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.
- Received 4 October 2013
- Publisher error corrected 11 February 2014
DOI:https://doi.org/10.1103/PhysRevX.4.011015
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Published by the American Physical Society
Corrections
11 February 2014
Popular Summary
When a laser drives a mechanical oscillator, quantum effects of this interaction can take on fascinating forms of manifestation. Cooling of micromechanical oscillators to their quantum ground states by a driving laser field, recently demonstrated experimentally, is just such an example. In this case, the frequency of the driving laser is tuned to be below the frequency of the mechanical oscillator. Tuning the former to go above the latter sends the dynamics of the system into a different—the so-called limit-cycle—regime, where the amplitude of the mechanical oscillator settles toward a finite value.
While numerical simulations of the full quantum-mechanical description of the “optomechanics” in this regime have been performed, predicting the generation of “nonclassical” states of the mechanical oscillator (those that do not have any counterpart in the state of classical oscillators), analytical theories that enable systematic, yet technically tractable investigations, and that can create a coherent picture unifying new and already existing physical insights, are still missing. In this paper, we present such a theory, report the confirmation of its validity by numerical results, and predict the occurrence of nonclassical states of mechanical oscillation under parameter conditions where such states were unexpected.
Our analytical model is based on the laser theory of Haake and Lewenstein. It treats optomechanical nonlinearity, a consequence of the nonlinearity inherent in radiation pressure associated with an optical field, in a new and effective way that interpolates between the existing treatments of the nonlinearity, the “dressed phonon” picture and the full photon-phonon coupling picture. Conditions on system parameters for the appearance of nonclassical signatures in the state of the mechanical oscillator can now be, and are, concretely obtained. In particular, we show that, rather surprisingly, it is possible to observe nonclassical negative Wigner density of the mechanical oscillator even outside the single-photon regime.
Our work fills a theoretical gap and should also guide experimental observations of nonclassical states of massive mechanical oscillators.