Continuous phase preserving measurement of a quantum N-level system*

We present an analysis of the continuous monitoring of a quantum N-level system using phase-preserving measurements. In recent times, many experiments, like those with superconducting qubits, involve interacting such a system to a resonator and then collecting signals via phase-preserving amplifiers. Using the measured quadratures at every time step, we construct a stochastic master equation, describing the quantum trajectories of the system. We use the dynamical equations that follow from the stochastic master equation, to discuss the realizations of the formalism for a few special quantum systems.