Quantum signatures of transitions from stable fixed points to limit cycles in optomechanical systems

Abstract

Optomechanical systems, due to its inherent nonlinear optomechanical coupling, owns rich nonlinear dynamics of different types of motion. The interesting question is that whether there exist some common quantum features to infer the nonlinear dynamical transitions from one type to another. In this paper, we have studied the quantum signatures of transitions from stable fixed points to limit cycles in an optomechanical phonon laser system. Our calculations show that the entanglement of stable fixed points in the long run does not change with time, however, it will oscillate periodically with time at the mechanical vibration frequency for the limit cycles. Most strikingly, the entanglement quite close to the boundary line keeps as a constant, and it is very robust to the thermal phonon noise, as strong indications of this particular classical transitions.

Nearly invariant boundary entanglement in optomechanical systems*

“…Since both the mechanism to generate entanglement and the parameter dependence of boundary entanglement are quite similar as in our threemode system, we believe that the two-mode system has captured the main features in the three-mode system. The studies in this paper can not only complement our former study on boundary entanglement, [11,12] but also extend to a more general conclusion that parametric down conversion plus weak mechanical dissipation can lead to nearly invariant boundary entanglement.…”

“…b is also utilized to generate entanglement in the three-mode system. [11,25,26] It involves one mechanical mode (i.e., b) and two optical supermodes (i.e., ĉ1 and ĉ2 ) separated by frequency difference 2J, where J denotes the tunneling rate of two local optical cavity modes constituting the supermodes. When the resonant condition, i.e., the frequency difference 2J equals the mechanical frequency ω m , is satisfied, driving the higher frequency supermode (i.e., ĉ1 ) with variable laser amplitude and frequency yields the phenomenon of invariant boundary entanglement.…”

“…[10] Based on a three-mode optomechanical system, we even recently discover that the steady-state entanglement along the boundary line remains unchanged, and it is very robust to thermal phonon noise, thus providing a strong quantum signature of transitions from stable fixed points to limit cycles. [11,12] However, due to the calculation complexity, we can only resort to numerical integrations and the parameter space involved is very limited. Therefore, several questions arise: Does this phenomenon of invari-ant boundary entanglement depend on the specific model?…”

“…This happens to be the region discussed in our previous three-mode system. [11,12] Most state-of-the-art experiments can reach the resolved sideband regime and the quality factor can range from 10 4 to 10 9 . [1] The nonlinear phenomena can be enhanced in this region with smaller mechanical damping rate γ m = ω m /Q, since it needs less driving power to get instability.…”

Quantum Zeno effect in self-sustaining systems: suppressing phase diffusion via repeated measurements

“…The exact solution of the dynamics of these quantum nonlinear systems is hard and in the literature various approximate treatments have been proposed: some meanfield treatments linearize the dynamics of the quantum fluctuations around the solution of the mean field classical nonlinear equations, so that the steady state of the system is a Gaussian state centered around the classical limit cycle [42][43][44][52][53][54][55]. However, such kind of state cannot be maintained after a transient, and inevitably exhibits non-Gaussianity, as was revealed in previous works by means of simulations [31,32,37,38,56].…”