# Enhanced entanglement and controlling quantum steering in a Laguerre-
Gaussian cavity optomechanical system with two rotating mirrors

Amjad Sohail<EMAIL_ADDRESS>Department of Physics, Government
College University, Allama Iqbal Road, Faisalabad 38000, Pakistan  Zaheer
Abbas  Department of Physics, Government College University, Allama Iqbal
Road, Faisalabad 38000, Pakistan  Rizwan Ahmed  Physics Division, Pakistan
Institute of Nuclear Science and Technology (PINSTECH), P. O. Nilore,
Islamabad 45650, Pakistan  Aamir Shahzad  Department of Physics, Government
College University, Allama Iqbal Road, Faisalabad 38000, Pakistan  Naeem
Akhtar  Department of Physics, Zhejiang Normal University, Jinhua 321004,
China  Jia-Xing Peng  State Key Laboratory of Precision Spectroscopy, Quantum
Institute for Light and Atoms, Department of Physics, East China Normal
University, Shanghai 200062, China

###### Abstract

Gaussian quantum steering is a type of quantum correlation in which two
entangled states exhibit asymmetry. We present an efficient theoretical scheme
for controlling quantum steering and enhancing entanglement in a Laguerre-
Gaussian (LG) rotating cavity optomechanical system with an optical parametric
amplifier (OPA) driven by coherent light. The numerical simulation results
show that manipulating system parameters such as parametric gain $\chi$,
parametric phase $\theta$, and rotating mirror frequency, among others,
significantly improves mirror-mirror and mirror-cavity entanglement. In
addition to bipartite entanglement, we achieve mirror-cavity-mirror tripartite
entanglement. Another intriguing discovery is the control of quantum steering,
for which we obtained several results by investigating it for various system
parameters. We show that the steering directivity is primarily determined by
the frequency of two rotating mirrors. Furthermore, for two rotating mirrors,
quantum steering is found to be asymmetric both one-way and two-way. As a
result, we can assert that the current proposal may help in the understanding
of non-local correlations and entanglement verification tasks.

##  I Introduction

Quantum entanglement is one of the most weirdest phenomenon of quantum
mechanics in which there exists a non-classical (non-local) correlation
between spatially separated states ent ; ent1 . Entanglement has been shown to
have numerous applications in quantum information processing tele ; ram ,
quantum teleportation tele ; ram , quantum algorithms algo , and quantum
computing ent2 ; comp . In recent years, many of the theoretical and
experimental studies focused on the generation of microscopic entanglement in
systems such as atoms, ions, and photons micro ; micro1 ; micro2 . However, it
remained a pipe dream for researchers to generate quantum entanglement for
macroscopic objects. As a result of hard efforts, macroscopic entanglement has
been experimetally demonstrated for superconducting qubits sup , atomic
ensembles atom  and entanglement between diamonds diam .

In addition to quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering
epr  is one of an important quantum correlation that sits between Bell’s non-
locality and entanglement bell ; bell1 ; bell2 . As there exists non-
exchangeable asymmetric roles between the two observers, Alice and Bob. This
phenomenon was first introduced by Schrödinger to show that the non-locality
in EPR states sch ; sch1 . Most distinctive and unique feature of quantum
steering is its directionality dir . It is interesting to mention here that
the steerable states of massive and macroscopic objects can be utilized to
test the foundational issues of quantum mechanics and the implementation of
quantum information processing. In recent years, many of the studies has
focused to achieve these goals in many optomechanical stee ; stee1 ; stee2
and in magnomechanical systems opent3 ; opent4 ; opent5 . It is because of
advancement in present day technology that the most promising feature of
quantum steering, one-way steering, has been experimentally demonstrated exp ;
exp1 .

Over the course of about last two decade, cavity optomechanical systems oms ;
oms2 ; oms3  have gained importance due to its academic significance and
applications in foundational issues of quantum mechanics found  and highly
precise measurements measure . Optomechanical systems have recently sparked
widespread interest in their discussion of quantumness wig1  via the Wigner
function naeem1 ; naeem2  and time-frequency analysis naeem3 . In addition to
above, optomechanical systems have been experimentally used because of the
development of optical micro-cavity technology. They provide an excellent test
bed macroscopic entangled state opent ; opent1 ; opent2 , coherent state SigJS
; coh , squeezed state sq ; Sin , ground state cooling cool  and
optomechanically induced transparency (OMIT) omit ; omit1 ; omit2 ; Sn1 ; Sn2
. In principle, nothing from the principles of quantum mechanics prohibit
macroscopic entanglement. This has been first investigated by Vitali and
collaborators in their seminal paper oms . Therefore, optomechanics is paving
a way towards the macroscopic non-local correlations at macroscopic scales.

