Steady-state entanglement in a hybrid optomechanical system enhanced by optical parametric amplifiers

Abraham Abebe Kibret; Abraham Abebe Kibret; Tewodros Yirgashewa Derge; Tesfay Gebremariam Tesfahannes

doi:10.1364/OPTCON.502349

1. Introduction

Quantum entanglement is an important resource in the fields of quantum information to realize quantum communication, quantum computation, quantum cryptography, and quantum teleportation [1–3]. Recently, entanglement has attracted great attention and experimental realization using different physical quantum system and information carriers [4–6]. Specifically, the mechanical resonators at mesoscopic scales are beginning to be available candidate systems to study various macroscopic quantum features, where in optomechanical systems have emerged as particularly promising platforms due to their versatility in design, fabrication, and control [7].

Optomechanics is one of the emerging research areas that focuses on the interaction of light with mechanical motion due to radiation pressure forces associated with momentum [8,9]. For the existing coherent phonon-photon interaction, the pertinent entanglement has been studied in various optomechanical systems by transferring the quantum behavior of photons to movable mirrors of coupled optical cavities. In this respect, different authors have studied the entanglement of two dielectric membranes suspended inside a cavity [10], optomechanical force-sensing [11,12], macroscopic quantum states in two superconducting qubits [13], dynamical quantum steering [14], nano-mechanical oscillators in a ring cavity by feeding squeezed light [15], and optomechanically induced transparency [16]. Furthermore, the bipartite entanglement between the cavity mode and the mode of oscillating mirrors has also been demonstrated with the help of an optomechanical array in which the optical cavities are connected to one oscillating end mirror via a photon hopping mechanism [17]. On the other hand, many authors have devoted their effort to studying the effect of the parametric amplification process on entanglement in a cavity where both ends of the mirrors are fixed [18,19]. In particular, there are also indications that placing an optical parametric amplifier inside the optomechanical cavity leads to cooling of the optomechanical oscillation [20], strengthening the optomechanical coupling [21], and splitting the normal mode [22]. In the same way, the optical parametric amplifier (OPAs) in a single optomechanical cavity has also been shown to improve entanglement between a cavity mode and a mechanical mode [23,24]. But, in this regard, we seek to study the effect of DOPA and the strength of coupling between two separate optomechanical cavities on the degree of entanglement.

In contrast to earlier considerations of linear optomechanical coupling schemes in two optomechanical cavities in the absence of DOPAs [25], in this paper, we seek to consider a hybrid optomechanical system whose cavities contain DOPA coupled to their cavities through cavity-cavity coupling. With this in mind, we strive to study steady-state bipartite entanglement in a hybrid optomechanical system by optical parametric amplifiers. To this aim, we obtain the dynamics of the optomechanical system by applying the corresponding Hamiltonian using the quantum Langevin equations in which the dissipation and fluctuation terms are added to the Heisenberg equations of motion. We then find the steady-state and the linearized quantum fluctuation of the system. Afterward, the steady-state bipartite entanglement is quantified by applying logarithmic negativity. Due to the presence of DOPA inside each cavity, the steady-state entanglement is found to significantly vary with the value of nonlinear gain of OPA. We also studied the change in entanglement owing to the strength of cavity-cavity coupling.

The paper is organized as follows. In Section 2, we introduce the model and details of the formalism that describes the system. In Section 3, we derive the quantum Langevin equation using the Hamiltonian of the system, linearize them around the steady-state and quantify the entanglement properties of the system by using the logarithmic negativity. In Section 4, we present and discuss the results of the steady-state bipartite entanglement between cavity-oscillator and cavity-cavity. The conclusions are summarized in Section 5.

2. Model and Hamiltonian of the system

We consider the system model as shown in Fig. 1 in which two cavity mirrors are fixed and the other two are movable with each of the optomechanical cavities contains OPA and effective mass m and frequency ${\omega _{mj}}$. Each cavity mode is assumed to be driven by a laser that interacts with the movable mirror via radiation pressure and J being the coupling parameter of the physical system that connects the optomechanical oscillators.

Fig. 1. The schematic model of a hybrid optomechanical system is under consideration. Thus, the separate cavities with degenerate optical parametric amplifier (DOPA) are externally coupled, while J being the coupling parameter, ${\beta _j}$ being the cavity decay rate, ${\gamma _j}$ as the mechanical damping rate, ${\omega _m}$ as the frequency of the mechanical oscillator, and the other symbols defined in the main text.

