Quantum Markov processes are quantum dynamical systems with a large range of applications in quantum optics and cavity QED. The paradigmatic example is that of an atom maser, in which atoms pass successively through an optical cavity, interact with the cavity field, such that the outgoing atoms carry information about the interaction.
We consider the problem of system identification for quantum Markov chains in the asymptotic set-up where the experimentalist has access to the output of the Markov chain and the number of `atoms' is large. In the special case of a one parameter model, we derive two asymptotic normality results, the first with respect to the quantum state of the output, the second with respect to the statistics of averages of simple measurement performed on the output. In particular we provide simple estimators whose Fisher information can be optimized over different choices of measured observables. These results can be extended to multiple parameter estimation and continuous time dynamics, opening up a new area of research in quantum statistics with direct relevance for quantum engineering.