Motivated by the progress in optmechanical systems, Bhattacharya and Meystre
proposed an optomechanical system comprising a rotating mirror (or
rovibrational mirror) which is directly coupled to a Laguerre-Gaussian (LG)
cavity mode via exchange of angular momentum bhatt ; bht . In their scheme,
they demonstrate that the rotating mirror can be sufficiently cooled to 8mK
right from the room temperature (300K), due to the incident radiation torque.
At such low temperature, one can observe various nonlinear and quantum
phenomena. In another interesting scheme, Bhattacharya and collaborators,
showed that one can generate entanglement between mechanical rotating mirror
and LG cavity mode bhatt1 . More recently, researchers found that orbital
angular momentum of LG cavity mode can be measured by utilizing OMIT window
width peng ; XHYM . Furthermore, researcher used hybrid rotational system to
study OMIT QingL ; peng3 ; YXW ; ZAAK ; Rao , cooling of rotating mirrors QioP
, and entanglement peng1 ; peng2 ; Singh ; FWK . Most recently, there is an
interesting study by Huang et al, in which they proposed a LG cavity
optomechanical scheme, assisted with an optical parametric amplifier (OPA)
whose pump frequency is double the frequency of frequency of anti-Stokes field
which is being generated by the external laser beam interacting with both
rotating mirrors, for exploring the entanglement between two rotating
mechanical mirrors huang  and showed that the maximum entanglement is
restricted by the gain of the OPA.

The current research focuses on the controllable quantum steering and enhanced
entanglement in a resolved-side-band of two rotating mechanical mirrors of a
single LG optomechanical cavity containing an OPA. We specifically
investigated mirror entanglement as well as quantum steering of rotating
mirrors, and our results show that parametric interactions haveÂ increased
entanglement. Moreover, we also discussed the dependence of entanglement upon
several system parameters such as parametric gain $\chi$, effective detuning
$\Delta$ and parametric phase $\theta$. It is also shown that there exists
tripartitite (mirror-cavity-mirror) entanglement, quantified by residual
cotangle. In addition to entanglement, we also studied the control of quantum
steering $\xi$, which has been found to be asymmetric both one-way and two-way
for rotating mirrors RM1 and RM2, respectively.

The paper is arranged as follows. In Section 2, we present the theoretical
model and the corresponding dynamical quantum langevin equations. The
discussion of methodology for measuring entanglement and quantum steering is
covered in Section 3. We present numerical results that discuss
bipartite/tripartite entanglement and quantum steering in Section 4 and
Section 5, respectively, then Section 6, shows the experimental feasibility of
the present scheme. Finally, Section 7 presents the conclusions.

##  II Theoretical Model

The system we considered comprises an optical cavity with length $L$ with two
rotating mirrors, as shown in Figure 1\. The cavity is driven by the input
Gaussian beam which has no topological charges. The Gaussian beam with
frequency $\omega_{L}$ and zero charge enters the cavity through the left
rotating mirror (RM1), which is partly transmissive. The reflected beam gets a
charge $-2l$ while the transmitted beam obtain zero charge. Moreover, the
transmitted beam is then incident on the right rotating mirror (RM2) and gets
charge $2l$ (zero) on reflection (transmission) from RM2. Thus a beam with
charge $2l$ interacts with the RM1 again. The transmitted beam from RM1 has
charge $2l$ while the reflected beam remains with no charge, demonstrating
that the net topological charge increment of light transmission back and forth
in the cavity is zero, ensuring the stability of the system. Moreover, the
coupling between the cavity and left/right rotating mirrors established due to
the exchange of orbital angular momentum. Therefore, we can expect the
entanglement between two rotating mirrors. The exchange of angular momentum
between each rotating mirror and each intracavity photon is $2l\hbar$ and time
for a round trip is $2L/c$, thus the radiation torque acting on both mirror is
the change in angular momentum per the time required for a round trip, that
is, $\frac{2l\hbar}{(2L/c)}=\frac{cl\hbar}{L}=\hbar g_{\phi}$, where
$g_{\phi}=cl/L$ optorotational coupling strength. For simplicity, we assumed
the same radius $R$ and the same mass $m$, and the same intrinsic damping rate
$\gamma_{m}$. In addition, a degenerate OPA is embedded into the cavity.
Finally, the beam is completely reflected from RM2. Moreover, we have ignored
the loss of incident light ASRA . In a frame rotating at the driving frequency
$\omega_{L}$, the Hamiltonian of the whole system takes the form

Figure 1: Diagrammatic scheme consists of two rotating mirrors RM1 and RM2
coupled to the Laguerre-Gaussian cavity mode via exchange of angular momentum.
The two rotating mirrors are mounted on support S1 and S2, respectively and
have the same direction along the cavity axis $z$. The angular deflections of
the two rotating mirror are denoted by the angle which are taken from their
equilibrium position. In addition, a degenrate OPA is embedded inside the LG-
cavity and pumped by a coherent light.

$\displaystyle H/\hbar$ $\displaystyle=$
$\displaystyle\left[\Delta_{0}+(g_{1}\varphi_{1}-g_{2}\varphi_{2})\right]a^{\dagger}a+\frac{1}{2}\sum_{j}^{2}\omega_{\varphi_{i}}\left(L_{z_{i}}^{2}+\varphi_{i}^{2}\right)$
(1) $\displaystyle+i\chi(a^{{\dagger}^{2}}e^{i\theta}-a^{2}e^{-i\theta})+\iota
E\left(a^{\dagger}-a\right),$

