# Hybrid Rotational Cavity Optomechanics Using Atomic Superfluid in a Ring

Sanket Das  Department of Physics, Indian Institute of Technology Guwahati,
Assam 781039, India  Pardeep Kumar<EMAIL_ADDRESS>Max Planck
Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany
M. Bhattacharya  School of Physics and Astronomy, Rochester Institute of
Technology, 84 Lomb Memorial Drive, Rochester, New York 14623, USA  Tarak N.
Dey  Department of Physics, Indian Institute of Technology Guwahati, Assam
781039, India

###### Abstract

We introduce a hybrid optomechanical system containing an annularly trapped
Bose-Einstein condensate (BEC) inside an optical cavity driven by Lauguerre-
Gaussian (LG) modes. Spiral phase elements serve as the end mirrors of the
cavity such that the rear mirror oscillates torsionally about the cavity axis
through a clamped support. As described earlier in a related system [P. Kumar
et. al., Phys. Rev. Lett. 127, 113601 (2021)], the condensate atoms interact
with the optical cavity modes carrying orbital angular momentum which create
two atomic side modes. We observe three peaks in the output noise spectrum
corresponding to the atomic side modes and rotating mirror frequencies,
respectively. We find that the trapped BEC’s rotation reduces quantum
fluctuations at the mirror’s resonance frequency. We also find that the atomic
side modes-cavity coupling and the optorotational coupling can produce
bipartite and tripartite entanglements between various constituents of our
hybrid system. We reduce the frequency difference between the side modes and
the mirror by tuning the drive field’s topological charge and the condensate
atoms’ rotation. When the atomic side modes become degenerate with the mirror,
the stationary entanglement between the cavity and the mirror mode diminishes
due to the suppression of cooling. Our proposal, which combines atomic
superfluid circulation with mechanical rotation, provides a versatile platform
for reducing quantum fluctuations and producing macroscopic entanglement with
experimentally realizable parameters.

##  I Introduction

Soon after its experimental realization [1, 2], Bose-Einstein condensate
(BEC), has emerged as a prominent and controllable system [3] to investigate
and mimic the persistent flow of superfluidity [4] and superconductivity [5].
Especially, when confined in multiply-connected geometries (toroidal traps)
[6] such systems exhibit persistent currents of superflow [7]. These
geometries can provide (i) topological protection to the quantum circulation
[8], (ii) longer dissipationless flow [9], and (iii) ‘supersonic’ rotations
[10]. Since the first experimental illustration of atomic persistent currents
in annularly-trapped BEC [11, 12], an incredible progress has been made in
this configuration to study atomic superflow for the investigation and
development of matter-wave interferometry [13], atomtronic circuits [14, 15,
16], topological excitations [17, 18], superfluid hydrodynamics [19, 20],
phase slips [21, 22], time crystals [23], gyroscopy [24] and cosmological
studies [25].

Therefore, in a toroidal geometry, it is very important to determine the
atomic circulation which involves the phase (winding number) measurement of a
rotational state. For detecting the winding number, the current state-of-the-
art technologies involve destructive methods [26], namely, optical absorption
imaging of the atoms in the ring which destroys its superfluid character.
Furthermore, in situ detection in the existing techniques is difficult due to
issues related to the optical resolution of the radius of the vortex which
demands time-of-flight expansion methods [27].

In recent studies, our group proposed a versatile method to detect the
magnitude [28, 29] as well as direction [30] of rotation of a bosonic ring
condensate with minimal destruction, in situ and in times. Specifically, the
method uses the tools of cavity optomechanics [31], a unique platform to
explore the radiation-pressure interaction of vibrating elements with the
photons confined inside an optical resonator. This method not only improves
the rotation sensing by three orders of magnitude but also provides a test bed
to manipulate the persistent currents by generating the optomechanical
entanglement between matter waves.

The aforementioned radiation-pressure interaction plays a dual role in cavity
optomechanics. On one side, it assists in manipulating the properties of the
mechanically pliable objects for applications such as quantum ground state
cooling [32, 33, 34] and generation of entanglement between macroscopic
objects [35, 36, 37, 38]. On the other end, it can also be used to control the
quantum properties of the light. For instance, optomechanical interactions can
generate squeezed states of light where the quantum fluctuations in one of the
optical quadratures are reduced below the shot-noise level. This comes with
increasing fluctuations in other orthogonal quadrature [39, 40, 41]. Such
engineered squeezed light states play a vital role in enhancing displacement
sensitivity in kilometer-sized gravitational wave observatories [42, 43],
optical communication [44], and metrology [45, 46].

Interestingly, the rotational analog of cavity optomechanics utilizes
radiation torque [47] from the angular momentum exchange between the Laguerre-
Gaussian (LG) cavity mode and a spiral phase plate as a rotating mirror [48].
Such systems have been investigated to cool the rotational degrees of freedom
to their quantum ground state [49] and for the realization of opto-rotational
entanglement [50, 51].

Nowadays, hybrid optomechanical systems [52, 53] pave a versatile pathway in
the development of quantum technologies [54]. Such systems take into account
of a mechanical oscillator interacting with an electromagnetic field [55] and
an additional physical system [56] or a degree of freedom [57]. In this paper,
we present a hybrid setup formed by confining an annularly trapped BEC inside
a LG cavity. The spiral phase elements serve as the end mirrors of the cavity
such that one mirror rotates about the cavity axis through a clamped support.
Specifically, in this hybrid system, we explore (i) ponderomotive squeezing
i.e., the reduction of quantum fluctuations of the output optical quadratures
below the shot noise level at various frequencies, and (ii) the generation of
bipartite and tripartite entanglement between the cavity and the matter waves
and the macroscopic rotating mirror. The main advantages of our hybrid setup
are: (a) From the perspective of toroidal BEC, the atomic rotation in it
provides a distinctive tool to correlate the optical amplitude and phase
quadratures and provide squeezing of about $87\%$ at a measurement angle of
$7^{\circ}$ around the frequencies of the Bragg-scattered sidemodes. On the
other hand, an optimum ponderomotive squeezing of $40$$\%$ below the shot
noise level occurs at the angular frequency of the rotating mirror that can
further be manipulated by the persistent currents of ring BEC. The same effect
also controls the bipartite and tripartite entanglement between the atomic
superfluids and the macroscopic object, paving a useful resource for quantum
information processing. (b) From the point of view of the LG cavity, it is
relatively easy to increase the orbital angular momentum (OAM) of the LG mode.
Moreover, in such a cavity setup, the optorotational effects scale with the
square of OAM [49] in contrast to the linear scaling of the conventional
cavity optomechanics. Using these facts, it is comparatively simple to
manipulate the optical interaction of the ring BEC and the rotating mirror
with a common LG mode. This in turn can be harnessed to increase the
ponderomotive squeezing and the simultaneous existence of bipartite and
tripartite entanglements. Our hybrid system represents a first proposal
involving matter wave rotation in hybrid optomechanics and can be exploited
for applications in optomechanical sensing, atomtronics and quantum
information processing.

The paper is organized as follows: In Sec. II, we provide details of our
hybrid system. In Sec. III, we then provide relevant equations for the quantum
dynamics and study the bistability and stability analysis. Sec. IV contains a
discussion on ponderomotive squeezing while Sec. V describes the bipartite and
tripartite entanglement in our hybrid setup. Finally, we conclude in Sec. VI.

##  II Model

The configuration under consideration is a hybrid setup consisting of a ring
BEC and an optical cavity. In the following, we provide details of each of
these elements.

Figure 1: (Color online) Schematic diagram of a hybrid setup consisting of a
ring BEC inside a cavity. The cavity comprises two spiral phase elements: one
is fixed and partially transmissive, while the other is rotating and highly
reflective. The colored arrows show the orbital angular momentum at different
positions: The purely extra cavity field, the purely intracavity field, and
the intracavity field arising from the intracavity field are green, red, and
blue arrows, respectively. The interference between the transmitted and the
reflected intracavity fields produces an optical lattice that probes the ring
BEC with winding number $L_{p}$.