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The total Hamiltonian that characterizes the system read

(1)$$\begin{aligned} H &= \sum\limits_{j = 1}^2 {\hbar {\omega _{cj}}{a_j}^{\dagger} {a_j} + \sum\limits_{j = 1}^2 {\hbar {G_{mj}}{a_j}^{\dagger} {a_j}{q_j}} + \sum\limits_{j = 1}^2 {\frac{1}{2}\hbar {\omega _{mj}}({{q_j}^2 + {p_j}^2} )} + \sum\limits_{j = 1}^2 {i\hbar {E_j}({{a_j}^{\dagger} {e^{ - i{\omega_{0j}}t}} - {a_j}{e^{i{\omega_{0j}}t}}} )} } \\ &+ \sum\limits_{j = 1}^2 {i\hbar \Omega ({{e^{i\theta }}{a_j}^{{\dagger} 2}{e^{ - i{\omega_{0j}}t}} - {e^{ - i\theta }}{a_j}^2{e^{i{\omega_{0j}}t}}} )} + \hbar J({{a_1}^{\dagger} {a_2} + {a_2}^{\dagger} {a_1}} ), \end{aligned}$$

where the first term describes the sum of the energy corresponding to the optical cavity modes with photon operators ${a_j}$ or ${a_j}^{\dagger} $ and frequency ${\omega _{cj}}$. While, the second term represents the interaction induced by radiation pressure, the third term is the energy of the mechanical oscillator with the position and momentum operators ${q_j}$ and ${p_j}$, the fourth term stands for driving field, the fifth term denotes the coupling between the OPA and the two cavity modes, $\Omega $ is the nonlinear gain of the OPA and $\theta $ is the phase of the optical field driving the OPA. The last term describe the coupling between the two cavity modes with the cavity-cavity coupling term of intensity $J$. The Hamiltonian of the whole system is written as follows after a frame rotating;

(2)$$\begin{aligned} H &= \sum\limits_{j = 1}^2 {\hbar {\Delta _j}{a_j}^{\dagger} {a_j} + \sum\limits_{j = 1}^2 {\hbar {G_{mj}}{a_j}^{\dagger} {a_j}{q_j}} + \sum\limits_{j = 1}^2 {\frac{1}{2}\hbar {\omega _{mj}}({{q_j}^2 + {p_j}^2} )} + \sum\limits_{j = 1}^2 {i\hbar {E_j}({{a_j}^{\dagger} - {a_j}} )} } \\ &+ \sum\limits_{j = 1}^2 {i\hbar \Omega ({{e^{i\theta }}{a_j}^{{\dagger} 2} - {e^{ - i\theta }}{a_j}^2} )} + \hbar J({{a_1}^{\dagger} {a_2} + {a_2}^{\dagger} {a_1}} ), \end{aligned}$$

where ${\Delta _j} = {\omega _{cj}} - {\omega _{0j}}$ is the optical detuning, the driving amplitude related to driving power is given by $|{{E_j}} |= \sqrt {{{2{P_j}\beta } / {\hbar {\omega _{0j}}}}} ,$ while the cavity decay rate ${\beta _1} = {\beta _2} = \beta = {{\pi c} / {2FL}}$, where F is the cavity finesse and c is the speed of light. In addition, ${G_{mj}} = ({{{{\omega_{cj}}} / L}} )\sqrt {{\hbar / {m{\omega _{mj}}}}}$ is the single-photon optomechanical coupling associated with the cavity mode with frequency ${\omega _{cj}}$, where L is the cavity length and m is the effective mass.

3. Dynamics of the system

The dynamics of the system are obtained using quantum Langevin equations. Thus, using Eq. (2), the time evolution of the system is obtained following nonlinear quantum Langevin equations:

(3)$$\begin{aligned} &\mathop {{a_1}}\limits^\cdot{=}{-} ({i{\Delta_1} + {\beta_1}} ){a_1} - i{G_{m1}}{q_1}{a_1} + {E_1} - iJ{a_2} + 2\varOmega {e^{i\theta }}{a_1}^{\dagger} + \sqrt {2{\beta _1}} {a_1}^{in},\\ &\mathop {{a_2}}\limits^\cdot{=}{-} ({i{\Delta_2} + {\beta_2}} ){a_2} - i{G_{m2}}{q_2}{a_2} + {E_2} - iJ{a_1} + 2\varOmega {e^{i\theta }}{a_2}^{\dagger} + \sqrt {2{\beta _2}} {a_2}^{in},\\ &\mathop {{q_1}}\limits^\cdot{=} {\omega _{m1}}{p_{_1}},\\ &\mathop {{p_1}}\limits^\cdot{=}{-} {\gamma _1}{p_1} - {G_{m1}}{a_1}^{\dagger} {a_1} - {\omega _{m1}}{q_1} + {\xi _1},\\ &\mathop {{q_2}}\limits^\cdot{=} {\omega _{m2}}{p_2},\\ &\mathop {{p_2}}\limits^\cdot{=}{-} {\gamma _2}{p_2} - {G_{m2}}{a_2}^{\dagger} {a_2} - {\omega _{m2}}{q_2} + {\xi _2}, \end{aligned}$$

where ${\beta _j}$and ${\gamma _j}$ represent the dissipation of optical and mechanical modes. The thermal Langevin force ${\xi _j}$ is resulting from the coupling of the oscillating mirror to the environment, which is auto-correlated, as [26]:

(4)$$\left\langle{\xi_j}(t ){\xi_j}({t^{\prime}}) \right\rangle = \frac{{{\gamma _j}}}{{{\omega _{mj}}}}\int {\frac{{d\omega }}{{2\pi }}{e^{ - i\omega ({t - {t^{\prime}}} )}}\omega } \left[ {\textrm{coth}\left( {\frac{{\hbar \omega }}{{2{k_B}T}}} \right) + 1} \right],$$

where ${k_B}$ is the Boltzmann constant, T is the temperature of the environment. $a_j^{in}$ is the vacuum noise operator for the cavity operator, in which the nonzero correlation function is

(5)$$\left\langle {{a_j}^{in}(t ){a_j}^{{\dagger} ,in}(t^{\prime})} \right\rangle = \delta ({t - {t^{\prime}}} ).$$

In the hope of reducing the involved rigor we opt to restrict the case when ${\omega _{m1}} = {\omega _{m2}}$. We assume that the optical cavity fields are intense, ${\alpha _{js}} \gg 1$. In this case, we seek to apply the linearization approach [27] by expanding each field operator as a sum of its steady-state mean values and fluctuation operator with zero-mean value, which can be treated separately as

(6)$${a_j} = {\alpha _{js}} + \delta {a_j},{p_j} = {p_{js}} + \delta {p_j},{q_j} = {q_{js}} + \delta {q_j}.$$

The steady-state mean values of the system can be obtained by setting the time derivatives to zero

(7)$$\begin{aligned} &{\alpha _{js}} = \frac{{{E_j}J}}{{({{\beta_j} - 2\Omega \cos \theta } )+ i({{\Delta^{0j}} - 2\varOmega \sin \theta } )+ {J^2}}},\\ &{p_{js}} = 0,\\ &{{{q_{js}} ={-} {G_{mj}}{{|{{\alpha_{js}}} |}^2}} / {{\omega _{mj}},}} \end{aligned}$$

where ${\Delta ^{0j}} = {\Delta _j} - {G_{mj}}{q_{js}}$ are the effective cavity detuning including the frequency shift due to the interaction with the mechanical resonator. It is straightforward to see that the OPAs lead to two effects: it modifies the cavity decay rate ${\beta _j} \Rightarrow {\beta _j} - 2\varOmega \cos \theta$ and also the effective detuning ${\Delta ^{0j}} \Rightarrow {\Delta ^{0j}} - 2\varOmega \sin \theta .$ In this case, the following linearized Langevin equations can be obtained:

(8)$$\begin{aligned} &\delta \mathop {{x_1}}\limits^\cdot{=}{-} ({{\beta_1} - 2\varOmega \cos \theta } )\delta {x_1} + ({{\Delta^{01}} + 2\varOmega \sin \theta } )\delta {y_1} + J\delta {y_2} + \sqrt {2{\beta _1}} \delta {x_1}^{in},\\ &\delta \mathop {{y_1}}\limits^\cdot{=}{-} ({{\beta_1} + 2\varOmega \cos \theta } )\delta {y_1} + ({{\Delta^{01}} - 2\varOmega \sin \theta } )\delta {x_1} + {g_1}\delta {q_1} - J\delta {x_2} + \sqrt {2{\beta _1}} \delta {y_1}^{in},\\ &\delta \mathop {{x_2}}\limits^\cdot{=}{-} ({{\beta_2} - 2\varOmega \cos \theta } )\delta {x_2} + ({{\Delta^{02}} + 2\varOmega \sin \theta } )\delta {y_2} + J\delta {y_1} + \sqrt {2{\beta _2}} \delta {x_2}^{in},\\ &\delta \mathop {{y_2}}\limits^\cdot{=}{-} ({{\beta_2} + 2\varOmega \cos \theta } )\delta {y_2} + ({{\Delta^{02}} - 2\varOmega \sin \theta } )\delta {x_2}\textrm{ + }{\textrm{g}_\textrm{2}}\delta {q_2} - J\delta {x_1} + \sqrt {2{\beta _2}} \delta {y_2}^{in},\\ &\delta \mathop {{q_1}}\limits^\cdot{=} {\omega _{m1}}\delta {p_1},\\ &\delta \mathop {{p_1}}\limits^\cdot{=}{-} {\gamma _1}\delta {p_1} - {g_1}\delta {x_1} - {\omega _{m1}}\delta {q_1} + {\xi _1},\\ &\delta \mathop {{q_2}}\limits^\cdot{=} {\omega _{m2}}\delta {p_2},\\ &\delta \mathop {{p_2}}\limits^\cdot{=}{-} {\gamma _2}\delta {p_2} - {g_2}\delta {x_2} - {\omega _{m2}}\delta {q_2} + {\xi _2}, \end{aligned}$$

where ${g_j} = \sqrt 2 {G_{mj}}{\alpha _{js}}$ is the effective optomechanical coupling strength, where we have taken ${\alpha _{js}}$ real by properly choosing the phase reference of the cavity fields. We have also defined the quadrature fluctuation operators of the cavity modes.