where $\Delta_{0}=\omega_{c}-\omega_{L}$ is the detuning of the cavity field
and $a$ ($a^{\dagger}$) is the annihilation (creation) operator of the cavity
field, and follow the relation $\left[a,a^{\dagger}\right]=1$. While
$\phi_{j}$ and $L_{z_{k}}$ are the angular displacement and angular momentum
of the rotating mirrors, respectively, with
$[\phi_{j},L_{z_{k}}]=i\delta_{jk}(j,k=1,2)$. In above Hamiltonian, the first
two terms are the bare Hamiltonian of the energy for the cavity field and the
two rotating mirrors, the third term describes the optomechanical coupling
between the cavity field and the two rotating mirrors with the coupling
constant given by
$g_{i}=(cl/L)\sqrt{\hbar/I\omega_{\phi_{i}}}\left(i=1,2\right)$ 11 , where $c$
is the velocity of light and $l$ is the orbital angular momentum quantum
number, $L$ is the length of the cavity and $I=MR^{2}/2$ is the moment of
inertia of the two rotating mirrors about the central axis of the cavity. The
next term interprets the interaction of the 2nd-order nonlinear optical
crystals, at $2\omega_{L}$, with the L-G cavity mode with $\chi$ ($\theta$)
being the parametric gain (phase) of the OPAs. The last term of Equation (1)
represents the interaction of the LG-cavity mode with the incoming coupling
field.

The dynamics of the system can be well described by the time evolution of the
operators and is given by the quantum Langevin equations (QLEs), which can be
written as:

$\displaystyle\dot{\phi}_{j}$ $\displaystyle=$
$\displaystyle\omega_{\phi_{j}}L_{z_{j}},(j=1,2)$ (2a)
$\displaystyle\dot{L}_{z_{j}}$ $\displaystyle=$
$\displaystyle-\omega_{\phi_{j}}\phi_{j}+g_{\alpha
j}a^{\dagger}a-\gamma_{m}L_{z_{j}}+\varepsilon^{in},$ (2b)
$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-[\kappa+i{\Delta_{\circ}+(g_{1}\phi_{1}-g_{2}\phi_{2})}]a+2\chi
a^{{\dagger}}e^{i\theta}$ (2c) $\displaystyle+E+\sqrt{2\kappa}a^{in},$ where
$g_{\alpha{1}}=-g_{1}$, $g_{\alpha{2}}=g_{2}$ and
$\Delta_{\circ}=\omega_{c}-\omega_{L}$. In addition, $\varepsilon^{in}$
account for the mechanical noise, and its fluctuation correlations which are
relevant to temperature $T$, can be described by
$\left\langle\delta\varepsilon^{in}(t)\delta\varepsilon^{in}(t^{{}^{\prime}})\right\rangle=\frac{\gamma_{m_{i}}}{\omega_{\phi_{i}}}\int\limits_{-\infty}^{\infty}\frac{d\omega}{2\pi}e^{-\iota\omega(t-t^{{}^{\prime}})}\omega\left[1+\Theta\right],$
(3)

where $\Theta=\coth\left(\frac{\hbar\omega}{2k_{B}T}\right)$, with $k_{B}$
being the Boltzmann’s constant. Similarly, $a^{in}$ is a noise operator that
of the incident laser beam on the optical cavity and whose delta-correlated
fluctuations are

$\left\langle
a^{in}(t)a^{in,\dagger}(t)\right\rangle=\delta(t-t^{{}^{\prime}}),$ (4)

To linearize the set of QLEs, i.e., Equation (2a-2c), we use the ansatz by
writing $\tau=\tau_{0}+\delta\tau$, $(\tau=\phi_{j},L_{z_{j}},a)$, where
$\tau_{s}$ ($\delta\tau$) is the steady-state (quantum fluctuation ) of the
operators. Under the long-term limit, the steady-state values of each operator
are:

$\displaystyle L_{z_{j}}$ $\displaystyle=$ $\displaystyle 0,$ (5a)
$\displaystyle\phi_{j0}$ $\displaystyle=$
$\displaystyle\frac{g_{\alpha{j}}\left|a_{0}\right|^{2}}{\omega_{\phi_{1}}},$
(5b) $\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{(\kappa-i\Delta+2\chi
e^{i\theta})}{(\kappa^{2}+\Delta^{2}+4\chi^{2})}E,$ (5c) where
$\Delta=\Delta_{0}+\sum_{j}^{2}g_{j}\phi_{j0}$ represents the effective cavity
detuning while $a_{0}$ denotes the effective field amplitude of cavity
modified by OPA. Since, $|a_{s}|\gg\delta a$, one can omit the unwanted
nonlinear terms, e.g., $\delta a\delta a^{\dagger}$. In this case, the
linearized set of QLEs takes the form:  $\displaystyle\delta\dot{\phi}_{j}$
$\displaystyle=$ $\displaystyle\omega_{\phi_{j}}\delta L_{z_{j}},$ (6a)
$\displaystyle\delta\dot{L}_{z_{j}}$ $\displaystyle=$
$\displaystyle-\omega_{\varphi_{j}}\delta\phi_{j}-\gamma_{m}\delta
L_{z_{j}}+G_{j}\delta X+\delta\varepsilon^{in},$ (6b)
$\displaystyle\delta\dot{X}$ $\displaystyle=$ $\displaystyle-\kappa\delta
X+\Delta\delta Y+\sqrt{2\kappa}X^{in},$ (6c) $\displaystyle\delta\dot{Y}$
$\displaystyle=$ $\displaystyle-\kappa\delta Y-\Delta\delta
X-G_{1}\delta\phi_{1}+G_{2}\delta\phi_{2}$ (6d)
$\displaystyle+\sqrt{2\kappa}Y^{in},$ where $G_{j}=\sqrt{2}g_{\alpha j}a_{0}$
is the effective optomechanical coupling strength. Here, we have introduced
the set of quadratures for the cavity field, that is, $\delta
X=\frac{1}{\sqrt{2}}(\delta a+\delta a^{{\dagger}})$, $\delta
Y=\frac{1}{\sqrt{2}i}(\delta a-\delta a^{{\dagger}})$, $\delta
X_{in}=\frac{1}{\sqrt{2}}(\delta a_{in}+\delta a_{in}^{{\dagger}})$ and
$\delta Y_{in}=\frac{1}{\sqrt{2}i}(\delta a_{in}-\delta a_{in}^{{\dagger}})$.
Hence, we rewrite the QLEs (Equation (6a-6d)) in a more compact form, as
follow  $\mathcal{\dot{Z}}(t)=\mathcal{A}\mathcal{Z}(t)+n(t),$ (7)

