; it is not being used to make up for alack of incident laser power (i.e. ācl too small).
An alternate use of squeezing which is equivalent to tuning the incident opticalpower was discussed in the seminal work (Caves, 1981). Imagine one wants to reach thequantum limit at mechanical resonance, but cannot achieve the optical power required
to balance the contributions from back-action and imprecision noises, i.e. achieve thecondition in Eq. (3.76). Squeezing can help in this situation. If one squeezes the phasequadrature of the incident light Ŷin (and thus necessarily amplifies the amplitude
quadrature X̂in ), the back-action noise is enhanced, and the imprecision noise reduced.
Hence, input squeezing in this case is equivalent to boosting the magnitude of the cavity
drive, i.e. increasing the cooperativity C. Squeezing in this case does not generate any
back-action-imprecision correlations.

4
Backaction evasion and conditionalsqueezing
In this chapter, we discuss a method for monitoring mechanical position that is not
subject to any fundamental quantum limit. As the quantum limit of the previous
chapter is indeed a true, unavoidable limit, the only way to do better is to somehowchange the rules of the game. Here, this is accomplished by being more modest in what
we choose to measure. Recall that a standard weak, continuous position measurement
one attempts to measure both quadrature components of the mechanical motion. i.e.
X(t) and Y (t) as defined in Eq. (4.1). Quantum mechanically these are conjugate, noncommuting observables; as a result, one cannot measure both quadratures perfectly,
as the measurement of X perturbs Y (and vice-versa). In this chapter, we will give up
trying to have full knowledge of x(t), and will instead attempt to monitor only one of
the two quadrature components. As we will show, this is something that can be donewith no fundamental limit coming from quantum mechanics. This opens the door toforce sensing with no fundamental quantum limit, as well as the possibility of using themeasurement to generate quantum squeezed states of the mechanical resonator. The
classic reference discussing such back-action evading single quadrature measurementsare (Braginsky et al., 1980), (Caves et al., 1980) and (Bocko and Onofrio, 1996), while
the full quantum theory (including the production of conditional quantum squeezed
states) was treated in (Clerk, Marquardt and Jacobs, 2008). Note that several recent
experiments have implemented this scheme using microwave circuit realizations ofoptomechanics: (Hertzberg et al., 2010; Suh et al., 2014; Lecocq et al., 2015).

4.1

Single quadrature measurements

Our goal is measure a single mechanical quadrature. The canonically-conjugate quadrature operators are defined in the Heisenberg picture (with respect to the mechanicalHamiltonian) as:√x̂(t) =
2xZPF X̂(t) cos(ωM t) + Ŷ (t) sin(ωM t).
(4.1)
with


1 
X̂ = √ b̂ + b̂† ,
2


−i 
Ŷ = √ b̂ − b̂†
2

(4.2)

We stress that in the Schrödinger picture, X̂(t) and Ŷ (t) are explicitly time-dependent
observables, and do not simply correspond to the position and momentum operatorsof the mechanical resonator. For example, in the Schrödinger picture, we have:

34

Backaction evasion and conditional squeezing11 X̂(t) = √ b̂eiωM t + b̂† e−iωM t =
xZPF
2



sin (ωM t)
cos (ωM t) x̂ −
p̂
mωM

(4.3)

In the second equality, x̂ (p̂) is the standard Schrödinger-picture position (momentum)
operator of the oscillator. For an undamped oscillator, both quadrature operators areconstants of the motion. Note that the definition of the quadrature operator necessarily
requires some external phase reference, or equivalently, some choice defining the zero oftime. With the above definition, X̂(t = 0) is proportional to the mechanical position,
while Ŷ (t = 0) is proportional to the mechanical momentum.
The goal is to measure say X̂, and have all the corresponding back-action of themeasurement drive the unmeasured quantity Ŷ . As X̂ is dynamically independentof Ŷ , the back-action will never come back to corrupt subsequent measurements ofX̂ at later times, and we can in principle make the measurement better and betterby increasing the measurement strength. Such a measurement is known as a backaction evading (BAE) measurement. It is also an example of a quantum non-demolition
measurement, as one is measuring an observable (i.e. X̂(t)) that is a constant of the
motion.
While the basic idea is clear, upon first glance implementation would seem to be achallenge. To measure X̂, we just need to couple this operator to some input operatorF̂ of our detector. From Eq. (4.3), we see that this would require time-dependent
couplings between the detector and both the mechanical position x̂ and mechanicalmomentum p̂. This would be extremely difficult to achieve (see, e.g., Y. Chen’s lectures
discussing the difficulties of coupling a detector to mechanical momentum).
Luckily, there is a simple trick to let us turn a standard coupling between detector and mechanical position (like we have in optomechanics) into the kind of singlequadrature coupling we need. The trick has two parts:• Start with a detector which couples to mechanical position, but modulate the coupling strength in time at the mechanical frequency ωM . Working in the Schrodinger
picture, we want to modify the basic system-detector interaction in Eq. (3.41) tonow have the form:Ĥint = Ā cos ωM t x̂ · F̂ = X̂(t) (1 + cos 2ωM t) + Ŷ (t) sin 2ωM t(4.4)
In the last line, we have re-expressed things in terms of the quadrature operators.
Note that the coupling to X̂ has a time-independent part, where the coupling toŶ is strictly oscillating.
• Next, imagine we work with a “slow” detector, one that cannot respond to perturbations occurring at frequencies ∼ 2ωM . In that case, the time-dependent oscillating terms in the above equation will average away, and we will be left witha time-independent coupling between the detector force operator F̂ and the X̂quadrature only.

