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. [Physical Review Letters, 109, 253601 (2012)]. Here we derive a Fokker-Planck equation describing mechanical limit cycles in the quantum regime which 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's 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 general approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.
Highlights
Optomechanical systems provide a test bed to study a broad range of paradigmatic quantum optical processes at so far unexplored meso- and macroscopic mass and length scales [1,2,3]
When the driving field is swept from the red to the blue side, the nonlinear dynamics sets in as a parametric amplification process where phonons and photons are created as correlated in pairs [12]
III, we introduce the main idea of Haake and Lewenstein’s laser theory in the context of optomechanics and apply it to derive the effective FokkerPlanck equation (FPE) for the mechanical oscillator
Summary
Optomechanical systems provide a test bed to study a broad range of paradigmatic quantum optical processes at so far unexplored meso- and macroscopic mass and length scales [1,2,3]. A recent numerical study of the full optomechanical master equation in the limit-cycle regime showed that the Wigner function of the mechanical oscillator can become strongly negative [27]: Negativities of the Wigner function occur for driving fields at the blue sidebands and—more pronouncedly— occur for resonant drive. The transition from parametric amplification to optomechanical limit cycles can be understood in analogy to the threshold behavior of a laser (or maser) cavity [30,31,32] where the roles of the laser cavity and the laser medium are played by, respectively, the mechanical oscillator and the optomechanical cavity [33] Along this line, a semiclassical rate-equation model was derived in Refs. The FPE predicted, in particular, a sub-Poissonian, or numbersqueezed, phonon statistics in the limit cycle when the driving field is blue detuned from the cavity resonance by the mechanical oscillation frequency. Readers who are interested only in one particular aspect are encouraged to jump directly to the respective section of interest
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