Quantum Langevin equations for optomechanical systems

Abstract

We provide a fully quantum description of a mechanical oscillator in the presence of thermal environmental noise by means of a quantum Langevin formulation based on quantum stochastic calculus. The system dynamics is determined by symmetry requirements and equipartition at equilibrium, while the environment is described by quantum Bose fields in a suitable non-Fock representation which allows for the introduction of temperature. A generic spectral density of the environment can be described by introducing its state through a suitable P-representation. Including interaction of the mechanical oscillator with a cavity mode via radiation pressure we obtain a description of a simple optomechanical system in which, besides the Langevin equations for the system, one has the exact input-output relations for the quantum noises. The whole theory is valid at arbitrarily low temperature. This allows the exact calculation of the stationary value of the mean energy of the mechanical oscillator, as well as both homodyne and heterodyne spectra. The present analysis allows in particular to study possible cooling scenarios and to obtain the exact connection between observed spectra and fluctuation spectra of the position of the mechanical oscillator.

Unitary unraveling for the dissipative continuous spontaneous localization model: Application to optomechanical experiments

“…This is in agreement with the results in Ref. [25], where it is also discussed how the presence of this noise, which in this context appears naturally, is required for having a well defined momentum operator.…”

“…For a more exhaustive description of stochastic Schrödinger equations under the action of a quantum noise, we refer to Ref. [25].…”

“…Specifically, we will construct a unitary unravelling of the dCSL master equation, following the approach described in Ref. [25], which has the advantage of greatly simplifying all the necessary calculations, while providing a rigorous approach to the quantification of the effects of the collapse mechanism. Such a unitary unravelling is built around a bosonic quantum noise instead of a standard classical noise, as custom to CSL.…”

No Thermalization without Correlations

“…(1) apply to many quantum phenomena including the translational motion of an excited atom in vacuum [1], Brownian motion in a dilute background gas [2], light-driven processes in semiconductor, nanoplasmonic and optomechanical systems [3][4][5], superconducting currents [6], quantum ratchets [7], energy transport in lowdimensional systems [8], dynamics of chemical reactions [9], two-dimensional vibrational spectroscopy and NMR signals [10,11] as well as more exotic entirely quantum dissipative effects [12,13]. The termF fr (p) in Eqs.…”

Stochastic wave-function unravelling of the generalized Lindblad equation

“…In this scenario, the master equation has non-Markovian form and to find the corresponding SSE, describing both the evolution of a quantum system and continuous monitoring, is a difficult task. However, a few strategies have been presented recently [9][10][11][12]. One such strategy is to start from the Markovian SSE and try to find the generalization for the non-Markovian case using colored noises and random co- * Electronic address: semin@ssau.ru † Electronic address: yusov@list.ru ‡ Electronic address: petruccione@ukzn.ac.za efficients [7,11,13].…”