4.2

Two-tone QND scheme

As first discussed in (Braginsky et al., 1980), the above “modulated” coupling scheme
for measuring a single mechanical quadrature can be achieved using a standard optomechanical setup if one works in the resolved sideband regime ωM  κ and drivesTwo-tone QND scheme

a)

35

b)



!cav

!M



!cav

!cav + !M!M!cav + !M

Fig. 4.1 (a) Cavity density of states and drive frequencies for the two-tone BAE measurement scheme; the frequencies of the classical drives are marked with large solid arrows.
Photons arriving at the cavity resonance could have been generated either by anti-Stokes
scattering from the red-detuned drive tone, or Stokes scattering from the blue-detuned drive
tone. As the amplitudes of these drives are equal, the cavity amplitude at resonance effects
a measurement of the mechanical X quadrature. One needs to be in the good cavity limit,
where the mechanical frequency ωM is much larger than the cavity linewidth κ. (b) The
mechanical motion can also generate weak amplitudes at frequencies ωcav ± 2ωM via non-resonant Raman processes. As the density of states for such processes is small, their amplitude
is weak. These processes effectively measure the mechanical Y quadrature, and thus causinga small back-action heating of X.

the cavity equally with coherent tones at both the red and blue mechanical sidebands,
i.e. at ωcav ± ωM . This is equivalent to amplitude modulating the laser drive on thecavity. We thus consider a cavity input field (i.e. laser drive) of the form:āin (t) ∼ e−iωcav t sin (ωM t + φ)

(4.5)

The phase φ of the amplitude modulation will directly determine the definition of themechanical quadrature which couples to the cavity field; we take φ = 0 for simplicity in what follows. As usual, this coherent-state drive will induce an average (timedependent) amplitude in the cavity α(t) that can be found by solving the classicalequations of motion. We will again work in a displaced frame, and thus write the cavity lowering operator as this classical amplitude plus a correction dˆ which describes
noise effects and the effects of the optomechanical coupling:ˆâ = ācl (t) + d,ācl (t) = α0 e−iωcav t cos ωM t

(4.6)

We can take α0 to be real without loss of generality.
Linearizing the optomechanical interaction in the usual way yields:†

Ĥint = gâ â

x̂
xZPF


'

g
xZPF

 

ˆ iωcav t + h.c.
α0 cos(ωM t) x̂ de≡ A(t)x̂ · F̂ (t)

(4.7)
(4.8)

36

Backaction evasion and conditional squeezing

We see that modulating the average cavity intensity naturally provides the modulated coupling we are after. Working in an interaction picture with respect to the freemechanical and cavity Hamiltonians Ĥ0 = ωM b̂† b̂ + ωcav dˆ† dˆ yields

i
G h
(4.9)
Ĥint = √ X̂ (1 + cos 2ωM t) + Ŷ sin 2ωM t dˆ + dˆ†2where we have defined the many-photon coupling G = gα0 . Note that in our interaction
picture, the mechanical quadrature operators have no explicit time-dependence, andare given by Eqs. (4.1).
We now make use of the resolved sideband, good-cavity condition ωM  κ, and thefact that we will be interested in coupling G  κ. In this limit the terms oscillating
at frequency 2ωM will average to zero on the timescales relevant to the dynamics(which will all be much longer than 1/κ). We can thus safely make the rotating-wave
approximation (RWA), and drop the oscillating terms, resulting in a time-independentinteraction Hamiltonian:!
dˆ + dˆ†√Ĥint = GX̂ ·≡ GX̂ · X̂cav(4.10)
2
We see that the mechanical X̂ quadrature is coupled only to the corresponding X
quadrature of the cavity. Without dissipation, both X̂ and X̂cav are constants of the
motion. Similar to the case of standard position detection, as the mechanics only
couples to the X cavity quadrature, information
on its motion will only drive theconjugate cavity quadrature Ŷ ∼ −i dˆ − dˆ† (i.e. the optical phase quadrature). By
measuring the output Y quadrature, we thus obtain a measurement of the mechanicalX̂ quadrature.
ˆTo see this explicitly, we first solve the Heisenberg Langevin equation for d:
√
d ˆ
κˆ
iG
d(t) = − d(t)− √ X̂(t) − κdˆin (t)dt
22(4.11)

Information on the mechanics will be at the cavity field near resonance, in a bandwidth∼ γ  κ. We thus only need to describe dˆ at frequencies  κ, and solve this equation
adiabatically, i.e. ignoring the d/dt term. This yields:2ˆ = −2iG√ X̂(t) − √ dˆin (t)
d(t)
κ
2κ

(4.12)

Information on the mechanical X quadrature is, as expected, encoded solely in theˆ and hence in the cavity Y quadrature. One thus measures theimaginary part of d,
Y quadrature of the output field. The measurement output (i.e. output homodyne
current) is then:
√
√
√
2 2G
ˆ
ˆ
I(t) = 2Ŷcav,out ∼ 2κŶcav ∼ √ X̂(t) + ξ(t)(4.13)
κ√
2 2G ≡ √X̂(t) + δ X̂imp (t)(4.14)
κTwo-tone QND scheme

37

where ξˆ = −i(dˆin − dˆ†in ) is the Y quadrature of the input noise driving the cavity, andis delta-correlated. In the last line, we have introduced the imprecision noise operatorˆδ X̂imp (t) in the usual way, by referring√ the intrinsic noise in I(t) back to the measured
mechanical quadrature. Note the 2 factor in the definition of the homodyne current
has just been included for convenience (the overall prefactor plays no role in our
analysis).
Using this form, one easily finds that symmetrized spectral density of the imprecision noise is given by:1κ≡(4.15)S̄XX,imp [ω] =8G2k̃The imprecision noise spectral density has the units of an inverse rate, and we havethus used it to define an effective measurement rate k̃.
The rate k̃ has a simple interpretation: it tells us how quickly the power signalto-noise (SNR) ratio grows for a measurement where we try to resolve whether themeasured mechanical quadrature has been displaced an amount ∼ xZPF , i.e. is X = 1
or 0? A simple estimator would be to simply integrate the output current I(t), i.e.
Z t
ˆ 0)m̂(t) ≡
dt0 I(t(4.16)
0

It is easy to then check that in the absence of mechanical dissipation (and to lowest
order in the coupling to the detector)
p
2
2
k̃t
−
0
signal power[hm̂(t)iX=1 − hm̂(t)iX=0 ]
SNR ≡
=
=
= k̃t
(4.17)
noise powerhhm̂2 (t)iitwhere the variance hhm̂2 (t)ii = hm̂2 (t)i − hm̂(t)i2 .
Having understood the basics of the measurement, we can now ask about the effectsof back-action. Consider first the ideal limit where κ/ωM → 0, and hence the cavity
has strictly no coupling to the mechanical Ŷ quadrature. In this case X̂ commuteswith the interaction Hamiltonian, implying that it is completely unaffected by the
coupling. By solving the mechanical Heisenberg-Langein equation, one finds that thefluctuations are the same as when G = 0, i.e.
Z
D
E

1 ∞
γ/2
S̄XX [ω] ≡dteiωt {X̂(t), X̂(0)} = 2
1 + 2n̄M(4.18)th2 −∞ω + (γ/2)2
Again, this is exactly what we would have without any coupling to the cavity.
In contrast, the mechanical Y quadrature is driven by the fluctuations in the cavity
X̂cav operator. This back-action does not change the damping rate of the quadrature,
but does heat it an amount corresponding to n̄BA quanta. One finds:S̄Y Y [ω] =γ/2
ω 2 + (γ/2)2
1 + 2n̄M
th + 2n̄BA ,n̄BA =

2G2≡C
κγ(4.19)We have introduced here again the optomechanical cooperativity C, which in thiscontext, can be viewed as the ratio of the measurement rate to the intrinsic mechanicaldamping rate.

