Large deviations and dynamical phase transitions in open quantum systems: from mathematical theory to physical applications

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We are currently entering a new technological era where quantum mechanics is used not only to predict physical behavior, but increasingly to exploit quantum resources in applications such as quantum communication and computation and quantum metrology. The last couple of decades have witnessed a revolution in the experimental realisation of quantum systems. Ultracold atomic gases are nowadays routinely created and used for the study of complex many body phenomena such as quantum phase transitions shedding light on open problems in condensed-matter physics.Real quantum systems are "open" in the sense that they interact with their environment (be it a thermal bath or other components of a larger quantum system). This leads to an irreversible loss of coherence and to energy dissipation. The simplest analysis applied to this kind of problems is based on the Markov approximation in which the environment possesses no memory and interacts weakly with the system. The Markov formalism has been successfully applied to many physical problems, such as the treatment of continuous-time measurements and the existence of "quantum jumps". This lead to the development of the "quantum trajectories" theory describing the stochastic dynamics of quantum systems, conditioned on the random outcomes of indirect observations. Current advances in quantum engineering drive theoretical and experimental research towards a new synthesis of quantum dynamics and classical control engineering, called quantum control theory. The Markov set-up is perfectly suited for applying feedback control techniques, where a classical or quantum output signal is processed in real time and used to act back on the system according to a control strategy. The quantum trajectories formalism is naturally connected to classical non-equilibrium statistical mechanics. Recently there has been much progress in our understanding of non-equilibrium systems, with many advances originating from the study of complex soft condensed-matter systems such as glasses. The mathematical language of statistical mechanics is large deviations (LD) theory, which was traditionally applied to the study of equilibrium phases and phase changes in many body systems. The LD formalism is now used to investigate the dynamical phases in non-equilibrium systems by treating ensembles of trajectories in the same way that equilibrium statistical mechanics treats ensembles of configurations. Important properties of classical non-equilibrium systems can be uncovered by exploiting this analogy. This new set of ideas and techniques has not yet been fully exploited in the quantum realm, and therefore much less is understood of quantum non-equilibrium dynamics. In this proposal we aim at building the mathematical foundation and identifying fundamental principles that will eventually allow us to construct a framework for a detailed and thorough description of quantum matter out of equilibrium. The central goal of this proposal is to develop the LD theory of open quantum systems and use it to explore the phenomena of metastability and dynamical phase transitions. We aim to bridge the gap in understanding that exists between classical and quantum non-equilibrium systems, by applying and extending the most novel methods developed in non-equilibrium statistical mechanics, the theory of quantum Markov processes and stochastic Schrödinger equations. In parallel to developing the mathematical theory we will perform a detailed analysis of physically relevant models such as driven many-body systems and the micromaser. By combining statistical and probabilistic methods of Markov processes with quantum feedback control theory, we will study the large and moderate deviations behaviour of systems coupled with classical or quantum controllers, and apply the theory to topics such as system identification and quantum metrology.