# Light propagation in a random three-dimensional ensemble of point scatterers
in a waveguide: size-dependent switching between diffuse radiation transfer
and Anderson localization of light

A. S. Kuraptsev<EMAIL_ADDRESS>Peter the Great St. Petersburg
Polytechnic University, 195251, St. Petersburg, Russia I. M. Sokolov
<EMAIL_ADDRESS>Peter the Great St. Petersburg Polytechnic University,
195251, St. Petersburg, Russia

###### Abstract

Light transport in a disordered ensemble of resonant atoms placed in a
waveguide is found to be very sensitive to the sizes of cross section of a
waveguide. Based on self-consistent quantum microscopic model treating atoms
as coherent radiating dipoles, we have shown that the nature of radiation
transfer changes from Anderson localization regime in a single-mode waveguide
to a traditional diffuse transfer in a multi-mode one. Moreover, the
transmittance magnitude undergoes complex step-like dependence on the
transverse sizes of a waveguide.

###### pacs:

31.70.Hq, 32.70.Jz, 42.50.Ct, 42.50.Nn

The transition from extended to localized eigenstates upon increasing disorder
in a quantum or wave system is named after Philip Anderson who was the first
to predict it for electrons in disordered solids 1 . This transition manifest
itself in sharp decrease of the diffusion coefficient for electrons down to
zero and associated suppression of the conductivity, thus a conducting medium
turns into a dielectric. Actually, the effect of phase transition induced by
disorder has a general nature. More recently, it was studied for various types
of quantum particles (cold atoms 2 , Bose-Einstein condensates 3 ) as well as
for classical waves (light 4 ; 5 ; 6 , ultrasound 7 ; 8 ). Anderson
localization of light may find applications in the design of future quantum-
information devices 9 , miniature lasers 10 , and solar cells 11 . Drawing a
parallel between Anderson localization predicted for electrons in solids and
localization of light in dispersive medium, the analog of metallic phase is
the regime of diffuse radiation transfer whereas the analog of non-conducting
phase is expressed in Anderson localization of light. The phenomenon of
Anderson localization of light can manifest itself in various physical
systems, but a special attention is paid to random ensembles of point
scatterers (such as cold atomic gases or impurity atoms embedded in a solid
transparent dielectric matrix). This is connected with the fact that these
objects represent an excellent playground for testing the theory.

In the literature, one can find different signs for Anderson localization of
light based on eigenstates diagram (mainly, lifetimes of eigenstates), on the
character of radiation trapping, on the inverse participation ratio, on the
Ioffe-Regel parameter, on the Thouless number, to name a few. However, these
signs represent the necessary conditions for Anderson localization but not
sufficient ones. The most reliable criterion comes from the exact definition
of localization, namely the transmittance in a stationary mode must
exponentially decrease with increasing of the thickness of a sample. There are
also some requirements, which have to be met if we want to use this criterion.
The first is the absence of any relaxation channels which lead to energy
losses of the joint atomic-field system. This circumstance is not a problem in
theoretical simulations, but it can represent a critical point for experiments
since the existence of such channels permits alternative explanations for
exponential behavior of the transmittance 12 ; 13 ; 14 . The second is that a
medium has to be optically dense, so the intensity of transmitted light is
determined mainly by its incoherent component. Actually, both these
requirements are valid not only in the case of transmission criterion but also
for all other signs for Anderson localization of light.

By now, it has been understood that Anderson localization of light in a random
three-dimensional ensemble of point scatterers without applying external
control fields is absent 15 . However, static magnetic field restores Anderson
localization of light in a cold-atom gas 16 . The situation essentially
changes when we deal with low-dimensional systems. It is known that in two
dimensions, there is no true metallic behavior of a disordered electronic
system 17 . Thus, reduced dimensionality facilitates the achievement of
Anderson localization. This also holds true when understanding Anderson
localization in a broad sense, including both metal-insulator transitions and
quantum-Hall-type transitions between phases with localized states 18 .
Focusing on the Anderson localization of light in atomic ensembles, it is
worth to note that the fabrication of low-dimensional ensembles usually
assumes their coupling with a cavity or waveguide structures. This imprints
the nature of interaction between atoms and electromagnetic field, that, in
turn, leads to a modification of interatomic dipole-dipole interaction 19 ; 20
; 21  and associated cooperative effects 22 ; 23 ; 24 ; 25 ; 26 . Correct
description of these effects requires a realistic three-dimensional picture
both because of 3D arrangement of atoms and, which is more fundamental,
because of vector nature of electromagnetic field. Moreover, we have recently
shown that polarization effects play a crucial role in spontaneous decay of an
excited atom in a single-mode waveguide 27 .

In this Letter we report the results of fully three-dimensional calculation of
light intensity transmitted through a random ensemble of point scatterers in a
waveguide. We show that the nature of light transport dramatically depends on
the transverse sizes of a waveguide. Thus, when a waveguide is single-mode
with respect to resonant transition wavelength, the regime of Anderson
localization is realized even for an arbitrarily low atomic density. An
increase in the transverse size turns a single-mode waveguide to a multimode
one, that, in turn, instantly cancels Anderson localization and restores
classical diffuse radiation transfer.

We consider an ensemble of $N\gg 1$ identical two-level atoms at random
position $\\{\textbf{r}_{i}\\}$ inside a waveguide, see Fig. 1. The resonant
frequency $\omega_{0}$ of atoms defines the natural length scale
$1/k_{0}=c/\omega_{0}$, where $c$ is the vacuum speed of light. The ground
state $|g_{i}\rangle$ of an isolated atom $i$ is nondegenerate with the total
angular momentum $J_{g}=0$, whereas the excited states $|e_{i}\rangle$ is
threefold degenerate with $J_{e}=1$ and natural free space linewidth
$\gamma_{0}$. The three degenerate substates $|e_{i,m_{J}}\rangle$ correspond
to the three possible projections $m_{J}=0,\pm 1$ of the total angular
momentum $\textbf{J}_{e}$ on the quantization axis $z$. For convenience, let
us choose the $z$ axis coinciding with the axis of a waveguide.

Figure 1:  Sketch of the waveguide and the atomic ensemble inside it.

The cross section of the waveguide is assumed to be rectangular having the
sizes $a$ and $b$. Atomic ensemble occupies a whole cross section of a
waveguide and has the longitudinal length $L$ along $z$ axis. Both input and
output of a waveguide are far-remote from the edges of atomic medium
(corresponding separations are much larger than $L$). Atomic ensemble is
illuminated by stationary probe radiation, which is considered to be
monochromatic with the frequency $\omega_{s}$. Transmitted radiation is
measured by a photodetector, which absorbs the whole output signal integrated
over the area of cross section indifferent to polarization state.

To describe the stationary regime of atomic excitation induced by external
radiation within the framework of consistent quantum-mechanical treatment, we
use the following technique. Let us suppose that probe radiation is created as
a result of spontaneous emission of some remote atom in a waveguide, which has
the same level structure as atoms from an ensemble but different resonance
transition frequency $\omega_{s}$ and narrow linewidth $\gamma_{s}$ (hereafter
we call it as "source-atom"). It allows us to consider the following initial
conditions: only source-atom is excited whereas all other atoms forming an
ensemble are in their ground state; electromagnetic field is in the vacuum
state at $t=0$. Thus, the problem considered here can be formally reduced to
the problem of collective spontaneous decay in a waveguide, which we studied
previously 27 .

Assuming the walls of a waveguide to be perfectly conductive (i.e., neglecting
the absorption), the dynamics of atomic-field system can be treated on the
basis of the non-steady-state Schrodinger equation with the following
Hamiltonian 28 :

$\displaystyle\widehat{H}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N+1}\sum_{m_{J}=-1}^{1}\hbar\omega_{0}|e_{i,m_{J}}\rangle\langle
e_{i,m_{J}}|$ (1) $\displaystyle+$
$\displaystyle\sum_{\textbf{k},\alpha}\hbar\omega_{k}\left(\widehat{a}_{\textbf{k},\alpha}^{\dagger}\widehat{a}_{\textbf{k},\alpha}+\frac{1}{2}\right)-\sum_{i=1}^{N+1}\widehat{\textbf{D}}_{i}\cdot\widehat{\textbf{E}}\left(\textbf{r}_{i}\right)$
$\displaystyle+$ $\displaystyle\frac{1}{2\epsilon_{0}}\sum_{i\neq
j}^{N+1}\widehat{\textbf{D}}_{i}\cdot\widehat{\textbf{D}}_{j}\delta\left(\textbf{r}_{i}-\textbf{r}_{j}\right),$

where the first two terms correspond to noninteracting atoms and the
electromagnetic field in an empty waveguide, respectively, the third term
describes the interaction between the atoms and the field in the dipole
approximation, and the last, contact term ensures the correct description of
the electromagnetic field radiated by the atoms 28 . In Eq. (1),
$\widehat{a}_{\textbf{k},\alpha}^{\dagger}$ and
$\widehat{a}_{\textbf{k},\alpha}$ are the operators of creation and
annihilation of a photon in the corresponding mode, $\omega_{k}$ is the photon
frequency, $\widehat{\textbf{d}}_{i}$ is the dipole operator of the atom $i$,
$\widehat{\textbf{E}}\left(\textbf{r}\right)$ is the electric displacement
vector in a waveguide, and $\textbf{r}_{i}$ is the position of the atom $i$.
The vacuum reservoir is also included in the atomic-field system described by
the Hamiltonian represented in Eq. (1).

Formally solving the Schrodinger equation for the joint system, which consists
of $N+1$ atom ($N$ atoms of an ensemble + source-atom) and the electromagnetic
field, and restricting ourselves by the states containing no more than one
photon (i.e. neglecting nonlinear effects), one obtains a system of equations
for the amplitudes $b_{e}$ of one-fold atomic excited states with the coupling
between atoms caused by dipole-dipole interaction. For Fourier components
$b_{e}(\omega)$ we have (at greater length see 29 )

$\sum_{e^{\prime}}\bigl{[}(\omega-\omega_{e})\delta_{ee^{\prime}}-\Sigma_{ee^{\prime}}(\omega)\bigl{]}b_{e^{\prime}}(\omega)=i\delta_{es}.$
(2)

The index $s$ as well as the indexes $e$ and $e^{\prime}$ contain information
both about the number of atom and about specific atomic sublevel excited in
the corresponding state.

The matrix $\Sigma_{ee^{\prime}}(\omega)$ describes both spontaneous decay and
photon exchange between the atoms. It is connected with the Green’s matrix
$G_{ee^{\prime}}(\omega)$ by a simple relation
$\Sigma_{ee^{\prime}}(\omega)=(-\gamma_{0}/2)G_{ee^{\prime}}(\omega)$. The
explicit expressions for the elements of the Green’s matrix corresponding to a
waveguide were derived in 27 .

By the inverse Fourier transform, we get the dynamics of the quantum
amplitudes in a time domain, $b_{e}(t)$. Further, we go to the stationary
limit of atomic excitation by external unaffected light source. For this, we
should remove the reverse influence of the atomic ensemble on the source-atom
and consider $\gamma_{s}\rightarrow 0$ and $t\rightarrow\infty$ assuming
$\gamma_{s}t\ll 1$. This leads to the following expression:

$b_{e}(t)=\exp(-i\omega_{s}t)\sum_{e^{\prime}\neq
s}R_{ee^{\prime}}(\omega_{s})\Sigma_{e^{\prime}s}(\omega_{s}),$ (3)

where $R_{ee^{\prime}}(\omega)$ is a resolvent operator of the considered
multi-atomic ensemble, which is defined as
$R_{ee^{\prime}}(\omega)=[(\omega-\omega_{e})\delta_{ee^{\prime}}-\Sigma_{ee^{\prime}}(\omega)]^{-1}$.

The transmission coefficient is defined as follows: $T=I/I_{0}$, where $I$ is
the intensity of transmitted light and $I_{0}$ is the intensity of incident
light; both intensities are averaged over the area of cross section of a
waveguide. Transmitted light intensity can be calculated in a straightforward
manner, as it was done in Refs. 30 ; 31 , or it can be simulated on the basis
of the alternative method, which considers the so-called "atom-detector". The
idea of this alternative approach is taking into consideration an imaginary
elusive "atom", which sense radiation emitted by the environment medium but do
not re-emits photons. So, it works as a point detector. It is also important
here that atom-detector must perceive any kind of polarization of
electromagnetic waves with equal susceptibility. The intensity of light at the
point of atom-detector is proportional to its excited state population. Thus,
the calculation of the transmission coefficient, $T$, is reduced to the
calculation of the population of excited state of atom-detector: consider the
presence of atomic ensemble to find $I$ and its absence to get $I_{0}$.

Figure 2 shows the transmission coefficient depending on the length of atomic
sample, $T(L)$, for both cases of single-mode waveguide, Fig. 2(a), and
multimode one, Fig. 2(b). The transverse sizes of a waveguide were chosen as
$a=4$, $b=2$ when plotting Fig. 2(a) and $a=b=8$ when plotting Fig. 2(b). The
mode composition of a waveguide can be easily justified on the basis of well-
known expression for cutoff frequencies of different modes,
$\omega_{c}=c\sqrt{(m\pi/a)^{2}+(n\pi/b)^{2}}$, where $m$ and $n$ are
transverse indexes of given mode. Thus, in the case of $a=4$, $b=2$, only
$\text{TE}_{10}$ mode can propagate in a waveguide at long distances as
oscillating wave at the transition frequency, while in the case of $a=b=8$,
there are 10 such modes: $\text{TE}_{10}$, $\text{TE}_{01}$, $\text{TE}_{20}$,
$\text{TE}_{02}$, $\text{TE}_{11}$, $\text{TE}_{12}$, $\text{TE}_{21}$,
$\text{TM}_{11}$, $\text{TM}_{12}$, $\text{TM}_{21}$.