38

Backaction evasion and conditional squeezing

a)

b)

Y

X

c)

Y
X

Y

X

Fig. 4.2 Mechanical state associated with various aspects of the BAE measurement, depicted as probability density in the phase space associated with the mechanical quadratureamplitudes X and Y . (a) Mechanical state without the measurement: a thermal state withequal uncertainty in the X and Y quadratures. (b) Unconditional mechanical state when themeasurement is on. The measured X quadrature is unaffected by the measurement, whereas
the back-action of the of the measurement heats the Y quadrature and thus increases its
uncertainty. (c) Conditional mechanical state. If we use the information in the measurement
record I(t) associated with a particular run of the experiment, we see that the mechanicalresonator is in a squeezed state, where the X quadrature uncertainty has been greatly reduced compared to its original value. However, the mean values of X and Y (i.e. centre of
the ellipse) undergoes a random walk which is completely correlated with the measurement
record. If we average over these fluctuations (e.g. discard the measurement record), we recover
the picture in panel (b).

What if we include the effects of κ/ωM ? The oscillating terms in Eq. (4.9) are now
not completely negligible. Heuristically, they give rise to scattering processes whereincident drive photons are scattered to frequencies ωcav ± 2ωM (see Fig. 4.1 (b)). Suchprocesses have a very small amplitude due to the very small cavity density of states atthese frequencies. They do however contain information on the mechanical Y quadrature, thus result in heating of the mechanical X quadrature. A careful calculation(see (Clerk et al., 2008)) finds that this additional heating results can be captured bymaking the replacement:

2
1
κ
M
n̄M
→
n̄
+
(4.20)thth
32 ωM
in Eqs. (4.18),(4.19) for the quadrature noise spectral densities

4.3

Conditional squeezing: heuristic description

We have seen that a perfect back-action evading measurement results in a heating of
the unmeasured, conjugate mechanical quadrature Y , while the measured mechanicalquadrature X is completely unaffected by the measurement, c.f. Eq. (4.18). This is in
keeping with the fact that X̂ commutes with the optomechanical coupling Hamiltonian,and is thus completely unaffected by the light field used for the measurement. We thus
expect the mechanical state (pictured as a phase space distribution) to be modified

Stochastic master equation description of a conditional measurement39

as shown in Fig. 4.2(b). Note that the total entropy of the state appears to have been
increased by the measurement.
While this seems simple enough, it would seem to contradict another aspect of ourtreatment. Namely, we showed that the added noise of the measurement (which has
no back-action contribution) could be made arbitrarily small by increasing the themeasurement strength (i.e. G), meaning that one could make an essentially perfectmeasurement of X. This seems to imply that the measurement will greatly reduce theuncertainty in the X quadrature, in stark contrast to what is depicted in Fig. 4.2(b).
There is of course no contradiction here. To fully describe the reduction in X uncertainty occurring during a measurement, we need to understand what happens during aparticular run of the experiment: given a particular history of the measurement output
I(t), what is the state of the mechanical resonator? We will show that there is a particularly simple picture for this “conditional” mechanical state, as is shown schematicallyin Fig. 4.2(c). Once transients have died away, the conditional mechanical state will
have greatly reduced X quadrature uncertainty– it will be a squeezed state. However,
the mean position of this state in phase space will undergo a random walk, a random
walk which is completely correlated with the seemingly random fluctuations of themeasurement record I(t). If we have access to the measurement record, we can follow these fluctuations, and thus to us, this random motion does not represent a true
uncertainty.
In contrast, for an observer who does not have access to I(t), the fluctuations in
the mean position of the mechanical squeezed state are just another uncertainty inthe mechanical state. For such an observer, the mechanical resonator is best describedby the unconditional state, where we average over all possible measurement outcomes,
and thus include the random walk done by the conditional squeezed state means in thestate uncertainty. Doing so, we return to the picture in Fig. 4.2(b): the fluctuations in
the mechanical resonator X quadrature to an observer who does not have access tothe measurement record are the same as they were without the measurement.
In what follows, we will develop in more detail the theoretical tools required todescribe such a conditional evolution and the conditional mechanical state. We willthen apply this formalism to our optomechanical back-action evading measurement tounderstand how one can generate squeezed mechanical states from the measurement.

4.4

Stochastic master equation description of a conditional
measurement

In this section, we will give a quick and dirty “derivation” of the stochastic masterequation approach that describes the conditional evolution of the mechanical state
during an ideal version of our back-action evading measurement. Our approach follows
closely the extremely pedagogical treatment in (Jacobs and Steck, 2006), which itself isrelated to the derivation in (Caves and Milburn, 1987). The derivation, interpretation
and use of such stochastic master equations are treated extensively in recent textbooks(Wiseman and Milburn, 2014; Jacobs, 2014).
4.4.1

Discretized measurement record

We start by writing the measurement output (Eq. (4.14)) as40

Backaction evasion and conditional squeezing

ˆ =
I(t)

p

ˆ
k̃ X̂(t) + ξ(t)(4.21)

As we already saw in the discussion surrounding Eq. (4.17), if we only look at I(t)over a very short time interval, we get almost no information on the state of the mechanical resonator: the noise ξ(t) in the measurement record completely dominates thecontribution from the signal X(t). Useful information is thus only acquired gradually
over time. To describe our measurement, we want to quantiatively describe the small
information gain that occurs over a given short time interval, and importantly, also
describe the corresponding change in the mechanical state.
We start by discretizing time into finite intervals of width ∆t and define tj ≡ j∆t;
we will eventually take the ∆t → 0 limit. During each of these finite intervals, we use
the information in I(t) to estimate the state of the mechanics. This is done analogously
to Eq. (4.17). The estimate of X derived from I(t) in the interval (tj−1 , tj ) is describedby the operator X̂tj , defined as
1