###  II.1 Ring BEC

The first ingredient of our hybrid setup is an atomic BEC confined in a ring
trap and located inside the optical cavity. In the toroidal trap, the harmonic
potential experienced by each condensate atom along the radial ($\rho$) and
axial ($z$) directions is given by [28]

$\displaystyle U(\rho,z)$
$\displaystyle=\frac{1}{2}m\omega_{\rho}\left(\rho-R\right)^{2}+\frac{1}{2}m\omega_{z}z^{2}\;,$
(1)

where $m$, $\omega_{\rho}$ ($\omega_{z}$) and $R$ represent the mass of the
condensate atom, harmonic trapping frequency along the radial (axial)
direction and radius of the ring trap, respectively. Due to the above
potential, we assume that the atomic evolution along radial and axial
directions remains unchanged. However, we consider the dynamical evolution
along the azimuthal direction, $\varphi$, as it is not subjected to any
potential. This one-dimensional description is possible to be applied within
the current state-of-the-art experiments if the number of atoms $N$ of the
condensate obeys the following constraint [58]

$\displaystyle N$
$\displaystyle<\frac{4R}{3a_{\mathrm{Na}}}\sqrt{\frac{\pi\omega_{\rho}}{\omega_{z}}}\;,$
(2)

where $a_{\mathrm{Na}}$ is the $s$-wave scattering length of the sodium atoms
in the condensate.

###  II.2 Optical Cavity

In our setup, an optical cavity is formed with two spiral phase elements with
the same handedness. In this arrangement, the input coupler (IC) is a fixed
partially transmissive mirror, and the rear mirror (RM) is perfectly
reflective. Now, IC is designed to preserve the OAM of the light while
transmitting. On the other hand, it removes OAM of $2l\hbar$ per photon in
reflection. The perfect reflection from RM removes $2l\hbar$ angular momentum
from each photon. In Fig. 1, we provide the mode buildup at the various
locations along the optic axis for an input Laguerre-Gaussian beam carrying
orbital angular momentum $+l\hbar$. With the above design, the radiation
torque per photon transferred to RM is written as $\xi_{\phi}=2lc\hbar/2L$,
where $c$ is the speed of light and $L$ is the cavity length, respectively.

##  III Quantum Dynamics

###  III.1 Hamiltonian

As described above, the modes carrying OAM $\pm l\hbar$ build up inside the
cavity, creating an angular lattice about the cavity axis. From such an
optical lattice, some of the condensate atoms get coherently Bragg scattered
from a macroscopically occupied rotational state with winding number $L_{p}$
to states with $L_{p}\pm 2nl$, with $n=1,2,3,\cdots$. In the following, we
consider a dipole potential weaker than the condensate’s chemical potential
and consider only first-order diffraction, $L_{p}\rightarrow L_{p}\pm 2l$.

In dimensionless units, the Hamiltonian for our hybrid configuration is
expressed as

$\displaystyle H$ $\displaystyle=H_{\mathrm{BEC}}+H_{\phi}\;.$ (3)

Here, $H_{\mathrm{BEC}}$ is one-dimensional Hamiltonian for the azimuthal
motion of the ring BEC and is governed by [28]

$\displaystyle H_{\mathrm{BEC}}$
$\displaystyle=\sum_{j=c,d}\Bigg{[}\frac{\hbar\omega_{j}}{2}\left(X_{j}^{2}+Y_{j}^{2}\right)+\hbar\left(GX_{j}+\frac{U_{0}N}{2}\right)a^{\dagger}a\Bigg{]}$
$\displaystyle+\sum_{j=c,d}2\hbar\tilde{g}N\Big{(}X_{j}^{2}+Y_{j}^{2}\Big{)}+2\hbar\tilde{g}N\Big{(}X_{c}X_{d}-Y_{c}Y_{d}\Big{)}\;.$
(4)

The first term in the square bracket in Eq. (4) denotes the energies of the
Bragg-scattered side modes [28] of frequencies
$\omega_{c}=\hbar(L_{p}+2l)^{2}/(2I_{a})$ and
$\omega_{d}=\hbar(L_{p}-2l)^{2}/(2I_{a})$. The moment of inertia of each atom
about the cavity axis is $I_{a}=mR^{2}$. The dimensionless position and
momentum quadratures are defined as
$X_{j}=\left(j^{\dagger}+j\right)/\sqrt{2}$ and
$Y_{j}=\left(j-j^{\dagger}\right)/i\sqrt{2}$, respectively. The second term in
the square bracket on the right-hand side of Eq. (4) governs the effective
optomechanical coupling between the side modes and the optical field with the
coupling strength $G=U_{0}\sqrt{N}/{2\sqrt{2}}$. Here
$U_{0}=g_{a}^{2}/\Delta_{a}$ such that $g_{a}$ gives the interaction between
single atom and single photon and $\Delta_{a}$ denotes the atomic detuning.
The final two terms arise due to two-body atomic interactions of strength
$g=(2\hbar\omega_{\rho}a_{Na})/R$, which can be scaled such that
$\tilde{g}=g/4\pi\hbar$. Interestingly, from the Bogoliubov analysis, the
actual side mode frequencies can be written as
$\omega_{j}^{\prime}=\sqrt{\omega_{j}^{2}+4\omega_{j}\tilde{g}N}$ [59].
However, for the rest of our analysis, we impose $\omega_{c,d}\gg 4\tilde{g}N$
and hence use $\omega_{c,d}^{\prime}\sim\omega_{c,d}$.

On the other hand, the Hamiltonian for the optical cavity in the rotating
frame of the driving laser is governed by

$\displaystyle
H_{\phi}=-\hbar\Delta_{o}a^{\dagger}a+\frac{\hbar\omega_{\phi}}{2}\left(L_{z}^{2}+\phi^{2}\right)+\hbar
g_{\phi}a^{\dagger}a\phi-i\hbar\eta\Big{(}a-a^{\dagger}\Big{)}\;,$ (5)

where the first two terms describe the free energy of the detuned cavity mode
with $\Delta_{0}=\omega_{l}-\omega_{0}$ and the rotating mirror with resonance
frequency $\omega_{\phi}$, respectively. Here $L_{z}$ and $\phi$ represent the
respective dimensionless angular momentum and angular displacement of RM about
the cavity axis and these conjugate variables satisfy $[L_{z},\phi]=-i$. The
third term in Eq. (5) governs the radiation torque on RM, giving rise to
optorotational coupling given by

$\displaystyle g_{\phi}$
$\displaystyle=\frac{cl}{L}\sqrt{\frac{\hbar}{I\omega_{\phi}}}\;.$ (6)

The moment of inertia of the rotating mirror about the cavity axis is
described as $I=MR_{m}^{2}/2$, where $M$ ($R_{m}$) is the mass (radius) of the
RM. Finally, the last term on the right-hand side of Eq. (5)represents the
cavity drive having amplitude
$\eta=\sqrt{P_{in}\gamma_{o}/(\hbar\omega_{o})}$, where $P_{in}$ is the input
drive power and $\gamma_{o}$ is the cavity linewidth.

