Statistical reduction for strongly driven simple quantum systems

We derive reduced equations of motion for simple quantum systems which are strongly driven by an external field and are modulated stochastically by a coupling to a bath. In the derivation we make use of the cumulant-expansion method of Kubo using two different time-ordering prescriptions. We demonstrate how the choice of the ordering prescription is related to the statistical properties of the bath, once the cumulant expansion is truncated. Our equations of motion are valid for arbitrary time scale for the motions of the bath relative to those of the system, and they change smoothly from the static to the Markov (motional narrowing) limit. As examples, we consider the problems of a randomly modulated and driven harmonic oscillator and a modulated and damped two-level system. In the Markovian limit both ordering prescriptions yield Bloch-type equations of motion; in general, however, the driving and modulation interfere and the different statistical properties of the bath, as determined by the two truncated ordering prescriptions, lead to different results.