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.
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.
In the first part of this thesis, we investigate the dynamics of open quantum systems coupled in a purely dephasing way to their environment, which is a paradigmatic example of decoherent dynamics. We focus more specifically on the emergence of non-Markovian effects in the dynamics of such open systems. After reviewing different measures of non-Markovianity, we obtain an analytical expression for the reduced density operator of a finite-dimensional open system coupled with its environment through time-dependent purely dephasing interactions. When the system is initially uncorrelated with its environment, we obtain the time-local master equation that describes its dynamics. In this case, we also derive an exact expression for the witness of non-Markovianity based on the volume of accessible states of the open system. These results are then applied to spin systems coupled to either a spin or a bosonic environment. For spin environments, we study non-Markovianity for different ranges of interaction and in different thermodynamical limits. Moreover, the generation of entanglement between the open system and its environment as well as between spins within the open system is investigated. For bosonic environments, we study the effect of the initial state of the environment on non-Markovianity. We also show that time-dependent modulations of the system-environment coupling can dramatically alter the non-Markovian behavior of the open system. Finally, we consider the case of an open quantum system simultaneously coupled to spin and bosonic baths. In that case, we study how the coupling between the different baths and of the presence of initial correlations in the environment affect the non-Markovianity displayed by the open quantum system. The second part of this thesis is a contribution to the investigation of how quantum systems can be used to process information. More precisely, we focus on the implementation of gate-based quantum computing using neutral atoms trapped in optical or magnetic lattices and interacting through dipole-dipole interactions when excited to Rydberg states. In this context, we present protocols implementing two-qubit quantum gates between atoms far apart in the lattice. To this end, we propose to use a chain of non-coding ancilla atoms to transfer the state of the control qubit to an atom near the target qubit. Before discussing entangling gates between distant qubits, we present two protocols that implement state transfer along a chain of two-level atoms relying respectively on single- and two-atom laser pulses combined with dipole blockade. After reviewing protocols implementing CZ and CNOT gates between nearby qubits using dipole-dipole interactions, we show how the state-transfer protocols previously presented can be used to perform entangling gates between distant qubits. The performance of the state-transfer and gate protocols are assessed using the process fidelity. Moreover, we provide a theoretical explanation for the behavior of the process fidelity with respect to dissipation and imperfect blockade.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.
In this thesis, we discuss aspects of the simulation, characterisation, and control of the dynamics of open quantum systems. The latter are systems governed by quantum mechanical laws and interacting with an environment, which is typically much larger than the system itself. According to the standard approach, one is usually able to perform a suitable average over the environmental degrees of freedom, resulting in an effective description of the main system dynamics, which accounts for the effects of the interaction with the surroundings. In standard scenarios, usually studied within the so-called Born-Markov approximation, the system-environment coupling is weak and such that we can perform a neat separation of timescales: the environmental dynamics is assumed to be fast compared to the typical evolution timescale of the system of interest. However, nowadays we are able to inspect physical scenarios where the usual Born-Markov approximation breaks down: the interaction between system and environment can be strong, likely leading to non-negligible memory (non-Markovian) effects. In this work, we discuss some instances in which we cannot work in the standard Born-Markov regime. We introduce and analyse some numerical and analytical techniques to characterise and simulate non-standard scenarios. We show that, even in the Markovian regime, the traditional formulation of thermodynamic irreversibility shows flaws and inconsistencies. We thus use phase-space methods to assess the role of initial correlations shared by the two parties of a bipartite harmonic system in the entropy production rate, including non-Markovian effects. Furthermore, we show that certain interactions enable to redraw the boundaries between the system and the environment in an effective manner, resulting in a new picture where the system degrees of freedom are augmented, while the residual environment is rearranged in such a way that the Born-Markov approximation is recovered. This analytical technique, known as reaction coordinate mapping approach, is employed in this work to show that, upon an accurate choice of the parameter regime, a spin-boson model can serve as a quantum analogue simulator of non-Markovian multiphoton Jaynes-Cummings models. These systematic studies shed light on the thermodynamic characterisation of open quantum systems, as well as on their numerical simulation.
Quantum Markov models are employed ubiqui-tously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approxi-mations for a wide range of quantum optical and meso scopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.