(9)$${{\delta {x_j} = ({\delta {a_j} + \delta {a_j}^{\dagger} } )} / {\sqrt 2 }},{{\delta {y_j} = i({\delta {a_j}^{\dagger} - \delta {a_j}} )} / {\sqrt 2 }}, $$

and the corresponding input noise operators

(10)$${{{x_j}^{in} = ({{a_j}^{in} + {a_j}^{in,{\dagger} }} )} / {\sqrt 2 }},{{{y_j}^{in} = i({{a_j}^{in,{\dagger} } - {a_j}^{in}} )} / {\sqrt 2 }}. $$

With this consideration, we express Eq. (8) in a more compact form as

(11)$$\mathop {S(t)}\limits^\cdot{=} AS(t) + \Theta (t), $$

with setting $S(t)$ as the column vector of the quantum fluctuations and $\Theta (t)$ being the column vector of the noise sources. Their transposed elements are thus turning out to be

(12)$$S{(t )^T} = ({\delta {q_1},\delta {p_1},\delta {q_2},\delta {p_2},\delta {x_1},\delta {y_1},\delta {x_2},\delta {y_2}} ),$$

(13)$$\Theta (t )= \left( {0,{\xi_1},0,{\xi_2},\sqrt {2{\beta_1}} \delta {x_1}^{in},\sqrt {2{\beta_1}} \delta {y_1}^{in},\sqrt {2{\beta_2}} \delta {x_2}^{in},\sqrt {2{\beta_2}} \delta {y_2}^{in}} \right).$$

The corresponding drift matrix A takes the form

(14)$$A = \left( {\begin{array}{cccccccc} 0&{{\omega_{m1}}}&0&0&0&0&0&0\\ { - {\omega_{m1}}}&{ - {\gamma_1}}&0&0&{ - {g_1}}&0&0&0\\ 0&0&0&{{\omega_{m2}}}&0&0&0&0\\ 0&0&{ - {\omega_{m2}}}&{ - {\gamma_2}}&0&0&{ - {g_2}}&0\\ 0&0&0&0&\vartheta &\phi &0&J\\ {{g_1}}&0&0&0&\varphi &\varsigma &{ - J}&0\\ 0&0&0&0&0&J&\psi &\sigma \\ 0&0&{{g_2}}&0&{ - J}&0&\chi &\zeta \end{array}} \right),$$

where

\begin{aligned} &\vartheta ={-} {\beta _1} + \rho ,\phi = {\Delta ^{01}} + \eta ,\varsigma ={-} {\beta _1} - \rho ,\varphi = {\Delta ^{01}} - \eta ,\psi ={-} {\beta _2} + \rho ,\sigma = {\Delta ^{02}} + \eta ,\\ &\zeta ={-} {\beta _2} - \rho ,\chi = {\Delta ^{02}} - \eta ,\rho = 2\Omega \cos \theta ,\eta = 2\Omega \sin \theta . \end{aligned}

The system would be stable and reaches a steady-state when the real parts of all eigenvalues of matrix A are negative. Due to the Gaussian nature of the quantum noise terms in Eq. (13) and the linearized dynamics, the steady-state quantum fluctuations of the system are fully characterized by $8 \times 8$ covariance matrix. In light of this, a steady-state covariance matrix can be obtained by solving the corresponding Lyapunov equation [28]

(15)$$AV + V{A^T} ={-} D,$$

in Eq. (15) $V$ with its entries defined as ${{{V_{ij}} = \left( {\left\langle {{u_i}(\infty ){u_j}(\infty )+ {u_j}(\infty ){u_i}(\infty )} \right\rangle } \right)} / 2}$, where $({i,j = 1,2} )$ and

(16)$${u^T}(\infty )= ({\delta {q_1}(\infty ),\delta {p_1}(\infty ),\delta {q_2}(\infty ),\delta {p_2}(\infty ),\delta {x_1}(\infty ),\delta {y_1}(\infty ),\delta {x_2}(\infty ),\delta {y_2}(\infty )} ),$$

is the vector of the fluctuation operators at the steady-state $({t \to \infty } ).$

