Continuous measurement of a statistic quantum ensemble

We consider an ensemble of quantum systems whose average evolution is descried by a density matrix solution of a Lindbladian differential equation. We will suppose that the decoherence is only due to a highly unstable excited state. We measure then the spontaneously emitted photons. Whenever we consider resonant laser fields, we can remove the fast dynamics associated to the excited state in order to obtain another differential equation for the slow part. We show that this slow differential equation is still of Lindblad type. The decoherence terms are then of second order and the measurement structure depends explicitly on the resonant laser field. The later can be adjusted to give information on a specific linear combination of the density matrix entries. The case of a 3-level system is treated in details and before the general case. On this 3-level system, we show how a simple PI regulator allows us to robustly control the 2-level slow subsystem.