Using Eq. (3), we derive the Heisenberg equations of motion and include
damping and noise appropriately to obtain the following quantum Langevin
equations

$\displaystyle\dot{a}-i\left[\tilde{\Delta}-G\left(X_{c}+X_{d}\right)-g_{\phi}\phi\right]a$
$\displaystyle=-\frac{\gamma_{o}}{2}a+\eta+\sqrt{\gamma_{o}}a_{in}\;,$
$\displaystyle\ddot{X}_{c}+\gamma_{m}\dot{X}_{c}+\Omega_{c}^{2}X_{c}$
$\displaystyle=-\tilde{\omega}_{c}Ga^{\dagger}a-\mathcal{A}X_{d}$
$\displaystyle+\Omega_{c}\epsilon_{c}\;,$
$\displaystyle\ddot{X}_{d}+\gamma_{m}\dot{X}_{d}+\Omega_{d}^{2}X_{d}$
$\displaystyle=-\tilde{\omega}_{d}Ga^{\dagger}a+\mathcal{A}X_{c}$
$\displaystyle+\Omega_{d}\epsilon_{d}\;,$
$\displaystyle\ddot{\phi}+\gamma_{\phi}\dot{\phi}+\omega_{\phi}^{2}\phi$
$\displaystyle=-\omega_{\phi}g_{\phi}a^{\dagger}a+\omega_{\phi}\epsilon_{\phi}\;,$
(7)

where $\tilde{\Delta}=\Delta_{o}-\frac{U_{0}N}{2}$ is the effective cavity
detuning and $\gamma_{m}$ ($\gamma_{\phi}$) is the damping of each condensate
side mode (RM). Further, we have written the quantities
$\Omega_{c,d}^{2}=\left(\omega_{c,d}+4\tilde{g}N\right)^{2}-4\tilde{g}^{2}N^{2}$,
$\tilde{\omega}_{c,d}=\omega_{c,d}+2\tilde{g}N$,
$\mathcal{A}=2\tilde{g}N\left(\omega_{c}-\omega_{d}\right)$. In the above
Heisenberg-Langevin equation, $a_{in}(t)$ represents the vacuum Gaussian noise
of average $\langle a_{in}(t)\rangle=0$, injected into the cavity, and its
fluctuations are delta-correlated as

$\displaystyle\langle\delta a_{in}(t)\delta
a_{in}^{\dagger}(t^{{}^{\prime}})\rangle$
$\displaystyle=\delta(t-t^{{}^{\prime}})\;.$ (8)

Additionally, $\epsilon_{j}~{}(j=c,d)$, and $\epsilon_{\phi}$ in the quantum
Langevin equation describe the Brownian noise in the condensate side modes and
the rotating mirror, respectively. Their average values are zero and
fluctuations at respective temperatures $T_{j}$ and $T_{\phi}$ obey following
correlations

$\displaystyle\langle\epsilon_{j}(t)\epsilon_{j}(t{{}^{\prime}})\rangle$
$\displaystyle=\frac{\gamma_{m}}{\Omega_{j}}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}e^{-i\omega(t-t{{}^{\prime}})}\omega\Big{[}1+\coth{\frac{\hbar\omega}{2k_{B}T_{j}}}\Big{]}\;,$
(9)
$\displaystyle\langle\epsilon_{\phi}(t)\epsilon_{\phi}(t{{}^{\prime}})\rangle$
$\displaystyle=\frac{\gamma_{\phi}}{\omega_{\phi}}\int_{-\infty}^{+\infty}\frac{d\omega}{2\pi}e^{-i\omega(t-t{{}^{\prime}})}\omega\Big{[}1+\coth{\frac{\hbar\omega}{2k_{B}T_{\phi}}}\Big{]}\;,$
(10)

where $k_{B}$ is Boltzmann’s constant.

###  III.2 Steady State

Following the linearization approach, each operator $\mathcal{O}(t)$ can be
decomposed into its steady-state values $\mathcal{O}_{s}$ and a small
fluctuation $\delta\mathcal{O}(t)$. The steady-state values of each operator
are

$\displaystyle a_{s}$
$\displaystyle=\frac{\eta}{\sqrt{{\Delta^{{}^{\prime}}}^{2}+\left(\frac{\gamma_{o}}{2}\right)^{2}}}\;,$
(11a) $\displaystyle X_{cs}$ $\displaystyle=-\tilde{\Omega}_{c}Ga_{s}^{2}\;,$
(11b) $\displaystyle X_{ds}$ $\displaystyle=-\tilde{\Omega}_{d}Ga_{s}^{2}\;,$
(11c) $\displaystyle\phi_{s}$
$\displaystyle=-\frac{g_{\phi}}{\omega_{\phi}}a_{s}^{2}\;,$ (11d)

where modified cavity detuning is given by
$\Delta^{{}^{\prime}}=\tilde{\Delta}+\tilde{\Omega}G^{2}a_{s}^{2}+\frac{g_{\phi}^{2}}{\omega_{\phi}}a_{s}^{2}$
and $\tilde{\Omega}=\tilde{\Omega}_{c}+\tilde{\Omega}_{d}$, such that
$\tilde{\Omega}_{c,d}=\left({\tilde{\omega}_{c,d}\Omega_{d,c}^{2}\mp\tilde{\omega}_{d,c}\mathcal{A}}\right)/\left({\mathcal{A}^{2}+\Omega_{c}^{2}\Omega_{d}^{2}}\right)$.
Here the phase of the cavity drive is chosen such that $a_{s}$ is real. When
the effective cavity detuning is larger than the critical value,
$\tilde{\Delta}_{cr}=-\sqrt{3}\gamma_{0}/2$, the above steady-state solution
of $|a_{s}|^{2}$ manifests a bistable response concerning the input drive
field intensity as depicted in Fig. 2(a). Additionally Fig. 2(b) suggests the
input drive field intensity ($P_{in}>P_{cr}$) can lead to the bistability
response in $|a_{s}|^{2}$ where the critical power is
$P_{cr}=\hbar\omega_{0}\gamma_{0}^{2}/3\sqrt{3}\left(\tilde{\Omega}G^{2}+g_{\phi}^{2}/\omega_{\phi}\right)$.
In the remainder of the paper, we choose the parameters to avoid the bistable
regime and keep our system monostable. This can also be achieved by using
electronic feedback [60, 61], which allows us to choose the modified detuning,
$\Delta^{{}^{\prime}}$, independently of the radiation pressure.

Figure 2: (Color online) Bistability plot as a function of (a) input power and
(b) cavity detuning. Here $\tilde{\Delta}_{\mathrm{cr}}/2\pi=-0.17$ MHz, and
$P_{\mathrm{cr}}=1$ pW. The parameters used are $R=12~{}\mu$m, $N=10^{4}$,
$\omega_{\rho}/2\pi=\omega_{z}/2\pi=840$ Hz, $\gamma_{o}/2\pi=0.2$ MHz,
$\Delta_{a}/2\pi=5.4$ GHz, $g_{a}/2\pi=0.7$ MHz, $U_{0}/2\pi=90.7$ Hz,
$L_{p}=1$, $l=10$, $G/2\pi=3.2$ kHz, $\gamma_{m}/2\pi=0.8$ Hz,
$\omega_{o}/2\pi=10^{15}$ Hz, $a_{\mathrm{Na}}=0.1$ nm,
$\tilde{g}=14\tilde{g}_{m}$, $\tilde{g}_{m}/2\pi=78.8~{}\mu$Hz, $L=4$ mm,
$M=3.08$ $\mu$g, $\omega_{\phi}/2\pi=653$ Hz, $\gamma_{\phi}=0.1\gamma_{m}$,
and $R_{m}=15~{}\mu$m.

###  III.3 Stability Analysis

The fluctuation part of Eq. (7) can be expressed as a set of linearized
equations

$\displaystyle\dot{u}(t)$ $\displaystyle=\tilde{F}u(t)+v(t)\;,$ (12)

where the fluctuation vector $u(t)=\left(\delta X_{c}(t),\delta
Y_{c}(t),\delta X_{d}(t),\delta
Y_{d}(t),\delta\mathcal{Q}(t),\delta\mathcal{P}(t),\delta\phi(t),\delta
L_{z}(t)\right)^{T}$ and the input noise vector
$v(t)=\left(0,\epsilon_{c}(t),0,\epsilon_{d}(t),\sqrt{\gamma_{0}}\delta\mathcal{Q}_{in}(t),\sqrt{\gamma_{0}}\delta\mathcal{P}_{in}(t),0,\epsilon_{\phi}(t)\right)^{T}$,
respectively. We have expressed the cavity field fluctuations in terms of
their amplitude and phase quadratures as $\delta\mathcal{Q}=(\delta a+\delta
a^{\dagger})/\sqrt{2}$, and $\delta\mathcal{P}=-i\left(\delta a-\delta
a^{\dagger}\right)/\sqrt{2}$. The explicit form of the drift matrix
$\tilde{F}$ is given by

