Mechanical $\mathcal{PT}$ symmetry in coupled optomechanical systems

We propose to realize mechanical parity-time $\left(\mathcal{PT}\right)$ symmetry in two coupled optomechanical systems. To provide gain to one mechanical resonator and the same amount of damping to the other, the two optical cavities should be driven by blue- and red-detuned laser fields, respectively. After adiabatically eliminating the degrees of freedom of the cavity modes, we derive a formula to describe the $\mathcal{PT}$ symmetry of two coupled mechanical resonators. Mechanical $\mathcal{PT}$-symmetric phase transition is demonstrated by the dynamical behavior of the mechanical resonators. Moreover, we study the effect of the quantum noises on the dynamical behavior of the mechanical resonators when the system is in the quantum regime.

Figure 1

(a) Schematic diagram of two coupled optomechanical systems with the cavities being driven by blue- and red-detuned laser fields, respectively. The possible realistic physical systems to implement the setup of coupled optomechanical systems for realizing mechanical parity-time $\left(\mathcal{PT}\right)$ symmetry: (b) the optical fiber cavity with a one-end-vibrating cantilever [71] and (c) and (d) the superconducting transmission line resonator coupled to a mechanical beam [39, 72, 73].

Figure 2

Tunable coupling of two vibrating cantilevers by (a) the piezoelectric effect and (b) the photothermal effect.

Figure 3

The dynamical behavior of the two mirrors for $J=0$: (a) left mirror $q_1$ (red curve) and (b) right mirror $q_2$ (green curve). The parameters used in the numerical calculation are ${\omega }_m=10\kappa ,-{\mathrm{\Delta }}_1={\mathrm{\Delta }}_2={\omega }_1={\omega }_2={\omega }_m,\gamma =\kappa /{10}^5,\mathrm{\Omega }=5000\kappa$, and $g=\kappa /10000$.

Figure 4

The dynamical behavior of the two vibrating mirrors $q_1$ [red (dark gray) curves] and $q_2$ [green (light gray) curves] given by Eqs. (5, 6, 7, 8) for $J=\kappa /100$. The other parameters used in the numerical calculation are the same as given in Fig. 3.

Figure 5

The dynamical behavior of the two vibrating mirrors $q_1$ [red (dark gray) curves] and $q_2$ [green (light gray) curves] given by (a), (c), and (e) Eqs. (5, 6, 7, 8) and (b), (d), and (f) Eqs. (20)–(23) for (a) $\mathrm{\Omega }/\kappa =5000$, (b) ${\gamma }_{\mathrm{eff}}=J$, (c) $\mathrm{\Omega }/\kappa =6700$, (d) ${\gamma }_{\mathrm{eff}}=1.8J$, (e) $\mathrm{\Omega }/\kappa =10000$, and (f) ${\gamma }_{\mathrm{eff}}=4J$. The other parameters are the same as those given in Fig. 4.

Figure 6

The dynamical behavior of the two vibrating mirrors $q_1$ [red (dark gray) curve] and $q_2$ [green (light gray) curve] given by Eqs. (13), (16), and (17) with $J=\kappa /100,{\mathrm{\Gamma }}_1-{\gamma }_1={\mathrm{\Gamma }}_2+{\gamma }_2={\gamma }_{\mathrm{eff}}=1.8J,{\omega }_1={\omega }_2={\omega }_m$, and $\delta {\omega }_i=2.25\times {10}^{-5}\kappa$.

Figure 7

The dynamical behaviors of the intensity of the optical powers in the cavities and the positions of the two vibrating mirrors given by Eqs. (5, 6, 7, 8): (a) $|a_1{|}^2$ (blue curve), (b) $|a_2{|}^2$ (cyan curve), (c) $q_1$ (red curve), and (d) $q_2$ (green curve). The parameters are the same as those given in Fig. 5.

Figure 8

The maximum value ${\left({\mathrm{\Lambda }}_i/\kappa \right)}_{\max }$ of all the eigenvalues of $M$ as a function of the effective optomechanical coupling rate $G_1=G_2\equiv G$. The parameters are ${\kappa }_1={\kappa }_2=\kappa ,J=\kappa /100,-{\mathrm{\Delta }}_1^{\prime }={\mathrm{\Delta }}_2^{\prime }={\omega }_1={\omega }_2=10\kappa$, and ${\gamma }_1={\gamma }_2=\kappa /{10}^5$.

Figure 9

The dynamical behaviors of the phonons from (a)–(c) spontaneous generation $n_1^{\mathrm{sp}}$ (blue solid line) and $n_2^{\mathrm{sp}}$ (blue dashed line), (d)–(f) stimulated generation $n_1^{\mathrm{st}}$ (red solid line) and $n_2^{\mathrm{st}}$ (red dashed line), and (g)–(i) the total phonons $n_1^{\mathrm{tt}}$ (black solid line) and $n_2^{\mathrm{tt}}$ (black dashed line) with different values of the effective optomechanical coupling rate $G_1=G_2\equiv G$: (a), (d), and (g) $G/\kappa =0.05$, (b), (e), and (h) $G/\kappa =0.069$, and (c), (f), and (i) $G/\kappa =0.08$. Here $\langle q_i^2\left(0\right)\rangle =\langle p_i^2\left(0\right)\rangle =3/2,T=0$, i.e., $n_i^{\mathrm{th}}=0$. The other parameters are the same as those given in Fig. 8.

Figure 10

The dynamical behaviors of the phonons generated by the quantum noises of the cavity modes $n_i^{\mathrm{cm}}$ and by the stochastic forces of the mechanical resonators $n_i^{\mathrm{mr}}$ with thermal phonon number: (a) and (b) $n_i^{\mathrm{th}}=0$ and (c) and (d) $n_i^{\mathrm{th}}=1000$. The effective optomechanical coupling rate $G_1=G_2\equiv G$ is (a) and (c) $G/\kappa =0.05$ and (b) and (d) $G/\kappa =0.08$. The other parameters are the same as those given in Fig. 8.

Figure 11

The dynamical behaviors of the phonons generated by the stimulated generation $n_1^{\mathrm{st}}$ (thin red line) and the total phonons $n_1^{\mathrm{tt}}$ (thick black line) with thermal phonon number $n_i^{\mathrm{th}}=1000$: (a) and (b) $\langle q_i^2\left(0\right)\rangle =\langle p_i^2\left(0\right)\rangle =3/2$ and (c) and (d) $\langle q_i^2\left(0\right)\rangle =\langle p_i^2\left(0\right)\rangle =100\frac{1}{2}$. The effective optomechanical coupling rate is $G/\kappa =0.05$ in (a) and (c) and is $G/\kappa =0.08$ in (b) and (d). The other parameters are the same as those given in Fig. 8.