# Bistability-assisted Mechanical Squeezing and Entanglement

Souvik Agasti<EMAIL_ADDRESS>IMOMEC division, IMEC,
Wetenschapspark 1, B-3590 Diepenbeek, Belgium  Institute for Materials
Research (IMO), Hasselt University, Wetenschapspark 1, B-3590 Diepenbeek,
Belgium  P. Djorwé<EMAIL_ADDRESS>Department of Physics, Faculty of
Science, University of Ngaoundere, P.O. Box 454, Ngaoundere, Cameroon

###### Abstract

We propose a scheme to squeeze mechanical motion and to entangle optical field
with mechanical motion in an optomechanical system containing a parametric
amplification. The scheme is based on optical bistability which emerges in the
system for a strong enough driving field. By considering the steady state’s
lower branch of the bistability, the system shows weak entanglement and almost
no mechanical squeezing. When the steady state is on the upper branch of the
bistable shape, both squeezing and entanglement are greatly enhanced.
Specifically, the entanglement shows three degrees of magnitude enhancement.
However, this giant entanglement is fragile against decoherence and thermal
fluctuation. Regarding the mechanical squeezing, it reaches the standard
quantum limit (SQL) in the upper branch of the bistability. Our proposal
provides a way to improve quantum effects in optomechanical systems by taking
advantage of nonlinearities. This scheme can be realized in similar systems
such as superconducting microwave, and hybrid optomechanical systems.

Optomechanics, bistability, entanglement, standard quantum limit, Wigner
function

††preprint: APS/123-QED

##  I Introduction

The precision measurement of the motion of a harmonic oscillator can be
minimized to ground-state quantum zero point fluctuation, which is limited by
the Heisenberg uncertainty principle ($\Delta X_{1}\Delta X_{2}\geq 1$, where
$X_{1}$ are the $X_{2}$ are its two motional quadratures). One can, in
principle, minimize the fluctuation in one quadrature with the cost of an
equivalent increment on the other one. This process is known as quantum
squeezing, and it was primarily generated on the squeezed laser in the early
80’s [1, 2]. Since that, quantum squeezing has been used in various
experiments including mechanical motion and optical field in cavity
optomechanical systems for instance [3, 4, 5]. Squeezed states are interesting
for a range of applications including force-sensing [6, 7], exceptional point
sensors [8, 9, 10, 11] and, overall interferometric detectors such as those in
LIGO [12] and VIRGO [13] experiments.

A lot of effort has been made over decades to squeeze mechanical oscillators
below standard quantum limit (SQL), for instance by using frequency modulation
[14], two-tone driving [4] and reservoir engineering [5, 15]. Besides, it is
also seen that periodical modulation of the amplitude of external drive can
induce a high degree of mechanical squeezing along with an optomechanical
entanglement in a cavity optomechanical system [16]. The ground-state cooling
of the mechanical mode, usually required to achieve this goal, has been
obtained through sideband cooling. Such schemes can be considered as an
example of reservoir engineering [15] where the cavity acts as a reservoir
whose force noise is squeezed. Squeezed states of light are typically
generated using Kerr nonlinear systems through parametric down-conversion
process [17]. In this direction, the dynamical Casimir effect (DCE), i.e.
quick movement of the mirror can generate photon pairs from a quantum vacuum,
has been implemented in cavity optomechanical systems [18, 19], where a
parametric conversion of mechanical energy to optical photons has been
observed. However, to observe the DCE, the mirror velocity is required to be
close to the velocity of light, which is very difficult to achieve in
practice. To bypass this problem, a few alternative schemes have been
proposed, e.g. modulating the ultrastrong light-matter coupling in cavity
quantum electrodynamics (QED) or driving an optical parametric oscillator. In
the present work, we explore both of them to reach the limit to obtain DCE.