$\displaystyle\tilde{F}$
$\displaystyle=\begin{pmatrix}0&\Omega_{c}&0&0&0&0&0&0\\\
-\Omega_{c}&-\gamma_{m}&-\frac{\mathcal{A}}{\Omega_{c}}&0&-\frac{\tilde{\omega}_{c}G_{r}}{\Omega_{c}}&0&0&0\\\
0&0&0&\Omega_{d}&0&0&0&0\\\
\frac{\mathcal{A}}{\Omega_{d}}&0&-\Omega_{d}&-\gamma_{m}&-\frac{\tilde{\omega}_{d}G_{r}}{\Omega_{d}}&0&0&0\\\
0&0&0&0&-\frac{\gamma_{o}}{2}&-\Delta^{\prime}&0&0\\\
-G_{r}&0&-G_{r}&0&\Delta^{\prime}&-\frac{\gamma_{o}}{2}&-g_{\phi}^{r}&0\\\
0&0&0&0&0&0&0&\omega_{\phi}\\\
0&0&0&0&-g_{\phi}^{r}&0&-\omega_{\phi}&-\gamma_{\phi}\end{pmatrix}\;.$ (13)

The enhanced side modes-cavity coupling strength and optorotational coupling
are described as $G_{r}=\sqrt{2}Ga_{s}$, and
$g_{\phi}^{r}=\sqrt{2}g_{\phi}a_{s}$, respectively. We derive the stability
condition for the hybrid rotational cavity optomechanical system by invoking
the Routh-Hurwitz criterion [62], which suggests all the eigenvalues of
$\tilde{F}$ must have negative real parts. From Fig. 3, it is evident that our
system lies in the stable region for the weaker single photon optorotational
coupling strength with respect to the side mode-cavity coupling strength.
Moreover, increasing $g_{\phi}$ results gradual decrease in the
characteristics frequency bound of the RM.

Figure 3: (Color online) The stable and unstable regions are determined as
functions of the normalized optorotational coupling $g_{\phi}/G$ and the
normalized frequency of the rotating mirror $\omega_{\phi}/\omega_{d}$. We
consider $\Delta^{{}^{\prime}}=\omega_{\phi}$, $P_{\mathrm{in}}=12.4$ fW. The
other parameters are the same as in Fig. 2.

##  IV Ponderomotive Squeezing

In this section, we analyze a hybrid system to manipulate the quantum
properties of the output light. Our interest, in particular, is to reduce the
quantum fluctuations in the optical quadratures well below the shot noise
level and to describe the influence of quantum circulation on the optical
squeezing.

###  IV.1 Quadrature Noise Spectrum

To describe the ponderomotive squeezing, we invoke the homodyne measurement
technique and obtain the quadrature noise spectrum of the output optical
field. The homodyne-detected signal can be expressed as [63, 64]

$\displaystyle\mathcal{Q}^{out}_{\theta}(\omega)$
$\displaystyle=\mathcal{Q}_{out}(\omega)\cos\theta+\mathcal{P}_{out}(\omega)\sin\theta\;,$
(14)

where $\theta$ determines the measured field quadrature and is adjusted
experimentally. The cavity relates the output and input field quadratures as
$\mathcal{Q}_{out}(\omega)=\sqrt{\gamma_{o}}\delta\mathcal{Q}(\omega)-\mathcal{Q}_{in}(\omega)$,
and
$\mathcal{P}_{out}(\omega)=\sqrt{\gamma_{o}}\delta\mathcal{P}(\omega)-\mathcal{P}_{in}(\omega)$.
The output quadrature noise spectrum is then calculated as

$\displaystyle S(\omega)$
$\displaystyle=|\xi_{1}(\omega)|^{2}+|\xi_{2}(\omega)|^{2}+i\left[\xi^{\ast}_{1}(\omega)\xi_{2}(\omega)-\xi^{\ast}_{2}(\omega)\xi_{1}(\omega)\right]$
$\displaystyle-2\gamma_{m}\omega\Bigg{[}\frac{|\xi_{3}(\omega)|^{2}}{\Omega_{c}}+\frac{|\xi_{4}(\omega)|^{2}}{\Omega_{d}}\Bigg{]}\Bigg{[}1-\coth\Big{(}\frac{\hbar\omega}{2k_{B}T}\Big{)}\Bigg{]}$
$\displaystyle-2\frac{\gamma_{\phi}\omega}{\omega_{\phi}}|\xi_{5}(\omega)|^{2}\Bigg{[}1-\coth\Big{(}\frac{\hbar\omega}{2k_{B}T_{\phi}}\Big{)}\Bigg{]}\;.$
(15)

A full derivation of the above spectrum and detailed expressions of
$\xi_{i}$’s are given in Appendix A. Note that the noise spectrum in Eq.
(IV.1) is normalized in such a way that $S(\omega)=1$ represents a shot-noise
level [65].

Figure 4: (Color online) (a) Power spectral density (PSD) of the output
optical quadrature as a function of the response frequency $\omega/2\pi$, for
the measurement angles $\theta=90^{\circ}$ (black solid line),
$\theta=30^{\circ}$ (red dashed line), and $\theta=5^{\circ}$ (blue dot-dashed
line). (b) PSDs as a function of the measurement angle $\theta$. Solid red,
dashed blue and solid green curves are produced by fixing the response
frequency near $\Omega_{c}$, $\Omega_{d}$ and $\omega_{\phi}$, respectively.
Also, in these plots the spectrum is normalized to the shot noise (black
dotted line). Parameters used are $T=10$ nK, $T_{\phi}=1$ mK and other
parameters are same as in Fig. 3.

In Fig. 4, we show the power spectral density (PSD) of the optical quadrature
as a function of the response frequency for different homodyne measurement
angles. Here the black dotted curve represents the shot noise level. For
$\theta=90^{\circ}$, the phase quadrature of the output optical field results
in three Lorentzian peaks on top of the shot noise background at the
characteristic frequencies, $\Omega_{d}/2\pi\sim 595$ Hz, and
$\Omega_{c}/2\pi\sim 717$ Hz corresponding to side modes of the rotating BEC,
and the rotating mirror, $\omega_{\phi}/2\pi\sim 653$ Hz, respectively. It is
evident that the fluctuations remain above the vacuum noise in this scenario.
However, decreasing the measurement angle $\theta$ reduces the value of
$S(\omega)$ well below the shot noise level near the atomic side mode
frequencies as presented by the blue dot-dashed curve of Fig. 4(a). It is a
clear signature of the ponderomotive squeezing which occurs due to the
existence of correlations between optical amplitude and phase quadratures.
For instance, at a measurement angle, $\theta=5^{\circ}$, the output optical
noise is reduced by $84\%$ below the shot noise within a bandwidth of $20$ Hz
around $\omega=\Omega_{c},\Omega_{d}$ and the amplitude quadratures become
asymmetric like the Fano lineshape [66]. Such a line profile arises due to the
interference effects generated by the optical quadratures and the resonant
process produced by the atomic density modulation driven by amplitude
quadrature [67]. In Fig. 4(a), we set the effective cavity detuning equal to
$\omega_{\phi}$. As a result, the correlation between amplitude and phase
quadratures of the input field produces a small squeezing at
$\theta=0^{\circ}$ [64], as shown in the inset of Fig. 4(a).

As discussed above, the measurement angle during the homodyne detection plays
a crucial role to reduce the spectral noise below the shot noise. Now, in Fig.
4(b), we display the PSD by tuning the measurement angle in the range
$[0,180^{\circ}]$ by fixing the response frequencies around the mechanical
side modes . For our parameters, the maximum noise reduction near the
mechanical side mode frequencies occurs around $\theta=7^{\circ}$ and the
maximum of $87\%$ ponderomotive squeezing is obtained.

###  IV.2 Optimum Squeezing

In the preceding section, we have described that in our hybrid model,
ponderomotive squeezing can be generated by choosing appropriate measurement
angle. This happens only around the frequencies of the mechanical side modes.
However, the fluctuations in the quadrature spectrum at the frequency of the
rotating mirror still remains well above the background noise floor. In order
to reduce the fluctuations, we optimize the optical quadrature squeezing
spectrum $S_{opt}(\omega)$ by choosing $\theta(\omega)$ in such a way that
$dS(\omega)/d\theta=0$ for all the frequencies [63]. This gives

$\displaystyle\theta_{opt}(\omega)=\frac{1}{2}\arctan\left[-\frac{B_{2}(\omega)}{B_{1}(\omega)}\right]\;,$
(16)

where the expressions of $B_{1}(\omega)$ and $B_{2}(\omega)$ are too involved
and are shown in the appendix A.