Figure 2:  Transmission depending on the length of atomic medium in a
waveguide. $n=2\times 10^{-3}$, $\Delta=1$. (a) $a=4$, $b=2$; (b) $a=b=8$.

Atomic density is chosen sufficiently small, $n=2\times 10^{-3}$, so that
cooperative effects in free space are negligible and, a fortiori, Anderson
localization of light in free space ensemble is impossible. The detuning of
the probe frequency related to the resonant frequency of atomic transition,
$\Delta=\omega_{s}-\omega_{0}$, is chosen $\Delta=1$. Under given condition,
the mean free path of a photon can be estimated on the basis of its free space
value,

$l_{ph}=\frac{1}{n\sigma_{0}}\times\frac{\Delta^{2}+(\gamma_{0}/2)^{2}}{(\gamma_{0}/2)^{2}},$
(4)

where $\sigma_{0}$ is single-atom resonant cross section, $\sigma_{0}=6\pi
k_{0}^{-2}$. Substituting the given values of $n$ and $\Delta$ in Eq. (4), we
get $l_{ph}\approx 133$. So, in order to study incoherent light transfer, we
should consider the values of $L$ several times larger than this estimation.
That is exactly we do when plotting Fig. 2.

Fig. 2(a) shows that in a single-mode waveguide, the dependence $T(L)$ is
exponential, $T(L)=T_{0}\exp(-L/l_{loc})$. This indicates on the regime of
Anderson localization of light, the parameter $l_{loc}$ means the localization
length and $T_{0}$ is normalization parameter. For given parameters,
$T_{0}=0.51$ and $l_{loc}=224$. With increasing in transverse sizes of a
waveguide, $a$ and $b$, both parameters $T_{0}$ and $l_{loc}$ undergo complex
step-like changes. Sharp steps correspond to the changes in the mode
compositions of a waveguide, e.g. when single-mode waveguide turns out to two-
mode. With further increasing in the transverse sizes, the nature of
transmittance scaling itself changes and ceases to be purely exponential.
Thus, in the case of multi-mode waveguide, the dependence $T(L)$ becomes close
to hyperbolic, $T(L)\propto 1/L$, see Fig. 2(b). This indicates on the regime
of classical diffuse radiation transfer.

In conclusion, we have calculated the transmission of disordered atomic
ensemble in a waveguide on the basis of self-consistent quantum microscopic
treatment taking into account tree-dimensional arrangement of atoms and
vectorial nature of electromagnetic field. We have found that the nature of
light transport essentially depends on the transverse sizes of a waveguide.
Single-mode waveguide with small transverse sizes exhibits Anderson
localization of light, which manifests itself in exponential decrease of the
transmittance with increasing in the thickness of atomic sample. An increase
of the transverse sizes breaks exponential law, so that in a few-mode
waveguide we observe complicated dependence of the transmission on the
thickness due to complex interplay between Anderson localization of light and
diffuse radiation transfer. With further increasing of the transverse sizes of
a waveguide, the transmission scaling obeys hyperbolic law, which indicates on
the regime of diffuse transfer in a multi-mode waveguide.

## Acknowledgments

A.S.K. appreciates financial support from the Foundation for the Advancement
of Theoretical Physics and Mathematics ’’BASIS’’. The results of the work were
obtained using computational resources of Peter the Great Saint-Petersburg
Polytechnic University Supercomputer Center (http://www.scc.spbstu.ru).

## References

  * (1) P.W. Anderson, Phys. Rev. 109, 1492 (1958).
  * (2) J. Chab, G. Lemari, B. Grmaud, D. Delande, P. Szriftgiser, and J.-C. Garreau, Phys. Rev. Lett. 101, 255702 (2008).
  * (3) F. Jendrzejewski, A. Bernard, K. Muller, P. Cheinet, V. Josse, M. Piraud, L. Pezze, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, Nat. Phys. 8, 398 (2012).
  * (4) D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, Nature (London) 390, 671 (1997).
  * (5) M. Strzer, P. Gross, C. M. Aegerter, and G. Maret, Phys. Rev. Lett. 96, 063904 (2006).
  * (6) T. Sperling, W. Bhrer, C.M. Aegerter, and G. Maret, Nat. Photonics 7, 48 (2013).
  * (7) H. Hu, A. Strybulevych, J. H. Page, S. E. Skipetrov, and B. A. van Tiggelen, Nat. Phys. 4, 945 (2008).
  * (8) A. Aubry, L. A. Cobus, S. E. Skipetrov, B. A. van Tiggelen, A. Derode, and J. H. Page, Phys. Rev. Lett. 112, 043903 (2014).
  * (9) L. Sapienza, H. Thyrrestrup, S. Stobbe, P. D. Garcia, S. Smolka, and P. Lodahl, Science 327, 1352 (2010).
  * (10) J. Liu, P. D. Garcia, S. Ek, N. Gregersen, T. Suhr, M. Schubert, J. Mork, S. Stobbe, and P. Lodahl, Nat. Nanotechnol. 9, 285 (2014).
  * (11) F. Pratesi, M. Burresi, F. Riboli, K. Vynck, and D. S. Wiersma, Opt. Express 21, A460 (2013).
  * (12) F. Scheffold, R. Lenke, R. Tweer, and G. Maret, Nature (London) 398, 206 (1999).
  * (13) T. van der Beek, P. Barthelemy, P.M. Johnson, D. S. Wiersma, and A. Lagendijk, Phys. Rev. B 85, 115401 (2012).
  * (14) F. Scheffold and D. Wiersma, Nat. Photonics 7, 934 (2013).
  * (15) S. E. Skipetrov and I. M. Sokolov, Phys. Rev. Lett. 112, 023905 (2014).
  * (16) S. E. Skipetrov and I. M. Sokolov, Phys. Rev. Lett. 114, 053902 (2015).
  * (17) E. Abrahams, P.W. Anderson, D. C. Licciardello, and T. V. Ramakrishnan, Phys. Rev. Lett. 42, 673 (1979).
  * (18) F. Evers and A. D. Mirlin, Rev. Mod. Phys. 80, 1355 (2008).
  * (19) T. Kobayashi, Q. Zheng, and T. Sekiguchi, Phys. Rev. A 52, 2835 (1995).
  * (20) E. V. Goldstein and P. Meystre, Phys. Rev. A 56, 5135 (1997).
  * (21) G. S. Agarwal and S. D. Gupta, Phys. Rev. A 57, 667 (1998).
  * (22) R. Rohlsberger, K. Schlage, B. Sahoo, S. Couet, and R. Ruffer, Science 328, 1248 (2010).
  * (23) Y.-Q. Zhang, L. Tan, and P. Barker, Phys. Rev. A 89, 043838 (2014).
  * (24) A. S. Kuraptsev and I. M. Sokolov, Journal of Experimental and Theoretical Physics 123, pp. 237248 (2016).
  * (25) A. S. Kuraptsev and I. M. Sokolov, Phys. Rev. A 94, 022511 (2016).
  * (26) M. D. Lee, S. D. Jenkins, Y. Bronstein, and J. Ruostekoski, Phys. Rev. A 96, 023855 (2017).
  * (27) A. S. Kuraptsev and I. M. Sokolov, Phys. Rev. A 101, 053852 (2020).
  * (28) O. Morice, Y. Castin, and J. Dalibard, Phys. Rev. A 51, 3896 (1995).
  * (29) I. M. Sokolov, D. V. Kupriyanov, and M. D. Havey, J. Exp. Theor. Phys. 112, 246 (2011).
  * (30) I. M. Sokolov, Laser Phys. 25, 065202 (2015).
  * (31) A. S. Kuraptsev, I. M. Sokolov, and M. D. Havey, Phys. Rev. A 96, 023830 (2017).


# Force sensing in an optomechanical system with feedback-controlled in-loop
light

F. Bemani<EMAIL_ADDRESS>Department of Optics, Palacký University,
17. listopadu 1192/12, 77146 Olomouc, Czechia  O. Černotík
<EMAIL_ADDRESS>Department of Optics, Palacký University, 17.
listopadu 1192/12, 77146 Olomouc, Czechia  L. Ruppert<EMAIL_ADDRESS>Department of Optics, Palacký University, 17. listopadu 1192/12, 77146
Olomouc, Czechia  D. Vitali<EMAIL_ADDRESS>Physics Division, School
of Science and Technology, University of Camerino, I-62032 Camerino (MC),
Italy  INFN, Sezione di Perugia, via A. Pascoli, Perugia, Italy  CNR-INO, L.go
Enrico Fermi 6, I-50125 Firenze, Italy  R. Filip<EMAIL_ADDRESS>Department of Optics, Palacký University, 17. listopadu 1192/12, 77146
Olomouc, Czechia

###### Abstract

Quantum control techniques applied at macroscopic scales provide us with
opportunities in fundamental physics and practical applications. Among them,
measurement-based feedback allows efficient control of optomechanical systems
and quantum-enhanced sensing. In this paper, we propose a near-resonant
narrow-band force sensor with extremely low optically added noise in an
optomechanical system subject to a feedback-controlled in-loop light. The
membrane’s intrinsic motion consisting of zero-point motion and thermal motion
is affected by the added noise of measurement due to the backaction noise and
imprecision noise. We show that, in the optimal low-noise regime, the system
is analogous to an optomechanical system containing a near quantum-limited
optical parametric amplifier coupled to an engineered reservoir interacting
with the cavity. Therefore, the feedback loop enhances the mechanical response
of the system to the input while keeping the optically added noise of
measurement below the standard quantum limit. Moreover, the system based on
feedback offers a much larger amplification bandwidth than the same system
with no feedback.

##  I  Introduction

Quantum control is a developing research area that explores techniques guiding
the system dynamics towards desired targets by external, time-dependent
manipulation. Advanced quantum control schemes are important when exploiting
coherent quantum effects in the emerging field of quantum technologies.
Feedback is a control technique where the outputs from the quantum system are
used as inputs that control the system dynamics and which enables us to evolve
the system towards some desired outcome. Feedback techniques show promise for
system stabilization and reducing decoherence and noise [1, 2]. Various
optical demonstrations of feedback techniques have been shown over the past
few years. Prominent among these are the coherent quantum feedback and
measurement-based feedback control [1, 2]. The latter is a powerful platform
for the state control of the mechanical oscillator and enhanced quantum
sensing [1, 2].

Combining ultralow-dissipation optical cavities with advanced nanofabrication
techniques for mechanical resonators has led to outstanding experimental
progress in the past decade [3]. These efforts have resulted in the coherent
interaction of light and mechanical motion of massive objects at the quantum
level and paved the way for a better understanding of light-matter interaction
in the rapidly developing area quantum optomechanics. Feedback techniques have
also been used in optomechanical systems. For instance, approaching the
quantum ground state of a kilogram-scale system in a gravity-wave
interferometer [4], quantum-enhanced feedback cooling of a mechanical
oscillator [5, 6], and force and position sensing via measurement-based
control of a mechanical oscillator [7, 8] have been investigated with a
feedback loop that operates directly on the mechanical element. More recently,
feedback-controlled in-loop light has been used to enhance the efficiency of
optomechanical systems, e.g., enhancing sideband cooling [9], normal-mode
splitting [10], improving the performance of an optomechanical heat engine
[11], and generating entanglement of two different mechanical resonators [12].

Any measurement of an observable that does not commute with itself at
different times has limits due to the Heisenberg uncertainty theorem [13]. The
standard quantum limit (SQL) is determined by a trade-off between shot noise
and radiation pressure backaction noise in force sensors based on quantum
optomechanics. Various approaches have been proposed and experimentally
realized to surpass the SQL: backaction-noise reduction to overcome the SQL
[14, 15, 16, 17], coherent quantum noise cancellation based on quantum
interference [18, 19, 20, 21, 22, 23] and exploitation of quantum correlations
to deviate from the SQL [24]. In force sensors based on the backaction-evasion
technique, the SQL is surpassed by producing a large signal without
suppressing the added noise while in the quantum noise cancellation technique,
the backaction noise is completely cancelled without amplifying the signal. It
has been also demonstrated that the parametric modulation of the spring
coefficient of a mechanical oscillator in an optomechanical system could be
used to perform single-quadrature detection of a force applied to the
mechanical oscillator with simultaneously suppressed optically added
measurement noise and amplified input signal [25].

In the red detuning regime, the optomechanical interaction causes a net energy
transfer from the mechanical mode into the cavity mode, hence, in effect,
cools the mechanical motion [26, 27, 28, 29, 30, 31]. In contrast, in the blue
detuning regime, the optomechanical interaction results in an energy transfer
into the mechanical motion; consequently, it results in optomechanical
amplification [32, 33, 34]. In this situation the mechanical noise is not
cooled and induces a significant added noise. Amplification can also be
realized in optomechanical systems subject to mechanical parametric driving
[25]. Both optical and mechanical responses of a linear amplifier for
optomechanical systems to quantum and classical fluctuations have been
developed in Ref. [35].