X̂tj ≡ p
k̃∆t

Z tj

ˆ 0 ).
dt0 I(t(4.22)

tj−1

For simplicity, we start by assuming that the instantaneous mechanical resonatorstate is pure and described by the wavefunction |ψ(t)i, and assume that it has no
intrinsic dynamics– we ignore its intrinsic dissipation and any external forces. We will
also assume that any back-action disturbance occurs instantaneously at the end of eachinterval (tj−1 , tj ), and hence |ψ(t)i is constant over the interval. As we are interested
in the ∆t → 0 limit, this approximation is not unreasonable. Using this fact and theˆfact that hξ(t)i= 0, we havehX̂tj i ' hX̂(t)i ≡ hψ(t)|X̂|ψ(t)i(4.23)

We can also calculate the variance of X̂tj :2  
D
E2 
X̂tj
≡
X̂tj − X̂tj
' p
=

Z tj

1
k̃∆t

2

1
≡σ
k̃∆t

tj−1

dt0

Z tj
tj−1

ˆ 0 )ξ(tˆ 00 )idt00 hξ(t(4.24)(4.25)

Note that we have ignored the contribution to the variance from the intrinsic uncertainty of X̂ in the mechanical resonator state |ψ(t)i. This is because the contribution
we keep (which is solely from the imprecision noise of the measurement, ξ(t)) completely dominates the intrinsic noise in the limit of small ∆t. This imprecision-noise
contribution scales like 1/∆t, while the intrinsic noise would tend to a constant.
We have thus discretized the measurement record I(t) that would be obtained ina given experimental run into a set of discrete X-quadrature estimates Xtj . Theseestimates can be viewed as an effective classical stochastic process, i.e.

Stochastic master equation description of a conditional measurement∆Wj
Xtj = hX̂(tj )i + p
k̃∆t

41

(4.26)

Here, the ∆Wj are random variables describing the fluctuations of each estimate dueto the imprecision noise ξ(t) in the measurement record. Equivalently, they represent
the difference in the outcome of the measurement during the given time interval fromthe mean of X̂ in the state |ψ(t)i. As ξ(t) is Gaussian white noise, it follows thatthe ∆Wj are each Gaussian random variables with zero mean, and further, are notcorrelated with one another. From Eq. (4.25), we have simply:∆Wj = 0∆Wj ∆Wj 0 = δj,j 0 ∆t

(4.27)The bar here represents an average over the classical stochastic process. We stressthat the classical stochastic process defined by Eq. (4.26) has been constructed sothat it yields the same statistics as the microscopic (quantum) theory describing ourmeasurement.
4.4.2

State evolution

Having come up with a simple way to think about the measurement record produced
in a single run of the experiment, we now return to the question of back-action: ina particular run of the experiment (described by a particular set of Xtj ), how will
the state |ψ(t)i of the mechanical resonator change over time? Again, we are only
considering here mechanical dynamics caused by the measurement, and are assumingthat these disturbances happen instantaneously between each measurement interval(tj−1 , tj ). At the end of a given measurement interval, the experimentalist will haveobtained a particular measurement result Xtj . If the measurement was a simple, strong
projective measurement, the mechanical state after the measurement would be completely determined by the outcome Xtj – the mechanics would just be projected into
the eigenstate of X̂ corresponding to this outcome.
In our system, we are however very far from the strong measurement limit, as
we only obtain a tiny bit of information on X during the given time interval. The
final mechanical state will thus depend both on the initial mechanical state and themeasurement outcome, and we can write:|ψ(tj + ∆t)i = M̂ Xtj |ψ(tj )i(4.28)
The operator M̂ Xtj describes the outcome-dependent disturbance of the mechanicalstate by the measurement in a given time interval. It is a so-called Krauss operator,
used in the POVM description of the kind of incomplete, weak measurements relevant
here (see, e.g., (Jacobs and Steck, 2006) for a more complete discussion). We assume
the ideal case where this state disturbance is as small as possible given the information
gain of the measurement. As the statistics of our measurement outcomes are Gaussian
distributed and dominated by imprecision noise, the corresponding Krauss operators
also have a Gaussian form:

2 
1 
,
(4.29)M̂ (x) ∝ exp − 2 x − X̂4σ

42

Backaction evasion and conditional squeezing

where σ = 1/(k̃∆t) is variance of each discrete measurement (c.f. Eq. (4.25)), andwe have dropped a normalization constant. Note that in extreme limit where σ →∞, M̂ (x) → 1: there is no information obtained in the measurement, and hence nomeasurement-induced change in the mechanical state. In the opposite case σ → 0, theM̂ (x) → |xihx|, i.e. the Krauss operators become projectors onto eigenstates of X̂,and we recover the standard description of a strong measurement.
Using Eq. (4.26), we can re-express the post-measurement mechanical state interms of the random variables ∆Wj as:
s

 

k̃
k̃∆t 2  k̃∆t(4.30)X̂ +hX̂(tj )i +∆Wj  X̂  |ψ(tj )i,|ψ(tj + ∆t)i ∝ exp −42
4
where again we have dropped purely constant prefactors affecting the normalizationof the state.
Next, as we are interested in the ∆t → 0 limit, all the terms in the argument of theabove exponential will become small, and we can thus Taylor expand keeping terms toorder ∆t only. This implies that we need terms that are up to second√ order in ∆Wj , asthis random variable has a variance ∆t, and hence a typical size ∆t. We thus find:
s




k̃
k̃∆t
k̃∆t
k̃
2
X̂ 2 + hX̂(tj )i +∆Wj  X̂ + (∆Wj ) X̂ 2  |ψ(tj )i|ψ(tj + ∆t)i ∝ 1 −4
2
4
8
(4.31)
Finally, we want to take the limit where the duration of our small time intervals
tends to an infinitesimal, ∆t → dt, which will give us a stochastic differential equation
for the evolution of the mechanical state. In this limit, we will follow standard convention, and label ∆W as dW , the so-called Wiener increment. In general, one would
think that dW 2 is itself a random variable which fluctuates. In the ∆t → 0 limit, thesefluctuations play no role, and we can rigorously replace dW 2 by its average value, dt(see, e.g., Ch. 3 of (Jacobs, 2010) for a detailed discussion). Using this, and also adding
terms to ensure that our state remains normalized, we find:d|ψi ≡ |ψ(t + dt)i − |ψ(t)i
s



