First-principles electronic structure theory explains properties of atoms, molecules and solids from underlying physical principles without input from empirical parameters. Time-dependent density functional theory (TDDFT) has emerged as arguably the most widely used first-principles method for describing the time-dependent quantum mechanics of many-electron systems. In this thesis, we will show how the fundamental principles of TDDFT can be extended and applied in two novel directions: The theory of open quantum systems (OQS) and quantum computation (QC). In the first part of this thesis, we prove theorems that establish the foundations of TDDFT for open quantum systems (OQS-TDDFT). OQS-TDDFT allows for a first principles description of non-equilibrium systems, in which the electronic degrees of freedom undergo relaxation and decoherence due to coupling with a thermal environment, such as a vibrational or photon bath. We then discuss properties of functionals in OQS-TDDFT and investigate how these differ from functionals in conventional TDDFT using an exactly solvable model system. Next, we formulate OQS-TDDFT in the linear-response regime, which gives access to environmentally broadened excitation spectra. Lastly, we present a hybrid approach in which TDDFT can be used to construct master equations from first-principles for describing energy transfer in condensed phase systems. In the second part of this thesis, we prove that the theorems of TDDFT can be extended to a class of qubit Hamiltonians that are universal for quantum computation. TDDFT applied to universal Hamiltonians implies that single-qubit expectation values can be used as the basic variables in quantum computation and information theory, rather than wavefunctions. This offers the possibility of simplifying computations by using the principles of TDDFT similar to how it is applied in electronic structure theory. Lastly, we discuss a related result; the computational complexity of TDDFT. / Physics

Open Effective Field Theories are a class of Effective Field Theories (EFTs) built using ideas from open quantum systems in which a measured sector (the system) interacts with an unmeasured sector (the environment). It is argued that Open EFTs are useful tools for any situation in which a quantum system couples to a gravitational background with an event horizon. The main reason for this is that for any EFT of gravity one generically expects perturbation theory to breakdown at late times (when interactions with the background persist indefinitely). It is shown that the tools of Open EFTs allow one to resum late-time perturbative breakdowns in order to make reliable late time predictions (without resorting to solving the dynamics exactly). To build evidence of their usefulness to these types of gravitational problems, Open EFT approximation methods are applied to two toy models relating to black hole physics. / Thesis / Doctor of Philosophy (PhD) / Open Effective Field Theories are a class of quantum theories in which a measured sector (the system) is used to make physical predictions with, while interacting with an unmeasured sector (the environment). In this thesis arguments are made that Open EFTs are useful for studying gravitating quantum systems, especially when there is an event horizon present (for example in gravitational fields like that of a black hole). Open EFTs are applied to simple toy problems in such settings to illustrate their usefulness.

Στην παρούσα έρευνα εξετάζουμε την αλληλεπίδραση ενός ανοικτού κβα- ντικού συστήματος ενός ή δύο αρμονικών ταλαντωτών με το περιβάλλον εκκι- νώντας από το μοντέλο κβαντικής κίνησης Brown. Υπολογίζουμε τις λύσεις της ομογενούς εξίσωσης κίνησης καθώς και τους πίνακες απωλειών και θορύβου, από τους οποίους κατασκευάζεται ο διαδότης του συστήματος. Στη βάση αυτών μας των υπολογισμών ελέγχουμε την αξιοπιστία της εξίσωσης Master δεύτερης τάξης, της οποίας τα αποτελέσματα τα βρίσκουμε αναξιόπιστα. Τα αποτελέσματα αυτά επιτρέπουν α). την ολοκλήρωση του θεωρητικού μοντέλου για την επικοι- νωνία δύο απομακρυσμένων κβαντικών συστημάτων και β).την εφαρμογή σε ζητήματα κβαντικής πληροφορίας. / In this research we examine the interaction of an open quantum system one or two harmonic oscillators with the environment starting from the quantum Brownian motion model. We compute the solutions of the homogeneous equation of motion and the dissipation and noise kernel, of which is constructed the propagator of the system. Based on these calculations we check the reliability of the Master equation of second order, whose the results are unreliable. These results allow a). the completion of the theoretical model for communication between two remote quantum systems and b). the application in issues quantum information.

This thesis is concerned with modelling the dynamics of open quantum systems in several different contexts. Of principal interest is the manner in which the environment can modify, or even dominate, a system’s quantum behaviour in order to facilitate the transport of energetic excitations. In the first research chapter, a time-local, non-Markovian quantum master equation is derived in a variationally defined reference frame, for networks of two-level systems coupled to bosonic environments. The predictions of this master equation are then compared with those derived using both weak-coupling and polaron approximations. The variational master equation is found to agree with these standard approaches in their regimes of validity, whilst interpolating between them in intermediate parameter regimes. The second research chapter focusses on the dynamics of a superconducting double quantum dot embedded in a resonant circuit. The device is considered in a regime where the ground state consists of a coherent spatial superposition of a single Cooper pair, which can be excited by a variety of interactions with the environment. The relevant transition rates are calculated and the processes responsible are identified. A numerical simulation of the system is then used to explain experimental data, and show that for certain parameters a significant fraction of excitations occur via absorption of photons from the environment. The final chapter considers a model for an olfactory receptor, in which odorant molecules are recognised by their vibrational modes. Electron transfer occurs in the receptor, dependent on the presence of a vibrational mode of the right frequency. A quantum master equation for the system is derived, and the resulting dynamics is compared to earlier semi-classical treatments. The behaviour of the receptor is found to be sensitive not only to the frequency of the vibrational mode, but also to the character of the surrounding environment. Increased dissipation on the odorant mode, as well as the presence of higher frequencies in the environment is found to improve the frequency resolution of the receptor.