Inspired by the above-mentioned studies on feedback control and optomechanical
force sensors, in this paper, we propose a high–precision near-resonance
narrow-band force sensor that is feasible with state-of-the-art experimental
setups [10, 9]. We show a close analogy between an optomechanical system with
in-loop feedback and an optomechanical amplifier with an engineered reservoir
interacting with the cavity. Consequently, the feedback loop enhances the
mechanical response of the system to the input force in the red detuning of
the resolved sideband regime. The force sensor introduced here exhibits
simultaneously quantum-limited signal amplification (due to the feedback loop)
and noise suppression (due to the cooling by the red-sideband laser drive) in
the resolved sideband regime. Moreover, the weak input signal is transduced
with higher gain to the optical output while the signal-to-noise ratio and
sensitivity of the system remain unchanged. The feedback loop also
significantly enhances the detection bandwidth.

The paper is organized as follows. In Sec. II, we describe the system under
consideration, i.e., a nanomechanical membrane placed inside an optical cavity
subject to a feedback-controlled in-loop light, and then we present the
quantum Langevin equations. In Sec. III, we demonstrate how to amplify the
input signal by enhancing the mechanical response of the system by feedback-
controlled in-loop light, at the same time suppressing the added noise of
measurement by red detuned laser drive. Then, we study the sensitivity as well
as the signal-to-noise ratio of the proposed force sensor. Finally, in Sec.
IV, we present our concluding remarks.

##  II  Theoretical description of the system

Figure 1: Schematic illustration of the proposed optomechanical force sensor:
A nanomechanical membrane is positioned inside a high–finesse optical cavity
with feedback-controlled in-loop light. The cavity is driven through the
mirror ${\bf M}_{1}$ and the amplitude of the transmitted beam at mirror ${\bf
M}_{2}$ is detected via homodyne detection which consists of a beam splitter
($\bf BS$), two photodiodes (${\bf D}_{1}$, ${\bf D}_{2}$) and a local
oscillator with a phase $\theta$ relative to the signal. The resulting signal
$i_{\rm fb}(t)$ is electronically processed, and fed back, through an acousto-
optic modulator (AOM) driver to the amplitude of the input field at mirror
${\bf M}_{1}$, thus closing the feedback loop.

Our scheme for the feedback-enhanced force sensor is depicted in Fig. 1. A
nanomechanical membrane is placed into a high–finesse double-sided Fabry–Perot
cavity with frequency $\omega_{c}$ which is driven through the mirror ${\bf
M}_{1}$. Compared with a single-sided optomechanical cavity, this setup
contains one additional parameter to optimize, allowing more efficient
detection. In addition, the position of the membrane can be measured by
monitoring the transmitted light using balanced homodyne detection. The
resulting photocurrent, $i_{\rm fb}(t)$, is fed back to the input at mirror
${\bf M}_{1}$ using an acousto-optic modulator (AOM) driver that modulates the
amplitude of the input. The magnitude of an external force on the sensor is
then estimated using standard linear response theory. We will show how to
amplify the force by applying the feedback loop in the system.

We assume a single mechanical mode coupled to another optical mode via
radiation pressure [3]. We consider a strongly driven optical cavity field and
weak optomechanical coupling. This allows us to linearize the quantum dynamics
of fluctuations around the semiclassical amplitudes [3, 36]. The linearized
optomechanical Hamiltonian describing the interaction is given by

$H=\hbar\Delta c^{\dagger}c+\hbar\omega_{\rm
m}b^{\dagger}b+\hbar\Lambda(c^{\dagger}+c)(b^{\dagger}+b)+{x_{{\rm{zpf}}}}F(t)(b^{\dagger}+b)\,.$
(1)

The first term describes the free energy of the cavity mode in a frame
rotating at laser frequency where $c$ ($c^{\dagger}$) is the annihilation
(creation) operator of the cavity mode and $\Delta=\omega_{\rm c}-\omega_{\rm
L}$ is the detuning between the cavity resonance $\omega_{\rm c}$ and the pump
laser frequency $\omega_{\rm L}$. The second term describes the mechanical
Hamiltonian where the operator $b$ ($b^{\dagger}$) is the annihilation
(creation) operator of the mechanical mode with natural frequency $\omega_{\rm
m}$. The interaction between the center-of-mass motion of the membrane and the
cavity mode is given by the third term in Eq. (1) where $\Lambda$ is the
interaction rate. Finally, the last term in the Hamiltonian accounts for the
coupling of the membrane to an input classical force $F(t)$ in the
$x$-direction, as depicted in Fig. 1. Moreover, $x_{\rm
zpf}=\sqrt{\hbar/{2m\omega_{m}}}$ is the zero-point posotion uncertainty of
the membrane with mass $m$.

Moving the operators to an interaction picture with respect to the Hamiltonian
$H_{0}=\hbar\Delta c^{\dagger}c+\hbar\omega_{\rm m}b^{\dagger}b$, we obtain
the linearized optomechanical Hamiltonian describing the interaction as

$H_{\rm
I}=\hbar\Lambda(c^{\dagger}b+b^{\dagger}c)+{x_{{\rm{zpf}}}}F(t)\big{(}b^{\dagger}e^{i\omega_{\rm{m}}t}+be^{-i\omega_{\rm{m}}t}\big{)}+H_{\rm
CR}\,.$ (2)

with $H_{\rm
CR}=\hbar\Lambda[cb\exp(-2i\omega_{\rm{m}}t)+b^{\dagger}c^{\dagger}\exp(2i\omega_{\rm{m}}t)]$
where we have also assumed $\Delta\simeq\omega_{m}$. In the resolved sideband
regime where the mechanical frequency is much larger than the damping rate of
the cavity ($\omega_{\rm{m}}\gg\kappa$), $H_{\rm CR}$ can be approximately
neglected. To account for the small correction to the dynamics caused by
$H_{\rm CR}$ in this regime, we use the sideband truncation approach described
in appendix A in our numerical calculations.

###  II.1 Applying the coherent feedback loop

To investigate the system dynamics, we now include fluctuation and dissipation
processes affecting the optical and the mechanical modes by adding for each of
them the corresponding damping and noise term and write the following quantum
Langevin equations

$\displaystyle\dot{b}=$ $\displaystyle-\gamma b+i\Lambda
c+\sqrt{2\gamma}\,\,b_{\rm in,f}(t)\,,$ (3) $\displaystyle\dot{c}=$
$\displaystyle-\kappa\,c+i\Lambda b+\sqrt{2\kappa_{1}}\,c_{\rm
in}^{(1)}(t)+\sqrt{2\kappa_{2}}\,c_{\rm in}^{(2)}(t)\,,$ (4)

where $c_{\rm in}^{(1)}(t)$ and $c_{\rm in}^{(2)}(t)$ are the noise operators
associated with the mirrors $\bf{M}_{1}$ and $\bf{M}_{2}$, respectively. The
nanomechanical membrane interacts with a thermal environment at a finite
temperature with a damping rate of $\gamma$. The input mechanical operator
$b_{\rm in,f}(t)$ consists of the external force to be detected, $F(t)$, and a
noise operator $b_{\rm in}(t)$,

$b_{\rm in,f}(t)=-\frac{i}{2\sqrt{\hbar m\gamma\omega_{\rm
m}}}F(t)e^{i\omega_{\rm{m}}t}+b_{\rm in}(t)\,.$ (5)

Here, the mechanical noise $b_{\rm in}$ affecting the nanomechanical membrane
satisfies the Markovian correlation functions

$\displaystyle\langle b_{\rm in}(t)\,b_{\rm in}^{\dagger}(t^{\prime})\rangle$
$\displaystyle=(1+\bar{n})\,\delta(t-t^{\prime})\,,$ (6) $\displaystyle\langle
b_{\rm in}^{\dagger}(t)\,b_{\rm in}(t^{\prime})\rangle$
$\displaystyle=\bar{n}\,\delta(t-t^{\prime})\,,$ (7)

where $\bar{n}=[\exp(\hbar\omega_{\rm m}/k_{B}T)-1]^{-1}$ is the mean number
of thermal excitations of the mechanical bath at temperature $T$. Moreoevre,
the nonzero noise correlators associated with the cavity input are given by
[3]

$\displaystyle\langle c_{\rm in,0}^{(1)}(t)\,c_{\rm
in,0}^{{(1)}{\dagger}}(t^{\prime})\rangle$
$\displaystyle=\delta(t-t^{\prime})\,,$ (8) $\displaystyle\langle c_{\rm
in}^{(2)}(t)\,c_{\rm in}^{{(2)}{\dagger}}(t^{\prime})\rangle$
$\displaystyle=\delta(t-t^{\prime})\,,$ (9)

By introducing the total cavity decay rate $\kappa=\kappa_{1}+\kappa_{2}$ in
terms of the corresponding decay rates of the mirrors, we can formally write
the total input noise operator for the cavity field as $\sqrt{2\kappa}c_{\rm
in}(t)=\sqrt{2\kappa_{1}}\,c_{\rm in}^{(1)}(t)+\sqrt{2\kappa_{2}}\,c_{\rm
in}^{(2)}(t)$.

We now modify the system dynamics by applying feedback-controlled in-loop
light [11, 10, 9, 37]. The operator $c_{\rm out}^{(2)}$ describes the
transmitted output field given by the input–output relation

$c_{\rm out}^{(2)}(t)=\sqrt{2\kappa_{2}}\,c(t)-c_{\rm in}^{(2)}(t).$ (10)

We then consider a phase-sensitive detection of an output quadrature at phase
$\theta$

$X_{\rm out,fb}^{\theta}(t)=\frac{{{e^{-i\theta}}c_{\rm
out}^{(2)}(t)+{e^{i\theta}}c_{\rm out}^{(2){\dagger}}(t)}}{{\sqrt{2}}}\,.$
(11)

The resulting photocurrent at the output of the second mirror is expressed as
[11, 37]

$i_{\rm fb}(t)=\sqrt{\eta}\,X_{{\rm
out,fb}}^{\theta}(t)+\sqrt{1-\eta}\,X_{\nu}(t),$ (12)

Here, we model the output light due to inefficient detection of the optical
field where $\eta$ is the detection efficiency and the operator ${X_{\nu}}$
accounts for the additional noise due to inefficient detection and fulfills
the relation $\langle
X_{\nu}(t)X_{\nu}(t^{\prime})\rangle=(1/2)\delta(t-t^{\prime})$.

As shown in Fig. 1, the photocurrent is fed back using an AOM driver to
modulate the amplitude of the input field at mirror ${\bf M}_{1}$. The overall
effect of the feedback loop can be modeled by the electronic noise $\Phi(t)$
modulating (displacing) the open-loop input noise operator $c_{\rm
in,0}^{(1)}(t)$. Therefore, the input noise operator applied to the mirror
$M_{1}$ is written as the superposition of the original input
$c_{{\rm{in,0}}}^{(1)}(t)$ and an additional term due to the feedback
$\Phi(t)$ according to

$c_{\rm in}^{(1)}(t)=c_{{\rm{in,0}}}^{(1)}(t)+\Phi(t).$ (13)

where $\Phi(t)=gi_{\rm fb}(t-\tau)$, $g$ is the feedback gain, and $\tau$ is
the signal time delay in the loop. We consider a feedback loop with a non-zero
delay caused by a latency of electronic processing. For high–quality
electronic circuits, we can neglect all dispersion effects. Therefore, we
write the delay-time dependence of the photocurrent in Eq. (12) as

$\displaystyle i_{\rm fb}(t-\tau)=$
$\displaystyle\sqrt{\eta/2}\left\\{e^{-i\theta}\left[\sqrt{2\kappa_{2}}\,c(t-\tau)-c_{\rm
in}^{(2)}(t-\tau)\right]\right.$
$\displaystyle+\left.e^{i\theta}\left[\sqrt{2\kappa_{2}}\,{c^{\dagger}}(t-\tau)-c_{\rm
in}^{(2){\dagger}}(t-\tau)\right]\right\\}$
$\displaystyle+\sqrt{1-\eta}\,X_{\nu}(t-\tau)\,.$ (14)

In the regime where pump detuning is much larger than the optomechanical
coupling and the cavity decay rate ($\Delta\gg\Lambda,\kappa$), it is
convenient to rewrite the cavity delayed time operators as a product of a
slowly varying term $\bar{c}_{{\rm{out}}}^{(2)}(t)$ and a fast oscillating one
as
$c_{{\rm{out}}}^{(2)}\left({t-\tau}\right)\simeq\bar{c}_{{\rm{out}}}^{(2)}(t-\tau){e^{-i\Delta(t-\tau)}}$.
If the feedback delay time is much shorter than both the characteristic time
of the optomechanical interaction and the decay time of the cavity, then the
delay time dependence of the slowly varying part,
$\bar{c}_{{\rm{out}}}^{(2)}(t-\tau),$ can be ignored, therefore we have
$\bar{c}_{{\rm{out}}}^{(2)}(t-\tau)\simeq\bar{c}_{{\rm{out}}}^{(2)}(t)$. We
thus can rewrite the output operator as