Our benchmark system is an optomechanical cavity containing a parametric
amplifier, which is used to induce optical bistability once the driving field
is strong enough. By applying a squeezing operator to diagonalize the
Hamiltonian, a DCE is induced in an observable range [20]. With experimentally
feasible parameters, both the DCE and the bistability are used to enhance
optical steady states variables in the upper branch of the bistable shape. By
using these enhanced optical steady states, we observed a giant enhancement of
mechanical squeezing together with a greatly enhancement of entanglement
between optical and mechanical modes. The generated entanglement is three
order of magnitude greater than the one generated by the lower branch optical
steady states. This work provides an alternative way of generating
nonclassical states by exploring nonlinear effects in optomechanical systems.
This scheme can be extended to similar setup such as superconducting
microwaves, and hybrid optomechanical systems.

The rest of this paper is organized as follows. In section II, we describe the
model and the dynamical equations. The bistability-assisted squeezing and
entanglement, together with the effect of the squeezing parameter, are
described in section III, while section IV concludes the work.

##  II MODEL: Dynamical Casimir Effect

We consider an optomechanical system detuned by a two-photon drive of
frequency $\omega_{L}$ and amplitude $E$, which eventually squeezes the cavity
mode. The system is sketched in Fig. 1(a) where the driving induces the
parametric down-conversion of mechanical phonons to generate correlated
cavity-photon pairs, which corresponds to the Dynamical Casimir Effect (DCE).
The cavity detuning $\Delta_{s}$ depends on the cavity frequency
$(\omega_{c})$ and driving frequency $(\omega_{L})$ as,
$\Delta_{s}=\omega_{c}-\omega_{L}/2$), which can be useful to tune the
parametric phonon-photon coupling into resonance. We also consider that the
mechanical mode is driven, which leads to the system’s Hamiltonian,

$H=H_{CD}+H_{OM}+H_{MD},$ (1)

with,

$\displaystyle H_{CD}$
$\displaystyle=\Delta_{s}a_{s}^{\dagger}a_{s}+\frac{E}{2}(a_{s}^{2}+{a_{s}^{\dagger}}^{2}),$
(2a) $\displaystyle H_{OM}$
$\displaystyle=\omega_{m}b^{\dagger}b-g_{0}a^{\dagger}_{s}a_{s}(b+b^{\dagger}),$
(2b) $\displaystyle H_{MD}$
$\displaystyle=\frac{F}{2}[e^{i\omega_{d}t}b+e^{-i\omega_{d}t}b^{\dagger}],$
(2c)

where $a_{s}(a_{s}^{\dagger})$ and $b(b^{\dagger})$ are the annihilation
(creation) operators of the cavity and mechanical modes, respectively. The
single-phonon mechanical drive has the frequency $\omega_{d}$ and amplitude
$F$. Under the influence of the parametric drive, the bare cavity mode is
squeezed by a squeezing parameter,

$r=\frac{1}{4}\ln\left(\frac{\Delta_{s}+E}{\Delta_{s}-E}\right),$ (3)

which transforms the cavity field operators to a squeezed mode, through the
Bogoliubov transformation,

$a=a_{s}\cosh r+a_{s}^{\dagger}\sinh r.$ (4)

This Bogoliubov transformation diagonalizes the cavity detuning Hamiltonian
$H_{CD}$ in Eq. (2a) to $\omega_{s}a^{\dagger}a$, where
$\omega_{s}=\sqrt{\Delta_{s}^{2}-E^{2}}$. The effective cavity frequency
$\omega_{s}$ and the squeezing parameter $(r)$ are depicted versus the
parametric drive in Fig. 1(b-I,II). It can be seen that $\omega_{s}$ decreases
over the parametric driving $E$, while the squeezing parameter $r$ increases.
We have considered $\Delta_{s}=\omega_{m}$ for this analysis, which is a usual
condition to witness optomechanical interaction in the red optomechanical
sideband. Similar methods have been used for enhancing light-matter
interactions in cavity optomechanics [21] and cavity QED [22]. After the
diagonalization, the Hamiltonian $H_{OM}$ yields,

