Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system

We study multipartite entanglement, the generation of Einstein-Podolsky-Rosen (EPR) states, and quantum steering in a three-mode optomechanical system composed of an atomic ensemble located inside a single-mode cavity with a movable mirror. The cavity mode is driven by a short laser pulse, has a nonlinear parametric-type interaction with the mirror and a linear beam-splitter-type interaction with the atomic ensemble. There is no direct interaction of the mirror with the atomic ensemble. A threshold effect for the dynamics of the system is found, above which the system works as an amplifier and below which as an attenuator of the output fields. The threshold is determined by the ratio of the coupling strengths of the cavity mode to the mirror and to the atomic ensemble. It is shown that above the threshold, the system effectively behaves as a two-mode system in which a perfect bipartite EPR state can be generated, while it is impossible below the threshold. Furthermore, a fully inseparable tripartite entanglement and even further a genuine tripartite entanglement can be produced above and below the threshold. In addition, we consider quantum steering and examine the monogamy relations that quantify the amount of bipartite steering that can be shared between different modes. It is found that the mirror is more capable for steering of entanglement than the cavity mode. The two-way steering is found between the mirror and the atomic ensemble despite the fact that they are not directly coupled to each other, while it is impossible between the output of cavity mode and the ensemble which are directly coupled to each other.

Figure 1

Schematic diagram of the system. An ensemble of identical two-level (TL) atoms, each of transition frequency ${\omega }_a$, is located inside a single-mode cavity of frequency ${\omega }_c$. The cavity mode is driven by a laser pulse of frequency${\omega }_L$ and the movable mirror oscillates with frequency ${\omega }_m$.

Figure 2

Variation of the inseparability parameter ${\Delta }_{a,m}$ with $r=G\tau$ for $n_0=0$ and several different values of $\alpha$: $\alpha =1$ (solid black line), $\alpha =2$ (dashed blue line), $\alpha =10$ (dotted-dashed green line).

Figure 3

Variation of the parameter ${\Delta }_{a,w}$ with $r=G\tau$ for several different values of $n_0$ and $\alpha$. Lower solid black line for $n_0=0,\alpha =1$. Lower dashed black line for $n_0=5,\alpha =1$. Upper solid blue line for $n_0=0,\alpha =1.5$. Upper dashed blue line for $n_0=5,\alpha =1.5$. Dotted line indicates position of ${\Delta }_{a,w}=2$, the threshold for entanglement.

Figure 4

Variation of the parameter ${\Delta }_{a,m}$ with $r=G\tau$ for the case $G_a>G$, $n_0=0$, and several values of ${\alpha }^{\prime }$: ${\alpha }^{\prime }=1.2$ (solid black line), ${\alpha }^{\prime }=1.5$ (dashed blue line), ${\alpha }^{\prime }=5$ (dotted-dashed green line).

Figure 5

Variation of the parameter ${{\rm Y}}_{m,c}$ with $r=G\tau$ for the case $G_a>G$, $n_0=0$, and several values of ${\alpha }^{\prime }$: ${\alpha }^{\prime }=1$ (solid black line), ${\alpha }^{\prime }=5$ (dashed blue line), ${\alpha }^{\prime }=10$ (dotted-dashed green line).

Figure 6

The separability parameters for the asymmetric combinations of the quadrature components are shown as a function of $r=G\tau$ for the case $G>G_a$ with $\alpha =2$ (a) and for the case $G_a>G$ with ${\alpha }^{\prime }=2$ (b). Black solid and black dotted-dashed lines represent ${\Delta }_{a,m}^g$ respectively for $n_0=100$ and $n_0=0$; while red dashed and red dotted lines represent ${{\rm Y}}_{m,c}^g$ respectively for $n_0=100$ and $n_0=0$.

Figure 7

Variation of the parameters ${\Delta }_{ij}$ and ${\Delta }_{\mathrm{sum}}$ with $r=G\tau$ is shown for the case $G>G_a$ with $n_0=0,\alpha =2$ for (a) symmetric combinations of the quadrature operators $(g_a=g_m=g_c=1)$, (b) asymmetric combinations of the quadrature operators with optimal weight factors. The dashed line shows ${\Delta }_{am}$ (black), ${\Delta }_{ac}$ (blue), and ${\Delta }_{mc}$ (red). The solid green line shows ${\Delta }_{\mathrm{sum}}$.

Figure 8

Same as in Fig. 7 but for the case $G_a>G$ with $n_0=0$, ${\alpha }^{\prime }=2$.

Figure 9

Variation of the steering parameters $E_{i|j}$ with $r$ for the case $G>G_a$, $n_0=0$, and $\alpha =2(G_a=0.75G)$. The solid lines show $E_{a|m}$ (black), $E_{m|c}$ (red), and $E_{a|w}$ (blue). The dashed lines show $E_{m|a}$ (black), $E_{c|m}$ (red), and $E_{w|a}$ (blue).

Figure 10

Variation of the steering parameters $E_{i|j}$ with $r$ for the case $G_a>G$, $n_0=0$, and ${\alpha }^{\prime }=2(G_a=4G/3)$. The black solid line shows $E_{a|m}$, the black dashed line $E_{m|a}$, the red solid line $E_{m|c}$, and the red dashed line $E_{c|m}$.