2


k̃
k̃
= −
X̂ − hX̂i dt +X̂ − hX̂i dW  |ψ(t)i
8
4
In the same limit, the measurement record can be represented as:
Z t
p
dI ≡
I(t0 ) → k̃hX̂(t)idt + dW

(4.32)(4.33)

t−∆t

Several comments are now in order:• Eq. (4.32) and Eq. (4.33) are coupled stochastic differential equations. The random
Wiener increment dW both dictates the evolution of the measurement record andStochastic master equation description of a conditional measurement

43

the evolution of the state of the mechanical resonator. Formally, it tells us thatthe noise in the measurement record directly reflects changes in the mechanicalstate.
• Eq. (4.32) is a nonlinear, as the expectation value hX̂i on the RHS is of course itself
a function of the mechanical state. It should be evaluated using the mechanicalstate at time t.
• It is useful to interpret the two terms Eq. (4.33) in a manner that connects to
Baysian probabilities. The first term is what we “expect” from our measurement,
given our current knowledge of the mechanical state (as represented by |ψ(t)i).
The second term (dW ) represents the “surprise” of our measurement. Just as in
Baysian probabilities, we should update our knowledge of the mechanical statebased on this new information. This is exactly the role of the dW term on the
RHS of Eq (4.32).
Finally, it is straightforward to calculate the corresponding equation of motion forthe density matrix describing the mechanical state, using ρ̂ = |ψihψ| and Eq. (4.32).
As usual, one needs to retain terms to order ∆W 2 , as in the ∆t → 0 limit, ∆W 2 →dW 2 = dt. One finds:
s
h
h
ii

k̃ 
k̃
X̂ ρ̂ + ρ̂X̂ − 2hX̂iρ̂ dW(4.34)dρ̂= − X̂, X̂, ρ̂ dt +84meas
Again, the last term on the RHS ∝ dW describes how the evolution of mechanicalstate is correlated with the noise in the measurement record; equivalently, it tells us
how our knowledge of the mechanical state (as encoded in ρ̂) is updated by the surprisedW of the measurement. Note that if ρ̂ is initially in a pure state, then Eq. (4.34) keeps
it in a pure state at all times (for the simple reason that the evolution described by
Eq. (4.32) also keeps the mechanical state pure). The evolution of the density matrix
clearly depends on the particular history and form of the measurement record (throughthe dW ) terms, and we call this the conditional density matrix. One could also ask
what the state of the mechanical resonator is averaged over all possible measurement
outcomes. Equivalently, if we don’t have access to I(t), how would we describe the mechanical state? Averaging over possible measurement records is equivalent to averaging
over the dW , which is simple, as dW = 0. Hence, the unconditional density matrix
evolves only under the first term on the RHS of Eq. (4.34). As can easily be checked,
this term causes off-diagonal elements of the the density matrix in the X̂-eigenstatebasis to decay exponentially. This is just the expected unconditional back-action of
the measurement, which is a heating of the unmeasured Y quadrature.
Note that while we have given a rather heuristic derivation of the conditional
master equation for our system, it is possible to give a more microscopic description,
one that starts with the photodetection involved in the homodyne measurement of thecavity output field, see (Clerk et al., 2008) for details.
Finally, we also need to include terms which correspond to the measurementindependent evolution of the mechanical resonator. In our case, the mechanical quadratures have no free Hamiltonian evolution, but will be subject to the heating and damping by the intrinsic sources of mechanical dissipation. We have discussed how these44

Backaction evasion and conditional squeezing

can be described using the Heisenberg-Langevin formalism. In the Markovian limit ofinterest, they can equivalently be described by Linblad terms in a quantum masterequation:





 †
1n † o1n † o†M= γ 1 + n̄M
b̂ρ̂
b̂
−
b̂
b̂,
ρ̂
dt
+
γ
n̄
b̂
ρ̂
b̂
−
b̂
b̂
,
ρ̂
dt (4.35)
dρ̂
th
th
2
2
diss
These describe the additional and removal of quanta from the mechanical resonatorby a thermal reservoir. The total evolution of the system in the presence of the measurement is then:(4.36)+ dρ̂dρ̂ = dρ̂
diss

4.5

meas

Conditional back-action evading measurement

We can now apply the general theory of the previous section to our ideal cavity optomechanical single quadrature measurement. We want to describe the conditional
state of the mechanical resonator, i.e. its state during a particular run of the measurement, with a particular measurement record I(t). As the RHS of the conditional
master equation has no terms that are involve more than two mechanical raising or
lowering operators, it has the property that Gaussian states (e.g. a thermal state, theground state) remain Gaussian under the evolution. We can thus reduce our conditional master equation to a set of ODE’s for the means and variances of the Gaussianstate. For the X̂ quadrature, we thus need to know:2X̄(t) ≡ hX̂(t)icond
VX (t) ≡ hX̂ 2 (t)icond − hX̂(t)icond(4.37)
where all averages are with respect to the conditional density matrix of the mechanicalresonator.
One finds straightforwardly that the evolution of the variances and covariances arecompletely deterministic, i.e. they do not involve dW . For the X quadrature, one finds:


dVX
1
2
M
= −k̃VX − γ VX − n̄th −(4.38)
dt
2This has an extremely simple interpretation. Without the measurement k̃ = 0, theintrinsic mechanical dissipation causes VX to relax exponentially (rate γ) to its thermal
equilibrium value. In contrast, with the measurement, the first term tries to relax VX
towards zero, i.e. prepare a squeezed state where the X quadrature variance is minimal.
Note that for our system, the measurement strength k̃ = 4Cγ, where the cooperativity is defined in Eq. (4.19). For the large cooperativity limit, the stationary valueof the variance is given by:rr
1 + 2n̄M
1
VX
th
→
≡
(4.39)
VX,zpt2C
2Cthwhere in the last line, we have introduced the so-called thermal cooperativity. We thus
see that achieving strong quantum squeezing requires Cth > 1/2; this is similar to theConditional back-action evading measurement45