$H_{OM}=\omega_{m}b^{\dagger}b+[-g_{OM}a^{\dagger}a+G_{0}(a^{2}+{a^{\dagger}}^{2})](b+b^{\dagger})$
(5)

where $g_{OM}=g_{0}\cosh(2r)$ is an effective single-photon optomechanical
coupling, and $G_{0}=g_{0}\sinh(2r)/2$ is a coupling associated with the DCE.
Under the rotating wave approximation and by moving to the frame of mechanical
drive, the Hamiltonian in Eq. 1 leads to an effective Hamiltonian,

${H}_{eff}=\Delta
a^{\dagger}a+\Omega_{M}b^{\dagger}b+G_{0}\left(b^{\dagger}a^{2}+b{a^{\dagger}}^{2}\right)+\frac{1}{2}(F^{*}b+Fb^{\dagger})$
(6)

where $\Delta=\omega_{s}-\omega_{d}/2$ and $\Omega_{M}=\omega_{m}-\omega_{d}$
are the effective cavity and mechanical resonance frequency, respectively. The
DCE in the squeezed frame can be understood as mechanically induced two-photon
hyper-Raman scattering in the laboratory frame. Fig. 1(c) shows that the
hyper-Raman scattering is an anti-Stokes process. The squeezing gives rise to
an anti-Stokes cavity-sideband at frequency $\omega_{s}+\omega_{L}/2$. The
driving at frequency $\omega_{L}$ produces photon pairs at frequency
$\omega_{L}/2$. The anti-Stokes scattering of driving photon pairs is
anticipated to happen by absorbing mechanical phonons which are at frequency
$\omega_{m}\approx 2\omega_{s}$.

Figure 1: (a) Bloc diagram of parametrically driven cavity optomechanical
system. (b-I) Effective cavity frequency in squeezed frame and (b-II)
squeezing factors with the variation of parametric drive, for
$\Delta_{s}=\omega_{m}$. (c) illustration in frequency-domain of mechanically
induced two-photon hyper-Raman scattering. The left arrow is the two-photon
driving ($\omega_{L}/2$), and the right arrow is the squeezing-induced anti-
Stokes cavity-sideband ($\omega_{s}+\omega_{L}/2$). Figure 2:  (a) Bistability
in the cavity population satisfying the resonant condition
$\Delta=\Omega_{M}/2$. Solid, dotted, and dashed lines correspond to upper,
unstable, and lower branches, respectively. The other parameters chosen here
are $\kappa=500\gamma_{m},\Omega_{M}=10^{4}\gamma_{m}$ and $n_{m}=0$. (b)
Entanglement in (I) upper and (II) lower branches where the system is operated
in the red detuning regime, i.e. $\Delta_{s}=\omega_{m}$. $g_{0}=1.15\times
10^{2}\gamma_{m}$ and the squeezing parameter is monitored by tuning cavity
drive $E=0.43\omega_{m},0.87\omega_{m},0.95\omega_{m}$ which also gives
$\omega_{s}=0.9\omega_{m},0.5\omega_{m},0.3\omega_{m}$, respectively, for
blue, red and yellow lines. Figure 3:  Entanglement between cavity and
mechanics for system saturating upper branch, with the variation of (a)
mechanical drive and optomechanical coupling, (b) cavity detuning and
linewidth, and (c) mechanical oscillation frequency and reservoir thermal
population. All other parameters remain the same with Fig. 2 Figure 4:
Squeezing of mechanical motion in different quadratures and Wigner functions
for different parameters, such as (a) strength of mechanical drive
($F/\gamma_{m}=6\times 10^{4};10\times 10^{4};20\times 10^{4};40\times
10^{4}$), (b) DCE generated optomechanical coupling (
$G_{0}/\gamma_{m}=20;40;80;150$), (c) cavity detuning (
$\Delta/\gamma_{m}=2500,6000,8000,10^{4}$), (d) cavity linewidth
($\kappa/\gamma_{m}=50;500;1000;1500$), (e) frequency of mechanical oscillator
( $\Omega_{M}/\gamma_{m}=0.2\times 10^{4};0.3\times 10^{4};0.4\times
10^{4};1.2\times 10^{4}$), and (f) thermal population of mechanical reservoir
$(n_{m}=25;50;75;100)$. All other parameters remain the same with Fig. 2. The
black dashed lines represent the boundary of SQL in the plot of fluctuation of
mechanical quadratures. White dashed lines in the plots of Wigner functions
represent half of the maxima, and black dashed lines are for the corresponding
coherent states, representing SQLs.

