Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

Abstract

Optomechanical systems can exhibit self-sustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown recently in a numerical study by Qian et al. [Phys. Rev. Lett. 109, 253601 (2012)]. Here, we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime that correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation and is based on techniques borrowed from the laser theory due to Haake and Lewenstein. We compare our analytical model with numerical solutions of the master equation based on Monte Carlo simulations and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e., outside the single-photon strong-coupling regime, for strong cavity drive and rather large limit-cycle amplitudes. The approach taken here provides a natural starting point for further studies of quantum effects in optomechanics

Noise-induced transitions in optomechanical synchronization

“…Optomechanical systems exhibit a Hopf bifurcation and can be optically driven into mechanical limit-cycle oscillations [34][35][36][37]. These self-oscillations have also been analyzed theoretically in the quantum regime [38][39][40]. The theoretical description of the synchronization dynamics in optomechanical arrays has initially focussed on the classical regime [31,41].…”

Antibunching in an optomechanical oscillator

“…The intrinsic optomechanical interaction is, however, nonlinear, which comes to the fore in the single-photon strong-coupling regime. The nonlinear nature of the optomechanical interaction gives rise to a variety of features previously explored in nonlinear quantum optics [18], including photon blockade effects [19], generation of non-Gaussian states [20], and nonclassical antibunched mechanical resonators [21][22][23][24].…”

Phase Synchronization of Two Anharmonic Nanomechanical Oscillators

“…Here we show that oscillators based on nanoelectromechanical systems (NEMS) can readily enable the resolution of such details, while providing many unique advantages for experimental studies of nonlinear dynamics [6][7][8]. In addition, nanomechanical systems might prove useful for exploring quantum synchronization [9,10].Nanomechanical oscillators also have been exploited for a variety of applications [11][12][13]. In particular, nanoscale mechanics exhibits enhanced nonlinearity [14,15] and tunability [16,17], which has been used to suppress feedback noise [18,19] and create new types of electromechanical oscillators [20][21][22].…”

“…Here we show that oscillators based on nanoelectromechanical systems (NEMS) can readily enable the resolution of such details, while providing many unique advantages for experimental studies of nonlinear dynamics [6][7][8]. In addition, nanomechanical systems might prove useful for exploring quantum synchronization [9,10].…”