Quantum optomechanics in the bistable regime

We study the simplest optomechanical system with a focus on the bistable regime. The covariance matrix formalism allows us to study both cooling and entanglement in a unified framework. We identify two key factors governing entanglement; namely, the bistability parameter (i.e., the distance from the end of a stable branch in the bistable regime) and the effective detuning, and we describe the optimum regime where entanglement is greatest. We also show that, in general, entanglement is a nonmonotonic function of optomechanical coupling. This is especially important in understanding the optomechanical entanglement of the second stable branch.

Figure 1

Mechanical restoring force and radiation pressure force around a cavity resonance. The leftmost and rightmost intersection points are stable equilibrium positions, whereas the middle one is unstable.

Figure 2

Bistability of the intracavity power with respect to input power. The solid and dotted lines correspond to the stable and unstable branches, respectively. The inset shows the bistability parameter $\eta$ for the two stable branches. The end of each stable branch corresponds to $\eta =0$.

Figure 3

Optomechanical entanglement (a) and optomechanical coupling constant (b) as a function of bistability parameter $\eta$ and normalized effective detuning $\Delta /{\omega }_m$ for a cavity decay rate $\kappa =1.4{\omega }_m$.

Figure 4

Plot of $\alpha$ and $\beta$ as a function of normalized detuning. The parameters are the same as in Fig. 3.

Figure 5

Plot of optomechanical entanglement as a function of input power for both stable branches. The dot dashed (dashed) line corresponds to the end of the first (second) stable branch. The parameters are the same as in Fig. 2.

Figure 6

Contour plot for bistability parameter (a) and entanglement (b) versus bare detuning ${\Delta }_0$ and input power $P_{\text{in}}$ in mW. The parameters are the same as in Fig. 2.

Figure 7

Plot of logarithmic negativity versus input power (mW) for different environment temperatures: $T=0.4$ K (solid line), $T=5$ K (dot dashed), and $T=10$ K (dashed). The other parameters are the same as in Fig. 2.