Relaxation of a driven harmonic oscillator

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Abstract
The quantum theory of a driven damped harmonic oscillator is presented. By using iteration methods the time development of the density operator for the oscillator is obtained and its normally ordered generating functional is written. The field of the oscillator appears to be the superposition of three fields. The first field, which shows the damping of the oscillator, is statistically similar to the one before the interaction; the second one is a coherent field that depends on the driving field; and the third describes a thermal field. In order to evaluate two-time averages by differentiation, a two-time generating functional of the oscillator is obtained. This functional allows two-time averages to be computed from one-time averages before the interaction.