This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. Our primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. In Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. We continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. The last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.

There are several reasons why exotic and artificial atoms attract the interest of different scientific communities.In exotic atoms, matter and antimatter can coexist for surprisingly long times. Thus, they present a unique natural laboratory for high precision antimatter studies. In artificial atoms, electrons can be confined in an externally controlled way. This aspect is crucial, as it opens new possibilities for high precision measurements and also makes artificial atoms promising potential candidates for qubits, i.e. the essential bricks for quantum computation.The first part of the thesis presents theoretical studies of resonant states in antiprotonic atoms and spherical two-electron quantum dots, where well established techniques, frequently used for conventional atomic systems, can be applied after moderate modifications. In the framework of Markovian master equations, it is then demonstrated that systems containing resonant states can be approached as open systems in which the resonance width determines the environmental coupling. The second part of the thesis focuses on possible quantum computational aspects of two kinds of artificial atoms, quantum dots and Penning traps. Environmentally induced decoherence, the main obstacle for a practical realization of a quantum computer based on these devices, is studied within a simple phenomenological model. As a result, the dependence of the decoherence timescales on the temperature of the heat bath and environmental scattering rates is obtained.

We investigate the some of the many subtleties in taking a microscopic approach to modelling the decoherence of an Open Quantum System. We use the RF-SQUID, which will be referred to as a simply a SQUID throughout this paper, as a non-linear example and consider different levels of approximation, with varied coupling, to show the potential consequences that may arise when characterising devices such as superconducting qubits in this manner. We first consider a SQUID inductively coupled to an Ohmic bath and derive a Lindblad master equation, to first and second order in the Baker-Campbell-Hausdorff expansion of the correlation-time-dependent flux operator. We then consider a SQUID both inductively and capacitively coupled to an Ohmic bath and derive a Lindblad master equation to better understand the effect of parasitic capacitance whilst shedding more light on the additions, cancellations and renormalisations that are attributed to a microscopic approach.

A quantum system which evolves in time while interacting with an external environ- ment is said to be an open quantum system (OQS), and the influence of the environment on the unperturbed unitary evolution of the system generally leads to non-unitary dynamics. This kind of open system dynamical evolution has been typically modeled by a Standard Prescription (SP) which assumes that the state of the OQS is initially uncorrelated with the environment state. It is here shown that when a minimal set of physically motivated assumptions are adopted, not only does there exist constraints on the reduced dynamics of an OQS such that this SP does not always accurately describe the possible initial cor- relations existing between the OQS and environment, but such initial correlations, and even entanglement, can be witnessed when observing a particular class of reduced state transformations termed purity extractions are observed. Furthermore, as part of a more fundamental investigation to better understand the minimal set of assumptions required to formulate well defined reduced dynamical quantum maps, it is demonstrated that there exists a one-to-one correspondence between the set of initial reduced states and the set of admissible initial system-environment composite states when G-consistency is enforced. Given the discussions surrounding the requirement of complete positivity and the reliance on the SP, the results presented here may well be found valuable for determining the ba- sic properties of reduced dynamical maps, and when restrictions on the OQS dynamics naturally emerge.

Semigroups of completely positive maps arise naturally both in noncommutative stochastic processes and in the dynamics of open quantum systems. Since its inception in the 1970's, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. In quantum dynamics, this amounts to embedding a given open system inside some closed system, while in noncommutative probability, it corresponds to the construction of a Markov process from its transition probabilities. In addition to the existence of dilations, one is interested in what properties of the original semigroup (unitality, various kinds of continuity) are preserved. Several authors have proved the existence of dilations, but in general, the dilation achieved has been non-unital; that is, the unit of the original algebra is embedded as a proper projection in the dilation algebra. A unique approach due to Jean-Luc Sauvageot overcomes this problem, but leaves unclear the continuity of the dilation semigroup. The major purpose of this thesis, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations. This existence is proved in Theorem 6.4.9, the central result of the thesis. The dilation depends on a modification of free probability theory, and in particular on a combinatorial property akin to free independence. This property is implicit in some Sauvageot's original calculations, but a secondary goal of this thesis is to present it as its own object of study, which we do in chapter 3.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18776号 / 理博第4034号 / 新制||理||1581(附属図書館) / 31727 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)准教授 武末 真二, 教授 佐々 真一, 教授 早川 尚男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM