Large deviations and dynamical phase transitions for quantum Markov processes - van Horssen, Merlijn (2014)
Quantum Markov processes are widely used models of the dynamics open quantum systems, a fundamental topic in theoretical and mathematical physics with important applications in experimental realisations of quantum systems such as ultracold atomic gases and new quantum information technologies such as quantum metrology and quantum control. In this thesis we present a mathematical framework which effectively characterises dynamical phase transitions in quantum Markov processes, using the theory of large deviations, by combining insights developed in non-equilibrium dynamics with techniques from quantum information and probability. We provide a natural decomposition for quantum Markov chains into phases, paving the way for the rigorous treatment of critical features of such systems such as phase transitions and phase purification. A full characterisation of dynamical phase transitions beyond properties of the steady state is described in terms of a dynamical perspective through critical behaviour of the quantum jump trajectories. We extend a fundamental result from large deviations for classical Markov chains, the Sanov theorem, to a quantum setting; we prove this Sanov theorem for the output of quantum Markov chains, a result which could be extended to a quantum Donsker-Varadhan theory. We perform an in-depth analysis of the atom maser, an infinite-dimensional quantum Markov process exhibiting various types of critical behaviour: for certain parameters it exhibits strong intermittency in the atom detection counts, and has a bistable stationary state. We show that the atom detection counts satisfy a large deviations principle, and therefore we deal with a phase cross-over rather than a genuine phase transition, although the latter occurs in the limit of infinite pumping rate. As a corollary, we obtain the Central Limit Theorem for the counting process.