• Open Access

Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime

Niels Lörch, Jiang Qian, Aashish Clerk, Florian Marquardt, and Klemens Hammerer
Phys. Rev. X 4, 011015 – Published 31 January 2014

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.

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  • Received 4 October 2013
  • Publisher error corrected 11 February 2014

DOI:https://doi.org/10.1103/PhysRevX.4.011015

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Corrections

11 February 2014

Authors & Affiliations

Niels Lörch1,2, Jiang Qian3, Aashish Clerk4, Florian Marquardt5,6, and Klemens Hammerer1,2

  • 1Institut für Gravitationsphysik, Leibniz Universität Hannover and Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Callinstraße 38, 30167 Hannover, Germany
  • 2Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
  • 3Arnold Sommerfeld Center for Theoretical Physics, Center for NanoScience and Department of Physics, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 München, Germany
  • 4Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada
  • 5Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, D-91058 Erlangen, Germany
  • 6Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany

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.

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Vol. 4, Iss. 1 — January - March 2014

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