Figure 5: (Color online) Optimized PSD of the optical quadrature as a function
of the response frequency for $G/2\pi=3.2$ kHz (black solid line),
$G/2\pi=9.6$ kHz (red solid line), and $G/2\pi=22.4$ kHz (blue solid line).
Parameters used are same as in Fig. 2.

The optimized squeezing spectrum is presented in Fig. 5. The exciting findings
of the optimized squeezing spectrum are: (i) there is a significant reduction
in the fluctuations below the shot noise level at the vicinity of the side
mode frequencies $\Omega_{c,d}$. (ii) Increasing the coupling strength between
the atomic side modes with the cavity enhances the ponderomotive squeezing
near the atomic side mode frequencies. Further, we observe a significant
enhancement in the optical squeezing spectrum at the side mode frequencies
$\Omega_{c,d}$. To understand its reason, we plot the optimized homodyne angle
as a function of the frequency as depicted in Fig. 5(b). The black-solid, red-
solid, and blue-solid curves suggest the optimized homodyne angle becomes zero
at the atomic side mode frequencies and at the mirror frequency. At these
specific frequencies, the major contribution to the optimized optical
squeezing spectrum originates from the first term of Eq. IV.1
($|\xi_{1}(\omega)|^{2}$), only. The vanishingly small functional values of
$\tilde{F_{2}}$ (Eq. A) at
$\Omega_{c},~{}\Omega_{d},~{}\textrm{and}~{}\omega_{\phi}$ lead the optimal
squeezing to the unity.

Figure 6: (Color online) Optimized PSD of the output optical quadrature
plotted at the frequency of the rotating mirror as a function of the winding
number $L_{p}$. Parameters used are same as in Fig. 2.

So far we have described the ponderomotive squeezing in our hybrid system by
tuning various parameters. However, such squeezing occurs only at the
frequencies of the Bragg-scattered mechanical side modes, whereas the optical
quadrature fluctuations at the frequency of rotating mirror reduced just to
the level of shot noise. In Fig. 6, we explore the influence of the persistent
flow in the ring BEC to manipulate the noise reduction and to achieve
ponderomotive squeezing at the resonance frequency of the rotating mirror. The
blue solid curve of Fig. 6, suggests the increase in the winding number of the
BEC enhances its interaction with the OAM carrying input field ($l=15$) to
obtain $40\%$ of ponderomotive squeezing. A weaker topological charge of the
input field requires a relatively higher winding number of the rotating BEC to
produce a similar amount of ponderomotive squeezing. Additionally, for lower
$L_{p}$ values, the atomic collisions dominate to give rise to the optical
mode squeezing.

##  V Entanglement

In the preceding section, we have exploited the radiation pressure force to
squeeze the quantum fluctuations of the output light field. The radiation
pressure also plays a crucial role in cooling down the rotational mirror to
its quantum ground state and producing entanglement. In particular, our hybrid
system sets a stage where the interactions between the atomic side modes with
the optical field and the radiation torque play a pivotal role in obtaining
the bipartite entanglements between various degrees of freedom.

###  V.1 Bipartite Entanglement

To quantify the entanglement between various subsystems, we use the linearized
dynamics of Eq.(12) and the Gaussian character of the quantum noise to achieve
the stationary Gaussian state, which can be fully characterized by a $8\times
8$ covariance matrix $V$, whose elements are written as
$V_{ij}=\frac{1}{2}\langle
u_{i}(\infty)u_{j}(\infty)+u_{j}(\infty)u_{i}(\infty)\rangle$. Under the
stable condition, the covariance matrix $V$ satisfies the Lyapunov equation
[31]

$\displaystyle\tilde{F}V+V\tilde{F}^{T}=-D,$ (17)

where the matrix of the stationary noise correlation function is
$D=\textrm{diag}\\{0,\gamma_{m}(2n_{c}+1),0,\gamma_{m}(2n_{d}+1),\gamma_{0}/2,\gamma_{0}/2,0,\gamma_{\phi}(2n_{m}+1)\\}$.
The mean thermal excitation for the BEC side modes and the mechanical
excitations of the rotating mirror are
$n_{i}=\left(\textrm{exp}\\{\hbar\Omega_{i}/k_{B}T\\}-1\right)^{-1}~{}(i=c,d)$,
and
$n_{m}=\left(\textrm{exp}\\{\hbar\omega_{\phi}/k_{B}T_{\phi}\\}-1\right)^{-1}$,
respectively. Using the above formalism, we study the two body entanglement in
the hybrid system by evaluating the logarithmic negativity $E$, defined as
[68]

$\displaystyle E$ $\displaystyle=\max[0,-\ln 2\eta^{-}]\;,$ (18)

where
$\eta^{-}=2^{-1/2}[{\Sigma(V)-[\Sigma(V)^{2}-4\det(V_{sub})]^{1/2}}]^{1/2}$,
with $\Sigma(V)=\det A+\det B-2\det C$. Here, $V_{sub}$ is a generic $4\times
4$ submatrix

$\displaystyle V_{sub}$ $\displaystyle=\begin{pmatrix}A&C\\\ C^{T}&B\\\
\end{pmatrix}\;,$ (19)

representing a particular bipartite system under consideration. $A,B$ and $C$
are $2\times 2$ blocks of the covariance matrix. The bipartite entanglement
exists if $E>0$ i.e., when $\eta^{-}<1/2$.

Figure 7: (Color online) The bipartite entanglement between the optical and
the mirror mode is plotted as a function of (a) cavity detuning when the
orbital angular momentum of the input field $l=243$ (b) the OAM of the driving
field for $L_{p}=1~{}\textrm{and}$ (c) the winding number of BEC for $l=243$
while the cavity detuning is $~{}\Delta^{\prime}=-1.2\omega_{\phi}$. We
consider the resonance frequency of the rotating mirror to be
$\omega_{\phi}=3$ MHz, mass of the mirror $M=0.1$ ng, radius $R_{m}=20$
$\mu$m, the cavity length $L=1$ mm, $\gamma_{0}/2\pi=0.48$ MHz,
$\gamma_{m}/2\pi=0.8$ Hz, $\gamma_{\phi}/2\pi=4.77$ Hz,
$\omega_{0}/2\pi=10^{15}$ Hz, $G/2\pi=7.67$ kHz, $U_{0}/2\pi=153.5$ Hz,
$\Delta_{a}=1.04$ GHz, $\omega_{\rho}/2\pi=8.4$ kHz, and $a_{Na}=2.5$ nm. The
input power of the driving field is $0.19$ nW. The bath temperature of the
rotating mirror is $T_{\phi}=5$ mK and the temperature of the atomic side
modes are $T=10$ nK.

In Fig. 7(a), we study the influence of the cavity detuning on the bipartite
entanglements in our hybrid system. $E_{am}$, $E_{ac}$, and $E_{ad}$ denote
the bipartite entanglements between cavity-mirror, cavity-atomic c mode, and
cavity-atomic d mode, respectively. The black solid curve suggests the optimal
cavity-mirror entanglement occurs around
$\Delta^{\prime}\approx-0.6\omega_{\phi}$. However, the blue and red solid
curves represent the optimal values of $E_{ac}$ and $E_{ad}$ at
$\Delta^{\prime}\approx-1.2\omega_{\phi}$. The entanglement between the cavity
and the atomic side modes sustains up to the bath temperature of the rotating
mirror $T_{\phi}\approx 13$ mK, as presented by the inset of Fig. 7(a). In
Fig. 7(b) and (c), we investigate the influence of the topological charge of
the input optical drive and atomic rotation on the entanglement properties
between various bipartite subsystems, respectively. The prominent interaction
strength between the optical and the acoustic modes produces a more
significant entanglement response than cavity-atomic side modes. More
interestingly, our study predicts diminishing entanglements at a specific
region of the OAM of the input beam. Also, the black solid curve of Fig. 7(c)
suggests for the topological charge of the input beam $l=243$ [69, 70, 71, 72,
73], the entanglement between the optical and the acoustic mode diminishes
when the $L_{p}$ values lie between 30 and 34. Moreover, the two Bragg
scattered atomic side modes produce distinct entanglement responses arising
from the optimal cavity detuning condition
$\Delta^{\prime}=-1.2\omega_{\phi}=-\Omega_{c,d}$. Now, to explain the
diminishing entanglements, we determine the energy of the rotating mirror,
which is given by

$\displaystyle U$
$\displaystyle=\frac{\hbar\omega_{\phi}}{2}\left[\langle\delta\phi^{2}\rangle+\langle\delta
L_{z}^{2}\rangle\right]=\frac{\hbar\omega_{\phi}}{2}\left(V_{77}+V_{88}\right)$
$\displaystyle=\hbar\omega_{\phi}\left(n_{eff}+\frac{1}{2}\right),$ (20)

where the steady-state mean phonon number ($n_{eff}$) is associated with the
effective mirror temperature ($T_{eff}$) by the relation
$n_{eff}=\left(\textrm{exp}({\hbar\omega_{\phi}/k_{B}T_{eff}})-1\right)^{-1}$.