$c_{{\rm{out}}}^{(2)}\left({t-\tau}\right)\simeq\bar{c}_{{\rm{out}}}^{(2)}(t){e^{i\Delta
t}}{e^{-i\Delta\tau}}=c_{{\rm{out}}}^{(2)}(t){e^{-i\Delta\tau}}$ (15)

By defining a global phase $\phi=\theta+\Delta\tau$, we then approximate Eq.
(II.1) as

$\begin{split}{i_{{\rm{fb}}}}(t-\tau)=&\sqrt{\eta/2}\left\\{{{e^{-i\phi}}\left[{\sqrt{2{\kappa_{2}}}c(t)-c_{{\rm{in}}}^{(2)}(t)}\right]}\right.\\\
&+\left.{{e^{i\phi}}\left[{\sqrt{2{\kappa_{2}}}{c^{\dagger}}(t)-c_{{\rm{in}}}^{(2){\dagger}}(t)}\right]}\right\\}+\sqrt{1-\eta}{X_{\nu}}(t)\,.\end{split}$
(16)

By using equations (13) and (16), we can rewrite the equation of motion for
the cavity field as

$\begin{split}\dot{c}=&-\kappa\,c+i\Lambda
b+g\,\sqrt{2\eta\kappa_{1}\kappa_{2}}\left[e^{-i\phi}c(t)+e^{i\phi}c^{\dagger}(t)\right]\\\
&+\sqrt{2\kappa_{\rm fb}}\,c_{\rm in,fb}(t),\end{split}$ (17)

where we have used the feedback-modified cavity decay rate

$\kappa_{\rm fb}=\kappa-g\sqrt{2\eta\kappa_{1}\kappa_{2}}\,,$ (18)

and the corresponding noise operator

$\begin{split}c_{\rm in,fb}&(t)=\frac{1}{\sqrt{2\kappa_{\rm
fb}}}\left\\{-g\sqrt{\kappa_{1}\eta}\left[{e^{-i\phi}c_{\rm
in}^{(2)}(t)+e^{i\phi}c_{\rm in}^{(2){\dagger}}(t)}\right]\right.\\\
&\left.+\sqrt{2\kappa_{2}}c_{\rm
in}^{(2)}(t)+g\sqrt{2\kappa_{1}(1-\eta)}X_{\nu}(t)+\sqrt{2\kappa_{1}}c_{\rm
in,0}^{(1)}(t)\right\\}\,.\end{split}$ (19)

The feedback-modified noise operator obeys the following correlation functions

$\displaystyle\langle c_{\rm in,fb}(t)c_{\rm in,fb}(t^{\prime})\rangle=$
$\displaystyle\left(n_{\rm fb}-m_{\rm fb}\right)\delta(t-t^{\prime})\,,$ (20)
$\displaystyle\langle{c_{\rm in,fb}^{\dagger}(t)c_{\rm
in,fb}^{\dagger}(t^{\prime})}\rangle=$ $\displaystyle\left(n_{\rm fb}-m_{\rm
fb}^{*}\right)\delta(t-t^{\prime})\,,$ (21) $\displaystyle\langle c_{\rm
in,fb}^{\dagger}(t)c_{\rm in,fb}(t^{\prime})\rangle=$ $\displaystyle n_{\rm
fb}\delta(t-t^{\prime})\,,$ (22) $\displaystyle\langle c_{\rm in,fb}(t)c_{\rm
in,fb}^{\dagger}(t^{\prime})\rangle=$
$\displaystyle\left({{n_{{\rm{fb}}}}\\!+{\kappa}/{{{\kappa_{{\rm{fb}}}}}}\\!-m_{\rm
fb}\\!-m_{\rm fb}^{*}}\right)\delta(t-t^{\prime})\,,$ (23)

where we have defined $n_{\rm fb}=\zeta^{2}/(4\eta\kappa_{2}\kappa_{\rm fb})$
as the feedback-mediated number of thermal excitations and
${m_{{\rm{fb}}}}=\left({\zeta/2{\kappa_{{\rm{fb}}}}}\right)\exp\left({i\phi}\right)$,
with feedback reduction of cavity linewidth
$\zeta=g\sqrt{2\eta\kappa_{1}\kappa_{2}}$. This reveals that the cavity decay
rate, as well as the cavity frequency (see Eq. (17)), can be controlled by
manipulating the feedback gain $g$ and the global phase $\phi$. Moreover, the
system can be viewed as an engineered reservoir interacting with the cavity
and mechanical mode. Together with correlations (6-7) and (20-23) the quantum
Langevin equations (3) and (17) describe the evolution of the cavity field and
the mechanical motion of the nanomechanical membrane, including a feedback-
controlled in-loop light and all fluctuation effects. We then summarize the
equations of motion in the following compact matrix form

$\frac{{{\rm d}{\bf{u}}(t)}}{{{\rm
d}t}}={\bf{Au}}(t)+{\bf{H}}{{\bf{n}}_{{\rm{in,fb}}}}(t)\,.$ (24)

Here, we have defined the vector of fields
${\bf{u}}(t)=[c(t),c^{\dagger}(t),b(t),b^{\dagger}(t)]^{T}$, matrix
${\bf{H}}={\rm{diag}}\left[{\sqrt{2\kappa_{fb}},\sqrt{2\kappa_{fb}},\sqrt{2\gamma},\sqrt{2\gamma}}\right]$
and the corresponding vector of noises
${{\bf{n}}_{{\rm{in,fb}}}}(t)=[{{c_{{\rm{in}},{\rm{fb}}}}(t),c_{{\rm{in}},{\rm{fb}}}^{\dagger}(t),{b_{{\rm{in,f}}}}(t),b_{{\rm{in,f}}}^{\dagger}(t)}]^{T}$.
We have also defined the drift matrix ${\bf A}$ as

${\bf{A}}=\left({\begin{array}[]{*{20}{c}}{\zeta{e^{-i\phi}}-\kappa}&{\zeta{e^{i\phi}}}&{i\Lambda}&{0}\\\
{\zeta{e^{-i\phi}}}&{\zeta{e^{i\phi}}-\kappa}&{0}&{-i\Lambda}\\\
{i\Lambda}&{0}&{-\gamma}&0\\\
{0}&{-i\Lambda}&0&{-\gamma}\end{array}}\right)\,.$ (25)

###  II.2 System response in the frequency domain

Now that we understand the system dynamics, we can investigate how the
feedback loop can be used for force sensing. For a detailed treatment and a
review of the basic statistical properties of quantum noise, including its
detection and a basic introduction to weak continuous measurements, see [13].
Equation (24) can be solved in the frequency space by taking the Fourier
transform of all the operators and noise sources introduced via
$o[\omega]=\smallint_{-\infty}^{+\infty}{dt{e^{i\omega t}}o(t)}$. We then
write the solution of Eq. (24) in the frequency domain as

${\bf{u}}[\omega]=-{\left({{\bf{A}}+i\omega{{\bf{I}}_{4\times
4}}}\right)^{-1}}{\bf{H}}{{\bf{n}}_{{\rm{in,fb}}}}[\omega]\,,$ (26)

where ${{\bf{I}}_{4\times 4}}$ is the identity matrix.

By substituting ${\bf u}\left[\omega\right]$ into the input-output relation
${{\bf{n}}_{{\rm{out}}}^{(2)}}[\omega]={\bf{Gu}}[\omega]-{{\bf{n}}_{{\rm{in}}}^{(2)}}[\omega]$,
which relates output and input fields, the fluctuations of the output fields
are obtained as

$\displaystyle{{\bf{n}}_{{\rm{out}}}^{(2)}}[\omega]$
$\displaystyle=-\left[{{{{\bf{G}}\left({{\bf{A}}+i\omega{{\bf{I}}_{4\times
4}}}\right)}^{-1}}{\bf{H}}}\right]{{\bf{n}}_{{\rm{in,fb}}}}[\omega]-{{\bf{n}}_{{\rm{in}}}^{(2)}}[\omega]$
$\displaystyle={\bf{s}}[\omega]{{\bf{n}}_{{\rm{in,fb}}}}[\omega]-{{\bf{n}}_{{\rm{in}}}^{(2)}}[\omega]\,,$
(27)

where have defined
${\bf{n}}_{{\rm{in}}}^{(2)}={[c_{{\rm{in}}}^{(2)},c_{{\rm{in}}}^{(2){\dagger}},{b_{{\rm{in,f}}}},b_{{\rm{in,f}}}^{\dagger}]^{T}}$,
${\bf{n}}_{{\rm{out}}}^{(2)}={[c_{{\rm{out}}}^{(2)},c_{{\rm{out}}}^{(2){\dagger}},{b_{{\rm{out}}}},b_{{\rm{out}}}^{\dagger}]^{T}}$
and
${\bf{G}}={\rm{diag}}\left[{\sqrt{2{\kappa_{2}}},\sqrt{2{\kappa_{2}}},\sqrt{2\gamma},\sqrt{2\gamma}}\right]$.
We have also introduced the scattering matrix ${\bf{s}}[\omega]$ as

${\bf{s}}[\omega]=-{\bf{G}}{\left({{\bf{A}}+i\omega{{\bf{I}}_{4\times
4}}}\right)^{-1}}{\bf{H}},$ (28)

Note that we use the convention
${o^{\dagger}}[\omega]={\smallint}_{-\infty}^{+\infty}dte^{i\omega
t}o^{\dagger}(t)$ which implies $o^{\dagger}[\omega]=(o[-\omega])^{\dagger}$.
Full expressions of the scattering matrix elements are given in appendix B.

##  III  Force measurement

In this section, we describe force sensing beyond the SQL in the presence of
the feedback loop. We start by deriving expressions for the mechanical
response and the optically added noise. Then we show that the feedback loop
can enhance the mechanical response and improve its bandwidth. We also show
how noise suppression and signal amplification can be achieved by tuning the
system parameters, particularly the asymmetry in the decay rates of the two
mirrors. Moreover, the optically added noise and the force sensitivity will
not change much compared to an open-loop system.

###  III.1 Spectral characteristics of force

An external force acting on the nanomechanical membrane displaces it from the
equilibrium point which, via the optomechanical coupling, affects the optical
cavity output. As a consequence, the signal associated with the external force
can be extracted by measuring the optical output generalized in-loop
quadrature, $X_{\rm out}^{\phi,(2)}$, given in terms of the input fields by

$X_{{\rm{out}}}^{\phi,(2)}=\frac{1}{{\sqrt{2}}}\left\\{{\Psi[\omega]{c_{{\rm{in}},{\rm{fb}}}}+\Phi[\omega]{b_{{\rm{in}},{\rm{f}}}}+{\rm{H}}{\rm{.c}}{\rm{.}}}\right\\}-X_{{\rm{in}}}^{\phi,(2)}{\mkern
1.0mu},$ (29)

where we have defined the functions $\Psi[\omega]$ and $\Phi[\omega]$ in terms
of the scattering matrix elements given in appendix B as

$\displaystyle\Psi[\omega]=$
$\displaystyle{s_{11}}{e^{-i\phi}}+{s_{21}}{e^{i\phi}}$ $\displaystyle=$
$\displaystyle\frac{{2{e^{-i\phi}}(\gamma-i\omega)\sqrt{{\kappa_{2}}{\kappa_{{\rm{fb}}}}}}}{{-2\zeta(\gamma-i\omega)\cos\phi+(\gamma-i\omega)(\kappa-i\omega)+{\Lambda^{2}}}}\,,$
(30) $\displaystyle\Phi[\omega]=$
$\displaystyle{s_{13}}{e^{-i\phi}}+{s_{23}}{e^{i\phi}}$ $\displaystyle=$
$\displaystyle\frac{{2i\Lambda{e^{-i\phi}}\sqrt{\gamma{\kappa_{\rm{2}}}}}}{{-2\zeta(\gamma-i\omega)\cos\phi+(\gamma-i\omega)(\kappa-i\omega)+{\Lambda^{2}}}}\,.$
(31)

We then use the output quadrature to estimate the applied external force on
the nanomechanical membrane. Using the expressions for the elements of the
scattering matrix, one finally obtains the following result for the
transmitted optical quadrature:

$X_{{\rm{out}}}^{\phi,(2)}={\left.{X_{{\rm{out}}}^{\phi,(2)}}\right|_{{F}=0}}+X_{F}^{\phi}\,,$
(32)

where the contribution of the quantum noise
${X_{{\rm{out}}}^{\phi,(2)}}|_{F=0}$ can be obtained by replacing
${b_{{\rm{in,f}}}}\to{b_{{\rm{in}}}}$ and $b_{{\rm{in,f}}}^{\dagger}\to
b_{{\rm{in}}}^{\dagger}$ in Eq. (29). Moreover, the contribution of the
external force is

$\displaystyle X_{F}^{\phi}=$ $\displaystyle-\frac{i}{{2\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}}}\left\\{({s_{13}}{e^{-i\phi}}+{s_{23}}{e^{i\phi}})F[\omega+{\omega_{\rm{m}}}]\right.$
$\displaystyle\left.-({s_{14}}{e^{-i\phi}}+{s_{24}}{e^{i\phi}})F[\omega-{\omega_{\rm{m}}}]\right\\}$
$\displaystyle=$ $\displaystyle\frac{i}{{\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}}}\bar{F}[\omega]\,,$ (33)

where we define the transduction force by the force estimator

$\bar{F}[\omega]=\frac{1}{2}\left({{\Phi^{*}}[\omega]F[\omega-{\omega_{\rm{m}}}]-\Phi[\omega]F[\omega+{\omega_{\rm{m}}}]}\right)\,.$
(34)

