† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 11247001), the Scientific Research Foundation of the Higher Education Institutions of Anhui Province, China (Grant No. KJ2012A083), and the Doctor (Master) Fund of Anhui University of Science and Technology, China.
A scheme to generate entanglement in a cavity optomechanical system filled with an optical parametric amplifier is proposed. With the help of the optical parametric amplifier, the stationary macroscopic entanglement between the movable mirror and the cavity field can be notably enhanced, and the entanglement increases when the parametric gain increases. Moreover, for a given parametric gain, the degree of entanglement of the cavity optomechanical system increases with increasing input laser power.
Entanglement is one of the most attractive topics of quantum mechanics, which has been widely applied in diverse embranchments of physics. It is of basic and practical significance that we can generate entanglement among mesoscopic and even macroscopic systems. With the advance of technology, the Schrödinger cat state which describes the superposition states of macroscopic systems has been realized in experiments.[1] A macroscopic entanglement state can be prepared by optomechanical coupling via radiation pressure. Since the larger number of photons can lead to stronger radiation pressure, enhancing the photon number to increase the radiation pressure can acquire robust entanglement of optomechanical systems. A cavity optomechanical system is an available candidate system to study the entanglement of macroscopic states. Recently, there has been a growing interest in investigating entanglement in macroscopic systems.[2–13] In optomechanical systems, the quantum measurement theory and its applications have been explored,[3] a large variety of nonclassical states of both the cavity field and the movable mirror can be generated,[4] and the robust stationary entanglement can be produced between the intracavity mode and the mechanical mirror.[6] Reference [8] obtained a steady state entanglement of the motion of two dielectric membranes, which are suspended inside a Fabry–Perot cavity. The properties of optomechanical entanglement in a coupled cavity array with a movable mirror have been studied in Ref. [10]. Barzanjeh et al.[11] proposed a scheme for the realization of stationary continuous-variable entanglement in a hybrid tripartite system formed by an optical cavity and a microwave cavity, both interacting with a mechanical resonator. Chiara et al.[12] investigated the mirror–light entanglement in a hybrid optomechanical device formed by a Bose–Einstein condensate. In recent years, the experiments have realized the macroscopic entanglement.[14,15]
With the improvement of the qualities of the nonlinear crystals, optical parametric amplifiers (OPAs) including degenerate OPA and nondegenerate OPA, have shown tremendous applications in generating squeezed and entangled states.[16–24] Agarwal[17] investigated interferences in the quantum fluctuations of the output of an OPA, and demonstrated that the interferences can be manipulated by choosing the squeezing of the input field. Reference [23] obtained broadband entangled light through cascading nondegenerate OPAs, and the application of the entangled light from the cascading nondegenerate OPAs to broadband teleportation has been discussed.
Recently, the enhancement of the entanglement has been investigated by introducing nondegenerate OPAs.[25–27] Chen et al.[25] theoretically showed that the degree of the correlation of the input entangled beams can be improved by the nondegenerate OPA inside an optical cavity. In Refs. [26] and [27], it was experimentally demonstrated that the entanglement degree of the entangled state can be improved and manipulated by using nondegenerate OPAs. In this paper, we propose a scheme to produce the stationary macroscopic entanglement between a cavity field and a movable mirror in an optomechanical system filled with OPA. With the existence of the OPA, the entanglement can be pronouncedly increased compared to that without OPA, and the entanglement increases with the increasing parametric gain of the OPA. Furthermore, for a fixed parametric gain, the larger entanglement of the optomechanical system corresponds to the larger input laser power.
As shown in Fig.
![](cpb_25_7_074202/cpb152917ieqn1.gif)
![](cpb_25_7_074202/cpb152917ieqn2.gif)
![]() | Fig. 1. Sketch of the system. An optical parametric amplifier is inserted in a Fabry–Perot cavity with a fixed mirror and a movable mirror. |
Considering all of the noise and damping terms, we can investigate the dynamics of the system by using the quantum Langevin equation. By introducing phenomenologically the damping and noise terms into the Heisenberg operator equation, the quantum Heisenberg–Langevin equation can be obtained as
![](cpb_25_7_074202/cpb152917ieqn3.gif)
![](cpb_25_7_074202/cpb152917ieqn4.gif)
In order to study the steady state entanglement of the system, we adopt the standard methods of quantum optics[28] to solve Eq. (
When the cavity is very intensely driven, i.e., as ≫ 1, each operator of the system can be expanded as the sum of its steady-state mean value and a small fluctuation with zero mean value, namely, the operator can be written as ô = os+δô, and the dynamics of the small fluctuations is around the steady state of the system. We have linearized Langevin equations as follows:
When the real parts of all the eigenvalues of matrix
We aim to investigate the properties of entanglement of the cavity optomechanical system containing OPA. We employ the logarithmic negativity to quantify the entanglement. The logarithmic negativity can be defined as[31]
![](cpb_25_7_074202/cpb152917ieqn5.gif)
We now discuss the properties of the entanglement of the cavity optomechanical system. The entanglement of the system is displayed in Figs.
Figure
We give a brief discussion on the experimental feasibility of our program. To numerically calculate the logarithmic negativity of the optomechanical system, we have chosen feasible parameters based on the present experiment.[32] In fact, it is an experimental challenge to detect the entanglement of macroscopic mechanical systems. However, some promising schemes have been proposed[6,33] to relatively easily detect the quantum correlation by homodyne measurement techniques, so quantum entanglement can be indirectly detected by quantum correlation. We believe that our scheme is realizable in experiments with the development of science and technology.
In conclusion, we have proposed a scheme to generate stationary macroscopic entanglement of an optomechanical device filled with OPA. Our study shows that the entanglement of the cavity optomechanical system can be pronouncedly enhanced due to the introduction of the OPA, and the larger entanglement corresponds to the larger parametric gain G. Moreover, for a given G, the degree of entanglement increases when the input laser power increases.
1 | |
2 | |
3 | |
4 | |
5 | |
6 | |
7 | |
8 | |
9 | |
10 | |
11 | |
12 | |
13 | |
14 | |
15 | |
16 | |
17 | |
18 | |
19 | |
20 | |
21 | |
22 | |
23 | |
24 | |
25 | |
26 | |
27 | |
28 | |
29 | |
30 | |
31 | |
32 | |
33 |