where $\mathcal{Z}^{T}=(\delta\phi_{1},\delta L_{z_{1}},\delta\phi_{2},\delta
L_{z_{2}},\delta X,\delta Y)$ is the fluctuation vector and
$n^{T}(t)=(0,\varepsilon^{in},0,\varepsilon^{in},\sqrt{2\kappa}X^{in},\sqrt{2\kappa}Y^{in})$
represents the noise vector. Therefore, the drift matrix of the system is
given by

$\mathcal{A}=\left(\begin{array}[]{cccccc}0&\omega_{\phi_{1}}&0&0&0&0\\\
-\omega_{\phi_{1}}&-\gamma_{m}&0&0&-G_{1}&0\\\ 0&0&0&\omega_{\phi_{2}}&0&0\\\
0&0&-\omega_{\phi_{2}}&-\gamma_{m}&G_{2}&0\\\ 0&0&0&0&\mu_{+}&\rho_{+}\\\
-G_{1}&0&G_{2}&0&\rho_{-}&\mu_{-}\end{array}\right),$ (8)

where $\mu_{\pm}=-\kappa\pm 2\chi\Re(e^{\theta})$ and
$\rho_{\pm}=\pm\Delta+\Im(e^{\theta})$. The above drift matrix of the system
determines the stability of the optorotational system, is stable and reaches
its steady-state if none of the eigenvalues of the above drift matrix
$\mathcal{A}$ has a positive real part RHC .

##  III Entanglement and Gaussina Steering measurement

Covariance Matrix. The foremost task is to find the stability of any system
which, in this case, is obtained by applying the Routh-Hurwitz criterion which
impose certain constraints on the system parameters. Once the constraints for
the current optorotational system are fulfilled, the steady state of the
fluctuations must be a Gaussian state. Since,
$\langle\varepsilon^{in}\rangle=\langle a^{in}\rangle=0$, and in this
situation, quantum correlations of the system can then be characterized by the
covariance matrix, which has the same order as the drift matrix, with
$V_{jk}=[\mathcal{Z}_{j}(\infty)\mathcal{Z}_{k}(\infty)+\mathcal{Z}_{k}(\infty)\mathcal{Z}_{j}(\infty)]/2$,
where

$\displaystyle\mathcal{Z}^{T}(\infty)$ $\displaystyle=$
$\displaystyle(\delta\phi_{1}(\infty),\delta
L_{z1}(\infty),\delta\phi_{2}(\infty),\delta L_{z2}(\infty),$ (9)
$\displaystyle\delta X(\infty),\delta Y(\infty))$

represents the fluctuation matrix. The elements of the covariance matrix take
the form CWSP

$V_{jk}=\sum_{k,l}\int_{0}^{\infty}dt\int_{0}^{\infty}dt^{\prime}\Lambda_{i,k}(t)\Lambda_{j,l}(t^{\prime})\Psi_{k,l}(t-t^{\prime}),$
(10)

where $\Lambda=\text{exp}(\mathcal{A}t)$ and

$\Psi_{k,l}(t-t^{\prime})=\langle
n_{k}(t)n_{l}(t^{\prime})n_{l}(t^{\prime})n_{k}(t)/2$ (11)

is noise correlation functions matrix. For large mechanical quality factor,
the quantum Brownian noise can be delta-correlated, that is

$\displaystyle\Psi_{k,l}(t-t^{\prime})$ $\displaystyle=$ $\displaystyle
D_{k,l}\delta(t-t^{\prime}),$ $\displaystyle=$
$\displaystyle\text{Diag}[0,\gamma_{m_{1}}(2n_{1}+1),0,\gamma_{m_{1}}(2n_{2}+1),\kappa,\kappa],$

which simplifies Equation (10) as follow

$V=\int_{0}^{\infty}dt\Lambda(t)D\Lambda^{T}(t).$ (12)

The covariance matrix $V$ of system satisfy steady-state Lypunov equation,

$\mathcal{A}V+\mathcal{A}V^{T}=-D.$ (13)

The covariance matrix $V$ which delineate the entanglement configuration of
the current two mirrors optorotational system, its form is

$V=\left(\begin{array}[]{ccc}M_{1}&N_{12}&N_{1m}\\\ N_{12}^{T}&M_{2}&N_{2m}\\\
N_{1m}^{T}&N_{2m}^{T}&M_{m}\end{array}\right).$ (14)

Each entry of above covariance matric $V$ is a $2\times 2$ block matrix. The
diagonal entries of the above covariance matrix $V$ describe the local
properties of the participants while the rest of the entries define the the
correlations between different modes.
Bipartite entanglement. The logarithmic energy which quantify the entanglement
is defined as

