Enhancement of mechanical entanglement in hybrid optomechanical system

Abstract

We present a theoretical scheme for the enhancement of entanglement between the two distant mechanical resonators in a hybrid optomechanical system which is composed of a single-mode optomechanical cavity with an atomic ensemble of two-level atoms and two mechanical resonators coupled via Coulomb field. Our results show that the atomic detuning, collective atomic coupling and Coulomb interaction play a vital role in obtaining the enhanced entanglement between the two distant mechanical resonators. We can interplay these controlling parameters to obtain optimal entanglement for the system. Interestingly, the numerical simulation result reveals that such bi-partite entanglement persists for bath temperature up to 600 K. In the end, we show that the system’s interaction with the atomic ensemble lead to an enhanced entanglement spectrum, which is robust against bath temperature as well. The present scheme will open new perspectives in constructing entanglement between two distant resonators and helpful to realize quantum memories for the continuous-variable quantum information processing.

Appendix A. Working principle of the covariance matrix

The solution of Eq. (10) can be solved as following form

$$\begin{aligned} F(t)=\varrho (t)F(0)+ \int \limits _{0}^{t}\varrho (\tau )\varLambda (t-\tau )d\tau \end{aligned}$$

(18)

where $\varrho (t)=e^{Mt}$. The initial condition for the above transformation $\varrho (0)=I$, where I is Identity matrix. The Hamiltonian which corresponds to linearized quantum Langevin Eq. (9) guarantees that when the hybrid system is stable, it results into a Gaussian state whose information properties, such as entanglement, can be delineated by the symmetric $8\times 8$ covariance matrix. The components of this covariance matrix can be defined as

$$\begin{aligned} V_{mn}=\frac{\left\langle F_{m}F_{n}+F_{n}F_{m}\right\rangle }{2} \end{aligned}$$

(19)

From Eqs. (10) and (19), one can easily obtain a linear differential equation for the hybrid system’s covariance matrix

$$\begin{aligned} \dot{V}=MV+VM^{T}+D, \end{aligned}$$

(20)

where D is a diffusion matrix. The components of D are associated with the noise correlation function Eq. (14)

$$\begin{aligned} D_{a,b}\delta (t-\acute{t})=\frac{\left\langle \varLambda _{a}(t)\varLambda _{b}( \acute{t})+\varLambda _{b}(\acute{t})\varLambda _{a}(t)\right\rangle }{2} \end{aligned}$$

(21)

From the above equation, it is easy to get that D is diagonal which is given in Eq. (13).

The covariance matrix V of the present hybrid optomechanical system is given by

$$\begin{aligned} V=\left( \begin{array}{cccc} V_{m_{L}} &{} V_{m_{L}m_{R}} &{} V_{m_{L}c} &{} V_{m_{L}a} \\ V_{m_{L}m_{R}}^{\dagger } &{} V_{m_{R}} &{} V_{m_{R}c} &{} V_{m_{R}a} \\ V_{m_{L}c}^{\dagger } &{} V_{m_{R}c}^{\dagger } &{} V_{c} &{} V_{ca} \\ V_{m_{L}a}^{\dagger } &{} V_{m_{R}a}^{\dagger } &{} V_{ca}^{\dagger } &{} V_{a} \end{array} \right) , \end{aligned}$$

(22)

where each element is a $2 \times 2$ matrix. The diagonal elements $V_{m_{L}}$,$V_{m_{R}}$, $V_{c}$ and $V_{a}$ of the above covariance matrix V depict the local properties of the left mechanical mode, right mechanical mode, the cavity mode and the collective atomic mode, respectively, while the non-diagonal elements referring to the intermode correlations. For example, $V_{m_{L}a}$ describe the left mechanical mode-atom correlation.

To compute the entanglement among different bi-partite subsystems of the hybrid optomechanical system, we have to reduce the original $8\times 8$ covariance matrix V into a $4\times 4$ submatrix $V_{sub}$. If the indices m and n for the matrix element $V_{mn}$ are restricted to the set $\{1, 2, 3, 4\}$, the submatrix $V_{sub}=[V_{mn}]$ can be formed by the first four columns and rows of the covariance matrix V. This corresponds to the covariance between the two mechanical modes. Similarly, if the indices m and n for the matrix element $V_{mn}$ run over $\{1, 2, 7, 8\}$, $V_{sub}$ will describe the covariance matrix of the left mechanical mode and collective atomic mode. This is how the covariance matrix works.