$\mathcal{PT}$-Symmetric Phonon Laser

By exploiting recent developments associated with coupled microcavities, we introduce the concept of the $\mathcal{PT}$-symmetric phonon laser with balanced gain and loss. This is accomplished by introducing gain to one of the microcavities such that it balances the passive loss of the other. In the vicinity of the gain-loss balance, a strong nonlinear relation emerges between the intracavity-photon intensity and the input power. This then leads to a giant enhancement of both optical pressure and mechanical gain, resulting in a highly efficient phonon-lasing action. These results provide a promising approach for manipulating optomechanical systems through $\mathcal{PT}$-symmetric concepts. Potential applications range from enhancing mechanical cooling to designing phonon-laser amplifiers.

Figure 1

Gain-enhanced optomechanics in compound whispering-gallery resonators. The optical tunneling rate $J$ is tuned by changing the distance between ${\mathbb{R}}_{\kappa }$ and ${\mathbb{R}}_{\gamma }$. The corresponding optical supermodes coupled by phonons are also plotted.

Figure 2

(a) Steady-state cavity-photon number $N$: for $P_{\mathrm{in}}<30\mu \mathrm{W}$, the passive system shows linear responses (dashed line); in contrast, nonlinear behavior emerges for the gain-loss balanced system ($\delta =1$). (b) Optomechanical amplification factor $\eta$ with tunable ratio $\delta \equiv \kappa /\gamma$.

Figure 3

Plot of the stimulated emitted phonon number $n[{\mathrm{\Gamma }}_m^{-1}]=\exp [2(G-{\mathrm{\Gamma }}_m)/{\mathrm{\Gamma }}_m]=\exp [2(\beta P_{\mathrm{in}}/{\omega }_{+}-1)]$ in the $\mathcal{PT}$-symmetric or the passive system, as a function of $P_{\mathrm{in}}$. Here, $\beta \equiv (g{x}_0|\mathcal{C}|/2{)}^2$, and to compare with Ref. [13], we choose $2J={\omega }_m$ (well within the $\mathcal{PT}$-symmetric regime). The threshold power $P_{\mathrm{th}}$ denoted by the thick points is obtained for $G={\mathrm{\Gamma }}_m$. The threshold value of the passive COM system is $\sim 7\mu \mathrm{W}$, agreeing well with the experiment [13], which can be significantly lowered for $\delta \to 1$.