$E_{N}=\max[0,\ln(2\nu^{-})],$ (15)

where $\nu^{-}=2^{-\frac{1}{2}}\sqrt{\aleph-\sqrt{\aleph^{2}-4\det\varsigma}}$
is the smallest symplectic eigenvalue of the partially transposed covariance
matrix (CM), $\varsigma=[\varsigma_{\alpha},\
\varsigma_{\alpha\beta};\varsigma_{\alpha\beta}^{T},\varsigma_{\beta}]$ with
each entry of the $\varsigma$ is a $2\times 2$ block matrix and therefore,
$\varsigma$ is the $4\times 4$ CM of the bipartite subsystem involving mode
$\alpha$ and $\beta$ and
$\aleph=\det\varsigma_{\alpha}+\det\varsigma_{\beta}-2\det\varsigma_{\alpha\beta}$.

Tripartite entanglement. We opt the minimum residual cotangle Adesso ; Adesso2
; Coffman  for the basic criteria to realize tripartite entanglement, is given
by

$\mathcal{R}_{\tau}^{min}\equiv\min[\mathcal{R}_{\tau}^{m_{1}|m_{2}a},\mathcal{R}_{\tau}^{m_{2}|m_{1}a},\mathcal{R}_{\tau}^{a|m_{1}m_{2}}],$
(16)

where $\mathcal{R}_{\tau}^{f|jk}\equiv C_{f|jk}-C_{f|j}-C_{f|k}\geq 0$
($f,j,k=m_{1},m_{2},a$) is the monogamy of quantum entanglement and this
condition assures the invariance of tripartite entanglement under possible
permutations of all modes. Hence, $\mathcal{R}_{\tau}^{min}$ and is describes
the genuine tripartite entanglement of any three-mode Gaussian state. In
Equation (16), $C_{f|j}$ is the contangle of subsystems of $f$ and $j$ ( $j$
incorporate one or two modes), and is equal to the squared logarithmic
negativity. To execute one-mode-vs-two-modes logarithmic negativity
$E_{f|jk}$, one must reform $\nu^{-}$ (see Equation (15)) as

$\nu^{-}=\min eig\ |\bigoplus_{j=1}^{3}(-\sigma_{y})\widetilde{V_{4}}|,$ (17)

where $\widetilde{V_{4}}=\mathcal{P}_{1|2}\mathcal{V}_{4}\mathcal{P}_{1|2}$
with
$\widetilde{\mathcal{Q}}=\mathcal{G}_{i|jk}\mathcal{Q}\mathcal{G}_{i|jk}$.
Furthermore Furthermore, the partial transposition matrices are given by

$\displaystyle\mathcal{G}_{1|23}$ $\displaystyle=$
$\displaystyle\text{diag}(1,-1,1,1,1,1),$ $\displaystyle\mathcal{G}_{2|13}$
$\displaystyle=$ $\displaystyle\text{diag}(1,1,1,-1,1,1),$
$\displaystyle\mathcal{G}_{3|12}$ $\displaystyle=$
$\displaystyle\text{diag}(1,1,1,1,1,-1).$

Parameters | Symbol | Value
---|---|---
Length of the cavity | $L$ |  $1$mm
Mass of rotating mirrors | $m$ |  $35$ng
Radius of rotating mirrors | $R$ |  $10\mu$m
Left mirror angular frequency | $\omega_{\phi_{1}}/2\pi$ |  $10^{7}$Hz
Right mirror angular frequency | $\omega_{\phi_{2}}$ | $(0.5-1.5)\omega_{\phi_{1}}$
Input laser power | $\wp$ |  $50$mW
Quality factor | $Q$ | $2\times 10^{7}$
Laser wavelength | $\lambda$ | $810nm$
Optical finesse | $F$ | $5\times 10^{3}$
Orbital angular momentum | $l$ | $100\hbar$
Temperature | T | 15 mK
Table 1: Experimental values of the parameters used in the paper bhatt ;
bhatt1 ; peng1  Figure 2: (a) The mirror-mirror entanglement $E_{N}^{m-m}$ and
(b) the mirror-cavity entanglement $E_{N}^{m-c}$ plotted as a function of the
normalized detuning $\Delta/\omega_{\phi 1}$.

Gaussian steering. Gaussian steering has asymmetric characteristics in nature
and is much different from the entanglement. For the two interacting mode
Gaussian state, the recommended measurements of the quantum steerability
between any two interacting parties in different directions can be calculated
as

$\displaystyle\zeta_{\alpha|\beta}$ $\displaystyle=$
$\displaystyle\max\\{0,\mathcal{S}(2\mathcal{V}_{m_{l}})-\mathcal{S}(2\mathcal{V}_{in})\\},$
(18a) $\displaystyle\zeta_{\beta|\alpha}$ $\displaystyle=$
$\displaystyle\max\\{0,\mathcal{S}(2\mathcal{V}_{m_{2}})-\mathcal{S}(2\mathcal{V}_{in})\\},$
(18b) where $\mathcal{S}(\nu)$ is the Rényi-2 entropy and is
$\mathcal{S}(\nu)=\frac{1}{2}\ln\det(\nu)$, and
$\mathcal{V}_{in}=\left[\begin{array}[]{cc}\mathcal{V}_{m_{1}}&\mathcal{V}_{m_{1}m_{2}}\\\
\mathcal{V}_{m_{1}m_{2}}^{T}&\mathcal{V}_{m_{2}}\end{array}\right],$ (19)