Figure 8: (Color online) The mean phonon number of the rotating mirror as a
function of the topological charge of the input field when $L_{p}=1$. The
inset presents the variation of effective phonon number as a function of the
winding number of BEC for $l=243$. The dotted lines correspond to the unity of
the effective phonon number. All the other parameters are the same as in Fig.
7.

In Fig. 8 and in its inset, we present the effect of the OAM of the driving
field and the angular momentum of atomic BEC on the steady-state phonon
number, respectively. The solid black curve anticipates two distinct peaks in
the effective phonon response stemming from the cooling inhibition associated
with the topological charge $l\approx 226$ and $227$, and the red solid curve
of the inset depicts the suppression of cooling when the winding number
$L_{p}$ lies between 30 and 34. In the subsequent discussion, we demonstrate
how the occurrence of the dark modes suppresses the cooling mechanism by
introducing a center of mass coordinate $(X_{1cm},P_{1cm})$ and a relative
coordinate $(X_{1r},P_{1r})$ as

$\displaystyle
X_{1cm}=\frac{GX_{d}+g_{\phi}\phi}{\sqrt{G^{2}+g_{\phi}^{2}}},P_{1cm}=\frac{GY_{d}+g_{\phi}L_{z}}{\sqrt{G^{2}+g_{\phi}^{2}}},$
$\displaystyle X_{1r}=\frac{G\phi-
g_{\phi}X_{d}}{\sqrt{G^{2}+g_{\phi}^{2}}},P_{1r}=\frac{GL_{z}-g_{\phi}Y_{d}}{\sqrt{G^{2}+g_{\phi}^{2}}}.$
(21)

Neglecting the atom-atom interaction, we can further express the Hamiltonian
of Eq. 3 as

$\displaystyle H$
$\displaystyle=-\hbar\left(\Delta_{0}-\frac{U_{0}N}{2}\right)a^{\dagger}a-i\hbar\eta(a-a^{\dagger})+\frac{\hbar\omega_{c}}{2}\left(X_{c}^{2}+Y_{c}^{2}\right)+\hbar
GX_{c}a^{\dagger}a+\frac{\hbar}{2}\left(\frac{\omega_{d}G^{2}+\omega_{\phi}g_{\phi}^{2}}{G^{2}+g_{\phi}^{2}}\right)\left(X_{1cm}^{2}+P_{1cm}^{2}\right)$
$\displaystyle+\hbar\sqrt{G^{2}+g_{\phi}^{2}}X_{1cm}a^{\dagger}a+\frac{\hbar}{2}\left(\frac{\omega_{d}g_{\phi}^{2}+\omega_{\phi}G^{2}}{G^{2}+g_{\phi}^{2}}\right)\left(X_{1r}^{2}+P_{1r}^{2}\right)+\frac{\hbar}{2}\frac{Gg_{\phi}}{G^{2}+g_{\phi}^{2}}(\omega_{\phi}-\omega_{d})(X_{1cm}X_{1r}+X_{1r}X_{1cm}+P_{1cm}P_{1r}$
$\displaystyle+P_{1r}P_{1cm}),$ (22)

where the sixth and the last terms correspond to the interaction of the center
of mass coordinates with the optical field and the relative coordinate,
respectively. The above analysis shows that when
$L_{p}=1~{}\textrm{and}~{}l\approx 226$, one of the atomic side mode frequency
$\omega_{d}$ matches with $\omega_{\phi}$ and the relative coordinate is
decoupled from the center of mass coordinate as well as the optical mode.
Nonetheless, it is straightforward to show the existence of another set of
center of mass and relative coordinates defined as

$\displaystyle
X_{2cm}=\frac{GX_{c}+g_{\phi}\phi}{\sqrt{G^{2}+g_{\phi}^{2}}},P_{2cm}=\frac{GY_{c}+g_{\phi}L_{z}}{\sqrt{G^{2}+g_{\phi}^{2}}},$
$\displaystyle X_{2r}=\frac{G\phi-
g_{\phi}X_{c}}{\sqrt{G^{2}+g_{\phi}^{2}}},P_{2r}=\frac{GL_{z}-g_{\phi}Y_{c}}{\sqrt{G^{2}+g_{\phi}^{2}}},$
(23)

such that when $\omega_{c}=\omega_{\phi}$, the relative coordinate decouples
from the center of mass and the cavity field. Hence, the radiation torque
cooling is suppressed when the two atomic side modes degenerate with the
acoustic mode ($\omega_{\phi}=\omega_{c,d}$) as the relative coordinate stays
in the initial thermal state. Comparing Fig. 7(b) and Fig. 8, we can say that
cooling a rotating mirror close to its quantum ground state ($l<225$ and
$l>230$) helps to maintain quantum correlations. As a result, entanglement
persists. However, between $l=226$ and $227$, the phonon number attains a
minimum value of $2.2$, which signifies the proximity of the quantum ground
state. Even at these low phonon numbers, the quantum fluctuations are
sufficiently strong to disrupt the coherence necessary to attain any
entanglement.

###  V.2 Tripartite Entanglement

Lastly, we study the tripartite entanglements between the cavity, mirror, and
atomic side modes by quantifying the minimum residual contangle as [74, 75]

$\displaystyle\mathcal{R}_{\tau}^{\textrm{min}}$
$\displaystyle=\textrm{min}\left[\mathcal{R}_{\tau}^{a|mk},\mathcal{R}_{\tau}^{m|ak},\mathcal{R}_{\tau}^{k|am}\right]\;,$
(24)

where, $\mathcal{R}_{\tau}^{i|jk}=C_{i|jk}-C_{i|j}-C_{i|k}\geq
0~{}~{}(i,j,k=\textrm{cavity}(a),\textrm{mirror}(m),~{}\textrm{atomic}~{}c~{}\textrm{or}~{}d~{}\textrm{modes})$,
is the residual contangle written in terms of contangle $C_{u|v}$ of
subsystems of $u~{}\textrm{and}~{}v$ (see Ref. [76] for more details of
calculating $\mathcal{R}_{\tau}^{\textrm{min}}$). The black solid and the red
solid curve of Fig. 9(a) suggest that the tripartite entanglements between the
three constituents of the model system (cavity, mirror, and c-or d-modes) are
optimized when the cavity detuning $\Delta^{\prime}=-1.9\omega_{\phi}$.

Figure 9: (Color online) Tripartite entanglement in terms of minimum residual
contangle versus (a) cavity detuning for $l=243$, (b) the OAM of the driving
field for $L_{p}=1~{}\textrm{and}~{}\Delta^{\prime}=-1.9\omega_{\phi}$. The
black solid curve accounts for the tripartite entanglement associated with the
cavity-mirror-atomic c mode. The red solid curve accounts for the tripartite
entanglement associated with the cavity-mirror-atomic d mode. All the other
parameters are same as in Fig. 7.

