A stochastic representation of the dynamics of open quantum systems, suitable for non-perturbative system-reservoir interaction, non-Markovian effects and arbitrarily driven systems is presented. It includes the case of driving on timescales comparable to or shorter than the reservoir correlation time, a notoriously difficult but relevant case in the context of quantum information processing and quantum thermodynamics. A previous stochastic approach is re-formulated for the case of finite reservoir correlation and response times, resulting in a numerical simulation strategy exceeding previous ones by orders of magnitude in efficiency. Although the approach is based on a memory formalism, the dynamical equations propagated in the simulations are time-local. This leaves a wide range of choices in selecting the system to be studied and the numerical method used for propagation. For a series of tests, the dynamics of the spin-boson system is computed in various settings including strong external driving and Landau-Zener transitions.
A stochastic representation of the dynamics of open quantum systems, suitable for non-perturbative system-reservoir interaction, non-Markovian effects and arbitrarily driven systems is presented. It includes the case of driving on timescales comparable to or shorter than the reservoir correlation time, a notoriously difficult but relevant case in the context of quantum information processing and quantum thermodynamics. A previous stochastic approach is re-formulated for the case of finite reservoir correlation and response times, resulting in a numerical simulation strategy exceeding previous ones by orders of magnitude in efficiency. Although the approach is based on a memory formalism, the dynamical equations propagated in the simulations are time-local. This leaves a wide range of choices in selecting the system to be studied and the numerical method used for propagation. For a series of tests, the dynamics of the spin-boson system is computed in various settings including strong external driving and Landau-Zener transitions.
This thesis investigates two thematic lines of research, both underpinned by non-Markovian system-reservoir interactions in quantum optics. The overarching focus is on modelling the open system dynamics in a non-perturbative fashion, broadly on - though not restricted to - instances when the environment is structured. A theory is developed by means of enlarging the open system over environmental degrees of freedom to include memory effects in its dynamics. This is achieved using an established technique that involves mapping a bosonic environment onto a 1D chain of harmonic oscillators. Within this setting, we apply a Heisenberg equation-of-motion approach to derive an exact set coupled differential equations for the open system and a single auxiliary oscillator of the chain. The combined equations are shown to have their interpretation rooted in a quantum Markov stochastic process. Including the auxiliary chain oscillator as part of the original system then enables us to obtain an exact master equation for the enlarged system, avoiding any need for the Born-Markov approximations. Our method is valid for a dissipative two-state system, with cases of multiple excitations and added driving discussed. Separately, we apply the framework of quantum Darwinism to an atom-cavity system, and, subsequently, to a more general multiple-environment model. In both cases, the time-dependent spread of correlations between the open system and fractions of the environment is analysed during the course of the decoherence process. The degree to which information is redundant across different fractions is checked to infer the emergence of classicality. In the second case, we go further and present a decomposition of information in terms of its quantum and classical correlations. A quantitative measure of redundancy is also studied with regard to its ability to witness non-Markovian behaviour. Besides fundamental interest, our results have application to quantum information processing and quantum technologies, keeping in mind the potential beneficial use of non-Markovian effects in reservoir engineering.
The definition of memory in operational approaches to quantum non-Markovianity depends on the statistical properties of different sets of outcomes related to successive measurement processes performed over the system of interest. Using projector techniques we develop a perturbation theory that enable to expressing both joint probabilities and outcome correlations in terms of the unperturbed system density-matrix propagator. This object defines the open system dynamics in the absence of measurement processes. Successive series terms, which are scaled by the system-environment interaction strength, consist in a convolution structure involving system propagators weighted by higher-order bath correlations. The formalism is corroborated by studying different dynamics that admit an exact description. Using the perturbative approach, unusual memory effects induced by the interplay between the system-environment interaction and measurement processes are found in finite temperature reservoirs.
All systems are open to an essentially uncontrollable environment that acts as a source of decoherence and dissipation. In some cases the environment's only effect is to add a weak relaxation mechanism and thus can be ignored for short timescales. In others, however, the presence of the environment can fundamentally alter the behavior of the system. Such is the case in mesoscopic superconductors where the environment can stabilize superconductivity and in spin-boson systems where the environment induces a localization transition. Likewise, in technological applications we are often interested in systems operating far from equilibrium. Here the environment might act as a particle reservoir or strong driving force. In all these examples, we need accurate methods to describe the influence of the environment on the system and to solve for the resulting dynamics or equilibrium states. In this thesis, we develop computational and conceptual approaches to efficiently simulate quantum systems in contact with an environment. Our starting point is the use of numerical renormalization techniques. Thus, we restrict our attention to one-dimensional lattices or small quantum systems coupled to an environment. We have developed several complementary algorithms: a superoperator renormalization algorithm for simulating real-time Markovian dynamics and for calculating states in thermal equilibrium; a blocking algorithm for simulating integro-differential equations with long-time memory; and a tensor network algorithm for branched lattices, which can be used to simulate strongly dissipative systems. Further, we provide support for an idea that to generically and accurately simulate the real-time dynamics of strongly dissipative systems, one has to include all or part of the environment within the simulation. In addition, we discuss applications and open questions.