The diagonal entries of $\mathcal{V}_{m_{1}}$ and $\mathcal{V}_{m_{2}}$
defines the reduced states of modes $\mathcal{V}_{m_{1}}$ and
$\mathcal{V}_{m_{2}}$, respectively. We quantify the quantum steering by
$\zeta_{\alpha|\beta}$ which translates the mode $\alpha$ steers mode $\beta$
and similarly $\zeta_{\beta|\alpha}$ defines the swaped direction. It is an
established fact that the entanglement is symmetric property. However, the EPR
steering is intrinsically different from entanglement, i.e., asymmetric
property which means that a quantum state may be steerable from Bob to Alice
but not vice versa. Therefore, there exist mainly three different
possibilities for quantum steering. (i)
$\zeta_{\alpha|\beta}=\zeta_{\beta|\alpha}=0$ as no-way (NW) steering, (ii)
$\zeta_{\alpha|\beta}>0$ and $\zeta_{\beta|\alpha}=0$ or
$\zeta_{\alpha|\beta}=0$ $\zeta_{\beta|\alpha}>0$ as one-way (OW) steering and
$\zeta_{\alpha|\beta}>0$ $\zeta_{\beta|\alpha}>0$ as two-way (TW) steering.
Asymmetric steerability between the two modes of the Gaussian states can be
checked by introducing the steering asymmetry, defined as

$\zeta_{M}=|\zeta_{\alpha|\beta}-\zeta_{\beta|\alpha}|.$ (20)

It is vital to mentioned here that we obtain one-way or two way steering as
long as steering asymmetry remains positive, i.e., $\zeta_{M}>0$ and no-way if
$\zeta_{M}=0$.

Figure 3: (Color Online) Density plot of (a) the mirror-mirror entanglement
$E_{N}^{m-m}$ versus $\chi$ and $\phi$. Figure 4: The logarithmic negativity
$E_{N}^{m-m}$ as a function of temperature in the absence and presence of the
OPA with optimum detuning. Figure 5: Genuine tripartite entanglement (mirror-
cavity-mirror) versus the normalized detuning. The parameters are same as
given in Table.

##  IV Bipartite and Tripartite Entanglement

In the following discussion, we demonstrate the results regarding the
generation of bipartite (and tripartite) entanglement and quantum steering for
the current system. Throughout our numerics, we have assumed experimentally
realizable parameters, similar to those in ref bhatt ; bhatt1 ; peng1 , listed
in Table 1. In this paper, we consider two bi-partitions, namely $E_{N}^{m-m}$
and $E_{N}^{m-c}$ to represent the degree of entanglement between two phonon
modes and between L–G cavity mode and phonon mode, respectively. The utmost
task of studying the entanglement in such a double rotational cavity
optomechanical system is to find optimal value of the cavity detuning
$\Delta$. In Figure 2 (a-b), the logarithmic negativity is plotted as a
function of the detuning in the absence and presence of parametric gain (with
different parametric phases). To ensure the stability of the system, we choose
the parameter gain $\chi=0.1\omega_{\phi_{1}}$. We find that the steady state
entanglement between two mirrors $E_{n}^{m-m}$ (between the mirror and the
cavity $E_{n}^{m-c}$) is obtained when $\Delta<0$ ($\Delta>0$) in the steady
state which is similar to bhatt1 ; peng1 . In addition, the maximal value of
$E_{n}^{m-m}$ ($E_{n}^{m-c}$) is obtained at about $\Delta=-\omega_{\phi_{1}}$
($\Delta=\omega_{\phi_{1}}$). Furthermore, the enhancement pattern of
entanglement $E_{n}^{m-m}$, by varying the phase of the parametric, behaves
almost in converse manner as compared to $E_{n}^{m-c}$ as shown by Figure 2
(a-b).

Next, we examine the combined effect of parametric gain and parametric phase
(associated with the gain), on the mirror-mirror entanglement $E_{n}^{m-m}$
and mirror-cavity entanglement $E_{n}^{m-c}$ as shown by the Figure 3\.
Physically, the gradual increase of parametric gain induces the nonlinearity
in the system and consequently, we observe the enhancement in entanglement.
One can see that bipartite entanglement $E_{n}^{m-m}$ increases as the
$\theta$ increases for a larger value of parametric gain. We can also see from
the Figure 3 that by ensuring the stability of the current system, the maximum
enhancement of $E_{n}^{m-m}$ appears around $\theta=3\pi/2$ for varied
parametric gain. For specific value of $\chi$, the amount of maximum
entanglement increase, is different. For example, keeping the fixed parametric
gain, i.e., $\chi=0.1\omega_{\phi_{1}}$, the amount of maximum entanglement
$E_{n}^{m-m}$ increased, is about $14\%$.

Figure 6: Plot of the mirror-mirror entanglement $E_{N}^{m-m}$ (solid orange
line), Gaussian steering $\zeta_{m_{1}|m_{2}}$ (dashed blue line),
$\zeta_{m_{2}|m_{1}}$ (dotted black line) and the steering asymmetry
$\zeta_{M}$ (dot-dashed green line) of the two rotational modes as a function
of the normalized detuning $\Delta/\omega_{\phi 1}$ when (a)
$\omega_{2}=0.5\omega_{1}$ and (b) $\omega_{2}=1.5\omega_{1}$. We set
$\chi=0$. The rest of the parameters are same as in Table. 1.

