Tunable optical-gain-induced chaotic dynamics in a hidden $\mathcal{PT}$-symmetric optomechanical system

Exceptional points (EPs) are the singular branch points of the non-Hermitian Hamiltonian spectrum. They lead to many exotic phenomena of non-Hermitian systems. However, it is hard to realize parity-time $\left(\mathcal{PT}\right)$ symmetry breaking at EPs by increasing the optical gain due to the limitation of the optical gain materials. Here we propose a scheme to realize gain-induced $\mathcal{PT}$ symmetry breaking in two coupled passive optical resonators with a Stokes gain in the optomechanical resonator. As the Stokes gain results from radiation pressure coupling, it depends on the strength of the pump field. For this reason, $\mathcal{PT}$ symmetry breaking can be triggered by increasing the strength of the pump field. Moreover, Stokes gain also relies on the effective detuning of optomechanics, which is linear to the photon number. Therefore, $\mathcal{PT}$ symmetry breaking can also be induced by increasing the coupling strength between two coupled resonators and thus decreasing the photon number of the optomechanical resonator. In addition, this work opens up a method to control hidden $\mathcal{PT}$ symmetry breaking and nonlinear dynamics with radiation pressure coupling.

Figure 1

The hidden $\mathcal{PT}$ symmetrical optomechanics. A MOR and a PMR, denoted with $a_{⥁}$ and $c_{⥀}$, are evanescently coupled each other. The MOR couples to an optical fiber, where the pump field can generate Stokes photons with the radiation pressure coupling via the Stokes processes. The vibration of the mechanical mode is represented by the double-sided arrows.

Figure 2

Spectra $lnS\left(\omega \right)$ of the MOR mode $I_1=|a_{⥁\mathrm{sr}}{|}^2+{\left|a_{⥁\mathrm{si}}\right|}^2$ and PMR mode $I_2=|c_{⥀\mathrm{sr}}{|}^2+{\left|c_{⥀\mathrm{si}}\right|}^2$ for (a) $\mathrm{U}\mathcal{PT}\mathrm{S}$ with $g=7{\gamma }_0$ and (b) $\mathrm{B}\mathcal{PT}\mathrm{S}$ with $g=400{\gamma }_0$. Here ${\gamma }_0/2\pi =1$ MHz, ${\mathrm{\Omega }}_{\mathrm{d}}=400{\gamma }_0$, ${\kappa }_{\mathrm{a}}={\kappa }_{\mathrm{c}}={\gamma }_0$ MHz, ${\omega }_{\mathrm{m}}=23{\gamma }_0$, $\chi =7.4\times {10}^{-5}{\gamma }_0$, and ${\gamma }_{\mathrm{m}}=0.038{\gamma }_0$.

Figure 3

(a) LE for $I_1$ and (b) its bifurcation diagram for time interval $6.4\to 8\mu \mathrm{s}$ on $g/{\gamma }_0$. The shadowed green region is for the $\mathrm{U}\mathcal{PT}\mathrm{S}$, and the rest of the region is for the $\mathrm{B}\mathcal{PT}\mathrm{S}$. The rest of the parameters are the same as Fig. 2.

Figure 4

LE for $I_1$ versus (a) ${\kappa }_{\mathrm{c}}/{\gamma }_0$ with ${\mathrm{\Omega }}_{\mathrm{d}}=400{\gamma }_0$ and $g=287{\gamma }_0$, and (b) ${\mathrm{\Omega }}_{\mathrm{d}}/{\gamma }_0$ with ${\kappa }_{\mathrm{c}}/2\pi =1$ MHz. The rest of the parameters are ${\gamma }_0/2\pi =1$ MHz, ${\kappa }_{\mathrm{c}}={\gamma }_0$, ${\omega }_{\mathrm{m}}=23{\gamma }_0$, $g_0=7.4\times {10}^{-5}{\gamma }_0$, ${\mathrm{\Delta }}_{\mathrm{c}}=-{\omega }_{\mathrm{m}}$, ${\mathrm{\Delta }}_{\mathrm{a}}=0.882{\mathrm{\Delta }}_{\mathrm{c}}$, $\chi =7.4\times {10}^{-5}{\gamma }_0$, and ${\gamma }_{\mathrm{m}}=0.038{\gamma }_0$.

Figure 5

(a) LE for $I_1={\left|a_{⥁}\right|}^2$ as a function of the pump field ${\mathrm{\Omega }}_{\mathrm{d}}$. Red and blue squares correspond to the pump field ${\mathrm{\Omega }}_{\mathrm{d}}=0.02\times {10}^7{\gamma }_0$ for $\mathrm{U}\mathcal{PT}\mathrm{S}$ and the one ${\mathrm{\Omega }}_{\mathrm{d}}=4\times {10}^7{\gamma }_0$ for $\mathrm{B}\mathcal{PT}\mathrm{S}$. (b and c) Cavity fields $I_1={\left|a_{⥁}\right|}^2$ and $I_2={\left|c_{⥀}\right|}^2$ versus time interval $0\mu \mathrm{s}\to 40\mu \mathrm{s}$. (d and e) Trajectories for $d{I}_1/dt$ versus $I_1$ in phase space for time interval $0\mu \mathrm{s}\to 40\mu \mathrm{s}$, and the trajectories for time interval $32\mu \mathrm{s}\to 40\mu \mathrm{s}$ are plotted in (f) and (g), respectively. Here ${\gamma }_0/2\pi =1MHz$, ${\kappa }_{\mathrm{c}}={\kappa }_{\mathrm{a}}={\gamma }_0$, ${\omega }_{\mathrm{m}}=23{\gamma }_0$, $\chi =7.4\times {10}^{-5}{\gamma }_0$, ${\gamma }_{\mathrm{m}}=0.038{\gamma }_0$, $g=40{\gamma }_0$, ${\mathrm{\Delta }}_{\mathrm{a}}=0.98{\mathrm{\Delta }}_{\mathrm{c}}$, and ${\mathrm{\Delta }}_{\mathrm{c}}=-{\omega }_{\mathrm{m}}$.