Steady-state one-way Einstein-Podolsky-Rosen steering in optomechanical interfaces

Einstein-Podolsky-Rosen (EPR) steering is a form of quantum correlation and its intrinsic asymmetry makes it distinct from entanglement and Bell nonlocality. We propose here a scheme for realizing one-way Gaussian steering of two electromagnetic fields mediated by a mechanical oscillator. We reveal that the steady-state one-way steering of the intracavity and output fields is obtainable with different cavity losses or strong mechanical damping. The conditions for achieving this asymmetric steering are found, and it shows that the steering is robust against thermal mechanical fluctuations. The present scheme can realize hybrid microwave-optical asymmetric steering by optoelectromechanics. In addition, our results are generic and can also be applied to other three-mode parametrically coupled bosonic systems.

Figure 1

(a) The schematic plot of double-cavity optomechanics in which two separate electromagnetic fields (e.g., microwave and optical fields) are mediated by a mechanical oscillator vibrating at frequency ${\omega }_m$. The two cavity fields are driven, respectively, on the red and blue sidebands of the driving fields. (b) After the squeezing transformation of Eq. (17), the composite mode ${\widehat{c}}_2$ is decoupled to the mechanical mode $\widehat{b}$ which interacts with the composite mode ${\widehat{c}}_1$ via a beam-splitter-like interaction and meanwhile the two composite modes are coupled to an effective reservoir in a two-mode squeezed vacuum. The mechanical damping drives the steady two-mode cavity field states to be asymmetric.

Figure 2

(a) Steady-state $S_{12}$, which is minimized with respect to $g_1/{\kappa }_1$ and ${\kappa }_2/{\kappa }_1$, versus $g_2/{\kappa }_1$ for ${\kappa }_2<{\kappa }_1$ and ${\gamma }_m=0.01{\kappa }_1$. From top to bottom, we have ${\overline{n}}_{\mathrm{th}}=1000$, 700, 500, 300, 100, and 0. The corresponding values of $S_{21}$ are larger than one and not plotted. (b) $S_{12}$ and $S_{21}$ versus time for ${\kappa }_2=0.4{\kappa }_1,g_1=10{\kappa }_1,g_2=20{\kappa }_1,{\gamma }_m=0.01{\kappa }_1$, and ${\overline{n}}_{\mathrm{th}}=0$. The values of ${\kappa }_2$ and $g_1$ in panel (b) are chosen such that $S_{12}$ is minimized.

Figure 3

(a) Steady-state $S_{21}$, which is minimized with respect to $g_1/{\kappa }_1$ and ${\kappa }_2/{\kappa }_1$, versus $g_2/{\kappa }_1$ for ${\kappa }_2>{\kappa }_1$ and ${\gamma }_m=0.01{\kappa }_1$. From top to bottom, we have ${\overline{n}}_{\mathrm{th}}=40$, 20, and 0. The corresponding values of $S_{12}$ are larger than one and not plotted. (b) $S_{12}$ and $S_{21}$ versus time for ${\kappa }_2=2.4{\kappa }_1,g_1=12{\kappa }_1,g_2=20{\kappa }_1,{\gamma }_m=0.01{\kappa }_1$, and ${\overline{n}}_{\mathrm{th}}=0$. In panel (b), the values of ${\kappa }_2$ and $g_1$ are chosen such that $S_{21}$ is minimized.

Figure 4

Steady-state $S_{12}$ and $S_{21}$ versus ${\gamma }_m/\kappa$ for ${\overline{n}}_{\mathrm{th}}=0$ (solid curves) and ${\overline{n}}_{\mathrm{th}}=0.3$ (dashed curves). The other parameters are $g_1=6\kappa$ and $g_2=10\kappa$.

Figure 5

The dependence of the minimized $S_{12}$ and $S_{21}$, with respect to $g_1/\kappa$ and $g_2/\kappa$, in the steady-state regime on the ${\gamma }_m/\kappa$, for ${\overline{n}}_{\mathrm{th}}=0$.

Figure 6

(a) The spectra of $S_{12}\left[\omega \right]$ and $S_{21}\left[\omega \right]$ for $g_1=6\kappa ,g_2=10\kappa ,{\gamma }_m=0.01\kappa$, and ${\overline{n}}_{\mathrm{th}}=0$. (b) $S_{12}\left[0\right]$ and $S_{21}\left[0\right]$ versus ${\overline{n}}_{\mathrm{th}}$.

Figure 7

(a) The spectra of $S_{12}\left[\omega \right]$ and $S_{21}\left[\omega \right]$ for $g_1=2\kappa ,g_2=3\kappa ,{\gamma }_m=9\kappa$, and ${\overline{n}}_{\mathrm{th}}=0$. (b) $S_{12}\left[0\right]$ and $S_{21}\left[0\right]$ versus ${\overline{n}}_{\mathrm{th}}$.

Figure 8

The spectra of $S_{12}\left[\omega \right]$ and $S_{21}\left[\omega \right]$ for different internal loss rates of the cavities. The same internal loss rates ${\kappa }_j^{\mathrm{int}}={\kappa }^{\mathrm{int}}$ are assumed, the parameters in (a) are the same as those in Fig. 6 (a), and the parameters in (b) are the same as in Fig. 7 (a). In (a) we have $S_{12}\left[\omega \right]=S_{21}\left[\omega \right]$.