The exact dynamics of a system coupled to an environment can be described by an integro-differential stochastic equation for the reduced density. The influence of the environment is incorporated through a mean field which is both stochastic and nonlocal in time and where the standard two-time correlation functions of the environment appear naturally. Since no approximation is made, the presented theory incorporates exactly dissipative and non-Markovian effects. Applications to the spin-boson model coupled to a heat bath with various coupling regimes and temperature show that the presented stochastic theory can be a valuable tool to simulate exactly the dynamics of open quantum systems. Links with the stochastic Schrödinger equation method and possible extensions to "imaginary time" propagation are discussed.
Theoretical investigations of open quantum systems have uncovered a wide variety of unique dynamical phenomena. For instance, current research in quantum information technology aims to control and utilize time-delayed system-reservoir interactions and phonon-induced dynamical dissipative processes for the design of novel chip-integrated quantum optical devices. This thesis develops and advances multiple theoretical approaches for the accurate and efficient description of open quantum systems, where a high degree of information compression is essential to extract crucial information from the open system degrees of freedom which scale exponentially with the system size. The presented methods range from correlation expansions to second-order perturbative master equations, real-time path integral formulations, numerical realizations of high-dimensional tensor networks and, based on the most recent theoretical developments, machine learning implementations in the form of artificial neural networks. We explore all of these techniques with respect to their compression efficiency, performance and representational limits by applying them to a variety of physical setups and scenarios. Firstly, emerging non-Markovian memory effects in open quantum systems are investigated, e.g., resulting from interactions with a structured phonon environment or time-discrete coherent quantum feedback, leading to information exchange and a time-delayed dynamical interplay between system and reservoir. Examined systems range from solid state p-wave topological superconductors coupled to a structured phonon environment, where a memory-induced stabilization and recovery of topological properties is observed, to semiconductor quantum dot nanostructures exhibiting complete population inversion and unidirectional quantum transport as a result of dissipation-induced non-Markovian system-reservoir interactions, and quantum emitters simultaneously subjected to coherent quantum feedback and a decoherence-inducing structured phonon reservoir, leading to the formation of dynamical dissipative structures and population trapping. Secondly, novel machine learning techniques based on artificial neural networks are investigated, enabling simulations of symmetric open quantum spin-1/2 systems with Markovian dynamics. Combining their potential for parallelization and efficient Hilbert space truncation, they facilitate unmatched degrees of information compression and numerical performance for large systems. In this thesis, we expand the representational limits of the restricted Boltzmann machine neural network architecture by developing novel hybrid sampling strategies for a highly customizable compression of configuration space, rendering accurate and efficient simulations of asymmetric systems feasible. Finally, the applicability of the neural network is extended beyond pure spin-1/2 systems by introducing a neural bit encoding scheme for Fock number states, facilitating high performing calculations of large hybrid open quantum systems with bosonic degrees of freedom.
Tracing out the environmental degrees of freedom is a necessary procedure when simulating open quantum systems. While being an essential step in deriving a tractable master equation it represents a loss of information. In situations where there is strong interplay between the system and environmental degrees of freedom this loss makes understanding the dynamics challenging. These dynamics, when viewed in isolation, have no time-local description: they are non-Markovian and memory effects induce complex features that are difficult to interpret. To address this problem, we here show how to use system correlations, calculated by any method, to infer any correlation function of a Gaussian environment, so long as the coupling between system and environment is linear. This not only allows reconstruction of the full dynamics of both system and environment, but also opens avenues into studying the effect of a system on its environment. In order to obtain accurate bath dynamics, we exploit a numerically exact approach to simulating the system dynamics, which is based on the construction and contraction of a tensor network that represents the process tensor of this open quantum system. Using this we are able to find any system correlation function exactly. To demonstrate the applicability of our method we show how heat moves between different modes of a bosonic bath when coupled to a two-level system that is subject to an off-resonant drive.