##  III Bistability induces mechanical squeezing and entanglement

The dynamical equations of our system can be derived through Heisenberg-
Langevin equations. From the effective Hamiltonian mentioned in Eq. (6), one
gets

$\displaystyle\dot{a}$
$\displaystyle=-(\kappa+i\Delta)a-2iG_{0}a^{\dagger}b+\sqrt{2\kappa}a_{in},$
(7a) $\displaystyle\dot{a}^{\dagger}$
$\displaystyle=-(\kappa-i\Delta)a^{\dagger}+2iG_{0}ab^{\dagger}+\sqrt{2\kappa}a^{\dagger}_{in},$
(7b) $\displaystyle\dot{b}$
$\displaystyle=-(\gamma_{m}+i\Omega_{M})b-iG_{0}a^{2}+\sqrt{2\gamma_{m}}b_{in}-iF/2,$
(7c) $\displaystyle\dot{b}^{\dagger}$
$\displaystyle=-(\gamma_{m}-i\Omega_{M})b^{\dagger}+iG_{0}{a^{\dagger}}^{2}+\sqrt{2\gamma_{m}}b_{in}^{\dagger}+iF^{*}/2,$
(7d)

where $\kappa$ ($\gamma_{m}$) is the optical (mechanical) dissipation rate,
and $a_{in}$ ($b_{in}$) is the reservoir noise operator for the cavity
(mechanics). Since the cavity oscillates at high frequency ($\omega_{c}$), the
thermal noise effect on the cavity is negligible,
$n_{c}(\omega_{c})\delta(t-t^{\prime})=\langle
a_{in}^{\dagger}(t)a_{in}(t^{\prime})\rangle\approx 0$. The contribution of
thermal noise to the mechanics is quantized by
$n_{m}(\omega_{m})\delta(t-t^{\prime})=\langle
b_{in}^{\dagger}(t)b_{in}(t^{\prime})\rangle$. By splitting the cavity and
mechanical fields into their steady-state and fluctuation ($a=\alpha+\delta a$
and $b=\beta+\delta b$), one can obtain steady states fields, by solving
zeroth order of the EOMs given in Eq (7), as

$\displaystyle\frac{d}{dt}\langle a^{\dagger}a\rangle$
$\displaystyle=-4G_{0}\Im[\langle a^{2}\rangle\langle
b\rangle^{*}]-2\kappa\langle a^{\dagger}a\rangle,$ (8a)
$\displaystyle\frac{d}{dt}\langle a^{2}\rangle$
$\displaystyle=-2i\Delta\langle a^{2}\rangle-2iG_{0}(2\langle
a^{\dagger}a\rangle+1)\langle b\rangle-2\kappa\langle a^{2}\rangle,$ (8b)
$\displaystyle\frac{d}{dt}\langle b\rangle$
$\displaystyle=-i\left(\Omega_{M}\langle b\rangle+G_{0}\langle
a^{2}\rangle+\frac{F}{2}\right)-\gamma_{m}\langle b\rangle,$ (8c)

which leads to the steady-state mean field of the cavity and the mechanics,

$\displaystyle\langle a^{2}\rangle_{ss}$
$\displaystyle=-\frac{FG_{0}z}{2\left(i\gamma_{m}\Delta-\Delta\Omega_{M}+G_{0}^{2}z+\gamma_{m}\kappa+i\kappa\Omega_{M}\right)},$
(9a) $\displaystyle\langle b\rangle_{ss}$
$\displaystyle=-\frac{F(ik-\Delta)}{2\left(i\gamma_{m}\Delta-\Delta\Omega_{M}+G_{0}^{2}z+\gamma_{m}\kappa+i\kappa\Omega_{M}\right)},$
(9b)

where $z=(2\langle a^{\dagger}a\rangle_{ss}+1)$ is obtained from the roots of
the cubic equation,