We then define the effective force operator as [38]

$F_{\rm eff}[\omega]=\frac{X_{\rm out}^{\phi,(2)}[\omega]}{\partial X_{\rm
out}^{\phi,(2)}[\omega]/\partial\bar{F}[\omega]}=N^{\phi}[\omega]+\bar{F}[\omega]\,,$
(35)

where the noise force operator is given by

${N^{\phi}}[\omega]=i\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}{\rm{}}{\left.X_{{\rm{out}}^{\phi,(2)}}\right|_{{F}=0}}.$
(36)

We then consider the spectral density of the noise force as

$\displaystyle\delta(\omega+\omega^{\prime})S_{NN}^{\phi}(\omega)=\frac{1}{{4\pi}}\langle{N^{\phi}}[\omega]{N^{\phi,\dagger}}[\omega^{\prime}]+(\omega\leftrightarrow\omega^{\prime})\rangle\,,$
(37)

where the notation $(\omega\leftrightarrow\omega^{\prime})$ denotes the
swapping of the arguments of the first term. After straightforward
calculations, the spectrum of the noise force operator, ${N^{\phi}}$, is given
by

$S_{NN}^{\phi}(\omega)=2\hbar m\gamma{\omega_{\rm{m}}}R_{\rm
m}[\omega]\left[{\left({\bar{n}+\frac{1}{2}}\right)+{n_{{\rm{add}}}}[\omega]}\right]\,,$
(38)

where we have used correlations in the frequency domain see appendix C and
${R_{\rm{m}}}[\omega]$ is the mechanical response defined by

${R_{\rm{m}}}[\omega]=\frac{1}{2}(\Phi[-\omega]{\Phi^{*}}[\omega]+\Phi[\omega]{\Phi^{*}}[-\omega])\,.$
(39)

In Eq. (38), ${n_{{\rm{add}}}}[\omega]$ is the noise of the measurement
process which can be interpreted as an effective increase in the number of
thermal excitations of the mechanical reservoir due to the backaction of the
optical mode. After evaluating lengthy but straightforward calculations, the
added measurement noise is given by

$\displaystyle{n_{{\rm{add}}}}[\omega]=$
$\displaystyle\frac{1}{{{2R_{\rm{m}}}[\omega]}}\Big{\\{}\big{[}2\Psi[\omega]\Psi[-\omega]\left({{n_{{\rm{fb}}}}-{m_{{\rm{fb}}}}}\right)+{\rm
c.c.}\big{]}\big{.}$
$\displaystyle+\big{.}\big{(}\Psi[\omega]{\Psi^{*}}[-\omega]+{\rm{c}}{\rm{.c.}}\big{)}\big{(}2{n_{{\rm{fb}}}}+\frac{\kappa}{{{\kappa_{{\rm{fb}}}}}}-{m_{{\rm{fb}}}}-m_{{\rm{fb}}}^{*}\big{)}\big{.}$
$\displaystyle\big{.}+2-\left({\Psi[\omega]+\Psi[-\omega]+{\rm{c}}{\rm{.c}}}\right)\left({{e^{i\phi}}p_{{\rm{fb}}}^{*}[\omega]+{\rm
c.c.}}\right)\big{.}$
$\displaystyle\big{.}-\left[{{e^{i\phi}}\left({\Psi[-\omega]+\Psi[\omega]}\right)+{\rm{c}}{\rm{.c}}{\rm{.}}}\right]\sqrt{{\kappa_{\rm{2}}}/{\kappa_{{\rm{fb}}}}}\Big{\\}},$
(40)

with
${p_{{\rm{fb}}}}=(-\zeta/\sqrt{4{\kappa_{2}}{\kappa_{{\rm{fb}}}}})\exp(i\phi)$.

The transduction of the external force and the mechanical inputs onto the
light field is characterized by Eq. (29). This transduction can be obtained in
terms of the corresponding scattering matrix elements given in Appendix B and
finally translated into functions $\Phi[\omega]$ and $\Phi^{*}[\omega]$. In
the case of an ideal resonant impulsive force, $F[\omega]=\textrm{const.}$,
the force estimator $\bar{F}[\omega]$ becomes

$\bar{F}[\omega]={R_{\rm{m}}}[\omega]\cos^{2}\phi F\,.$ (41)

###  III.2 Mechanical response and the added noise

We next calculate the added noise and the mechanical response given by Eqs.
(III.1) and (39), respectively, in the resolved sideband regime in the limit
where the mechanical frequency is much larger than the cavity decay rate,
${\omega_{\rm{m}}}\gg\kappa$. The frequency-dependent mechanical response and
the optically added noise in the rotating wave approximation (RWA) (with
calculations beyond the RWA delegated to the appendix A) are represented by

${R_{\rm{m}}}[\omega]=\frac{4{\cal C}\gamma^{2}\kappa\kappa_{2}}{|({\cal
C}+1)\gamma\kappa-2\zeta(\gamma-i\omega)\cos\phi-i\omega(\gamma+\kappa)-\omega^{2}|^{2}}\,,$
(42) $\displaystyle{n_{{\rm{add}}}}\\![\omega]\\!=$
$\displaystyle\\!\frac{\omega^{2}\\!\left(\gamma^{2}\\!-2{\cal
C}\gamma\kappa\\!+\kappa^{2}\right)\\!+\\!({\cal
C}\\!+1\\!)^{2}\\!\gamma^{2}\kappa^{2}\\!-\\!4{\cal
C}\gamma^{2}\kappa\kappa_{2}+\omega^{4}}{4{\cal C}\gamma^{2}\kappa\kappa_{2}}$
$\displaystyle+\frac{{\left({{\omega^{2}}+{\gamma^{2}}}\right)(1-\eta){\zeta^{2}}}}{{2{\cal
C}{\gamma^{2}}\eta\kappa{\kappa_{2}}}}\,,$ (43)

where ${\cal C}=\Lambda^{2}/\kappa\gamma$ is the optomechanical cooperativity.
Eqs. (42), (III.2) are expressed in the interaction picture, where $\omega=0$
means that we look at detection at the cavity resonance frequency. In this
case, we are looking at forces that are quasi-resonant with the mechanical
resonator. The on-resonance mechanical response and the optically added noise
are given by

$\displaystyle R_{\rm{m}}^{{\rm{fb}}}[\omega=0]$ $\displaystyle=\frac{4{\cal
C}\kappa\kappa_{2}}{({\cal C}\kappa-2\zeta\cos\phi+\kappa)^{2}}\,,$ (44)
$\displaystyle n_{{\rm{add}}}^{{\rm{fb}}}[\omega=0]$
$\displaystyle=\frac{{{{\left({{\cal C}-1}\right)}^{2}}}}{{4{\cal
C}}}+\frac{({\cal C}+1)^{2}\kappa_{1}}{4{\cal
C}\kappa_{2}}+\frac{{(1-\eta){\zeta^{2}}}}{{2{\cal
C}\eta\kappa{\kappa_{2}}}}\,.$ (45)

A remarkable result from Eq. (44) is that the on-resonance mechanical response
is never larger than unity for an open loop system. That is, a single phonon
at the mechanical input can be converted to only one photon at the output
which is a limitation in the functionality of the open-loop system in the red
detuned regime. Therefore according to Eq. (44), the open-loop system provides
only direct transduction of the force and any force-detection gain is
impossible.

We next turn to the numerical calculations of the added noise and mechanical
response by using parameters feasible in a typical experimental setup of
membrane-in-the-middle cavity optomechanical systems [10, 9]. We consider a
mode of vibration with an effective mass of $m=10^{-12}\,\rm kg$ inside an
optical cavity with decay rate $\kappa=\kappa_{1}+\kappa_{2}=0.06\omega_{\rm
m}$. The mechanical decay rate is $\gamma/\omega_{\rm m}=3.4\times{10^{-6}}$
and the mechanical frequency is $\omega_{\rm m}=2\pi\times
343.13\,{\rm{kHz}}$. The optomechanical coupling rate, the local oscillator
phase, the cavity detuning and the feedback gain could be tuned in situ.

Figure 2: The regions of standard quantum limit suppression $n_{\rm
add}[0]<1/2$ (blue honeycomb pattern), signal amplification $R_{\rm m}[0]>1$
(gray hatched pattern) and the instability (red solid area) as functions of
the optomechanical cooperativity and feedback gain for detection efficiency
$\eta=0.6$. Symmetric cavity with $\kappa_{2}/\kappa=0.5$ (a) and asymmetric
cavity with $\kappa_{2}/\kappa=0.75$ (b) and $\kappa_{2}/\kappa=0.90$ (c). We
consider an optical cavity with decay rate
$\kappa=\kappa_{1}+\kappa_{2}=0.06\omega_{\rm m}$, the mechanical decay rate
$\gamma/\omega_{\rm m}=3.4\times{10^{-6}}$ and the mechanical frequency
$\omega_{\rm m}=2\pi\times 343.13\,{\rm{kHz}}$.

In Fig. 2, we show the regions of SQL suppression $n_{\rm add}[0]<1/2$ [13,
39, 40] (blue honeycomb pattern) and signal amplification $R_{\rm m}[0]>1$
(gray-hatched pattern) versus optomechanical cooperativity and feedback gain
for three different cavity configurations. The denominator of the mechanical
response given by Eq. (44) must tend to its smallest possible value to achieve
signal amplification. This puts a constraint on the angle $-\pi/2<\phi<\pi/2$.
Hereafter, we take the global phase $\phi$ such that $\cos\phi=1$ to have
maximum amplification. Signal amplification starts when we set
$\zeta\simeq\kappa/4$ and ${\cal C}=0.5$ for a symmetric cavity configuration.
As is evident from Fig. 2, the sub-SQL region increases as we increase the
ratio of $\kappa_{2}/\kappa$. For an asymmetric configuration,
$\kappa_{2}>\kappa_{1}$, the regions are much wider than for a symmetric
configuration. Generally, simultaneous noise suppression and signal
amplification can not be achieved for relatively small values of
optomechanical cooperativity and feedback gain. What is noticeable is that the
feedback plays its role in signal amplification in the red detuning regime.
The additional requirement for performing the force measurement is that the
system under consideration should be stable. Here, the unstable region is
shown by a red solid region (see appendix D for details on the system’s
stability conditions).

Figure 3: (a) The on-resonant optically added noise of measurement and (b) the
mechanical response as a function of the optomechanical cooperativity for
different ratios of $\kappa_{2}/\kappa$, fixed values of the feedback gain
$\zeta/\kappa=0.49$ and the detection efficiency $\eta=1$. (c) The on-resonant
optically added noise and (d) the mechanical response as a function of the
optomechanical cooperativity for different values of feedback gain, fixed
$\kappa_{2}/\kappa=0.95$ and detection efficiency $\eta=0.6$. The line with
$\zeta/\kappa=0$ corresponds to the no feedback scenario. Other parameters
given in caption of Fig. 2. The noise suppression (beyond SQL, $n_{\rm
add}[0]<1/2$) and signal amplification ($R_{\rm m}[0]>1$) regions are shown in
blue and red, respectively.

Based on Eqs. (44) and (45), the decay rates of the cavity mirrors influence
the system dynamics. We drive the system through the mirror ${\bf M}_{1}$
which is highly reflective since we want to reduce any optical losses on this
cavity end. This high reflectivity then requires strong drive and high
feedback gain but, at the same time, ensures that the added noise remains low.
On the other hand, the signal we detect has to be sufficiently large so the
mirror ${\bf M}_{2}$ has to be more transmissive. Indeed, in the symmetric
case, $\kappa_{1}=\kappa_{2}$, it is clear that in the limit of ${\cal
C}\rightarrow 1$, the added optical noise in Eq. (45) is
${n_{{\rm{add}}}}[\omega=0]=1$, and the maximum value of the mechanical
response is $R_{\rm{m}}^{{\rm{fb}}}[\omega=0]=2$ (assuming unit detection
efficiency). The minimum added noise is obtained by using an asymmetric cavity
with the mirror ${\bf M}_{1}$ _less transmissive_ than the mirror ${\bf
M}_{2}$. Therefore, this proposal will be optimal in situations where we use
an asymmetric cavity $\kappa_{1}<\kappa_{2}$. In the strongly asymmetric case,
$\kappa_{1}\ll\kappa_{2}$, with unit detection efficiency, in the limit of
${\cal C}\rightarrow 1$, the added optical noise is,
${n_{{\rm{add}}}}[\omega=0]\simeq{\kappa_{1}}/{\kappa_{2}}$, and the maximum
achievable mechanical response is $R_{\rm{m}}^{{\rm{fb}}}[\omega=0]=4$.