Moreover, the tripartite entanglement arises when there’s frequent optical
interaction with matter waves, particularly in proportion to the number of
lattice maxima $(2l)$. It means that the likelihood of optical interaction
increases as the number of lattice maxima increases. When such interactions
become frequent, they can lead to the emergence of tripartite entanglement, as
shown in Fig. 9(b). In Fig. 9(c), we present the tripartite entanglement
between the cavity, mirror, and atomic side modes as a function of the winding
number of the condensate atoms when the topological charge of the driving
field is $l=243$. The black solid curve presents the diminishing effect of the
three body entanglement around $L_{p}=30$ due to the thermal fluctuations of
the rotating mirror. The red solid curve depicts the tripartite entanglement
response between the cavity, mirror, and the atomic d-mode. The presence of
the atomic d-mode produces a wider range in $L_{p}$, where the tripartite
entanglement does not persist. Moreover, the very distinct three-body
entanglement response can be utilized to distinguish clearly between the two
atomic side modes.

##  VI Conclusion

We have presented an unique hybrid opto-rotational system consisting of a ring
BEC confined inside a LG cavity. The system shows a distinctive way to squeeze
the quantum fluctuations of the output light field quadratures below the shot
noise. With the experimentally feasible parameters and for a measurement angle
of $7^{\circ}$, we achieved 87$\%$ of the ponderomotive squeezing around the
frequencies of Bragg-scattered sidemodes. Furthermore, two very distinct
systems in our hybrid configuration get coupled to a common cavity mode. As a
consequence, optical squeezing of about $40\%$ occurs even at the rotating
mirror frequency and can further be manipulated by the persistent currents in
the annularly trapped BEC. Our scheme also provides a versatile pathway to
utilize the atomic interactions and the radiation pressure force to produce
bipartite and tripartite entanglement between different physical entities of
the hybrid system. Interestingly, the quantum correlations in the system last
under the cooling conditions of the rotating end mirror and correspondingly
there exists a parameter regime where the entanglement survives. We expect our
results find interesting applications in atomtronics [16], optomechanics [77],
sensing [78] and quantum information processing [79].

## Acknowledgments

We thank R. Kanamoto for discussions. P. K. acknowledges the financial support
from the Max Planck Society. M.B. thanks the Air Force Office of Scientific
Research (FA9550-23-1-0259) for support.

##  Appendix A Power Spectral Density

To obtain the output quadrature noise spectrum, we use following input-output
relation,

$\displaystyle a_{out}(\omega)$ $\displaystyle=\sqrt{\gamma_{o}}\delta
a(\omega)-\delta a_{in}(\omega)\;.$ (25)

From Eq. (25), we can write following quadrature relations

$\displaystyle\mathcal{Q}_{out}(\omega)$
$\displaystyle=\sqrt{\gamma_{o}}\delta\mathcal{Q}(\omega)-\mathcal{Q}_{in}(\omega)\;,$
(26) $\displaystyle\mathcal{P}_{out}(\omega)$
$\displaystyle=\sqrt{\gamma_{o}}\delta\mathcal{P}(\omega)-\mathcal{P}_{in}(\omega)\;,$
(27)

By Fourier transforming the Eq. (12), we obtain the output optical quadratures
as

$\displaystyle\mathcal{Q}_{out}(\omega)$
$\displaystyle=\Big{[}\sqrt{\gamma_{o}}\tilde{F}_{2}(\omega)-1\Big{]}\mathcal{Q}_{in}(\omega)+i\sqrt{\gamma_{o}}\tilde{F}_{3}(\omega)\mathcal{P}_{in}(\omega)+\sqrt{2\gamma_{o}}\tilde{F}_{5}(\omega)\epsilon_{c}(\omega)+\sqrt{2\gamma_{o}}\tilde{F}_{7}(\omega)\epsilon_{d}(\omega)+\sqrt{2\gamma_{o}}\tilde{F}_{9}(\omega)\epsilon_{\phi}(\omega)\;,$
$\displaystyle\mathcal{P}_{out}(\omega)$
$\displaystyle=-i\sqrt{\gamma_{o}}\Big{[}2\tilde{F}_{1}(\omega)+\tilde{F}_{3}(\omega)\Big{]}\mathcal{Q}_{in}(\omega)+\Big{[}\sqrt{\gamma_{o}}\tilde{F}_{2}(\omega)-1\Big{]}\mathcal{P}_{in}(\omega)-i\sqrt{2\gamma_{o}}\tilde{F}_{4}(\omega)\epsilon_{c}(\omega)-i\sqrt{2\gamma_{o}}\tilde{F}_{6}(\omega)\epsilon_{d}(\omega)$
$\displaystyle-i\sqrt{2\gamma_{o}}\tilde{F}_{8}(\omega)\epsilon_{\phi}(\omega)\;,$
(28)

where $\tilde{F}_{i}(\omega)$’s are complicated complex functions written as
below.

$\displaystyle\tilde{F_{1}}(\omega)$
$\displaystyle=\frac{ia_{s}^{2}\sqrt{\gamma_{0}}}{D(\omega)}\left[G^{2}A_{\phi}(\omega)\left[\tilde{\omega}_{c}\left(\mathcal{A}+\tilde{A_{d}}(\omega)\right)+\tilde{\omega}_{d}\left(\tilde{A}_{c}(\omega)-\mathcal{A}\right)\right]+\omega_{\phi}g_{\phi}^{2}\left[\mathcal{A}^{2}+\tilde{A}_{c}(\omega)\tilde{A}_{d}(\omega)\right]\right],$
$\displaystyle\tilde{F_{2}}(\omega)$
$\displaystyle=\frac{\sqrt{\gamma_{0}}}{D(\omega)}\left[\mathcal{A}^{2}+\tilde{A}_{c}(\omega)\tilde{A}_{d}(\omega)\right]A_{3}(\omega)A_{\phi}(\omega),~{}\tilde{F_{3}}(\omega)=\frac{i\Delta^{\prime}\sqrt{\gamma_{0}}}{D(\omega)}A_{\phi}(\omega)\left[\mathcal{A}^{2}+\tilde{A}_{c}(\omega)\tilde{A}_{d}(\omega)\right],$
$\displaystyle\tilde{F_{4}}(\omega)$
$\displaystyle=-\frac{iGa_{s}\Omega_{c}}{D(\omega)}A_{3}(\omega)A_{\phi}(\omega)\left[\mathcal{A}+\tilde{A}_{d}(\omega)\right],~{}\tilde{F_{5}}(\omega)=\frac{\Delta^{\prime}\Omega_{c}Ga_{s}}{D(\omega)}A_{\phi}(\omega)\left[\mathcal{A}+\tilde{A}_{d}(\omega)\right],$
$\displaystyle\tilde{F_{6}}(\omega)$
$\displaystyle=-\frac{iGa_{s}\Omega_{d}}{D(\omega)}A_{3}(\omega)A_{\phi}(\omega)\left[\tilde{A}_{c}(\omega)-\mathcal{A}\right],~{}\tilde{F_{7}}(\omega)=\frac{\Delta^{\prime}\Omega_{d}Ga_{s}}{D(\omega)}A_{\phi}(\omega)\left[-\mathcal{A}+\tilde{A}_{c}(\omega)\right],$
$\displaystyle\tilde{F_{8}}(\omega)$
$\displaystyle=-\frac{ig_{\phi}a_{s}\omega_{\phi}}{D(\omega)}A_{3}(\omega)\left[\mathcal{A}^{2}+\tilde{A}_{c}(\omega)\tilde{A}_{d}(\omega)\right],~{}\tilde{F_{9}}(\omega)=\frac{\Delta^{\prime}\omega_{\phi}g_{\phi}a_{s}}{D(\omega)}\left[\mathcal{A}^{2}+\tilde{A}_{c}(\omega)\tilde{A}_{d}(\omega)\right].$
(29)