condition required for ground state cavity cooling with a system in the good cavity
limit.
We can also ask about how we recover the unconditional picture of the measurement, where the X quadrature is unaffected. Recall the picture in Fig. 4.2(c): the X
quadrature is in a squeezed state whose mean fluctuates in a way that is correlatedwith the measurement record. If we don’t have access to the measurement record, weshould include these fluctuations of the mean in the uncertainty of X, and expect that
this will offset the squeezing found before. To make this quantitative, we can calculate
the equation of motion for the average of X in the conditional mechanical state fromthe conditional master equation. One finds:pγ(4.40)dX̄(t) = − X̄dt + k̃VX (t)dW
2
The evolution of the mean of X is indeed fluctuating, in a way that is completely
correlated with the measurement record (i.e. dW both determines X̄ and the measurement record I(t)). If one averages over these fluctuations, one can explicitly checkthat:
1
VX + hX̄(t)2 i = + n̄M(4.41)
th2
i.e. one recovers the results of the unconditional theory, where the X quadrature variance is the thermal equilibrium value, the same as though there were no measurement.
What about the unmeasured Y quadrature, how does it evolve? Letting C(t) denote the covariance 21 h{X̂(t), Ŷ (t)}i, we have again that the remaining variances andcovariances also evolve deterministically:


dVY
1
k̃
+
= −k̃C 2 − γ VY − n̄M(4.42)
−
th
dt
2
4
dC
= −k̃VX C − γC
(4.43)
dt
The last term on the RHS of Eq. (4.42) describes the expected heating of the measurement. The first term tells us that if there are correlations between the quadratures,
then our measurement of X also has the effect of reducing the uncertainty in Y . Finally, the last equation tells us that under the measurement, any initial correlations
between the quadratures will decay away.
Finally, we have considered so far the case of a perfect measurement. It is of course
also important to understand what happens to the conditional squeezing generated
by the measurement when things are not so perfect. One key imperfection is that the
final measurement of the cavity output quadrature is not perfect. This could be due tolosses (which replace signal by vacuum noise), or due to unwanted added noise in the
final homodyne measurement (e.g. due to a following non-quantum-limited amplifier
in a microwave frequency optomechanics experiment). In either case, the next effectis to reduce the size of the signal term in Eq. (4.14) while keeping the noise term√
the same, i.e. k̃ → k̃ η, where the efficiency η ≤ 1. Including this imperfection, one
finds that the conditioning of the X quadrature (first term in Eq. (4.41) is reduced
by a factor of η, whereas the back-action heating of the Y quadrature is unchanged

46

Backaction evasion and conditional squeezing(last term in Eq. (4.42)). As a result, in the large cooperativity limit,the condition for
quantum squeezing becomes more stringent:
!
1
2G2
 ≥
Cth ≡
(4.44)
M2η
κγ 1 + 2n̄th

4.6

Feedback to create unconditional squeezing

As discussed in (Clerk et al., 2008), one can convert the conditional squeezing described above into true squeezing: one needs to use information in the measurement
record to apply an appropriate feedback force which suppresses the fluctuations in theconditional mean X̄(t). The feedback force should create an extra damping of the X̄(t);we thus need to apply a linear force which couples to the mechanical Ŷ quadrature:γ
Ĥfb = −α X̄(t) · Ŷ2

(4.45)

Here, α(γ/2) is the strength of the applied feedback. It is easy to check that this feedback force increases the damping of X̄ in Eq. (4.40) by an amount αγ/2. For large α,
this feedback-induced damping will suppress the X̄ fluctuations, and the unconditional
state variance of X will be the same as the conditional (squeezed) variance.
This might still seem mysterious: how does the experimentalist know what X̄(t)is? To answer this, we re-write Eq. (4.40) (including the feedback force) in terms ofthe measurement record dI(t) defined in Eq. (4.33). We find:
p
p 

d
γ
X̄ = − (1 + α) X̄ + k̃VX dI(t) − k̃ X̄
(4.46)
dt
2
We can now solve this equation, expressing X̄(t) in terms of the measurement record
at earlier times. Assuming VX has achieved its stationary value, we have
Z tp0
X̄(t) = k̃VX
dt0 e−Γ(t−t ) dI(t0 )(4.47)−∞γΓ = (1 + α) + k̃VX
2

(4.48)

Thus, the conditional mean X̄(t) (i.e. our best estimate for the value of the mechanical
X quadrature) is determined by filtering the measurement record obtained at earlier
times. The optimal filter is just exponentially decaying (i.e. the measurement record
at recent times influence our estimate more than the record at earlier times), with atime constant 1/Γ that depends on both the measurement strength (through k̃) andon the feedback strength (through α). As has been discussed extensively, the optimalfilter here coincides with the classical Kalman filter (see, e.g. (Jacobs, 2014)).
We now have a concrete prescription for how to get unconditional squeezing via
measurement plus feedback. At each instant in time, one first constructs the optimal
estimate of X̄(t) from the measurement record as per the above equation. Next, one
uses this estimate to apply the appropriate linear feedback force which damps X̄.
More details on feedback-induced squeezing in this system are provided in (Clerk
et al., 2008).

Extensions of back-action evasion techniques

4.7

47

Extensions of back-action evasion techniquesMore elaborate versions of back-action evasion measurements in quantum optomechanics are possible. In a system where two mechanical resonators are coupled to asingle cavity, one can extend the two-tone driving approach to make a back-action
free measurement of two commuting collective mechanical quadratures (Woolley and
Clerk, 2013) (e.g. the sum of the X quadratures of mechanical resonator 1 and 2,
and the difference of their Y quadratures). This is allows the possibility to measureboth quadratures of an applied force with absolutely no quantum limit; it is intimately
connected to ideas developed in (Tsang and Caves, 2010) and (Wasilewski et al., 2010).
Another interesting possibility is to “break” the QND measurement described in
the previous sections by slightly imbalancing the amplitudes of the two cavity drive
tones. This imbalance ruins the back-action evasion nature of the measurement, as
the mechanical X quadrature no longer commutes with the Hamiltonian. However,
the resulting measurement backaction on X can be harnessed to directly squeeze themechanical resonator. As discussed extensively in (Kronwald et al., 2013), this is an
example of coherent feedback, where the driven cavity both “measures” the mechanical
X quadrature, and also applied exactly the correct feedback force needed to squeeze
the mechanical resonator. This coherent feedback approach has the strong advantage
of not requiring the experimentalist both to make a shot-noise limited measurement of
the cavity output, nor to process a classical measurement record and generate an ideal
feedback force. In a sense, the driven cavity does all the work. This scheme has been
implemented in three recent experiments to achieve true quantum squeezing, where theX-quadrature mechanical uncertainty drops below the zero-point value (even though
one starts from a thermal state) (Wollman et al., 2015; Pirkkalainen et al., 2015;
Lecocq et al., 2015). This general idea of coherent feedback can also be extended to
the “two-mode” backaction-evasion scheme described above, thus providing a means
for generating mechanical entanglement (Woolley and Clerk, 2014).

