Quantum Filtering Equation for System Driven by Field in a Mixture of Vacuum and Coherent State

We consider the problem of optimal estimation of a state of a quantum system interacting with the Bose field being in a mixture of the vacuum and coherent state. The system and electromagnetic field are described by making use of quantum stochastic unitary evolution. We derive stochastic master equation using Gardiner and Collet's input-output theory and the concept of cascade systems. To find the stochastic evolution of the system conditioned on the result of the quadrature measurement of the output field, we extend the compound system by an ancilla driven by the vacuum and generating the desired non-classical state of light.