The on-resonance optically added noise and the mechanical response versus
optomechanical cooperativity for different cavity asymmetries and a fixed
value of feedback gain are shown in Figs. 3 (a) and (b). We see that
asymmetric cavities with $\kappa_{2}>\kappa_{1}$ are preferable for
suppressing noise below SQL and amplifying the signal. Although the backaction
and the feedback loop introduce measurement noise to the force measurement,
with an appropriate choice of parameters, the measurement noise can be
significantly reduced below the SQL. For instance, by using a high–efficiency
detector ($\eta\simeq 1$), the last term in Eq. (45), which is due to the
feedback loop, can be neglected. In Figs. 3 (c) and (d), we show the optically
added noise and the mechanical response as a function of the optomechanical
cooperativity for different values of feedback gain and fixed value of the
cavity decay ratio. We have a large gain for small cooperativities at higher
feedback gain, but the cost of the strong signal amplification is an increase
in the added measurement noise. Nevertheless, the added noise is not
excessively large in the region of interest (overlap of the amplification,
sub-SQL and stable regions) to influence a precise force measurement.

Figure 4: (a) The off-resonant optically added noise of measurement and (b)
the mechanical response as a function of the normalized frequency for
different ratios of $\kappa_{2}/\kappa$ for fixed values of the feedback gain
$\zeta/\kappa=0.49$ and the optomechanical cooperativity ${\cal C}=1$. (c) The
off-resonant optically added noise of measurement and (d) the mechanical
response as a function of the normalized frequency for different values of
feedback gain and fixed $\kappa_{2}/\kappa=0.95$, optomechanical cooperativity
${\cal C}=1$ and detection efficiency $\eta=0.6$. The line with
$\zeta/\kappa=0$ corresponds to the no feedback scenario. The noise
suppression (beyond SQL, $n_{\rm add}<1/2$) and signal amplification ($R_{\rm
m}>1$) regions are shown by blue and red area, respectively. (e) FWHM as a
function of the feedback gain. Other parameters given in the caption of Fig.
2.

The optically added noise and the mechanical response away from the resonance
($\omega\neq 0$) for different ratios of $\kappa_{2}/\kappa$ and for a fixed
value of feedback gain and optomechanical cooperativity ${\cal C}=1$ are
depicted in Figs. 4 (a) and (b). Again it helps to have a largely asymmetric
cavity and high feedback gain to achieve strong mechanical response. In Figs.
4 (c) and (d), we plot the off-resonant optically added noise and the
mechanical response as a function of the normalized frequency. One can see
that the range of signal frequencies over which there is amplification
increase as we increase the feedback gain. Outside a certain frequency band,
the mechanical response to the input signal drops compared to the response at
resonance. We quantify the bandwidth of the amplification by the full width at
half maximum (FWHM) of the mechanical response. That is, we solve the equation
$R_{\rm m}[\omega]=R_{\rm m}[0]/2$ and the difference between the two values
of the frequencies at which the mechanical response is equal to half of its
maximum value is chosen to be FWHM. In Fig. 4 (e), we plot FWHM as a function
of feedback gain to show the increase in amplification bandwidth. This way,
the total detection bandwidth is improved with high gain since the spectrum of
the added noise is almost unaffected by the feedback. Moreover, there is no
limitation on the gain-bandwidth product. That is, as we increase the
mechanical response, we also increase the amplification bandwidth.

###  III.3 Force sensitivity

We use the homodyne photocurrent of Eq. (29) to perform force measurements
[41]. In the case of stationary spectral measurements the signal $S(\omega)$
is

$\displaystyle{\cal
S}[\omega]=\left|{\left\langle{X_{{\rm{out}}}^{\phi,(2)}[\omega]}\right\rangle}\right|\,.$
(46)

The noise associated with the signal $S\left(\omega\right)$ is given by its
standard deviation

$\displaystyle{\cal
N}[\omega]={\left[{\frac{1}{2}{{\left\langle{\left[{X_{{\rm{out}}}^{\phi,(2)}\left(\omega\right)X_{{\rm{out}}}^{\phi,(2)}\left({-\omega}\right)+\left({\omega\leftrightarrow-\omega}\right)}\right]}\right\rangle}_{F=0}}}\right]^{1/2}}\,,$
(47)

and it is evaluated in the absence of the external force $F=0$. The signal-to-
noise ratio (SNR) is defined to be the ratio of the signal, Eq. (46), to the
variance of all the noises, Eq. (47), of a given measured signal and it
quantifies how much the signal is corrupted by the noise introduced by the
system [42, 41]. Here, we are interested in spectral measurements that are
stationary, that is, in the limit in which the measurement time approaches
infinity [42, 41]. We define SNR as

$\displaystyle{\rm SNR}[\omega]$ $\displaystyle=\frac{{\cal S}[\omega]}{{\cal
N}[\omega]}=\frac{{\left|{\bar{F}[\omega]}\right|}}{{\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}{{R_{\rm
m}[\omega]\left({\bar{n}+1/2+{n_{{\rm{add}}}}[\omega]}\right)}^{1/2}}}}$
$\displaystyle=\frac{{\left|{\left({{\Phi^{*}}[\omega]F[\omega-{\omega_{\rm{m}}}]-\Phi[\omega]F[\omega+{\omega_{\rm{m}}}]}\right)/2}\right|}}{{\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}R_{\rm
m}[\omega]{{\left({\bar{n}+1/2+{n_{{\rm{add}}}}[\omega]}\right)}^{1/2}}}}\,,$
(48)

In the case of an ideal resonant impulsive force, $F[\omega]=\textrm{const.}$,
SNR of the system becomes

${\rm SNR}[\omega]=\frac{{\left|{\cos\phi}\right|F}}{{\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}{{\left({\bar{n}+1/2+{n_{{\rm{add}}}}[\omega]}\right)}^{1/2}}}}\,.$
(49)

As expected, the SNR of the system depends on the optically added noise, the
thermal noise and the global phase. Since the signal enters into the system
through the same channel as the thermal noise, the mechanical gain amplifies
both the input signal and the mechanical thermal and quantum noises. That is,
the enhancement in the mechanical gain does not lead to the improvement of the
force sensitivity and the SNR. Although the added noise and the thermal noise
can be reduced for an open-loop system, the mechanical response never becomes
greater than unity. In contrast, in closed-loop dynamics, we can
simultaneously increase the mechanical response by the feedback loop and
reduce the added noise by the red-detuned laser drive.

We now turn to the investigation of the force sensitivity of the system. We
quantify the sensitivity of the measurement by the system SNR [42], given by
Eq (49). Performing a more sensitive measurement requires a greater associated
SNR [42]. There is a minimum value of SNR that allows the detection of the
force signal. The minimum detectable input of the device is the minimum signal
required to produce an output with ${\rm SNR}[\omega]=1$ [43, 44, 38, 42].
Therefore based on Eq (III.3), the sensitivity of a detector describes the
minimum magnitude of the input signal that can be faithfully detected and
measured, and it is defined as

${\rm S}[\omega]=\sqrt{2\hbar
m\gamma{\omega_{\rm{m}}}}{\left[{\left({\bar{n}+\frac{1}{2}}\right)+{n_{{\rm{add}}}}[\omega]}\right]^{1/2}}\,.$
(50)

In Fig. 5 (a), we plot the force sensitivity as a function of the normalized
frequency for different values of the feedback parameter and inefficent
detection $\eta=0.6$. We see that although we have nearly the same sensitivity
for all feedback parameters, inefficent detection reduces the force
sensitivity. As can be deduced from Eq. (50), the less noise we have, the
better the sensitivity we can achieve. As a result, Eq. (50) suggests that we
can improve the sensitivity by setting the value of the added noise as small
as possible. As shown in Fig. 5 (b), thermal noise dramatically reduces the
sensitivity of the system. Therefore, for a high-precision force sensing, we
need to cool the environment to keep the thermal noise $\bar{n}$ as small as
possible. For instance, for a system at room temperature $T=300\,K$
($\bar{n}=1.8\times 10^{7}$), the value of sensitivity is ${\rm S}=2.5\times
10^{-16}\,\,{\rm N}/\sqrt{\rm Hz}$, which can be improved to ${\rm
S}=4.1\times 10^{-20}\,\,{\rm N}/\sqrt{\rm Hz}$ by cooling the mechanical
motion to zero temperature using cooling techniques.

Figure 5:  (a) Force sensitivity at zero temperature as a function of the
normalized frequency and (b) force sensitivity at finite temperature for
different values of feedback gain and fixed $\kappa_{2}/\kappa=0.95$,
optomechanical cooperativity ${\cal C}=1$ and detection efficiency $\eta=0.6$.
The line with $\zeta/\kappa=0$ corresponds to the no-feedback scenario. Other
parameters given in the caption of Fig. 2

##  IV  Summary and Conclusion

We present a scheme for highly sensitive detection of forces achievable with
state-of-the-art optomechanical systems [10, 9, 24]. The scheme manifests the
potential of current experimental setups of membrane-in-the-middle cavity
optomechanical systems to be employed as high-precision force sensors. The
proposed force sensor is based on the optomechanical interaction between an
optical mode and a nanomechanical membrane assisted by feedback-controlled in-
loop light. We have shown that the optical cavity subjected to the feedback
loop is a powerful tool for manipulating mechanical systems. An asymmetric
optomechanical cavity in the resolved sideband regime ensures low measurement
noise; the feedback loop provides a high mechanical gain and large detection
bandwidth.

The applicability of our method depends strongly on the used parameter regime.
In the classical regime, where thermal noise is the main limiting factor,
amplifying the mechanical response using measurement feedback does not provide
an advantage. As the SNR and the sensitivity of the system are not improved
compared to open-loop dynamics, the closed-loop system performs equivalently.
In the regime of quantum sensors, where thermal noise is small ($\bar{n}$ is
of order one or less), the situation is, however, completely different. First,
if one decreases the measurement noise $n_{\rm add}$, its effect will be more
significant on a relative scale: one can go below the SQL by reducing the
added noise to a negligible level. Second, it is also useful to amplify the
mechanical motion to reduce the risk of the homodyne detector not registering
any signal at all. With that one amplifies thermal noise as well, but as its
negative effect is not significant for $\bar{n}\simeq 1$, it is a price worth
paying. In summary, in this regime, noise reduction and signal amplification
can be both extremely useful, which is impossible to achieve in the open-loop
case.

In the future, the proposed optomechanical setup could be modified to a
single-sided cavity with an optical isolator. Such architecture would exhibit
reduced optical loss, thus enhancing the effect of feedback on the system, and
allow combination with squeezed vacuum injection for further increase of the
measurement sensitivity. At the same time, such a system has one parameter
less for optimization and would require an optical isolator to separate the
output signal from input driving field. A detailed comparison of the
performance of the two cavities could then provide important insight into the
performance of feedback schemes in optomechanical systems.

###### Acknowledgements.

We would like to thank Stefano Zippilli and Ali Motazedifard for their useful
conversations. F. B. and L. R. acknowledge the support of project 19-22950Y of
the Czech Science Foundation. O. Č. and R. F. acknowledge the project of
19-17765S of Czech Science Foundation and the project
CZ.02.1.01/0.0/0.0/16_026/0008460 of MEYS CR. D. V. acknowledges the support
of the European Union Horizon 2020 Programme for Research and Innovation
through the Project QuaSeRT funded by the QuantERA ERA-NET Cofund in Quantum
Technologies and the Project No. 862644 (FET Open QUARTET).

##  Appendix A  Beyond the rotating wave approximation

We investigate the system dynamics beyond RWA by using the Floquet approach
[45, 46]. When including counter-rotating terms, the time-dependent equations
of motion arise as

$\frac{{{\rm d}{\bf{u}}(t)}}{{{\rm
d}t}}={\bf{A}}(t){\bf{u}}(t)+{\bf{H}}{{\bf{n}}_{{\rm{in,fb}}}}(t)\,.$ (51)

We can write
${\bf{A}}(t)=\sum\nolimits_{n=-1}^{1}{{{\bf{A}}^{(n)}}{e^{-2in{\omega_{\rm{m}}}t}}}={{\bf{A}}^{(-1)}}{e^{-2i{\omega_{\rm{m}}}t}}+{{\bf{A}}^{(0)}}+{{\bf{A}}^{(1)}}{e^{2i{\omega_{\rm{m}}}t}}$
where ${{\bf{A}}^{(0)}}={{\bf{A}}}$ in Eq. (25) and

${{\bf{A}}^{(1)}}=\left({\begin{array}[]{*{20}{c}}0&0&0&{i\Lambda}\\\
0&0&0&0\\\ 0&{i\Lambda}&0&0\\\ 0&0&0&0\end{array}}\right)\,,\\\
\quad{{\bf{A}}^{(-1)}}=\left({\begin{array}[]{*{20}{c}}0&0&0&0\\\
0&0&{-i\Lambda}&0\\\ 0&0&0&0\\\ {-i\Lambda}&0&0&0\end{array}}\right)\,.$ (52)

By defining the system in terms of its Fourier components as

${\bf{u}}(t)=\sum\nolimits_{n=-\infty}^{\infty}{{{\bf{u}}^{(n)}}(t){e^{-2in{\omega_{\rm{m}}}t}}}\,,$
(53)

we can write Eq. (51) in the frequency domain in the following matrix form

$\displaystyle\left({-i\omega-2in{\omega_{\rm{m}}}}\right){{\bf{u}}^{(n)}}[\omega]$
$\displaystyle={{\bf{A}}^{(-1)}}{{\bf{u}}^{(n-1)}}[\omega]+{{\bf{A}}^{(0)}}{{\bf{u}}^{(n)}}[\omega]$
$\displaystyle+{{\bf{A}}^{(1)}}{{\bf{u}}^{(n+1)}}[\omega]+{\delta_{n,0}}{\bf{H}}{{\bf{n}}_{{\rm{in}},{\rm{fb}}}}[\omega].$
(54)