Appendix A
Derivation of power gain expression
To be able to say that our detector truly amplifies the motion of the oscillator, it is
not sufficient to simply say the response function χIF must be large (note that χIFis not dimensionless!). Instead, true amplification requires that the power deliveredby the detector to a following amplifier be much larger than the power drawn by thedetector at its input– i.e., the detector must have a dimensionless power gain GP [ω]much larger than one. If the power gain was not large, we would need to worry about
the next stage in the amplification of our signal, and how much noise is added inthat process. Having a large power gain means that by the time our signal reaches
the following amplifier, it is so large that the added noise of this following amplifier isunimportant
To make the above more precise, we start with the ideal case of no reverse gain,
χF I = 0. We will define the power gain GP [ω] of our generic position detector ina way that is analogous to the power gain of a voltage amplifier. Imagine we drive
the oscillator we are trying to measure (whose position is x) with a force 2FD cos ωt;ˆthis will cause the output of our detector hI(t)i
to also oscillate at frequency ω. To
optimally detect this signal in the detector output, we further couple the detector
output I to a second oscillator with natural frequency ω, mass M , and position y:0
there is a new coupling term in our Hamiltonian, Hint
= B Iˆ · ŷ, where B is a couplingstrength. The oscillations in hI(t)i will now act as a driving force on the auxiliary
oscillator y (see Fig 3.1). We can consider the auxiliary oscillator y as a “load” we aretrying to drive with the output of our detector.
To find the power gain, we need to consider both Pout , the power supplied to the
output oscillator y from the detector, and Pin , the power fed into the input of theamplifier. Consider first Pin . This is simply the time-averaged power dissipation of the
input oscillator x caused by the back-action damping γBA [ω]. Using a bar to denote atime average, we have2
Pin ≡ M γBA [ω] · ẋ2 = M γBA [ω]ω 2 |χxx [ω]|2 FD
.

(A.1)

Note that the oscillator susceptibility χxx [ω] includes the effects of γBA , c.f. Eq. (3.52).
Next, we need to consider the power supplied to the “load” oscillator y at the detector output. This oscillator will have some intrinsic, detector-independent damping γld ,as well as a back-action damping γout . In the same way that the back-action damping
γBA of the input oscillator x is determined by the quantum noise in F̂ (cf. Eq. (3.20)),the back-action damping of the load oscillator y is determined by the quantum noiseˆin the output operator I:

Derivation of power gain expression

B2[−Im χII [ω]]Mω 2
SII [ω] − SII [−ω]
B
,
=
M ~ω
2

49

γout [ω] =(A.2)

ˆ responds towhere χII is the linear-response susceptibility which determines how hIi
ˆa perturbation coupling to I:
Z
iE
i ∞ Dh ˆˆχII [ω] = −dt I(t), I(0)eiωt .
(A.3)~ 0
As the oscillator y is being driven on resonance, the relation between y and I is givenby y[ω] = χyy [ω]I[ω] with χyy [ω] = −i[ωM γout [ω]]−1 . From conservation of energy, we
have that the net power flow into the output oscillator from the detector is equal tothe power dissipated out of the oscillator through the intrinsic damping γld . We thus
have
Pout ≡ M γld · ẏ 2

= M γld ω 2 |χyy [ω]|2 · |BAχIF χxx [ω]FD |21γld=· |BAχIF χxx [ω]FD |2 .
M (γld + γout [ω])2(A.4)

Using the above definitions, we find that the ratio between Pout and Pin is independent of γ0 , but depends on γld :Pout
1 A2 B 2 |χIF [ω]|2γld /γout [ω]
= 2 2
.
Pin
M ω γout [ω]γBA [ω] (1 + γld /γout [ω])2(A.5)

We now define the detector power gain GP [ω] as the value of this ratio maximized
over the choice of γld . The maximum occurs for γld = γout [ω] (i.e. the load oscillator
is “matched” to the output of the detector), resulting in:Pout
GP [ω] ≡ max
Pin
1A2 B 2 |χIF |2=
4M 2 ω 2 γout γBA|χIF [ω]|2=(A.6)
4Im χF F [ω] · Im χII [ω]In the last line, we have used the relation between the damping rates γBA [ω] andγout [ω] and the linear-response susceptibilities χF F [ω] and χII [ω], c.f. Eq. (3.24). We
thus find that the power gain is a simple dimensionless ratio formed by the threedifferent response coefficients characterizing the detector, and is independent of thecoupling constants A and B. As we will see, it is completely analogous to the power
gain of a voltage amplifier, which is also determined by three parameters: the voltage
gain, the input impedance and the output impedance.

50

Derivation of power gain expression

Finally, we note that the above results can be generalized to include a non-zerodetector reverse gain, χF I , see (Clerk et al., 2010). In the case of a perfectly symmetricdetector (i.e. χF I = χ∗IF ), one can show that the power gain is at most equal to one:true amplification is never possible in this case.

