Metastability, resets and large deviations in open quantum systems

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

Training neural network ensembles via trajectory sampling

In machine learning, there is renewed interest in neural network ensembles (NNEs), whereby predictions are obtained as an aggregate from a diverse set of smaller models, rather than from a single larger model. Here, we show how to define and train a NNE using techniques from the study of rare trajectories in stochastic systems. We define an NNE in terms of the trajectory of the model parameters under a simple, and discrete in time, diffusive dynamics, and train the NNE by biasing these trajectories towards a small time-integrated loss, as controlled by appropriate counting fields which act as hyperparameters. We demonstrate the viability of this technique on a range of simple supervised learning tasks. We discuss potential advantages of our trajectory sampling approach compared with more conventional gradient based methods.Comment: 12 pages, 5 figures, 1 appendi

Spectral properties of simple classical and quantum reset processes

We study the spectral properties of classical and quantum Markovian processes that are reset at random times to a specific configuration or state with a reset rate that is independent of the current state of the system. We demonstrate that this simple reset dynamics causes a uniform shift in the eigenvalues of the Markov generator, excluding the zero mode corresponding to the stationary state, which has the effect of accelerating or even inducing relaxation to a stationary state. Based on this result, we provide expressions for the stationary state and probability current of the reset process in terms of weighted sums over dynamical modes of the reset-free process. We also discuss the effect of resets on processes that display metastability. We illustrate our results with two classical stochastic processes, the totally asymmetric random walk and the one-dimensional Brownian motion, as well as two quantum models: a particle coherently hopping on a chain and the dissipative transverse field Ising model, known to exhibit metastability.Comment: 11 pages, 2 figure

Entanglement and localization in long-range quadratic Lindbladians

Existence of Anderson localization is considered a manifestation of coherence of classical and quantum waves in disordered systems. Signatures of localization have been observed in condensed matter and cold atomic systems where the coupling to the environment can be significantly suppressed but not eliminated. In this work we explore the phenomena of localization in random Lindbladian dynamics describing open quantum systems. We propose a model of one-dimensional chain of non-interacting, spinless fermions coupled to a local ensemble of baths. The jump operator mediating the interaction with the bath linked to each site has a power-law tail with an exponent $p$. We show that the steady state of the system undergoes a localization entanglement phase transition by tuning $p$ which remains stable in the presence of coherent hopping. Unlike the entanglement transition in the quantum trajectories of open systems, this transition is exhibited by the averaged steady state density matrix of the Lindbladian. The steady state in the localized phase is characterised by a heterogeneity in local population imbalance, while the jump operators exhibit a constant participation ratio of the sites they affect. Our work provides a novel realisation of localization physics in open quantum systems.Comment: 12 pages and 13 figure

Hierarchical classical metastability in an open quantum East model

We study in detail an open quantum generalization of a classical kinetically constrained model - the East model - known to exhibit slow glassy dynamics stemming from a complex hierarchy of metastable states with distinct lifetimes. Using the recently introduced theory of classical metastability for open quantum systems, we show that the driven open quantum East model features a hierarchy of classical metastabilities at low temperature and weak driving field. We find that the effective long-time description of its dynamics not only is classical, but shares many properties with the classical East model, such as obeying an effective detailed balance condition and lacking static interactions between excitations, but with this occurring within a modified set of metastable phases which are coherent, and with an effective temperature that is dependent on the coherent drive

Quantum jump Monte Carlo approach simplified: Abelian symmetries

We consider Markovian dynamics of a finitely dimensional open quantum system featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry on the space of density matrices commutes with the master operator governing the dynamics. We show how to encode the weak symmetry in quantum stochastic dynamics of the system by constructing a weakly symmetric representation of the master operator: a symmetric Hamiltonian, and jump operators connecting only the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this representation simplifies both the construction of the master operator as well as quantum jump Monte Carlo simulations, where, for a symmetric initial state, stochastic trajectories of the system state are supported within a single symmetry eigenspace at a time, which is changed only by the action of an asymmetric jump operator. Our results generalize directly to the case of multiple Abelian weak symmetries

Phase transitions in electron spin resonance under continuous microwave driving

We study an ensemble of strongly coupled electrons under continuous microwave irradiation interacting with a dissipative environment, a problem of relevance to the creation of highly polarized non-equilibrium states in nuclear magnetic resonance. We analyze the stationary states of the dynamics, described within a Lindblad master equation framework, at the mean-field approximation level. This approach allows us to identify steady state phase transitions between phases of high and low polarization controlled by the distribution of disordered electronic interactions. We compare the mean-field predictions to numerically exact simulations of small systems and find good agreement. Our study highlights the possibility of observing collective phenomena, such as metastable states, phase transitions and critical behaviour in appropriately designed paramagnetic systems. These phenomena occur in a low-temperature regime which is not theoretically tractable by conventional methods, e.g., the spin-temperature approach

A reinforcement learning approach to rare trajectory sampling

Very often when studying non-equilibrium systems one is interested in analysing dynamical behaviour that occurs with very low probability, so called rare events. In practice, since rare events are by definition atypical, they are often difficult to access in a statistically significant way. What are required are strategies to "make rare events typical" so that they can be generated on demand. Here we present such a general approach to adaptively construct a dynamics that efficiently samples atypical events. We do so by exploiting the methods of reinforcement learning (RL), which refers to the set of machine learning techniques aimed at finding the optimal behaviour to maximise a reward associated with the dynamics. We consider the general perspective of dynamical trajectory ensembles, whereby rare events are described in terms of ensemble reweighting. By minimising the distance between a reweighted ensemble and that of a suitably parametrised controlled dynamics we arrive at a set of methods similar to those of RL to numerically approximate the optimal dynamics that realises the rare behaviour of interest. As simple illustrations we consider in detail the problem of excursions of a random walker, for the case of rare events with a finite time horizon; and the problem of a studying current statistics of a particle hopping in a ring geometry, for the case of an infinite time horizon. We discuss natural extensions of the ideas presented here, including to continuous-time Markov systems, first passage time problems and non-Markovian dynamics