Stochastic Schrödinger equation for homodyne measurements of strongly correlated systems

We derive a stochastic Schrödinger equation that describes the homodyne measurement record of a strongly interacting atomic system. We derive this equation for a general system, where we use the rotating wave approximation in the linear atom-light interaction part, and the resulting equation is expressed in terms of the atomic operators only. Weak measurements are theoretically described in terms of positive operator-valued measures. Among different weak measurement schemes, several earlier references studied the Gaussian quantum continuous measurement in detail. Here we consider a homodyne measurement setup. We then demonstrate that the derived equation for this setup in the appropriate limit is the same as the one obtained while performing a Gaussian quantum continuous measurement.