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.
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.
We describe how to characterize dynamical phase transitions in open quantum systems from a purely dynamical perspective, namely, through the statistical behavior of quantum jump trajectories. This approach goes beyond considering only properties of the steady state. While in small quantum systems dynamical transitions can only occur trivially at limiting values of the controlling parameters, in many-body systems they arise as collective phenomena and within this perspective they are reminiscent of thermodynamic phase transitions. We illustrate this in open models of increasing complexity: a three-level system, the micromaser, and a dissipative version of the quantum Ising model. In these examples dynamical transitions are accompanied by clear changes in static behavior. This is however not always the case, and, in general, dynamical phases need to be uncovered by observables which are strictly dynamical, e.g., dynamical counting fields. We demonstrate this via the example of a class of models of dissipative quantum glasses, whose dynamics can vary widely despite having identical (and trivial) stationary states.
The theory of quantum jump trajectories provides a new framework for understanding dynamical phase transitions in open systems. A candidate for such transitions is the atom maser, which for certain parameters exhibits strong intermittency in the atom detection counts and has a bistable stationary state. Although previous numerical results suggested that the “free energy” may not be a smooth function, we show that the atom detection counts satisfy a large deviations principle and, therefore, we deal with a phase crossover rather than a genuine phase transition. We argue, however, that the latter occurs in the limit of an infinite pumping rate. As a corollary, we obtain the central limit theorem for the counting process. The proof relies on the analysis of a certain deformed generator whose spectral bound is the limiting cumulant generating function. The latter is shown to be smooth so that a large deviations principle holds by the Gärtner–Ellis theorem. One of the main ingredients is the Krein–Rutman theory, which extends the Perron–Frobenius theorem to a general class of positive compact semigroups.
In this paper, we consider the statistics of repeated measurements on the output of a quantum Markov chain. We establish a large deviations result analogous to Sanov’s theorem for the multi-site empirical measure associated to finite sequences of consecutive outcomes of a classical stochastic process. Our result relies on the construction of an extended quantum transition operator (which keeps track of previous outcomes) in terms of which we compute moment generating functions, and whose spectral radius is related to the large deviations rate function. As a corollary to this, we obtain a central limit theorem for the empirical measure. Such higher level statistics may be used to uncover critical behaviour such as dynamical phase transitions, which are not captured by lower level statistics such as the sample mean. As a step in this direction, we give an example of a finite system whose level-1 (empirical mean) rate function is independent of a model parameter while the level-2 (empirical measure) rate is not.
The work presented in this thesis is divided into two main parts: strong zero modes in quantum spin chains with open boundaries and large deviations in classical spin systems. The results concerning the strong zero modes are presented in chapters 2 and 3, while chapter 4 contains the results about large deviations in systems of many, but independent, classical degrees of freedom. These two distinct classes of problems in spin chains are both about aspects of slow relaxation to equilibrium in many-body systems. Nowadays slow dynamics and non-ergodicity in many-body quantum systems are receiving much attention due to their notable technological potential for storing and processing, for very long times, the quantum information encoded in boundary degrees of freedom. In the first part of the work presented in this thesis we mainly focus on quantum spin chains with open boundaries, which can display coherence times for edge spins, that diverge with the system size, as a consequence of the emergence, in the thermodynamic limit, of conserved local quantities, called strong zero modes. First, we discuss the fate of these coherence times when dynamics becomes dissipative. Understanding the behavior of these long correlation times in open quantum systems, undergoing Markovian (memoryless) dynamics, would allow for the possibility of exploiting the protected information encoded in the boundary spins for applications in quantum devices in realistic non-equilibrium settings. Second, we study one possible simple generalization of the transverse field Ising model to a class of ladder spin systems with plaquette interactions in the presence of a transverse field. Also for these systems one can explicitly construct the strong zero mode operators and understand the behavior of the consequential long coherence times for the edge degrees of freedom under isolated (unitary) dynamics. This is possible because the models we consider, while appearing complex superficially, can be brought to simple forms by exploiting an extensive number of conserved quantities they possess. The results presented in chapters 2 and 3 are closely related, even if they focus on different models and investigate different aspects of their dynamics. Nevertheless, the underlying concept they are based on is the same one, that is, an eigenstate phase transition for the strong zero modes that has interesting consequences for the full energy spectrum. In the second part of the thesis we focus on a different type of phase transitions. Here we address the issue of dynamical large deviation phase transitions from a disordered phase to an ordered one, in classical systems which consist in a collection of independent Ising spins evolving stochastically. Surprisingly, by reweighing trajectories accordingly to a certain type of time-integrated observable, even a system of independently evolving spins can display some critical behavior, at the fluctuation level, in trajectory space in the large size and long time limit. By making use of the tilted generator techniques we study the statistics of two particular time-integrated observables. Their associated tilted generators have an exact Kramers-Wannier duality and they look like, up to a sign and an additive constant, some specific well known quantum Hamiltonians. In this sense, the dynamical large deviation phase transitions we observe in these classical stochastic systems translate in ground state quantum phase transitions taking place at the self dual point of the tilted generators we study.
We analyze dynamical large deviations of quantum trajectories in Markovian open quantum systems in their full generality. We derive a quantum level-2.5 large deviation principle for these systems, which describes the joint fluctuations of time-averaged quantum jump rates and of the time-averaged quantum state for long times. Like its level-2.5 counterpart for classical continuous-time Markov chains (which it contains as a special case), this description is both explicit and complete, as the statistics of arbitrary time-extensive dynamical observables can be obtained by contraction from the explicit level-2.5 rate functional we derive. Our approach uses an unraveled representation of the quantum dynamics which allows these statistics to be obtained by analyzing a classical stochastic process in the space of pure states. For quantum reset processes we show that the unraveled dynamics is semi-Markovian and derive bounds on the asymptotic variance of the number of quantum jumps which generalize classical thermodynamic uncertainty relations. We finish by discussing how our level-2.5 approach can be used to study large deviations of nonlinear functions of the state, such as measures of entanglement.
Controlling dynamical fluctuations in open quantum systems is essential both for our comprehension of quantum nonequilibrium behavior and for its possible application in near-term quantum technologies. However, understanding these fluctuations is extremely challenging due, to a large extent, to a lack of efficient important sampling methods for quantum systems. Here, we devise a unified framework-based on population-dynamics methods-for the evaluation of the full probability distribution of generic time-integrated observables in Markovian quantum jump processes. These include quantities carrying information about genuine quantum features, such as quantum superposition or entanglement, not accessible with existing numerical techniques. The algorithm we propose provides dynamical free-energy and entropy functionals which, akin to their equilibrium counterpart, permit one to unveil intriguing phase-transition behavior in quantum trajectories. We discuss some applications and further disclose coexistence and hysteresis, between a highly entangled phase and a low entangled one, in large fluctuations of a strongly interacting few-body system.