Fluctuation-induced quenching of chaos in quantum optics

TL;DR

The paper investigates fluctuation-induced quenching of chaos in quantum-optical systems, focusing on an optomechanical cavity with Kerr nonlinearity driven by a laser at frequencies in the $10^5$ to $10^7$ Hz range. It analyzes both semiclassical dynamics via stochastic Langevin equations and full quantum dynamics via the Lindblad master equation, revealing that room-temperature fluctuations with occupancy $n_b$ around $10^{7}$ can suppress chaotic behavior in expectation values, while nonlinearity lowers the noise threshold toward vacuum fluctuations. The full quantum analysis shows that chaotic signatures disappear in the mean values but the quantum state acquires non-Gaussian features with negative Wigner function regions, illustrating a quantum limit to chaos. The results establish a universal mechanism for quantum suppression of chaos and illuminate the quantum-classical crossover in nonlinear systems, with implications for other chaotic quantum platforms and quantum information processing.

Abstract

Recent studies have extensively explored chaotic dynamics in quantum optical systems through the mean-field approximation, which corresponds to an ideal, fluctuation-free scenario. However, the inherent sensitivity of chaos to initial conditions implies that even minute fluctuations can be amplified, thereby questioning the applicability of this approximation. Here, we analyze these chaotic effects using stochastic Langevin equations or the Lindblad master equation. For systems operating at frequencies of $10^5$ to $10^7$ Hz, we demonstrate that room-temperature thermal fluctuations are sufficient to suppress chaos at the level of expectation values, even under weak nonlinearity. Furthermore, nonlinearity induces deviations from Gaussian phase-space distributions of the quantum state, revealing attractor-like features in the Wigner function. With increasing nonlinearity, the noise threshold for chaos suppression decreases, approaching the scale of vacuum fluctuations. These results provide a bidirectional validation of the quantum mechanical suppression of chaos.

Figures (5)

• Figure 1: Schematic of the optomechanical system, consisting of a Fabry-Pérot cavity with one movable end mirror coupled to a mechanical oscillator. The cavity is filled with a Kerr nonlinear medium.
• Figure 2: (a),(c): Time evolution of the intracavity light intensity obtained from the mean-field Eq. \ref{['mean_value']} for the weakly nonlinear ($g/2\pi = 1$ Hz) and strongly nonlinear regimes $g/2\pi = 25$ Hz), respectively. (b),(d) Phase-space projections of the trajectory segments highlighted in orange in (a) and (c). The parameters used for this simulation are $\omega_m/2\pi = 525$ kHz, $Q = \omega_m/\gamma = 10^{7}$, $P = 0.4$ mW, $\chi/2\pi = 1.625 \times 10^{-3}$ Hz, $\kappa/2\pi = 220$ kHz, $\kappa_{\rm in}/2\pi = 100$ kHz, and $\omega_d/2\pi = 10^{14}$ Hz. The corresponding dimensionless parameter are: $\gamma=10^{-7}$, $\chi \simeq 3.095 \times 10^{-9}$, $\kappa \simeq 0.4190$, $E\simeq 26404$ by setting $\omega_m=1$ as the unit.
• Figure 3: (a): Comparison between the mean-field solution and individual stochastic trajectories. The blue curve shows the intracavity light intensity obtained from the mean-field equation \ref{['mean_value']}, while the red and yellow curves depict two trajectories obtained from the stochastic Langevin equation \ref{['eq:cle']} with $n_b = 10^{7}$. The inset is the magnified view of the late-time evolution. (b) and (c): Intracavity light intensity obtained from $250,000$ realizations of Eq. \ref{['eq:cle']} for $n_b = 10^{-1}$ and $n_b = 10^{7}$, respectively. The Fourier transform of the red curves in (b) and (c) is plotted in (e) corresponding to the first half of the evolution ($t=0$ to $0.6\times 10^{-4}$ s) and (f) correspondin to the second half ($0.6\times 10^{-4}$ to $1.2\times 10^{-4}$ s). All other parameters are as in Fig. \ref{['fig:2']}.
• Figure 4: The blue line reproduces the mean-field light intensity from Fig. 3(a), while the red line shows the averaged light intensity from $250,000$ stochastic trajectories with $n_b=10^7$ [cf. Fig. 3(c)]. Eight representative time points, labeled (b)--(i), are selected across the dynamical evolution; for each, the normalized probability distribution of the optical field $a$ is displayed in the corresponding subpanel. All other parameters are as in Fig. \ref{['fig:3']}.
• Figure 5: (a): The expected value of the light intensity, $\langle\psi\vert \hat{a}^\dagger\hat{a}\vert\psi\rangle$, calculated from the quantum trajectories corresponding to three quantum jumps. (b): The expected value of the light intensity, $\text{Tr}( \hat{a}^\dagger\hat{a}\rho)$, calculated from the density matrix obtained from $5000$ trajectories. (c)-(f): Wigner functions of the cavity field state at times $t/10^{-4}=0.15$ s, $t/10^{-4}=0.3$ s, $t/10^{-4}=0.45$ s, and $t/10^{-4}=0.6$ s. Here, while keeping $C_1$ and $C_2$ unchanged, we set $g/2\pi=200$ kHz, i.e., $g/\omega_m= 0.38$. Other parameters are the same as in Fig. \ref{['fig:2']}.