The robustness of such a mirror-mirror entanglement in two rotating cavity
optomechanical system with respect to the temperature $T$ as shown in Figure
4\. It is clear that for a fixed value of the parametric gain, the amount of
entanglement decreases monotonically as the temperature of the environment
increases. and eventually cease to exist. However, the critical temperature of
mirror-mirror entanglement fluctuate by varying the parametric phase. On the
other hand, the curve shown in the inset of Figure 4 have dissimilar tendency
i.e., by choosing the optimal value of the parametric phase, the degree of
mirror-mirror entanglement $E_{n}^{m-m}$ increases with the rise of parametric
gain. However, the temperature decreases, implying that the system with larger
parametric gain mass possess weaker capability to refrain from decoherence of
the thermal environment.

One of the most vital contribution in the present double rotating system is to
find the possibility of exploring the genuine tripartite entanglement. In
Figure 5, we investigate the genuine tripartite entanglement versus normalized
detuning. It is clear that the genuine tripartite entanglement among the
participants (i.e., phonon-photon-phonon) exists in the current double
rotating system if the cavity mode is in blue sideband. The solid blue curve
shows the genuine tripartite entanglement when $\chi=0$. However, in the
presence of parametric gain $\chi=0.1\omega_{\phi_{1}}$, we plot tripartite
entanglement for different value of phase. It is clear from Figure 5 that
maximum enhancement in tripartite entanglement is obtained for $\theta=\pi/2$.
It is also noticeable that this enhancement go along with an increase in
domain of genuine tripartite entanglement over a relatively wider range of
effective cavity detuning. Furthermore, the maximum of genuine tripartite
entanglement with parametric phase increases by $40\%$. Hence, the parametric
phase of the OPA plays a vital role in enhancing the genuine tripartite
entanglement as exhibited in Figure 5.

##  V Controlling Gaussian Steering

Figure 7: The mirror-mirror entanglement (solid orange line), Gaussian
steering $\zeta_{m_{1}|m_{2}}$ (solid green line), $\zeta_{m_{2}|m_{1}}$ (dot-
dashed black line) and the steering asymmetry $\zeta_{M}$ (dashed magenta
line) as a function of the normalized detuning $\Delta/\omega_{\phi 1}$ when
(a) $\omega_{\phi_{2}}=0.9\omega_{\phi_{1}}$, (b)
$\omega_{\phi_{2}}=0.95\omega_{\phi_{1}}$, (c)
$\omega_{\phi_{2}}=1.05\omega_{\phi_{1}}$ and (d)
$\omega_{\phi_{2}}=1.1\omega_{\phi_{1}}$. The rest of the parameters are same
as in Fig. 6 and in Table. 1. Figure 8: Quantum steering as a function of
$\chi/\omega_{\phi 1}$ for different values of $\theta$: (a)-(e) for
$\theta=0$, (b)-(f) is for $\theta=\pi/2$, (c)-(g) represents the cases when
$\theta=\pi$, and (d)-(h) with $\theta=3\pi/2$. In addition, we set
$\omega_{\phi_{2}}=0.5\omega_{\phi_{1}}$ in (a)-(d) and
$\omega_{\phi_{2}}=1.5\omega_{\phi_{1}}$ in (e)-(h).We set the optimal value
of the detuning. The rest of the parameters are kept same as given in Table.
1.

Two-way (TW) control. In the following, our purpose is to find the
controllable quantum steering in the double rotational cavity optomechanical
system. Since the entanglement and the quantum steering are two different
facets of inseparable quantum correlations, therefore, it is better to study
them simultaneously under the same conditions. For this purpose, we employ the
Rényi-2 entropy to quantify the steerability between the two rotational modes.
We first demonstrate this idea in Figure 6, the peak values of the
entanglement and quantum steering of the two mechanical modes. It is evident
from Figure 6\. that steerable states are always entangled, however, the
reverse order is not generally true, i.e., entangled states are not usually
steerable. This property reflects the idea that stronger quantum correlations,
between the two rotational modes, are indispensable for realizing the Gaussian
steering than that for the entanglement. In such a two-rotating cavity
optomechanical system, there may exist (one-) two-way quantum steering between
the two rotational modes due to frequency difference. To demonstrate the
Gaussian steering more clearly, we use the pink area to area to depict the
presence of two-way steering, green and yellow areas for one-way steering, and
the blue area to represent the presence of no-way steering. It is clear from
Figure 6 (a-b), that both the mirror steer each other, however, the
steerability mainly depends on the frequency of each mirror. The mirror which
has high frequency steers more as compared to the mirror which has less
frequency as clearly demonstrated by the pink area in Figure 6 (a-b). The two-
way steering implies that both Alice and Bob can convince each other that the
state they shared, is entangled. Furthermore, Figure 6\. reveal situation
where $\zeta_{m_{1}|m_{2}}>0$, $\zeta_{m_{2}|m_{1}}=0$ and $E_{N}^{m-m}>0$ (or
$\zeta_{m_{1}|m_{2}}=0$, $\zeta_{m_{2}|m_{1}}>0$ and $E_{N}^{m-m}>0$)
witnesses the Gaussian one-way (OW) steering region as shown by the green
(yellow) region. One-way (OW) steering in the current system translates that
the states of the two rotational modes are steerable in single direction even
though they are entangled. Such a kind of response is a direct manifestation
of the problem which has been reported in dir . Furthermore, Figure 6 also
shows that not only the two steerabilities $\zeta_{m_{1}|m_{2}}$ and
$\zeta_{m_{2}|m_{1}}$ but also the entanglement $E_{N}^{m-m}$ are strongly
sensitive to the frequencies of the two mechanical modes. Moreover, it can be
seen that when the value of the frequency of mirror-2 is greater than the
mirror-1, the region for the one-way (two-way) steering gets broaden
(shorten).

