$\mathcal{P}\mathcal{T}$-Symmetry-Breaking Chaos in Optomechanics

We demonstrate $\mathcal{P}\mathcal{T}$-symmetry-breaking chaos in an optomechanical system, which features an ultralow driving threshold. In principle, this chaos will emerge once a driving laser is applied to the cavity mode and lasts for a period of time. The driving strength is inversely proportional to the starting time of chaos. This originally comes from the dynamical enhancement of nonlinearity by field localization in the $\mathcal{P}\mathcal{T}$-symmetry-breaking phase. Moreover, this chaos is switchable by tuning the system parameters so that a $\mathcal{P}\mathcal{T}$-symmetry phase transition occurs. This work may fundamentally broaden the regimes of cavity optomechanics and nonlinear optics. It offers the prospect of exploring ultralow-power-laser-triggered chaos and its potential applications in secret communication.

Figure 1

(a) A schematic illustration of a $\mathcal{P}\mathcal{T}$-symmetric OMS including a passive OMS (with cavity mode ${\widehat{a}}_1$ and a mechanical mode $\widehat{b}$) coupled to an active cavity ${\widehat{a}}_2$ with tunneling strength $J$. Here ${\widehat{a}}_1^{\text{in}}$ and ${\widehat{a}}_1^{\mathrm{out}}$ are the input and output, respectively, of a driving field with frequency ${\omega }_d$; $\gamma$ (${\gamma }_m$) and $\kappa$ are, respectively, the optical (mechanical) decay rate and gain rate. (b) The realization of a $\mathcal{P}\mathcal{T}$-symmetric OMS in coupled microtoroidal resonators. (c) The phase diagram in terms of $J/\gamma$ and $\kappa /\gamma$ indicates the $\mathcal{P}\mathcal{T}\mathrm{SP}$ and $\mathcal{P}\mathcal{T}\mathrm{BP}$ in the passive-active system, as well as the parameter regime corresponding to the passive-passive system.

Figure 2

Power spectrum $\mathrm{Ln}\mathrm{S}(\omega )$ of the intracavity field $I_1=|a_1{|}^2$ in $\mathcal{P}\mathcal{T}\mathrm{SP}$ (a) and $\mathcal{P}\mathcal{T}\mathrm{BP}$ (b). The inset in (b): Bifurcation diagram versus $J/\gamma$. Lyapunov exponent versus (c) $J/\gamma$, inset of (c) $\kappa /\gamma$, and (d) ${\mathrm{\Omega }}_d/\gamma$ corresponding to a fixed time interval with $0.5\mu \mathrm{s}$. The shadowed blue region of the inset in (b) and (c) corresponds to $\mathcal{P}\mathcal{T}\mathrm{BP}$, and the red arrows indicate the EPs. The inset in (d) corresponds to a case of weak driving, i.e., the black square in (d). According to recent microcavity experiments [35], the parameters are ${\omega }_c=190\mathrm{THz}$, ${\omega }_m/2\pi =23\mathrm{MHz}$, $\gamma /2\pi =1\mathrm{MHz}$, $g_0=7.4\times {10}^{-5}\gamma$, ${\gamma }_m/2\pi =0.038\gamma$, $\kappa =0.8\gamma$, ${\mathrm{\Delta }}_c={\omega }_m$, and (a)–(c) $P=1\mu \mathrm{W}$ (or ${\mathrm{\Omega }}_d/\gamma \approx 4000$) (d) $J=\gamma$ and $J=0.2\gamma$ correspond to the blue circle and red square, respectively.

Figure 3

Evolutions of (a) the intracavity intensities $I_1$ (passive cavity), $I_2$ (active cavity), and (b) the dimensionless mechanical displacement $x$ in $\mathcal{P}\mathcal{T}\mathrm{SP}$, i.e., $J=\gamma$. The insets: The trajectories of (a) the optical mode, i.e., the first derivation of $I_1$ versus $I_1$, and (b) the mechanical mode, i.e., $p$ versus $x$, in phase space corresponding to a fixed time interval $8\to 9\mu \mathrm{s}$. The purple dashed lines indicate the amplitudes of $I_1$, $I_2$, and $x$ at $1.6\mu \mathrm{s}$. The purple arrows indicate the approximate evolution direction of the trajectories. The parameters are the same as Fig. 2 except for $P\approx 0.02\mathrm{pW}$, i.e., ${\mathrm{\Omega }}_d=0.5\gamma$.

Figure 4

Evolutions of (a) $I_1$, $I_2$, and (b) $x$ in $\mathcal{P}\mathcal{T}\mathrm{BP}$, i.e., $J=0.2\gamma$. The optical and mechanical trajectories in phase space are presented in the insets in (a) and (b). (c) The optical trajectory corresponding to time interval (iii). The inset in (c) presents the chaos dynamics in the long-time limit. The purple arrow in (c) indicates the approximate evolution direction of trajectory. The parameters are the same as Fig. 3.

Figure 5

The evolutions of $I_1$ and $I_2$ in the passive-passive system when (a) $P\approx 0.02\mathrm{pW}$ or ${\mathrm{\Omega }}_d\approx 0.5\gamma$ and (b) $P\approx 500\mathrm{mW}$ or ${\mathrm{\Omega }}_d\approx 3\times {10}^6\gamma$. The insets present the corresponding optical trajectories in phase space. The black arrow indicates the approximate evolution direction of trajectory. (c) The approximate starting time $\tau$ of chaotic motion versus driving strength ${\mathrm{\Omega }}_d/\gamma$ in the active-passive system. The parameters are the same as Fig. 2 except for (a),(b) $\kappa =-0.8\gamma$.