In RWA, ${{\bf{A}}^{(n\neq 0)}}=0$, we neglect terms containing $\omega\pm
2\omega_{\rm{m}}$ in these equations. Counter-rotating components introduce
sidebands shifted by $\pm 2\omega_{\rm{m}}$. We can then write Eq. (A) as

${\bf{\bar{u}}}[\omega]=-{\left({i\omega{{\bf{I}}_{12\times
12}}+{\bf{\bar{A}}}}\right)^{-1}}{\bf{\bar{H}}}{{{\bf{\bar{n}}}}_{{\rm{in}},{\rm{fb}}}}[\omega]\,,$
(55)

where ${{\bf{I}}_{12\times 12}}$ is the identity matrix and we define the
vector of fluctuation as $\bar{\bf
u}=\left[{{{\bf{u}}^{(-1)}}[\omega],{{\bf{u}}^{(0)}}[\omega],{{\bf{u}}^{(1)}}[\omega]}\right]$
and its corresponding noise vector as
${{{\bf{\bar{n}}}}_{{\rm{in}},{\rm{fb}}}}[\omega]=\left[0,{{\bf{n}}_{{\rm{in}},{\rm{fb}}}}[\omega],0\right]$.
Moreover, ${\bf{\bar{H}}}={\rm{diag}}[0,{\bf{H}},0]$ and the matrix
${\bf{\bar{A}}}$ is given by

${\bf{\bar{A}}}=\left({\begin{array}[]{*{20}{c}}{{{\bf{A}}^{(0)}}-2i{\omega_{\rm{m}}}{{\bf{I}}_{4\times
4}}}&{{{\bf{A}}^{(1)}}}&0\\\
{{{\bf{A}}^{(-1)}}}&{{{\bf{A}}^{(0)}}}&{{{\bf{A}}^{(1)}}}\\\
0&{{{\bf{A}}^{(-1)}}}&{{{\bf{A}}^{(0)}}+2i{\omega_{\rm{m}}}{{\bf{I}}_{4\times
4}}}\end{array}}\right)\,.$ (56)

By substituting $\bar{\bf u}\left[\omega\right]$ into the input-output
relation the fluctuations of the output fields can be obtained.

In Fig. 6 (a), we show the optically added noise for a system with
$\kappa=2\omega_{\rm m}$ to which RWA does not apply. The dashed lines
correspond to the RWA in the system dynamics. One can see that the influence
of the counter-rotating terms appears as we increase the optomechanical
cooperativity. The minimum of the added noise increases by considering
counter-rotating term and hence the system sensitivity decreases. In Fig. 6
(b), we plot the mechanical response as a function of the optomechanical
cooperativity for different values of feedback gain. We can also see that the
counter-rotating terms contribute to the signal amplification in the system.
Therefore, the price paid for signal amplification beyond RWA is sensitivity
reduction.

Figure 6: (a) The optically added noise of measurement and (b) the mechanical
response as a function of the optomechanical cooperativity for different
values of feedback gain and for fixed $\kappa_{2}/\kappa=0.95$ and
$\kappa=2\omega_{\rm{m}}$. Solid lines and dashed lines correspond to beyond
the RWA and RWA, respectively. Other parameters same as before. The noise
suppression and the signal amplification regions are shown in blue and red,
respectively.

##  Appendix B  Scattering matrix elements

The output and input fields are related via the scattering matrix ${\bf s}$ as

$\displaystyle{{\bf{n}}_{{\rm{out}}}^{(2)}}[\omega]={\bf{s}}[\omega]{{\bf{n}}_{{\rm{in,fb}}}}[\omega]-{{\bf{n}}_{{\rm{in}}}^{(2)}}[\omega]\,,$
(57)

where have defined
${{\bf{n}}_{{\rm{in,fb}}}}={[{c_{{\rm{in,fb}}}},c_{{\rm{in,fb}}}^{\dagger},{b_{{\rm{in,f}}}},b_{{\rm{in,f}}}^{\dagger}]^{T}}$,
${\bf{n}}_{{\rm{in}}}^{(2)}={[c_{{\rm{in}}}^{(2)},c_{{\rm{in}}}^{(2){\dagger}},{b_{{\rm{in}}}},b_{{\rm{in}}}^{\dagger}]^{T}}$,
${\bf{n}}_{{\rm{out}}}^{(2)}={[c_{{\rm{out}}}^{(2)},c_{{\rm{out}}}^{(2){\dagger}},{b_{{\rm{out}}}},b_{{\rm{out}}}^{\dagger}]^{T}}$.
Hereafter, the frequency dependence of these coefficients, represented by
$\omega$ is implicit for conciseness of notation. By introducing the
susceptibilities of the cavity field and the mechanical oscillator as

$\displaystyle\chi_{c}^{-1}[\omega]=$ $\displaystyle-\kappa+i\omega+\zeta
e^{-i\phi}\,,$ (58) $\displaystyle\chi_{m}^{-1}[\omega]=$
$\displaystyle{-\gamma+i\omega}\,,$ (59)

with ${\chi_{c}^{*}}[\omega]=({\chi_{c}}[-\omega])^{*}$ and
${\chi_{m}^{*}}[\omega]=({\chi_{m}}[-\omega])^{*}$ and after straightforward
calculations, we obtain the matrix elements $s_{ij}$ with $i,j=1,\dots,4$ as

$\displaystyle{s_{11}}=$
$\displaystyle-2\sqrt{{\kappa_{2}}{\kappa_{{\rm{fb}}}}}{\chi_{c}}\Xi\left({{\Lambda^{2}}\chi_{c}^{*}\chi_{m}^{*}+1}\right)\,,$
(60a) $\displaystyle{s_{12}}=$ $\displaystyle
2\sqrt{{\kappa_{2}}{\kappa_{{\rm{fb}}}}}\zeta{e^{i\phi}}{\chi_{c}}\chi_{c}^{*}\Xi\,,$
(60b) $\displaystyle{s_{13}}=$ $\displaystyle
2i\sqrt{\gamma{\kappa_{2}}}{\chi_{c}}{\chi_{m}}\Lambda\Xi\left({{\Lambda^{2}}\chi_{c}^{*}\chi_{m}^{*}+1}\right)\,,$
(60c) $\displaystyle{s_{14}}=$ $\displaystyle
2i\sqrt{\gamma{\kappa_{2}}}\zeta{e^{i\phi}}\Lambda{\chi_{c}}\chi_{c}^{*}\chi_{m}^{*}\Xi\,,$
(60d) $\displaystyle{s_{21}}=$ $\displaystyle
2\sqrt{{\kappa_{2}}{\kappa_{{\rm{fb}}}}}\zeta{e^{-i\phi}}{\chi_{c}}\chi_{c}^{*}\Xi\,,$
(60e) $\displaystyle{s_{22}}=$
$\displaystyle-2\sqrt{{\kappa_{2}}{\kappa_{{\rm{fb}}}}}\chi_{c}^{*}\Xi\left({{\Lambda^{2}}{\chi_{c}}{\chi_{m}}+1}\right)\,,$
(60f) $\displaystyle{s_{23}}=$
$\displaystyle-2i\sqrt{\gamma{\kappa_{2}}}\zeta{e^{-i\phi}}\Lambda{\chi_{c}}\chi_{c}^{*}{\chi_{m}}\Xi\,,$
(60g) $\displaystyle{s_{24}}=$
$\displaystyle-2i\sqrt{\gamma{\kappa_{2}}}\chi_{c}^{*}\chi_{m}^{*}\Lambda\Xi\left({{\Lambda^{2}}{\chi_{c}}{\chi_{m}}+1}\right)\,,$
(60h) $\displaystyle{s_{31}}=$ $\displaystyle
2i\sqrt{\gamma{\kappa_{{\rm{fb}}}}}{\chi_{c}}{\chi_{m}}\Lambda\Xi\left({{\Lambda^{2}}\chi_{c}^{*}\chi_{m}^{*}+1}\right)\,,$
(60i) $\displaystyle{s_{32}}=$
$\displaystyle-2i\sqrt{\gamma{\kappa_{{\rm{fb}}}}}\zeta{e^{i\phi}}\Lambda{\chi_{c}}\chi_{c}^{*}{\chi_{m}}\Xi\,,$
(60j) $\displaystyle{s_{33}}=$ $\displaystyle
2\gamma{\chi_{m}}\Xi\left[{\chi_{c}^{*}\left({{\zeta^{2}}{\chi_{c}}-{\Lambda^{2}}\chi_{m}^{*}}\right)-1}\right]\,,$
(60k) $\displaystyle{s_{34}}=$ $\displaystyle
2\gamma\zeta{e^{i\phi}}{\Lambda^{2}}{\chi_{c}}\chi_{c}^{*}{\chi_{m}}\chi_{m}^{*}\Xi\,,$
(60l) $\displaystyle{s_{41}}=$ $\displaystyle
2i\sqrt{\gamma{\kappa_{{\rm{fb}}}}}\zeta{e^{-i\phi}}\Lambda{\chi_{c}}\chi_{c}^{*}\chi_{m}^{*}\Xi\,,$
(60m) $\displaystyle{s_{42}}=$
$\displaystyle-2i\sqrt{\gamma{\kappa_{{\rm{fb}}}}}\chi_{c}^{*}\chi_{m}^{*}\Lambda\Xi\left({{\Lambda^{2}}{\chi_{c}}{\chi_{m}}+1}\right)\,,$
(60n) $\displaystyle{s_{43}}=$ $\displaystyle
2\gamma\zeta{\Lambda^{2}}{e^{-i\phi}}{\chi_{c}}\chi_{c}^{*}{\chi_{m}}\chi_{m}^{*}\Xi\,,$
(60o) $\displaystyle{s_{44}}=$ $\displaystyle
2\gamma\chi_{m}^{*}\Xi\left[{{\chi_{c}}\left({{\zeta^{2}}\chi_{c}^{*}-{\Lambda^{2}}{\chi_{m}}}\right)-1}\right]\,,$
(60p)

where we have defined

$\displaystyle{\Xi^{-1}}={\Lambda^{2}}\left({{\Lambda^{2}}{\chi_{c}}\chi_{c}^{*}{\chi_{m}}\chi_{m}^{*}+{\chi_{c}}{\chi_{m}}+\chi_{c}^{*}\chi_{m}^{*}}\right)+1-{\zeta^{2}}{\chi_{c}}\chi_{c}^{*}\,,$
(61)

Evidently, the output measurement quadrature of the field depends on the phase
of the local oscillator, the feedback gain and the optomechanical coupling
rate. In the case of zero optomechanical interaction $\Lambda=0$, the elements
$s_{13}$, $s_{14}$, $s_{23}$, $s_{24}$, $s_{31}$, $s_{32}$, $s_{41}$ and
$s_{42}$ become zero as one should expect.

##  Appendix C  Correlations in the frequency domain

Here, we transform correlations given by Eqs. (20)-(23) into the frequency-
domain

$\displaystyle\langle c_{\rm in,fb}[\omega]c_{\rm
in,fb}[\omega^{\prime}]\rangle=$ $\displaystyle 2\pi\left(n_{\rm fb}-m_{\rm
fb}\right)\delta(\omega+\omega^{\prime})\,,$ (62) $\displaystyle\langle c_{\rm
in,fb}^{\dagger}[\omega]c_{\rm in,fb}^{\dagger}[\omega^{\prime}]\rangle=$
$\displaystyle 2\pi\left(n_{\rm fb}-m_{\rm
fb}^{*}\right)\delta(\omega+\omega^{\prime})\,,$ (63) $\displaystyle\langle
c_{\rm in,fb}^{\dagger}[\omega]c_{\rm in,fb}[\omega^{\prime}]\rangle=$
$\displaystyle 2\pi n_{\rm fb}\delta(\omega+\omega^{\prime})\,,$ (64)
$\displaystyle\langle c_{\rm in,fb}[\omega]c_{\rm
in,fb}^{\dagger}[\omega^{\prime}]\rangle=$ $\displaystyle 2\pi\big{(}n_{\rm
fb}\\!+\\!\frac{\kappa}{{\kappa_{\rm fb}}}\\!-\\!m_{\rm fb}\\!-\\!m_{\rm
fb}^{*}\big{)}\delta(\omega+\omega^{\prime})\,,$ (65)