Further, using Eq. (A) in Eq. (14), we obtain

$\displaystyle\mathcal{Q}_{\theta}^{out}(\omega)$
$\displaystyle=\xi_{1}(\omega)\mathcal{Q}_{in}(\omega)+\xi_{2}(\omega)\mathcal{P}_{in}(\omega)+\xi_{3}(\omega)\epsilon_{c}(\omega)$
$\displaystyle+\xi_{4}(\omega)\epsilon_{d}(\omega)+\xi_{5}(\omega)\epsilon_{\phi}(\omega)\;,$
(30)

where, $\xi_{i}$’s can be expressed as

$\displaystyle\xi_{1}(\omega)$
$\displaystyle=-i\sqrt{\gamma_{o}}\left[2\tilde{F}_{1}(\omega)+\tilde{F}_{3}(\omega)\right]\sin\theta$
$\displaystyle+\left[\sqrt{\gamma_{o}}\tilde{F}_{2}(\omega)-1\right]\cos\theta\;,$
$\displaystyle\xi_{2}(\omega)$
$\displaystyle=\left[\sqrt{\gamma_{o}}\tilde{F}_{2}(\omega)-1\right]\sin\theta+i\sqrt{\gamma_{o}}\tilde{F}_{3}(\omega)\cos\theta\;,$
$\displaystyle\xi_{3}(\omega)$
$\displaystyle=-i\sqrt{2\gamma_{o}}\tilde{F}_{4}(\omega)\sin\theta+\sqrt{2\gamma_{o}}\tilde{F}_{5}(\omega)\cos\theta\;,$
$\displaystyle\xi_{4}(\omega)$
$\displaystyle=-i\sqrt{2\gamma_{o}}\tilde{F}_{6}(\omega)\sin\theta+\sqrt{2\gamma_{o}}\tilde{F}_{7}(\omega)\cos\theta\;,$
$\displaystyle\xi_{5}(\omega)$
$\displaystyle=-i\sqrt{2\gamma_{o}}\tilde{F}_{8}(\omega)\sin\theta+\sqrt{2\gamma_{o}}\tilde{F}_{9}(\omega)\cos\theta\;.$
(31)

The output quadrature spectrum is obtained from the definition

$\displaystyle S(\omega)$
$\displaystyle=\frac{1}{\pi}\int_{-\infty}^{\infty}d\omega^{\prime}\langle\mathcal{Q}_{\theta}^{out}(\omega^{\prime})\mathcal{Q}_{\theta}^{out}(\omega)\rangle\;.$

By using the following correlation relations

$\displaystyle\langle\mathcal{Q}_{in}(\omega^{{}^{\prime}})\mathcal{Q}_{in}(\omega)\rangle$
$\displaystyle=\langle\mathcal{P}_{in}(\omega^{{}^{\prime}})\mathcal{P}_{in}(\omega)\rangle=\pi\delta(\omega^{{}^{\prime}}+\omega)\;,$
$\displaystyle\langle\mathcal{Q}_{in}(\omega^{{}^{\prime}})\mathcal{P}_{in}(\omega)\rangle$
$\displaystyle=-\langle\mathcal{P}_{in}(\omega^{{}^{\prime}})\mathcal{Q}_{in}(\omega)\rangle=i\pi\delta(\omega^{{}^{\prime}}+\omega)\;,$
$\displaystyle\langle\epsilon_{c,d}(\omega^{{}^{\prime}})\epsilon_{c,d}(\omega)\rangle$
$\displaystyle=2\pi\frac{\gamma_{m}\omega^{{}^{\prime}}}{\Omega_{c,d}}\Bigg{[}1+\coth\Big{(}\frac{\hbar\omega^{{}^{\prime}}}{2k_{B}T}\Big{)}\Bigg{]}\delta(\omega^{{}^{\prime}}+\omega)\;,$
$\displaystyle\langle\epsilon_{\phi}(\omega^{{}^{\prime}})\epsilon_{\phi}(\omega)\rangle$
$\displaystyle=2\pi\frac{\gamma_{\phi}\omega^{{}^{\prime}}}{\omega_{\phi}}\Bigg{[}1+\coth\Big{(}\frac{\hbar\omega^{{}^{\prime}}}{2k_{B}T_{\phi}}\Big{)}\Bigg{]}\delta(\omega^{{}^{\prime}}+\omega)\;,$
(33)

we obtain the analytical expression of Eq. IV.1.

##  Appendix B Optimized Squeezing

The optimized angle in Eq. (16) is expressed in terms of $B_{1}(\omega)$ and
$B_{2}(\omega)$ and their expressions are given below

$\displaystyle B_{1}(\omega)$
$\displaystyle=|\kappa_{1}(\omega)|^{2}-|\kappa_{3}(\omega)|^{2}+i\left(\kappa_{1}^{*}(\omega)\kappa_{2}(\omega)-\kappa_{1}(\omega)\kappa_{2}^{*}(\omega)-\kappa_{2}^{*}(\omega)\kappa_{3}(\omega)+\kappa_{3}^{*}(\omega)\kappa_{2}(\omega)\right)$
$\displaystyle-2\gamma_{m}\frac{\omega}{\Omega_{c}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\right]\left(|\kappa_{4}(\omega)|^{2}-|\kappa_{5}(\omega)|^{2}\right)-2\gamma_{m}\frac{\omega}{\Omega_{d}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\right]\left(|\kappa_{6}(\omega)|^{2}-|\kappa_{7}(\omega)|^{2}\right)$
$\displaystyle-2\gamma_{\phi}\frac{\omega}{\omega_{\phi}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T_{\phi}}\right)\right]\left(|\kappa_{8}(\omega)|^{2}-|\kappa_{9}(\omega)|^{2}\right),$
$\displaystyle B_{2}(\omega)=$
$\displaystyle\kappa_{1}^{*}(\omega)\kappa_{2}(\omega)+\kappa_{2}^{*}(\omega)\kappa_{1}(\omega)+\kappa_{2}^{*}(\omega)\kappa_{3}(\omega)+\kappa_{2}(\omega)\kappa_{3}^{*}(\omega)+i\left(\kappa_{1}^{*}(\omega)\kappa_{3}(\omega)-\kappa_{3}^{*}(\omega)\kappa_{1}(\omega)\right)$
$\displaystyle-2\gamma_{m}\frac{\omega}{\Omega_{c}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\right]\left(\kappa_{4}^{*}(\omega)\kappa_{5}(\omega)+\kappa_{4}(\omega)\kappa_{5}^{*}(\omega)\right)$
$\displaystyle-2\gamma_{m}\frac{\omega}{\Omega_{d}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)\right]\left(\kappa_{6}^{*}(\omega)\kappa_{7}(\omega)+\kappa_{6}(\omega)\kappa_{7}^{*}(\omega)\right)$
$\displaystyle-2\gamma_{\phi}\frac{\omega}{\omega_{\phi}}\left[1-\coth\left(\frac{\hbar\omega}{2k_{B}T_{\phi}}\right)\right]\left(\kappa_{9}(\omega)\kappa_{8}^{*}(\omega)+\kappa_{9}^{*}(\omega)\kappa_{8}(\omega)\right)$
(35)

where $\kappa_{i}$’s are written as

$\displaystyle\kappa_{1}(\omega)=-i\sqrt{\gamma_{o}}\Big{[}2\tilde{F}_{1}(\omega)+\tilde{F}_{3}(\omega)\Big{]}\;,\kappa_{2}(\omega)=\sqrt{\gamma_{o}}\tilde{F}_{2}(\omega)-1\;,\kappa_{3}(\omega)=i\sqrt{\gamma_{o}}\tilde{F}_{3}(\omega)\;,$
$\displaystyle\kappa_{4}(\omega)=-i\sqrt{2\gamma_{o}}\tilde{F}_{4}(\omega)\;,\kappa_{5}(\omega)=\sqrt{2\gamma_{o}}\tilde{F}_{5}(\omega)\;,\kappa_{6}(\omega)=-i\sqrt{2\gamma_{o}}\tilde{F}_{6}(\omega)\;,$
$\displaystyle\kappa_{7}(\omega)=\sqrt{2\gamma_{o}}\tilde{F}_{7}(\omega)\;,\kappa_{8}(\omega)=-i\sqrt{2\gamma_{o}}\tilde{F}_{8}(\omega)\;,\kappa_{9}(\omega)=\sqrt{2\gamma_{o}}\tilde{F}_{9}(\omega)\;.$
(36)

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