$F^{2}G_{0}^{2}z-(z-1)\left(2G_{0}^{2}z(\gamma_{m}\kappa-\Delta\Omega_{M})+G_{0}^{4}z^{2}+(\gamma_{m}^{2}+\Omega_{M}^{2})(\Delta^{2}+\kappa^{2})\right)=0.$
(10)

To get an insight into the DCE, we determine the steady states for the
realistic parameters used in [20]. However, the cavity and mechanical drives
used in that work are not strong enough to induce bistability which is of
interest. Therefore, we have increased the strength of both drives, which
enhances the DCE in our proposal. Besides that, it is also required to satisfy
the resonant cavity detuning condition $\omega_{m}=2\omega_{c}$, therefore
$\Delta\approx\Omega_{m}/2$, to ensure the parametric energy of the mechanical
motion to be delivered to the electromagnetic field [20]. The cubic equation
given in Eq. (10) induces DCE-generated bistability of the population of the
cavity mode depicted in Fig. 2(a). In order to enhance mechanical squeezing
and entanglement, we will use the optical steady states lying on the upper
branch of this $S$-like the shape of this bistable behavior. The DCE-generated
bistability in optomechanical systems has also been reported before in [20].
Also, such non-linearity generated bistability is often observed in Kerr
nonlinear systems [23, 24], and the hysteresis has also been observed within
the range of transition region [25, 26].

Both squeezing and entanglement are characterized by mean of the amplitude and
phase quadratures of the cavity $(X=(\delta a+\delta
a^{\dagger})/\sqrt{2},Y=-i(\delta a-\delta a^{\dagger})/\sqrt{2})$ and
mechanics ($Q=(\delta b+\delta b^{\dagger})/\sqrt{2},P=-i(\delta b-\delta
b^{\dagger})/\sqrt{2}$ ). These quadratures can be deduced from the first
order of the field operators in Eq. (7) as,

$\dot{\mathbf{u}}=\mathbf{A_{M}}\mathbf{u}+\mathbf{n},$ (11)

where $\rm{u}=[X,Y,Q,P]^{T}$ and
$\rm{n}=[\sqrt{2\kappa}X_{in},\sqrt{2\kappa}Y_{in},\sqrt{2\gamma}Q_{in},\sqrt{2\gamma}P_{in}]^{T}$
are the vectors of system and noise quadrature operators, respectively, with
$(X_{in}=(a_{in}+a_{in}^{\dagger})/\sqrt{2},Y_{in}=-i(a_{in}-a_{in}^{\dagger})/\sqrt{2})$
and
($Q_{in}=(b_{in}+b_{in}^{\dagger})/\sqrt{2},P_{in}=-i(b_{in}-b_{in}^{\dagger})/\sqrt{2}$)
the external noise operators of the cavity and mechanical quadratures. The
stability of the system can be analyzed by determining eigenvalues of the
matrix $\mathbf{A_{M}}$,

$\mathbf{A_{M}}=\left(\begin{array}[]{cccc}-\kappa+2G_{0}\Im(\beta)&\Delta-2G_{0}\Re(\beta)&-2G_{0}\Im(\alpha)&2G_{0}\Re(\alpha)\\\
-\Delta-2G_{0}\Re(\beta)&-\kappa-2G_{0}\Im(\beta)&-2G_{0}\Re(\alpha)&-2G_{0}\Im(\alpha)\\\
2G_{0}\Im(\alpha)&2G_{0}\Re(\alpha)&-\gamma_{m}&\Omega_{M}\\\
-2G_{0}\Re(\alpha)&2G_{0}\Im(\alpha)&-\Omega_{M}&-\gamma_{m}\\\
\end{array}\right).$ (12)

According to the Routh–Hurwitz criteria, the real part of the eigenvalues of
$\mathbf{A_{M}}$ must be negative for the system to be stable. Following this
criterion, we found that the lower and upper branches of the bistability shown
in Fig. 2(a) are stables, while the middle branch is unstable. This allows us
to determine the steady-state correlation matrix between the position and
momentum operators of the optomechanical system
$(\mathbf{V}=\langle\\{\mathbf{u},\mathbf{u}^{T}\\}\rangle/2)$ by solving
Lyapunov equation as,

$\rm{A_{M}}\rm{V}+\rm{V}\rm{A_{M}}^{T}=-\rm{D},$ (13)

where
$\rm{D}=\langle\\{\rm{n},\rm{n}^{T}\\}\rangle/2=\text{diag}[\kappa,\kappa,(n_{m}+1/2)2\gamma_{m},(n_{m}+1/2)2\gamma_{m}]$.

