In this thesis, we have explored the commonalities and connections between different classes of quantum systems that do not thermalize. Specifically, we have (1) shown that localized systems possess conservation laws like integrable systems, which can be constructed in a systematic way and used to detect localization-delocalization transitions , (2) studied the phenomenon of many-body localization in a model with a single particle mobility edge, (3) shown that interesting finite-size scaling emerges, with universal exponents, when athermal quantum systems are forced to thermalize through the application of perturbations and (4) shown that these scaling laws also arise when a perturbation causes a crossover between quantum systems described by different random matrix ensembles. We conclude with a brief summary of each chapter. In Chapter 2, we have investigated the effects of finite size on the crossover between quantum integrable systems and non-integrable systems. Using exact diagonalization of finite-sized systems, we have studied this crossover by obtaining the energy level statistics and Drude weight associated with transport. Our results reinforce the idea that for system size L → ∞, non-integrability sets in for an arbitrarily small integrabilitybreaking perturbation. The crossover value of the perturbation scales as a power law ∼ L−3 when the integrable system is gapless and the scaling appears to be robust to microscopic details and the precise form of the perturbation. In Chapter 3, we have studied the crossover among different random matrix ensembles CHAPTER 6. CONCLUSION 127 [Poissonian, Gaussian Orthogonal Ensemble (GOE), Gaussian Unitary Ensemble (GUE) and Gaussian Symplectic Ensemble (GSE)] realized in different microscopic models. We have found that the perturbation causing the crossover among the different ensembles scales to zero with system size as a power law with an exponent that depends on the ensembles between which the crossover takes place. This exponent is independent of microscopic details of the perturbation. We have also found that the crossover from the Poissonian ensemble to the other three is dominated by the Poissonian to GOE crossover which introduces level repulsion while the crossover from GOE to GUE or GOE to GSE associated with symmetry breaking introduces a subdominant contribution. Finally,we have conjectured that the exponent is dependent on whether the system contains interactions among the elementary degrees of freedom or not and is independent of the dimensionality of the system. In Chapter 4, we have outlined a procedure to construct conservation laws for Anderson localized systems. These conservation laws are found as power series in the hopping parameters. We have also obtained the conservation laws for the disorder free Aubry-Andre model, where the states are either localized or extended depending on the strength of a coupling constant. We have formulated a specific procedure for averaging over disorder, in order to examine the convergence of the power series. Using this procedure for the Aubry-Andre model, we show that integrals of motion given by our construction are well-defined in the localized phase but not so in the extended phase. Finally, we also obtain the integrals of motion for a model with interactions to lowest order in the interaction. In Chapter 5, we have studied many body localization and investigated its nature in the presence of a single particle mobility edge. Employing the technique of exact diagonalization for finite-sized systems, we have calculated the level spacing distribution, time evolution of entanglement entropy, optical conductivity and return probability to characterize the nature of localization. The localization that develops in the presence of interactions in these systems appears to be different from regular Many-Body Localization (MBL) in that the growth of entanglement entropy with time is linear (like in CHAPTER 6. CONCLUSION 128 a thermal phase) instead of logarithmic but saturates to a value much smaller than the thermal value (like for MBL). All other diagnostics seem consistent with regular MBL