(17)$$D = \textrm{ diag}({0,{\gamma_1}({2\overline n + 1} ),0,{\gamma_2}({2\overline n + 1} ),\beta ,\beta ,\beta ,\beta } ),$$

is the diffusion matrix. We have supposed the mechanical resonator is of high quality factor $Q = {{{\omega _{mj}}} / {{\gamma _j}}} \gg 1$, which is satisfied under the experiment [29]. The correlation function of the noise ${\xi _j}(t )$ can be obtained as

(18)$${{\left\langle {{\xi_j}(t ){\xi_j}({{t^{\prime}}} )+ {\xi_j}({{t^{\prime}}} ){\xi_j}(t )} \right\rangle } {\big /} 2} \simeq {\gamma _j}({2\overline n + 1} )\delta (t - t^{\prime}),$$

where $\overline n = {[{\textrm{exp}({{{\hbar {\omega_{mj}}} / {{k_B}T}}} )- 1} ]^{ - 1}}$ is the mean thermal phonon number of the resonator in which T is the environmental temperature. The covariance matrix (CM) can also be written in the form of a block matrix:

(19)$$V = \left( {\begin{array}{*{20}{c}} {{R_{a1}}}&{K{a_1}{a_2}}&{K{a_1}{a_3}}&{{K_{ma1}}}\\ {K{a_1}{a_2}}&{R{a_2}}&{Km{a_3}}&{Km{a_2}}\\ {K{a_1}{a_3}}&{Km{a_3}}&{R{a_3}}&{Km{a_3}}\\ {Km{a_1}}&{Km{a_2}}&{Km{a_3}}&{{R_m}} \end{array}} \right),$$

where each element of this matrix are $2 \times 2$ matrix.

Logarithmic negativity is a convenient entanglement measure because it is the only one which can always be explicitly computed and it is also additive. Once the covariance matrix V is obtained, we can then calculate the steady-state bipartite entanglement. To compute the bipartite entanglement, we choose logarithmic negativity [30], which is defined as

(20)$$EN = \textrm{max}[{0, - \ln 2\overline \eta } ], $$

where $\overline \eta $ is the minimum simplestic eigenvalue of partially transposed CM corresponding to the bipartite system of interest, which is given by

(21)$$\overline \eta = \sqrt {\frac{{\sum {({{L_m}} )- \sqrt {\sum {{{({{L_m}} )}^2} - 4\textrm{det}{L_m}} } } }}{2}} . $$

${L_m}$ can be expressed in a form of a block matrix as:

(22)$$\begin{aligned} &{L_m} = \left( {\begin{array}{cc} B&C\\ {{C^T}}&I \end{array}} \right),\\ &\sum {({{L_m}} )} = \textrm{det}B + \textrm{det}I - 2\textrm{det}C, \end{aligned}$$

where $B$,$C$, and $I$ are being $2 \times 2$ blocks of the correlation matrix. Once the covariance matrix is obtained, one can then reduce $8 \times 8$ to $4 \times 4$ sub-matrix. The CM is taken to characterize the system, since the blocks B and I are associated with the oscillating mirror and cavity modes, respectively. The block matrix C describes the optomechanical correlation in the presence of degenerate OPA inside the cavity.

4. Results and discussion

In this section, we discuss and study the nature of steady-state bipartite entanglement, such as the entanglement between the first cavity-mechanical oscillator modes, the second cavity-mechanical oscillator modes, and the two cavity modes. For such a system without OPA, the entanglement properties of bipartite modes have been studied [17,25,31,32]. We opt to focus on the effects of the nonlinear gain of OPA and cavity-cavity coupling strength on logarithmic negativity. In doing so, we choose experimental parameters and the parameters of the two optomechanical cavities to be the same: ${\beta _1} = {\beta _2} = \beta $, ${\gamma _1} = {\gamma _2} = \gamma $, and ${\omega _{m1}} = {\omega _{m2}} = {\omega _m}$ [33,34]. Besides, the results are obtained by adopting the parameters analogous to Refs. [35–37].

Firstly, we investigate steady-state bipartite entanglement between the cavity-mechanical oscillating modes, the second cavity-mechanical modes, and the two cavity modes as a function of the phase $\theta $.

In Fig. 2(a), the entanglement of the first cavity-mechanical oscillator modes as a function of the phase $\theta $ for various optimal values of nonlinear gain is plotted. We observe that the steady-state entanglement generated and the nature of entanglement significantly varies with the value of nonlinear gain of the OPA. Specifically, from Fig. 2(a), we see that when the nonlinear gain of OPA medium increases, the entanglement measurement also increases.