Furthermore, we study entanglement between mechanics and cavity modes when the
cavity saturates to the upper branch. The steady-state entanglement between
two systems is calculated employing logarithmic negativity, defined by,

$E_{N}=\max[0,-\ln 2\eta^{-}],$ (14)

where

$\eta^{-}=\sqrt{\frac{1}{2}\left(\Sigma(\mathbf{V})-\sqrt{\Sigma(\mathbf{V})^{2}-4\det(\mathbf{V})}\right)},$
(15)

with
$\Sigma(\mathbf{V})=\det\mathbf{V_{C}}+\det\mathbf{V_{M}}-2\det\mathbf{V_{CM}}$
and

$\mathbf{V}=\begin{bmatrix}\mathbf{V_{C}}&\mathbf{V_{CM}}\\\
\mathbf{V_{CM}}^{T}&\mathbf{V_{M}}\end{bmatrix}.$ (16)

It is interesting to mention that the elements of the $\rm{V^{\prime}s}$
matrix are given by $\rm{V_{ij}}=\frac{1}{2}\langle
u_{i}u_{j}+u_{j}u_{i}\rangle$. From the aforementioned definition of the
logarithmic negativity, a Gaussian state is entangled $(\rm{E_{N}}>0)$ if and
only if $\rm{\eta}^{-}<\frac{1}{2}$, which is equivalent to Simon’s necessary
and sufficient entanglement nonpositive partial trace criteria for Gaussian
states [27]. We study the cavity-mechanical entanglement for the upper
Bistability branch in Fig 2(b-I). We observe that, while increasing cavity
drive (which also increases cavity squeezing), the entanglement decreases
rapidly. However, the entanglement is moreover seen to increase in the lower
branch in Fig. 2(b-II), even though it overall remains weak. Therefore, we
target on upper branch to generate optomechanical entanglement and to observe
entanglement-assisted quantum phenomenon.

Fig. 3 shows the logarithmic negativity plotted in $\rm{3D}-$space versus
certain parameters. In Fig. 3(a), the entanglement is shown versus both
mechanical drive and effective coupling strength. It can be seen that the
entanglement decreases quickly as these parameters increase. Fig. 3(b) shows
how the logarithmic negativity increases with the cavity detuning frequency
while it abruptly drops for large optical linewidth. The phenomenon remains
consistent even though the change of cavity detuning frequency does not meet
the parametric resonant condition $(\Delta=\Omega_{M}/2)$. This also justifies
the fact the increase in the strength of parametric cavity drive $(E)$ reduces
the cavity detuning and raises the squeezing parameter $(r)$. This in turn
enhances the DEC-generated optomechanical coupling, which leads to a reduction
of entanglement between cavity and mechanics. Finally, Fig. 3(c) depicts the
robustness of the entanglement against mechanical oscillation frequency and
thermal noise. Therefore, the overall generated entangled in the upper
bistability is three orders of magnitude greater than the one induced on the
lower branch. However, this entanglement is somehow fragile, and this may be
justified by decoherence coming from the nonlinear nature of the enhancement
process. Moreover, this entanglement in the upper branch transfers the
squeezing of cavity mode to the mechanical oscillator. To get insight of the,
we define a generalized quadrature of mechanics as
$Q_{M}^{\theta}=(e^{-i\theta}b+e^{i\theta}b^{\dagger})/\sqrt{2}$, and we
calculate the fluctuation of the mechanical motion
$(S_{M}^{\theta}=1/2\langle\\{Q_{M}^{\theta},Q_{M}^{\theta}\\}\rangle)$ as
shown in Fig. 4. The plot exhibits a clear dependency of the squeezing of
mechanical motion on the quadrature phase angle $\theta$. This phenomenon can
be also understood by analyzing the Wigner quasi-probability distribution in
the phase spaces of the mechanical quadratures
$(W(\mathbf{\tilde{\beta}})=2W(\mathbf{u_{m}})$ where
$\mathbf{\tilde{\beta}}=[\Re\\{\beta\\},\Im\\{\beta\\}]^{T}+\sqrt{2}\mathbf{u_{m}}$
and $\mathbf{u_{M}}=[Q,P]^{T})$. The Brownian noise acting on the mechanical
oscillator is considered to be a zero-mean Gaussian quantum stochastic
process. Therefore, as a consequence, the linearized dynamics make the steady
state of the system a zero-mean bipartite Gaussian state. The single mode
correlation function which characterizes the motion of mechanics
$(\mathbf{V_{M}}=\langle\\{\mathbf{u_{M}},\mathbf{u_{M}}^{T}\\}\rangle/2)$,
determines the Wigner function as,

