Non-adiabatic elimination of auxiliary modes in continuous quantum measurements

When measuring a complex quantum system, we are often interested in only a few degrees of freedom-the plant, while the rest of them are collected as auxiliary modes-the bath. The bath can have finite memory (non-Markovian), and simply ignoring its dynamics, i.e., adiabatically eliminating it, will prevent us from predicting the true quantum behavior of the plant. We generalize the technique introduced by Strunz et. al. [Phys. Rev. Lett 82, 1801 (1999)], and develop a formalism that allows us to eliminate the bath non-adiabatically in continuous quantum measurements, and obtain a non-Markovian stochastic master equation for the plant which we focus on. We apply this formalism to three interesting examples relevant to current experiments.