References
Andrews, R W, Peterson, R W, Purdy, T P, Cicak, K, Simmonds, R W, Regal, C A,
and Lehnert, K W (2014, March). Bidirectional and efficient conversion between
microwave and optical light. Nat. Phys., 10(4), 321–326.
Aspelmeyer, Markus, Kippenberg, Tobias J., and Marquardt, Florian (2014, Dec).
Cavity optomechanics. Rev. Mod. Phys., 86, 1391–1452.
Averin, D. (2003). Linear quantum measurements. In Quantum Noise in Mesoscopic
Systems (ed. Y. Nazarov), pp. 205–228. Kluwer, Amsterdam.
Bocko, Mark F. and Onofrio, Roberto (1996). On the measurement of a weak classicalforce coupled to a harmonic oscillator: experimental progress. Rev. Mod. Phys., 68,
755–799.
Braginsky, V. B. and Khalili, F. Y. (1992). Quantum Measurement. Cambridge
University Press, Cambridge.
Braginsky, V. B. and Khalili, F. Y. (1996). Quantum nondemolition measurements:
the route from toys to tools. Rev. Mod. Phys., 68, 1.
Braginsky, V. B., Vorontsov, Y. I., and Thorne, K. P. (1980). Quantum nondemolition
measurements. Science, 209, 547–557.
Bruus, Henrik and Flensberg, Karsten (2004). Many-body quantum theory in condensed matter physics: a introduction. Oxford University Press, Oxford.
Caves, C.M. and Milburn, G J (1987). Quantum-Mechanical Model for Continuous
Position Measurements. Phys. Rev. A, 36(12), 5543–5555.
Caves, Carlton M. (1981). Quantum-mechanical noise in an interferometer. Phys.
Rev. D, 23, 1693–1708.
Caves, Carlton M., Thorne, Kip S., Drever, Ronald W. P., Sandberg, Vernon D., andZimmermann, Mark (1980). On the measurement of a weak classical force coupled
to a quantum-mechanical oscillator. i. issues of principle. Rev. Mod. Phys., 52,
341–392.
Clerk, A. A. (2004). Quantum-limited position detection and amplification: A linear
response perspective. Phys. Rev. B , 70, 245306.
Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F., and Schoelkopf, R. J.
(2010). Introduction to quantum noise, measurement and amplification. Rev. Mod.
Phys., 82, 1155.
Clerk, A A and Marquardt, F. (2014). Cavity Optomechanics, Chapter Basic Theoryof Cavity Optomechanics, pp. 5–23. Springer, Berlin.
Clerk, A. A., Marquardt, F., and Jacobs, K. (2008). Back-action evasion and squeezing of a mechanical resonator using a cavity detector. New J. Phys., 10, 095010.
Dong, C, Fiore, V, Kuzyk, M C, and Wang, H (2012, December). Optomechanical
Dark Mode. Science, 338(6114), 1609–1613.
Gerry, C. and Knight, P. (2005). Introductory Quantum Optics. Cambridge University

52

References

Press, Cambridge.
Gottfried, K. (1966). Quantum Mechanics, Volume I: Fundamentals. W. A. Benjamin, Inc.
Hertzberg, J. B., Rocheleau, T., Ndukum, T., Savva, M., Clerk, A. A., and Schwab,
K. C. (2010). Back-action evading measurements of nanomechanical motion. Nature
Phys., 6, 213.
Hill, Jeff T, Safavi-Naeini, Amir H, Chan, Jasper, and Painter, Oskar (2012, October).
Coherent optical wavelength conversion via cavity optomechanics. Nat. Comm., 3,
1196–7.
Jacobs, K. (2010). Stochastic Processes for Physicists. Cambridge University Press,
Cambridge.
Jacobs, Kurt (2014). Quantum Measurement Theory and its Applications. Cambridge
University Press, Cambridge.
Jacobs, K. and Steck, D. A. (2006). A straightforward introduction to continuous
quantum measurement. Contemporary Physics, 47, 279.
Khalili, Farid, Miao, Haixing, Yang, Huan, Safavi-Naeini, Amir H, Painter, Oskar,and Chen, Yanbei (2012, September). Quantum back-action in measurements ofzero-point mechanical oscillations. Phys. Rev. A, 86(3), 033840.
Kronwald, Andreas, Marquardt, Florian, and Clerk, Aashish A. (2013, Dec). Arbitrarily large steady-state bosonic squeezing via dissipation. Phys. Rev. A, 88,
063833.
Law, C. K. (1995, March). Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. Phys. Rev. A, 51(3), 2537–2541.
Lecocq, F, Clark, J B, Simmonds, R W, Aumentado, J, and Teufel, J D (2015,
December). Quantum Nondemolition Measurement of a Nonclassical State of a
Massive Object. Phys. Rev. X , 5(4), 041037.
Lemonde, Marc-Antoine and Clerk, Aashish A. (2015, Mar). Real photons fromvacuum fluctuations in optomechanics: The role of polariton interactions. Phys.
Rev. A, 91, 033836.
Marquardt, F., Chen, J. P., Clerk, A. A., and Girvin, S. M. (2007). Quantum theory ofcavity-assisted sideband cooling of mechanical motion. Phys. Rev. Let., 99, 093902.
Pace, A F, Collett, M J, and Walls, D F (1993, April). Quantum Limits in Interferometric Detection of Gravitational-Radiation. Phys. Rev. A, 47(4), 3173–3189.
Pirkkalainen, J.-M., Damskägg, E., Brandt, M., Massel, F., and Sillanpää, M. A.
(2015, Dec). Squeezing of quantum noise of motion in a micromechanical resonator.
Phys. Rev. Lett., 115, 243601.
Suh, J, Weinstein, A J, Lei, C U, Wollman, E E, Steinke, S K, Meystre, P, Clerk,
Aashish, and Schwab, K. C. (2014, June). Mechanically detecting and avoiding thequantum fluctuations of a microwave field. Science, 344(6189), 1262–1265.
Tsang, Mankei and Caves, Carlton M. (2010, Sep). Coherent quantum-noise cancellation for optomechanical sensors. Phys. Rev. Lett., 105, 123601.
Vyatchanin, S.P. and Zubova, E.A. (1995). Quantum variation measurement of aforce. Physics Letters A, 201(4), 269 – 274.
Vyatchanin, S. P. and Matsko, A. B. (1996). Quantum variational measurements offorce and compensation of the nonlinear backaction in an interferometric displace-References

53

ment transducer. JETP , 83(4), 690.
Walls, D. F. and Milburn, G. J. (2008). Quantum Optics (2nd edn). Springer, Berlin.
Wasilewski, W., Jensen, K., Krauter, H., Renema, J. J., Balabas, M. V., and Polzik,
E. S. (2010, Mar). Quantum noise limited and entanglement-assisted magnetometry.
Phys. Rev. Lett., 104, 133601.
Wilson-Rae, I., Nooshi, N., Zwerger, W., and Kippenberg, T. J. (2007). Theory ofground state cooling of a mechanical oscillator using dynamical back-action. Phys.
Rev. Lett., 99, 093901.
Wiseman, H. M. and Milburn, G. J. (2014). Quantum Measurement and Control.
Cambridge University Press, Cambridge.
Wollman, E. E., Lei, C. U., Weinstein, A. J., Suh, J., Kronwald, A., Marquardt,
F., Clerk, A. A., and Schwab, K. C. (2015). Quantum squeezing of motion in amechanical resonator. Science, 349(6251), 952–955.
Woolley, M. J. and Clerk, A. A. (2013, Jun). Two-mode back-action-evading measurements in cavity optomechanics. Phys. Rev. A, 87, 063846.
Woolley, M. J. and Clerk, A. A. (2014, Jun). Two-mode squeezed states in cavity
optomechanics via engineering of a single reservoir. Phys. Rev. A, 89, 063805.