One-way (OW) control. As we discussed, the steerability and entanglement
between the rotational modes are sensitive to their frequencies. We observe
that when the two rotational modes are about to synchronize, i.e., the ratio
of the frequency of the two rotating mirrors approaches unity, the two-way
steering completely vanishes. Furthermore, Figure 7(a) and 7(d) present a
fascinating situation where the states of the two rotational modes are
entangled; though they show steerability in one direction, which indicates the
perfect asymmetry of quantum correlations (one-way steering). Such an
asymmetry of quantum correlations describes that both Alice and Bob can
execute the same Gaussian measurements on their side of the entangled system,
however, they find completely different results. One-way steering can be
explained by the steering asymmetry introduced in Eq. (20). Figure 7(a) and
7(d) clearly show that one-way steering can be observe until
$\omega_{\phi_{2}}/\omega_{\phi_{1}}=0.9$ or
$\omega_{\phi_{2}}/\omega_{\phi_{1}}=1.1$. The one-way steering has been
experimentally realized in dir ; exp1  and imparts one-side device independent
quantum key distribution (QKD). However, we observe no-way steering when
$\omega_{\phi_{2}}/\omega_{\phi_{1}}\simeq 0.95$ or
$\omega_{\phi_{2}}/\omega_{\phi_{1}}\simeq 1.05$. Furthermore, when the two
rotating mirrors rotate with exactly the same frequency (complectly
synchronize), even the entanglement between rotating mirrors completely cease
to exist (not shown).

Effect of Parametric Gain and Phase on quantum steering. Finally, the degree
of entanglemnt entanglement and dynamics quantum steering can be studied under
influence of the parametric gain for different value of phase as shown in
Figure 8 around the optimum value of the cavity detuning. Firstly, it can be
seen in Figure 8\. that quantum steering remains upper bounded by the
entanglement irrespective of which rotational mirror has higher frequency.
Figure 8 (a-d) shows that with gradual increase of the parametric gain, the
two rotational modes exhibit two-way steering except the case when
$\theta=\pi/2$, where two-way steering converts to one-way steering around
$\chi/\omega_{\phi_{1}}=0.13$. This is because we obtained reduced
entanglement at this parametric phase (can be seen in Figure 2(a)). The same
situation can also be observe in lower panel of Figure 8, where the one-way
steering vanishes around $\chi/\omega_{\phi_{1}}=0.16$. Owing to the gradual
enhancement of entanglement for $\theta=3\pi/2$, one-way steering converts to
two-way steering for large value of parametric gain. Hence, the current scheme
may provide motivation for handling quantum steering in two rotational cavity
optomechanical system.

##  VI Experimental implementations

The parameters for the current scheme is given in Table. 1. Keeping these
parameters in mind, we now talk about the feasibility of the current two
rotating system based on the recent experiments. In recent experiment, the
value of the angular momentum is taken very high because it can easily be
realizable via spiral phase elements. In addition, the azimuthal structure of
beam of light can be altered via transmission or reflection from the spiral
phase elements SYC . It is shown that low mass of the rotational mirrors with
high precision can be well fabricated. Futhermore, topological charge of the
LG laser beams can be taken as high as $1000$ SYC . With the fast growing
nanotechnology, the mechanical oscillators, with low effective mass
($m=37pg$), high-quality factor ($Q_{m}=10^{8}$) and and high frequency (a few
MHz), has been experimentally reported KHG . This implies that the current
system is an experimentally feasible system under the present-day technology.

##  VII Conclusions

We have presented an efficient and effective method for improving the
entanglement and one-way (two-way) steering characteristics of two rotating
mirrors in a Laguerre-Gaussian (LG) rotating cavity, which composed of two
rotational mirrors coupled to a cavity via orbital angular momentum exchange
and an optical parametric amplifier (OPA) driven by coherent light. We have
shown that the frequencies of two rotational mirrors play a vital role in
modulating the entanglement and controlling the quantum steering. For example,
when $\omega_{\phi 2}=0.5\omega_{\phi 1}$ ($\omega_{\phi 2}=1.5\omega_{\phi
1}$), the steerability of two mirrors is enhanced as compared to the case when
the two mirrors are near to synchronized, that is, near the resonance
frequency, two-way steering converts to one-way steering (or no-way steering)
with a reduced degree of entanglement. It is interesting to mention that the
switch from two-way steering to one-way (or no-way) steering is practicable by
changing the frequencies of two rotational mirrors, parametric gain and phase.
Hence, we argued that the steering directivity in the current scheme would
support understanding the quantum correlation and may have major applications
in one-way (two-way) quantum computing, quantum secret sharing, and quantum
key distribution.

## Data availability

All numerical data that support the findings in this study is available within
the article

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