Fig. 2. Plots of logarithmic negativity EN between (a) the first cavity-mechanical oscillator modes (C1-M1), (b) the second cavity-mechanical oscillator modes (C2-M2), and (c) the two cavity modes (C1-C2) at $\Omega = \beta$ (red solid curve), $\Omega = 0.95\beta$ (green dashed curve), and $\Omega = 0.9\beta$ (blue solid curve) as a function of the phase $\theta$ for various optimal values of nonlinear gain at temperature $T = 400mK$, $\beta = 44MHz$, and coupling strength $J = 0.5{\omega _m}$. The rest of the parameters are given in Table 1.

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Table 1. List of experimental parameters for the system

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In Fig. 2(b), we also plot the entanglement of the second cavity-mechanical oscillator modes as a function of the phase $\theta $ for various values of the nonlinear gain of OPA and the other parameters that have been used are the same as Fig. 2. Similarly, from this figure, we can see that the entanglement increases with increasing nonlinear gain of OPA medium. But, comparing Fig. 2(a) and (b), the maximum entanglement between the first cavity-mechanical oscillating modes is greater than the maximum entanglement between the second cavity-mechanical modes. The reason is that increasing the input laser power increases the parametric gain and this causes a stronger coupling between the movable mirror and the cavity field, leading to increase the entanglement.

Furthermore, Fig. 2(c) illustrates the entanglement between two cavity modes as a function of phase $\theta $ and the other parameters are the same as in Fig. 2. We also observe that the entanglement between two cavity modes is generated in a steady state. Specifically, the entanglement increases with the nonlinear gain of the OPA medium until certain $\theta $, then it decreases. The minimum value of entanglement between the two cavity modes at a reasonable temperature is non-zero; this shows that there is an entanglement transfer in the system due to cavity-cavity coupling strength.

From the three Figs. 2(a), (b), and c, we observe that the steady-state entanglement generated and the entanglement measurement increases with increasing the value of nonlinear gain $\Omega $. The fact is increasing the nonlinear gain of the medium causes a stronger coupling between the cavity and mechanical oscillator modes. Furthermore, the minimum value of entanglement at a reasonable temperature is non-zero; this shows that there is an entanglement transfer in the system. More importantly, from Figs. 2(a), (b), and c, we can see that the generation of steady-state entanglement and entanglement transfer strongly depends on cavity-cavity coupling strength and OPA medium. In light of the observed result, it is worth noting that the emerging entanglement is induced due to the coupling of degenerate OPA with their cavity and the presence of cavity-cavity coupling strength. Comparing the behavior of entanglement measurement of a Fabry-Perot cavity without OPA of an oscillating micro-mirror and driven by coherent light [28] with the presence of OPA, the entanglement is more enhanced. Additionally, in the present paper, the system is different from the standard optomechanical system reported in Refs. [17,23,24]. Therefore, such a hybrid optomechanical system can utilize immediate practical interest in the investigation of remote entanglement measurements and is suitable for future information processing applications.

Secondly, we discuss the steady-state bipartite entanglement as a function of the normalized detuning ${\Delta / {{\omega _m}}}$ for different values of the coupling parameter $J$. The main reason for the choices of the parameter is to see an evident difference in the entanglement in coupled system and in what way the nature of entanglement would be influenced by the strength of cavity-cavity coupling parameter. With this respect, we plot the bipartite entanglement as a function of the normalized detuning ${\Delta / {{\omega _m}}}$ for different values of $J$.

Specifically, in Fig. 3(a), we plot the entanglement of the first cavity-mechanical oscillator modes, second cavity-mechanical oscillator modes, and the two cavity modes as a function of ${\Delta / {{\omega _m}}}$ with coupling strength $J = 0.3{\omega _m}$ and the other parameters are the same as in Fig. 2. We can see that the entanglement of the first cavity-mechanical oscillator modes and second cavity-mechanical oscillator modes increases with normalized detuning ${\Delta / {{\omega _m}}}$ until a certain range, and then it decreases.

Fig. 3. Plots of logarithmic negativity EN of the first cavity-mechanical oscillator modes (red dashed curve), the second cavity-mechanical oscillator modes (blue solid curve), and the two cavity modes (green dashed curve) as a function of the normalized detuning ${\Delta / {{\omega _m}}}$ for coupling strength (a) $J = 0.3{\omega _m}$, (b) $J = 0.4{\omega _m}$, (c) $J = 0.5{\omega _m}$, and (d$J = 0.6{\omega _m}$, with fixed value of $\Omega = 0.95\beta$, where the other parameters are taken to be the same as in Fig. 2.

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In Fig. 3(b), we also plot the bipartite entanglement as a function of the normalized detuning ${\Delta / {{\omega _m}}}$ for coupling strength $J = 0.4{\omega _m}$ and the other parameters are taken to be the same as in Fig. 2. The logarithmic negativity that accounts for the entanglement of cavity-mechanical oscillator modes also increases until a certain detuning, and then it decreases. But the entanglement between the two cavity modes almost decreases from a certain maximum entanglement. It is possible to see from Fig. 3(b) and (c) that continuous variable entanglement witnessed for the parameter ${\Delta / {{\omega _m}}}$ ranging from 0 to 4.