$W(\mathbf{u_{m}})=\frac{1}{2\pi\sqrt{\det[\mathbf{V_{M}}]}}\exp\left[-\frac{1}{2}\mathbf{u_{M}}^{T}\mathbf{V_{M}}^{-1}\mathbf{u_{M}}\right].$
(17)

The white dashed lines indicate the boundary of the Wigner function where the
amplitude becomes half of the maximum possible amplitude, and the black dashed
line corresponds to that for an ideal non-squeezed coherent state, and
therefore, representing SQL. The fact that the boundary of the state,
presented by white dashed lines going inside the boundary of the black one
ensures the state to be squeezed below SQL. Fig. 4(a) shows that the
quadrature of squeezing is itself dependent on the mechanical drive, which
also neutralizes while increasing. The sharp squeezed quadrature observed in
the lower drive flattens while increasing its strength. Besides, the
fluctuation in the quadrature of maximum variance reduces significantly in
higher mechanical drive, bringing up a stable squeezing. This is also hinted
from the plots of Wigner functions, where we see the direction of the squeezed
quadrature rotates in phase space. A similar phenomenon is observed for the
variation of DCE-generated optomechanical coupling in Fig. 4(b). This happens
due to a better coherent transfer of cavity squeezing to mechanical mode with
the increment of mechanical drive and optomechanical coupling. The enhancement
of the influence of cavity squeezed mode on the mechanical oscillator
saturates with further incrementation. The phenomenon is confirmed in Appendix
A where we determined the quadrature fluctuations in two different ways and
compared them. However, an opposite circumstance is experienced while
increasing cavity detuning in Fig. 4(c), as the squeezing sharpen, which is
also witnessed from the plots of corresponding Wigner functions. This also
concludes that both an increase of mechanical and cavity drives, have a
similar impact on the dynamics. As a stronger cavity drive reduces cavity
detuning and enhances cavity squeezing which increases effective
optomechanical coupling, it causes strengthening mechanical squeezing.
Furthermore, Fig. 4(d) shows that the squeezing quadrature changes steadily
with linewidth broadening. Therefore, the Wigner function shows the squeezed
quadrature rotates in phase space, with a minimum change in squeezing
amplitude. Opposite to cavity detuning, Fig. 4(e) shows that the squeezing
weakens with the increment of the mechanical oscillation frequency.
Furthermore, Fig. 4(f) exhibits an increment of fluctuations in mechanical
quadratures due to the injection of more thermal noise into the mechanical
oscillator. The Wigner phase space boundary is finally seen to exceed SQL
limits for higher thermal noise. Note, importantly, that the system parameters
are chosen to be in the experimentally executable range which can be
applicable in realistic optomechanical setup. As the parameters have been used
before in [20] and they satisfy the system’s stability condition for the
feasible input cavity and mechanical drives, our results can then be
implemented.

