In this thesis we apply, and build upon, a comprehensive theory of metastability in open quantum systems. We gain a full understanding of the long time states and the dynamics between them in a 1D open quantum Ising model, and a glassy, dissipative quantum East model: in order to do this, we develop algorithmic approaches to conducting such analyses in generic models exhibiting a metastability which can be considered classical. In addition, we provide an extensive discussion of the use of symmetries in the numerical study of open quantum systems, containing results of substantial use throughout this work. We finish with a general set of results about the spectra of operators describing the evolution of classical or quantum models undergoing stochastic resetting, allowing a precise analysis of the effect of resets on metastability in the 1D open quantum Ising model.
In this thesis we apply, and build upon, a comprehensive theory of metastability in open quantum systems. We gain a full understanding of the long time states and the dynamics between them in a 1D open quantum Ising model, and a glassy, dissipative quantum East model: in order to do this, we develop algorithmic approaches to conducting such analyses in generic models exhibiting a metastability which can be considered classical. In addition, we provide an extensive discussion of the use of symmetries in the numerical study of open quantum systems, containing results of substantial use throughout this work. We finish with a general set of results about the spectra of operators describing the evolution of classical or quantum models undergoing stochastic resetting, allowing a precise analysis of the effect of resets on metastability in the 1D open quantum Ising model.
We present a general theory of classical metastability in open quantum systems. Metastability is a consequence of a large separation in timescales in the dynamics, leading to the existence of a regime when states of the system appear stationary, before eventual relaxation toward a true stationary state at much larger times. In this work, we focus on the emergence of classical metastability, i.e., when metastable states of an open quantum system with separation of timescales can be approximated as probabilistic mixtures of a finite number of states. We find that a number of classical features follow from this approximation, for the manifold of metastable states, long-time dynamics between them, and symmetries of the dynamics. Namely, those states are approximately disjoint and thus play the role of metastable phases, the relaxation toward the stationary state is approximated by a classical stochastic dynamics between them, and weak symmetries correspond to their permutations. Importantly, the classical dynamics is observed not only on average but also at the level of individual quantum trajectories: We show that time coarse-grained continuous measurement records can be viewed as noisy classical trajectories, while their statistics can be approximated by that of the classical dynamics. Among others, this explains how first-order dynamical phase transitions arise from metastability. Finally, to verify the presence of classical metastability in a given open quantum system, we develop an efficient numerical approach that delivers the set of metastable phases together with the effective classical dynamics. Since the proximity to a first-order dissipative phase transition manifests as metastability, the theory and tools introduced in this work can be used to investigate such transitions---which occur in the large size limit---through the metastable behavior of many-body systems of moderate sizes accessible to numerics.
We apply a recently developed theory for metastability in open quantum systems to a one-dimensional dissipative quantum Ising model. Earlier results suggest this model features either a nonequilibrium phase transition or a smooth but sharp crossover, where the stationary state changes from paramagnetic to ferromagnetic, accompanied by strongly intermittent emission dynamics characteristic of first-order coexistence between dynamical phases. We show that for a range of parameters close to this transition or crossover point the dynamics of the finite system displays pronounced metastability, i.e., the system relaxes first to long-lived metastable states before eventual relaxation to the true stationary state. From the spectral properties of the quantum master operator we characterize the low-dimensional manifold of metastable states, which are shown to be probability mixtures of two, paramagnetic and ferromagnetic, metastable phases. We also show that for long times the dynamics can be approximated by a classical stochastic dynamics between the metastable phases that is directly related to the intermittent dynamics observed in quantum trajectories and thus the dynamical phases.
We analyze the dynamics of metastable Markovian open quantum systems by unraveling their average dynamics into stochastic trajectories. We use quantum reset processes as examples to illustrate metastable phenomenology, including a simple three-state model whose metastability is of classical type, and a two-qubit model that features a metastable decoherence-free subspace. In the three-state model, the trajectories exhibit classical metastable phenomenology: fast relaxation into distinct phases and slow transitions between them. This extends the existing correspondence between classical and quantum metastability. It enables the computation of committors for the quantum phases, and the mechanisms of rare transitions between them. For the two-qubit model, the decoherence-free subspace appears in the unraveled trajectories as a slow manifold on which the quantum state has a continuous slow evolution. This provides a classical (nonmetastable) analog of this quantum effect. We discuss the general implications of these results, and we highlight the useful role of quantum reset processes for analysis of quantum trajectories in metastable systems. Published by the American Physical Society 2024
Abstract We consider a closed quantum system subject to a stochastic resetting process. The generic expression for the resulting density operator is formulated for arbitrary resetting dynamics, fully characterised by the distribution of times between consecutive reset events. We analyse the behaviour of the state in the long-time regime, as well as the evolution of relevant quantities in the study of quantum coherence and closed- versus open-system dynamics. Our general results are complemented with examples involving particular resetting distributions, and special attention is paid to the two-level (qubit) system, in which we elucidate the effects of the renewal process on the speed of evolution toward an orthogonal state, and gain insight into the resetting applied to open systems.
Metastability in open system dynamics describes the phenomena of initial relaxation to long-lived metastable states before decaying to the asymptotic stable states. It has been found in continuous-time stochastic dynamics of both classical and quantum systems. However, many cases of open quantum system dynamics are intrinsically discrete, and the evolution within each discrete time interval is described by an arbitrary quantum channel, which often cannot be generated by continuous-time master equations. Here we develop a general theory of metastability in discrete-time open quantum dynamics, described by sequential repetitive quantum channels. We apply the general metastability theory to a typical class of quantum channels on a target system, induced by an ancilla qubit with a pure-dephasing coupling to the target system and under Ramsey sequences. Interesting metastable behaviors are predicted and numerically demonstrated by decomposing the average dynamics of sequential quantum channels into stochastic trajectories. We also present examples of applications in quantum state and dynamics engineering of a target quantum system with an ancilla qubit.
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.