We also use the following correlations in the derivation of the noise spectrum

$\displaystyle\langle{c_{{\rm{in}}}^{(2)}[\omega]{c_{{\rm{in}},{\rm{fb}}}}[\omega^{\prime}]}\rangle$
$\displaystyle=2\pi{p_{{\rm{fb}}}}\delta(\omega+\omega^{\prime})\,,$ (66)
$\displaystyle\langle{c_{{\rm{in}}}^{(2)}[\omega]c_{{\rm{in,fb}}}^{\dagger}[\omega^{\prime}]}\rangle$
$\displaystyle=2\pi\left({{p_{{\rm{fb}}}}+\sqrt{{\kappa_{\rm{2}}}/{\kappa_{{\rm{fb}}}}}}\right)\delta(\omega+\omega^{\prime})\,,$
(67)
$\displaystyle\langle{c_{{\rm{in}}}^{(2){\dagger}}[\omega]c_{{\rm{in,fb}}}^{\dagger}[\omega^{\prime}]}\rangle$
$\displaystyle=\langle{c_{{\rm{in}}}^{(2){\dagger}}[\omega]{c_{{\rm{in,fb}}}}(t^{\prime})}\rangle=0\,,$
(68)
$\displaystyle\langle{{c_{{\rm{in,fb}}}}[\omega]c_{{\rm{in}}}^{(2){\dagger}}[\omega^{\prime}]}\rangle$
$\displaystyle=2\pi\left({p_{{\rm{fb}}}^{*}[\omega]+\sqrt{{\kappa_{\rm{2}}}/{\kappa_{{\rm{fb}}}}}}\right)\delta(\omega+\omega^{\prime})\,,$
(69)
$\displaystyle\langle{c_{{\rm{in,fb}}}^{\dagger}[\omega]c_{{\rm{in}}}^{(2){\dagger}}[\omega^{\prime}]}\rangle$
$\displaystyle=2\pi p_{{\rm{fb}}}^{*}\delta(\omega+\omega^{\prime})\,,$ (70)
$\displaystyle\langle{{c_{{\rm{in}},{\rm{fb}}}}[\omega]c_{{\rm{in}}}^{(2)}[\omega^{\prime}]}\rangle$
$\displaystyle=\langle{c_{{\rm{in,fb}}}^{\dagger}c_{{\rm{in}}}^{(2)}[\omega^{\prime}]}\rangle=0\,,$
(71)

##  Appendix D  System stability

In this appendix, we use the Routh–Hurwitz criterion to investigate the
stability of the system. The system reaches its steady state when all the
eigenvalues of the drift matrix ${\bf A}$ have negative real parts [47]. The
system is stable if the following conditions are satisfied:

$\displaystyle{C_{1}}=$ $\displaystyle\gamma-\zeta\cos\phi+\kappa>0\,,$ (72)
$\displaystyle{C_{2}}=$
$\displaystyle\zeta\cos\phi\left[{-4{\gamma^{2}}-({\cal
C}+8)\gamma\kappa+2\zeta(2\gamma+\kappa)\cos\phi-3{\kappa^{2}}}\right]$
$\displaystyle+(\gamma+\kappa)\left[{{\gamma^{2}}+({\cal
C}+3)\gamma\kappa+{\kappa^{2}}}\right]>0\,,$ (73) $\displaystyle{C_{3}}=$
$\displaystyle\big{\\{}{\zeta\cos\phi\left[{\zeta\cos\phi({\cal
C}\kappa+2\kappa+2\gamma)-(\gamma+\kappa)(2{\cal
C}\kappa+3\kappa+\gamma)}\right]}\big{.}$ $\displaystyle\big{.}{+\kappa({\cal
C}+1){{(\gamma+\kappa)}^{2}}}\big{\\}}(\gamma-2\zeta\cos\phi+\kappa)>0\,,$
(74) $\displaystyle{C_{4}}=$ $\displaystyle{\cal
C}\kappa-2\zeta\cos\phi+\kappa>0\,.$ (75)

## References

  * Zhang  _et al._ [2017] J. Zhang, Y.-x. Liu, R.-B. Wu, K. Jacobs, and F. Nori, Quantum feedback: theory, experiments, and applications, Phys. Rep. 679, 1 (2017).
  * Serafini [2012] A. Serafini, Feedback control in quantum optics: An overview of experimental breakthroughs and areas of application, ISRN Optics 2012, 1 (2012).
  * Aspelmeyer  _et al._ [2014] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  * LIGO Scientific Collaboration [2009] LIGO Scientific Collaboration, Observation of a kilogram-scale oscillator near its quantum ground state, New J. Phys. 11, 073032 (2009).
  * Schäfermeier  _et al._ [2016] C. Schäfermeier, H. Kerdoncuff, U. B. Hoff, H. Fu, A. Huck, J. Bilek, G. I. Harris, W. P. Bowen, T. Gehring, and U. L. Andersen, Quantum enhanced feedback cooling of a mechanical oscillator using nonclassical light, Nat. Commun. 7, 1 (2016).
  * Poggio  _et al._ [2007] M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar, Feedback cooling of a cantilever’s fundamental mode below 5 mk, Phys. Rev. Lett. 99, 017201 (2007).
  * Wilson  _et al._ [2015] D. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, Measurement-based control of a mechanical oscillator at its thermal decoherence rate, Nature 524, 325 (2015).
  * Rossi  _et al._ [2018a] M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, Measurement-based quantum control of mechanical motion, Nature 563, 53 (2018a).
  * Rossi  _et al._ [2017] M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali, Enhancing sideband cooling by feedback-controlled light, Phys. Rev. Lett. 119, 123603 (2017).
  * Rossi  _et al._ [2018b] M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali, Normal-mode splitting in a weakly coupled optomechanical system, Phys. Rev. Lett. 120, 073601 (2018b).
  * Abari  _et al._ [2019] N. E. Abari, G. V. D. Angelis, S. Zippilli, and D. Vitali, An optomechanical heat engine with feedback-controlled in-loop light, New J. Phys. 21, 093051 (2019).
  * Li  _et al._ [2017] J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, Enhanced entanglement of two different mechanical resonators via coherent feedback, Phys. Rev. A 95, 043819 (2017).
  * Clerk  _et al._ [2010] A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, Introduction to quantum noise, measurement, and amplification, Rev. Mod. Phys. 82, 1155 (2010).
  * Khalili [2010] F. Y. Khalili, Optimal configurations of filter cavity in future gravitational-wave detectors, Phys. Rev. D 81, 122002 (2010).
  * Caniard  _et al._ [2007] T. Caniard, P. Verlot, T. Briant, P.-F. Cohadon, and A. Heidmann, Observation of back-action noise cancellation in interferometric and weak force measurements, Phys. Rev. Lett. 99, 110801 (2007).
  * Pontin  _et al._ [2014] A. Pontin, C. Biancofiore, E. Serra, A. Borrielli, F. S. Cataliotti, F. Marino, G. A. Prodi, M. Bonaldi, F. Marin, and D. Vitali, Frequency-noise cancellation in optomechanical systems for ponderomotive squeezing, Phys. Rev. A 89, 033810 (2014).
  * Lecocq  _et al._ [2015] F. Lecocq, J. B. Clark, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Quantum nondemolition measurement of a nonclassical state of a massive object, Phys. Rev. X 5, 041037 (2015).
  * Bariani  _et al._ [2015] F. Bariani, H. Seok, S. Singh, M. Vengalattore, and P. Meystre, Atom-based coherent quantum-noise cancellation in optomechanics, Phys. Rev. A 92, 043817 (2015).
  * Wimmer  _et al._ [2014] M. H. Wimmer, D. Steinmeyer, K. Hammerer, and M. Heurs, Coherent cancellation of backaction noise in optomechanical force measurements, Phys. Rev. A 89, 053836 (2014).
  * Motazedifard  _et al._ [2016] A. Motazedifard, F. Bemani, M. H. Naderi, R. Roknizadeh, and D. Vitali, Force sensing based on coherent quantum noise cancellation in a hybrid optomechanical cavity with squeezed-vacuum injection, New J. Phys. 18, 073040 (2016).
  * Buchmann  _et al._ [2016] L. F. Buchmann, S. Schreppler, J. Kohler, N. Spethmann, and D. M. Stamper-Kurn, Complex squeezing and force measurement beyond the standard quantum limit, Phys. Rev. Lett. 117, 030801 (2016).
  * Khalili and Polzik [2018] F. Y. Khalili and E. S. Polzik, Overcoming the standard quantum limit in gravitational wave detectors using spin systems with a negative effective mass, Phys. Rev. Lett. 121, 031101 (2018).
  * Møller  _et al._ [2017] C. B. Møller, R. A. Thomas, G. Vasilakis, E. Zeuthen, Y. Tsaturyan, M. Balabas, K. Jensen, A. Schliesser, K. Hammerer, and E. S. Polzik, Quantum back-action-evading measurement of motion in a negative mass reference frame, Nature 547, 191 (2017).
  * Mason  _et al._ [2019] D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, Continuous force and displacement measurement below the standard quantum limit, Nat. Phys. 15, 745 (2019).
  * Levitan  _et al._ [2016] B. A. Levitan, A. Metelmann, and A. A. Clerk, Optomechanics with two-phonon driving, New J. Phys. 18, 093014 (2016).
  * Arcizet  _et al._ [2006] O. Arcizet, P.-F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Radiation-pressure cooling and optomechanical instability of a micromirror, Nature 444, 71 (2006).
  * Gigan  _et al._ [2006] S. Gigan, H. R. Böhm, M. Paternostro, F. Blaser, G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bäuerle, M. Aspelmeyer, and A. Zeilinger, Self-cooling of a micromirror by radiation pressure, Nature 444, 67 (2006).
  * Schliesser  _et al._ [2006] A. Schliesser, P. Del’Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Radiation pressure cooling of a micromechanical oscillator using dynamical backaction, Phys. Rev. Lett. 97, 243905 (2006).
  * Wilson-Rae  _et al._ [2007] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Theory of ground state cooling of a mechanical oscillator using dynamical backaction, Phys. Rev. Lett. 99, 093901 (2007).
  * Marquardt  _et al._ [2007] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Quantum theory of cavity-assisted sideband cooling of mechanical motion, Phys. Rev. Lett. 99, 093902 (2007).
  * Schliesser  _et al._ [2008] A. Schliesser, R. Rivière, G. Anetsberger, O. Arcizet, and T. J. Kippenberg, Resolved-sideband cooling of a micromechanical oscillator, Nat. Phys. 4, 415 (2008).
  * Massel  _et al._ [2011] F. Massel, T. T. Heikkilä, J.-M. Pirkkalainen, S. U. Cho, H. Saloniemi, P. J. Hakonen, and M. A. Sillanpää, Microwave amplification with nanomechanical resonators, Nature 480, 351 (2011).
  * McRae and Bowen [2012] T. G. McRae and W. P. Bowen, Near threshold all-optical backaction amplifier, Appl. Phys. Lett. 100, 201101 (2012).
  * Kippenberg  _et al._ [2005] T. J. Kippenberg, H. Rokhsari, T. Carmon, A. Scherer, and K. J. Vahala, Analysis of radiation-pressure induced mechanical oscillation of an optical microcavity, Phys. Rev. Lett. 95, 033901 (2005).
  * Botter  _et al._ [2012] T. Botter, D. W. C. Brooks, N. Brahms, S. Schreppler, and D. M. Stamper-Kurn, Linear amplifier model for optomechanical systems, Phys. Rev. A 85, 013812 (2012).
  * Kippenberg and Vahala [2007] T. Kippenberg and K. Vahala, Cavity opto-mechanics, Opt. Express 15, 17172 (2007).
  * Zippilli  _et al._ [2018] S. Zippilli, N. Kralj, M. Rossi, G. Di Giuseppe, and D. Vitali, Cavity optomechanics with feedback-controlled in-loop light, Phys. Rev. A 98, 023828 (2018).
  * Xu and Taylor [2014] X. Xu and J. M. Taylor, Squeezing in a coupled two-mode optomechanical system for force sensing below the standard quantum limit, Phys. Rev. A 90, 043848 (2014).
  * Hertzberg  _et al._ [2010] J. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A. Clerk, and K. Schwab, Back-action-evading measurements of nanomechanical motion, Nat. Phys. 6, 213 (2010).
  * Schreppler  _et al._ [2014] S. Schreppler, N. Spethmann, N. Brahms, T. Botter, M. Barrios, and D. M. Stamper-Kurn, Optically measuring force near the standard quantum limit, Science 344, 1486 (2014).
  * Vitali  _et al._ [2002] D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, Mirror quiescence and high-sensitivity position measurements with feedback, Phys. Rev. A 65, 063803 (2002).
  * Bernal-García  _et al._ [2020] D. N. Bernal-García, H. Vinck-Posada, and M. J. Woolley, Nonstationary force sensing under dissipative mechanical quantum squeezing, Phys. Rev. A 102, 053515 (2020).
  * Lucamarini  _et al._ [2006] M. Lucamarini, D. Vitali, and P. Tombesi, Scheme for a quantum-limited force measurement with an optomechanical device, Phys. Rev. A 74, 063816 (2006).
  * Motazedifard  _et al._ [2019] A. Motazedifard, A. Dalafi, F. Bemani, and M. H. Naderi, Force sensing in hybrid bose-einstein-condensate optomechanics based on parametric amplification, Phys. Rev. A 100, 023815 (2019).
  * Malz and Nunnenkamp [2016a] D. Malz and A. Nunnenkamp, Floquet approach to bichromatically driven cavity-optomechanical systems, Phys. Rev. A 94, 023803 (2016a).
  * Malz and Nunnenkamp [2016b] D. Malz and A. Nunnenkamp, Optomechanical dual-beam backaction-evading measurement beyond the rotating-wave approximation, Phys. Rev. A 94, 053820 (2016b).
  * DeJesus and Kaufman [1987] E. X. DeJesus and C. Kaufman, Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A 35, 5288 (1987).