Similarly, in Fig. 3(c) and (d), we plot the entanglement of the first cavity-mechanical oscillator modes, second cavity-mechanical oscillator modes, and the two cavity modes as a function of ${\Delta / {{\omega _m}}}$ for coupling strength $J = 0.5{\omega _m}$ and $J = 0.6{\omega _m}$, respectively. We also observe the same behavior as Fig. 3(a) and (b). We see from Fig. 3(c) that continuous variable entanglement witnessed for the parameter ${\Delta / {{\omega _m}}}$ ranging from 0 to 5, which entails that the greater possibility of entanglement can be easily realized in an experiment. This shows the behavior of the generation of our bipartite entanglement consistent with the photonic-crystal optomechanical cavity [35].

Additionally, we observe the effect of coupling rate $J$ on the bipartite entanglement from Table 2,

Table 2. The maximum value of EN concerning the strength of the coupling parameter

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In Table 2, we see that the entanglement of the first cavity-mechanical oscillator modes and second cavity-mechanical oscillator coupled modes increases with increasing the value of cavity-cavity coupling strength $J$. However, the entanglement between the two cavity modes is almost consistent with increasing the strength of the coupling parameter. In addition, these results can be compared with the situation discussed in Ref. [17]. From this point of view, the bipartite entanglement with our model can be generated and significantly enhanced by a variation in the cavity-cavity coupling strength process. The generation of entanglement between the two cavity modes can be transferred entirely due to the coupling strengths. It might then be possible to deduce that the strength of coupling rate J would be very crucial in transferring quantum features in a hybrid optomechanical system.

Thirdly, we examine the effect of the coupling rate on the entanglement. Figure 4 exhibits the logarithmic negativity EN of optomechanical systems for different input laser power when the nonlinear gain of OPA $\Omega = 0.95\beta$. From Figs. 4 (a), (b), and (c), we observe that the entanglement of the coupled cavity optomechanical systems increase when the laser power is increased. That means, increasing the input laser power increases the mean photon number, leading to increase the coupling rate ${g_j}$. The reason is that the real part of the effective coupling rate is given by ${g_j} = \sqrt 2 {G_{mj}}{\alpha _{js}}$. Since the steady-state amplitudes ${\alpha _{js}}$ of the cavity fields (see Eq. (13)) are proportional to laser driving amplitude ${E_j}$, then it should be realized that the ${g_j}$ factor in the drift matrix A is essentially determined by ${g_j} \propto {G_{mj}}{E_j}$. This shows that the behavior of coupling rate ${g_j}$ on the entanglement consistent with recent experiment [38]. Therefore, one can show that our discussion for steady-state entanglement is actually experimentally realistic.

Fig. 4. Plots of logarithmic negativity EN between (a) the first cavity-mechanical oscillator modes, (b) the second cavity-mechanical oscillator modes, and (c) the two cavity modes as a function of the normalized detuning ${\Delta / {{\omega _m}}}$, $\Omega = 0.95\beta$, blue solid and red solid lines correspond to the parameters ${P_1} = 100mW$ and ${P_2} = 50mW$, respectively. The rest of the parameters are taken to be the same as in Fig. 2.

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In Fig. 5, we observe that the bipartite entanglement between the cavity-mechanical oscillator modes increases as the cavity field decay rate decrease. In otherworld’s, when the decay rate increases the bipartite entanglement degrades.

Fig. 5. Plots of logarithmic negativity EN of the first cavity-mechanical oscillator modes (red solid line) and the second cavity-mechanical oscillator modes (blue solid line) as a function of the decay rate $\beta$, for $T = 100mK\;\Omega = 6 \times {10^6}{s^{ - 1}}$, where the other parameters are taken to be the same as in Fig. 2.

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In fact, from Fig. 5, we also see that the optical cavity losses destroy entanglement. Therefore, decay rates decreases the bipartite entanglement, and produce different amounts of bipartite entanglement on the subsystem. Additionally, the bipartite entanglement between the mechanical oscillators is not a monotonic function of the decay rate.

Moreover, we plot the entanglement versus the nonlinear gain of Ω for various temperatures. From Fig. 6, we observe that as the temperature of the environment increases, the bipartite entanglement between the subsystem decreases. This implies that at the moderate value of nonlinear gain, the bipartite entanglement increases. Furthermore, at a large value of nonlinear gain, the entanglement decreases. This shows that our system would be preferred to work at cryogenic temperatures.

Fig. 6. Plots of logarithmic negativity EN as a function of the nonlinear gain of Ω for various temperatures and the other parameters are taken to be the same as in Fig. 2.

