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.