Diffusive Stochastic Master Equation (SME) with dispersive qubit/cavity coupling

A detailed analysis of the diffusive Stochastic Master Equation (SME) for qubit/cavity systems with dispersive coupling is provided. This analysis incorporates classical input signals and output signals (measurement outcomes through homodyne detection). The dynamics of the qubit/cavity density operator is shown to converge exponentially towards a slow invariant manifold, parameterized via a time-varying deterministic Kraus map by the density operator of a fictitious qubit. This fictitious qubit is governed by a SME incorporating the classical input/output signals. Extension where the qubit is replaced by any qudit dispersively coupled to an arbitrary set of modes with collective input/output classical signals.