Distinguishing photon blockade in a $\mathcal{PT}$-symmetric optomechanical system

We study the effects of parity-time ($\mathcal{PT}$) symmetry on the photon blockade and distinguish the different blockade mechanisms in a double-cavity optomechanical system. By studying the light statistics of the system, we find the completely different photon blockade behaviors when the $\mathcal{PT}$ symmetry is broken or unbroken, which is related to the $\mathcal{PT}$ phase transition. Furthermore, an interesting phenomenon that the two cavities are blocked at the same time is found with the appropriate system parameters. Those statistical phenomena are all analyzed in detail and demonstrated by analytically solving the Schrödinger equation and numerically simulating the master equation, respectively. Finally, we also consider the non-$\mathcal{PT}$-symmetric situations which further reveal the physical essence of the photon blockade by comparing those results. Different from the usual photon blockade, our proposal is feasible even with weak parameter mechanism, i.e., the proposal neither requires the strong optomechanical coupling nor the large tunneling coupling between cavities.

Figure 1

Schematic of the $\mathcal{PT}$-symmetric double-cavity optomechanical system composed of the whispering-gallery resonators.

Figure 2

The equal-time second-order correlation functions versus the detuning (left column) or versus both the detuning and the optomechanical coupling strength (right column). In the left column, the solid black lines are the analytical solutions of the correlation functions defined as Eq. (11), and the red squares represent the numerical simulation results, respectively, by the master function Eq. (12) with the Hamiltonian $H_1$. In the right column, the dashed white line is the optimal detuning changing with the optomechanical coupling strength. The system parameters are set as $T=1\mathrm{mK},{\kappa }_1=2\pi \mathrm{MHz},{\kappa }_2={\kappa }_1,{\mathrm{\Delta }}_2={\mathrm{\Delta }}_1,{\omega }_m=100{\kappa }_1,{\gamma }_m={\omega }_m/{10}^6,J=0.5{\kappa }_1,g=3{\kappa }_1$, and $E=0.01{\kappa }_1$.

Figure 3

The perfect photon blockade with the different detunings.

Figure 4

The delayed second-order correlation functions for different cavities versus the time-delay ${\kappa }_1\tau$ where the detuning is set as ${\mathrm{\Delta }}_1=g^2/\left(2{\omega }_m\right)$ for $g_1^{\left(2\right)}\left(\tau \right)$ and ${\mathrm{\Delta }}_1=g^2/{\omega }_m$ for $g_2^{\left(2\right)}\left(\tau \right)$ and $g_{12}^{\left(2\right)}\left(\tau \right)$, respectively. The other parameters are the same as in Fig. 2.

Figure 5

The equal-time correlation function of the passive cavity 1 in the different $\mathcal{PT}$-symmetric phases. (a) and (b) represent the correlation function changing with the detuning for different $J$'s, which correspond to the broken $\mathcal{PT}$-symmetric region and the unbroken $\mathcal{PT}$-symmetric region, respectively. The solid black line is the analytical solution of the correlation function $g_1^{\left(2\right)}\left(0\right)$ defined in Eq. (11), and the red squares represent the numerical simulation result by the master function Eq. (12) with the Hamiltonian $H_1$. (c) The correlation function $g_1^{\left(2\right)}\left(0\right)$ versus both the detuning and the coupling strength between the two cavities. The dashed white line is the optimal parameters, which come from the theory of CPB. The solid blue line represents the $\mathcal{PT}$ phase transition, which divides the system into the broken and unbroken $\mathcal{PT}$-symmetric regions. (d) The distributed probabilities of states $|1,0\rangle$ and $|2,0\rangle$ versus the detuning for $J=0.7{\kappa }_1$. The other parameters are the same as in Fig. 2.

Figure 6

Level diagram of the photon state for the $\mathcal{PT}$-symmetric optomechanical system. (a) Dressed state. (b) Bare state.

Figure 7

The equal-time correlation function of the passive cavity 1 versus the detuning with different $J$'s. The other parameters are the same as in Fig. 2.

Figure 8

The equal-time correlation function of the passive cavity 1 versus the detuning in a non-$\mathcal{PT}$-symmetric optomechanical system. (a) ${\kappa }_1\ne {\kappa }_2$ and ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2$. (b) ${\kappa }_1={\kappa }_2$ and ${\mathrm{\Delta }}_1\ne {\mathrm{\Delta }}_2$. The other parameters are the same as in Fig. 2.