##  IV Conclussion

We investigated the possibilities of inducing the squeezing of mechanical
motion and generating entanglement based on bistability in optomechanics.
These nonclassical states have been engineered by using optical steady states
lying on the lower and upper branches of the bistable shape. We found that the
generated nonclassical states are weak when the optical steady states are on
the lower branch. By using the steady states on the upper branch, we observed
a giant entanglement and a large squeezing going beyond the SQL. We observe
that the optomechanical entanglement decreases while increasing optomechanical
squeezing and mechanical drive due to mechanical decoherence. The overall
entanglement seems to be fragile due to the nonlinear effect involved in its
enhancement process. The squeezing depends on the phase quadrature and shows
robustness against thermal fluctuations. This work opens up a new way to
generate nonclassical states by exploring nonlinear effects in optomechanics.
This scheme can be also extended to similar fields such as in superconducting
microwave and hybrid optomechanical systems.

###### Acknowledgements.

SA wishes to acknowledge the European Union; Project number: 101065991
(acronym: SingletSQL) for supporting the work.

Figure 5:  Analytically and numerically determined squeezing with the
variation of (a) mechanical drive and (b) effective optomechanical coupling.
The rest of all other parameters remain the same with Fig. 2

##  Appendix A Appendix: Numerical vs Analytical estimation of mechanical
squeezing

In the presence of a strong coherent tone, the dynamics of the system are
affected by the fluctuations introduced by the nonlinear Hamiltonian. We
determine the fluctuation spectra analytically from the linearized Heisenberg
EOM over nonlinear stationary field amplitude.

$\tilde{S}_{M}^{\theta}(\omega)=\frac{1}{2}\langle\left\\{Q_{M}^{\theta}(\omega),Q_{M}^{\theta}(-\omega)\right\\}\rangle,$
(18)

with
$\tilde{Q}_{M}^{\theta}(\omega)=1/\sqrt{2}\left(\tilde{b}_{-\omega}^{\dagger}e^{i\theta}+\tilde{b}_{\omega}e^{-i\theta}\right)$,
with the usual definition of the Fourier transform:

$\displaystyle\tilde{b}_{\omega}=\int_{-\infty}^{\infty}\mathrm{d}te^{i\omega
t}\tilde{b}(t),\qquad\qquad\tilde{b}_{-\omega}^{\dagger}=\int_{-\infty}^{\infty}\mathrm{d}te^{i\omega
t}\tilde{b}^{\dagger}(t)$

Similarly, the Fourier transform of cavity modes can also be defined in this
way. Therefore, the equation of motion given in Eq. (11), can be written in
the Fourier domain as

$(i\omega+\mathbf{A_{M}})\tilde{\mathbf{u}}=\tilde{\mathbf{n}}(\omega)$ (19)

which determines the spectra of the correlation matrix

$\tilde{\mathbf{V}}(\omega)=\int\mathrm{d}\omega^{\prime}\,\mathbf{\tilde{M}}(\omega)\mathbf{\tilde{D}}(\omega,\omega^{\prime})\mathbf{\tilde{M}}^{\dagger}(\omega)$
(20)

where

$\displaystyle\mathbf{\tilde{M}}(\omega)$
$\displaystyle=(i\omega+\mathbf{A_{M}})^{-1}\qquad\qquad\text{and}$ (21a)
$\displaystyle\mathbf{\tilde{D}}(\omega,\omega^{\prime})$
$\displaystyle=\mathbf{D}\delta(\omega-\omega^{\prime})$ (21b)

The variance of a mechanical quadrature can be determined by integrating over
the fluctuation spectra:
$S_{M}^{\theta}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{d}\omega\,\,\tilde{S}_{M}^{\theta}(\omega)$.
A comparison between analytically and numerically determined quadrature
variance is presented in Fig. 5 where we see good agreement of both plots. The
squeezing is seen to be shifted from phase to amplitude quadrature while
increasing both the mechanical drive and effective optomechanical coupling.
This also shows stability in squeezing in higher mechanical drive and
optomechanical coupling which happens from the fact that higher optomechanical
coupling or mechanical drive enables a stable coherent transfer of cavity
optomechanical squeezed mode to mechanics.

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