Brownian motion of a quantum oscillator

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Abstract
The theory of Brownian motion of a quantum oscillator is presented. The equation of motion for the density operator of the oscillator is obtained by a model Hamiltonian which characterizes the interaction between this oscillator and a reservoir. By applying iteration methods the equation of motion is solved and the time development of the density operator is obtained. The solution is then used to write the normally ordered generating functional for the field of the oscillator executing Brownian motion. This field appears to be the superposition of a thermal field in equilibrium with the reservoir, a chaotic field, and a field statistically similar to the one before the interaction. Next, it is shown that for large times the oscillator approaches equilibrium; moreover, it is shown that, if the reservoir is at zero temperature and initially the state of the oscillator is a coherent state, then it remains in a coherent state for later time. In order to obtain by differentiation two-time modal densities, a suitable two-time generating functional is introduced. This functional allows two-time averages to be computed from one-time averages before the interaction. Finally, some time-dependent statistical properties of the oscillator are studied.

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