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Furthermore, from our theoretical point of view, we investigated the entanglement between the cavity and mechanical oscillator modes, and the two cavity modes are analyzed through the applicable choice of nonlinear gain of OPA, optical cavity detuning, and cavity-cavity coupling strength. However, it is possible to generate the entanglement for the optomechanical subsystem between the two mechanical modes. To achieve this, we have to fix the cavity–laser detuning associated with the mechanical resonator. Therefore, under this regime and by switching the roles of the cavity modes and the mechanical mode, comparable techniques were utilized to generate entanglement between the subsystems such as two mechanical resonators [24,39].

Subsequently, there is a state transfer between the mechanical resonator and the cavity mode. This confirms that still there is a possibility of generating entanglement between subsystems. Such entanglement between the two mechanical oscillators is very sensitive to the choice of cavity-cavity coupling strength. Since, in our model, there is no direct interaction between two mechanical oscillators, and the entanglement is entirely transferred due to cavity-cavity coupling strength. This confirms we can generate the entanglement by fixing the value of cavity-cavity coupling strength, and one can measure the optomechanical entanglement between two oscillators. From the above, we can conclude that by exchanging the roles of the cavity modes, the mechanical modes, and cavity-cavity coupling strength, a similar mechanism can be used to prepare the entanglement between the two mechanical oscillators interacting with the cavity modes.

5. Conclusion

We have studied the steady-state entanglement between the first cavity-mechanical oscillator modes, the second cavity-mechanical modes, and the two cavity modes in a hybrid optomechanical system whose cavities containing two separate degenerate OPA are externally coupled. In the steady-state regime, we obtained the linearized quantum Langevin equations and quantified the steady-state entanglement because of the measure of logarithmic negativity. In this respect, the degenerate OPA generates the bipartite entanglement at steady-state. It has been found that the entanglement increases with the nonlinear gain of the OPA medium. It has also shown that the strength of cavity-cavity coupling significantly affects the nature of entanglement between cavity-mechanical oscillator modes. Moreover, we have shown that the minimum value of entanglement between the two cavity modes at a reasonable temperature is non-zero, the reason is there is entanglement transfer in the system, which is experimentally relevant. Hopefully, the possibility of entangling the modes of mechanical oscillators externally coupled via sharing the quantum properties of the accompanying OPA may have potential applications in the realization of continuous-variable quantum information processing applications.

Acknowledgment

We gratefully acknowledge Adama Science and Technology University, (ASTU) for their support during this work.

Disclosures

The authors declare no conflicts of interest.

Data availability

All data generated and analyzed are presented in the research presented.

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Parameters	Symbol	Value
Frequency of mechanical oscillator	$ω_{m}$	$2 π \times 10 M H z$
Mechanical quality factor	$Q$	$10^{5}$
Optical laser wavelength	$λ$	$810 n m$
Length of the cavity	$L$	$1 m m$
Mass of mechanical oscillator	$m$	$5 \times 10^{- 12} k g$
Driven laser power	$P_{1}$	100 mW
	$P_{2}$	$50 m W$
Cavity finesse	$F$	$1.07 \times 10^{4}$
Mechanical damping rate	$γ$	$2 π \times 100 H z$

The strength of coupling rate $J$	EN¹	EN²	EN³
$0.3 ω_{m}$	0.27	0.19	0.26
$0.4 ω_{m}$	0.29	0.194	0.27
$0.5 ω_{m}$	0.30	0.19	0.28
$0.6 ω_{m}$	0.31	0.19	0.29

Parameters	Symbol	Value
Frequency of mechanical oscillator	$ω_{m}$	$2 π \times 10 M H z$
Mechanical quality factor	$Q$	$10^{5}$
Optical laser wavelength	$λ$	$810 n m$
Length of the cavity	$L$	$1 m m$
Mass of mechanical oscillator	$m$	$5 \times 10^{- 12} k g$
Driven laser power	$P_{1}$	100 mW
	$P_{2}$	$50 m W$
Cavity finesse	$F$	$1.07 \times 10^{4}$
Mechanical damping rate	$γ$	$2 π \times 100 H z$

The strength of coupling rate $J$	EN¹	EN²	EN³
$0.3 ω_{m}$	0.27	0.19	0.26
$0.4 ω_{m}$	0.29	0.194	0.27
$0.5 ω_{m}$	0.30	0.19	0.28
$0.6 ω_{m}$	0.31	0.19	0.29

Steady-state entanglement in a hybrid optomechanical system enhanced by optical parametric amplifiers

Abstract

1. Introduction

2. Model and Hamiltonian of the system

3. Dynamics of the system

4. Results and discussion

5. Conclusion

Acknowledgment

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (2)

Equations (23)

Optics Continuum