The study of open quantum systems forms one of the main problems of modern physics. This thesis aims at contributing to this field, and focuses on the investigation of non-Markovian dynamics of open quantum systems. Here, as a special case, quantum many-body systems provide for a rich field of research. With the ability to control them being of essential importance for the development of quantum technologies, the characterization of their dynamical properties has also drawn a lot of interest lately. One of the main methodological tasks in the study of open quantum systems is to reduce the infinite degrees of freedom of the total system. In order to achieve this, the thesis makes use of the tensor network method matrix product states and combines it with the picture of the quantum stochastic Schrödinger equation. However, when computing the dynamics of many-body systems of system size N, even the reduced system dynamics scale with 2N, making larger system sizes difficult to access. Therefore, especially when computing non-Markovian time-dynamics where the information backflow from the reservoir has to be taken into account, efficiently constructed algorithms are required. This thesis proposes these type of algorithms and demonstrates their application. They enable an access to larger system sizes, thus making it possible to investigate non-equilibrium steady-states of quantum many-body systems in a non-Markovian interaction for system sizes up to N=30. This thesis consists of three parts. The first part provides an introduction to the theory of open quantum systems and to the methodology for modeling them. The second part examines many-body system-reservoir interaction in the non-Markovian regime by investigating the case of fully coherent self-feedback. It uses the one-dimensional Heisenberg spin-1/2 chain as a paradigmatic and recently intensely studied many-body system. It is demonstrated that this gives rise to new conditions for the feedback phase and to highly non-trivial states within the system as well as to the possibility of partially characterizing the system non-invasively and storing excitation within it. Next, a novel approach to the boundary-driven Heisenberg chain is presented, where the chain is exposed to a coherent driving field, thus enabling a comparison to the well-studied incoherently driven model. The third part demonstrates the use of an external pump to control the feedback phase of a Lambda-type three-level system, where the external pump pulse is shown to be a new control parameter. The thesis concludes with the proposition to enhance the indistinguishability of two photons by feedback control.
The theory of open quantum systems represents one of the major developments of quantum mechanics over the last 50 years. Its central object of study is constituted by quantum systems that cannot be regarded as fully isolated, but instead interact with additional degrees of freedom, giving rise to irreversible phenomena such as dissipation and decoherence. The tools provided by this framework are nowadays of paramount importance across many active areas of research, including quantum computation, thermodynamics, metrology and biology. In this thesis, we investigate three main topics within the field. The first contribution concerns the microscopic derivation of the evolution equations, socalled quantum master equations, for open systems which are also subject to external driving fields. Standard techniques developed for time-independent Hamiltonians encounter several difficulties in the time-dependent case, largely restricting their applicability to specific instances, such as periodic or adiabatic drivings. In contrast, a core result of the derivation presented in the thesis is that, when the environment is sufficiently Markovian (essentially, memoryless), the non-adiabatic contributions to the driving field can be neglected in a controlled fashion, yielding an accurate master equation with time-dependent generator, capable of correctly reproducing the open dynamics even for strong drivings. This result is further supported by benchmarks against numerically exact simulations. The second contribution addresses the long-time behavior of general types of (Markovian) quantum master equations, including those of the form introduced above. While this problem has been extensively analyzed in the time-independent setting, virtually no results are known for time-dependent generators. In particular, we establish the foundations of a general theory of relaxation, that is, convergence towards an asymptotic trajectory in the long-time limit, independently of the initial state. The main results include a theorem guaranteeing sufficient conditions for relaxation and the derivation of a rate function governing the convergence to the asymptotic trajectory. The analysis is further extended to a simple non-Markovian case. Moreover, we treat in detail the specific master equation derived in the first part, connecting the aforementioned relaxation property directly to the underlying microscopic dynamics. The third contribution investigates chirality-induced spin selectivity (CISS) effects from the perspective of open quantum systems. Although the majority of theoretical studies aiming to model CISS are based on a coherent description of the spin dynamics, such effects have been observed also for electrons subject to incoherent transport across molecular networks. In this regard, we provide a rigorous ab initio derivation of the microscopic Hamiltonian describing the electronic transport, incorporating electron spin effects and contributions stemming from the nuclear vibrations. Within this result, we critically revise a master equation approach aiming to explain the origin of CISS effects observed in photosynthetic complexes. Taken together, these contributions offer fresh insights and original results on the interplay between driving fields and dissipation, relaxation phenomena, and electronic spin transport in the presence of incoherent dynamics. More broadly, these results provide new theoretical tools with several potential applications in quantum technologies, from modeling of control pulses in noisy environments to quantum reservoir engineering, and contribute to a deeper understanding of the role of quantum effects in biological systems.