Orientador: Antonio Vidiella Barranco / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-31T02:56:15Z (GMT). No. of bitstreams: 1 Decordi_GustavoLazero_D.pdf: 2456755 bytes, checksum: 42a17eacb2e1a86e81888e48272ed08e (MD5) Previous issue date: 2016 / Resumo: Estudamos nesta tese, sistemas quânticos compostos sob a influência de ambientes externos. Na primeira parte do trabalho, investigamos um sistema de dois qubits interagentes, estando um deles isolado e o outro acoplado a um banho térmico. Analisamos os efeitos da temperatura do banho sobre a dinâmica do sistema de dois qubits. Com essa finalidade, empregamos dois modelos distintos da interação sistema-ambiente: i) um modelo microscópico, no qual a equação mestra é obtida levando-se em conta o acoplamento entre os qubits na dedução do termo dissipativo, ii) um modelo fenomenológico, no qual o termo dissipativo é simplesmente adicionado ao termo unitário da equação de evolução do operador densidade. Obtemos soluções analíticas para os modelos, o que permitiu estudá-los em um intervalo considerável do acoplamento entre os qubits. Dedicamos a segunda parte do trabalho ao estudo de um sistema quântico em particular acoplado a um pequeno ambiente. Neste contexto, resolvemos exatamente o modelo da interação radiação-matéria conhecido como modelo de Tavis-Cummings a dois átomos. De posse das soluções, obtidas em circunstâncias bastante gerais e até então não encontradas na literatura, investigamos os efeitos oriundos da interação de um pequeno ambiente (átomo em estado de mistura estatística) sobre a dinâmica do subsistema composto pelo outro átomo acoplado ao modo do campo eletromagnético. Nós mostramos que propriedades não-clássicas associadas ao sistema principal podem ser significativamente degradadas pela ação do ambiente quando o átomo 2 está acoplado de maneira resonante ao campo. Encontramos que o comportamento não-clássico do sistema pode ser restaurado a medida que dessintonizamos o campo da frequência de transição do átomo 2, o ambiente / Abstract: We study in this thesis composite quantum systems under the influence of external environments. In the first part of this work, we investigate a two qubit interacting system having one of them isolated and the other coupled to a thermal bath. We analyze the effect of the temperature of the bath on the dynamics of the two qubit system. In order to do that, we consider two different models of system-reservoir interaction: i) a "microscopic" model, in which the master equation is derived taking into account the interaction between the two subsystems (qubits), ii) a "phenomenological" approach, in which the master equation consists of a dissipative term added to the unitary evolution term. We show that in the strong coupling regime between the subsystems (qubits), the expected thermal equilibrium steady state for the two-qubit system naturally arises in the framework of the microscopic model, while in the phenomenological approach it is obtained a steady state density operator which is not correct. Furthermore, the differences are even more profound in the weak coupling regime, when the models give rise to opposite behaviors with regard to the linear entropy of qubit 1. At the context of quantum systems coupled to environments with few degrees of freedom, we solve analytically the matter-radiation interaction model known as two atom Tavis-Cummings Model. With the solutions at hand, achieved in general circumstances until the present not found in literature, in which the constituent atoms may be coupled with different strengths to the field and also have different frequency detunings, we study the effects that arise from the interaction of a small environment (atom in a statistical mixture state) with the other atom coupled to an oscillator (cavity mode). We show that nonclassical features associated to the main system may be significantly degraded by the action of the small environment, if atom 2 is resonantly coupled to the field. We also demonstrate that the nonclassical behaviour of the system may be restored if we detune the field from the transition frequency of atom 2, the environment / Doutorado / Física / Doutor em Ciências / 899872/2011 / CAPES

Orientadores: Amir Ordacgi Caldeira, Eduardo Peres Novais de Sá / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-09-01T01:58:28Z (GMT). No. of bitstreams: 1 LopezDelgado_DanielAntonio_D.pdf: 831710 bytes, checksum: 17fbe60b2052b9d8534b963d0e85fe0e (MD5) Previous issue date: 2016 / Resumo: A criação de um computador quântico é um projeto que guia, ao mesmo tempo, avanços tecnológicos e um melhor entendimento das propriedades de sistemas quânticos e da Mecânica Quântica em geral. O teorema do limiar é derivado da teoria quântica de correção de erros e garante que, se o ruido estocástico que afeta os componentes de um computador quântico encontra-se abaixo de um valor limite, podemos operar esse computador quântico confiavelmente. Investigamos como esse teorema é modificado quando consideramos uma memória quântica (a qual usa o código de superfície para corrigir erros) acoplada a um ambiente correlacionado. O limiar de erros nesse caso é relacionado à transição de fase ordem-desordem de um sistema de spin equivalente / Abstract: The design of a quantum computer is a project which drives, at the same time, technological advancement and a better understanding of the properties of quantum systems and of Quantum Mechanics in general. The threshold theorem comes from quantum error correction theory and it guarantees that, if stochastic noise affecting the components of a quantum computer is below some threshold value, we can operate this quantum computer reliably. We investigate how this theorem is modified when we consider a quantum memory (which uses the surface code to correct errors) coupled to a correlated environment. The error threshold in this case is related the order-disorder phase transition of an equivalent spin system / Doutorado / Física / Doutor em Ciências

We utilize the multi Davydov-Ansatz, an Ansatz of the bosonic many-body wave function in terms of moving Gaussian basis functions, to illuminate several aspects of open quantum system dynamics and quantum many-body theory. By two artifices alongside the time-dependent variational principle we extract from this Ansatz, commonly considered ill-behaved and not converging, a highly stable and converging method. Its extremely favourable scaling of the numerical effort with the number of degrees of freedom facilitates exploration of the zero and non-zero temperature physics of both system and environment of open quantum systems in the strong coupling regime, even in cases where the system is laser-driven. The discovery that strongly coupling a system to an environment may, apart from the introduction of dissipation and decoherence also serve as a resource for the system has fuelled the research on strongly correlated open quantum systems. Although the advent of ultra powerful data processors enables advanced methods to tackle these systems, their explicit treatment without further assumptions remains an eminently challenging task. With the multi Davydov-Ansatz we numerically exactly calculate the dynamics of various open systems coupled strongly to an environment. In particular, we illuminate diverse aspects of laser-driven molecular dynamics in dissipative environments. Based on a rigorous investigation of the time-dependent variational principle for moving Gaussian basis functions, we systematically develop a linear algebra formulation of the system of equations of motion for the Ansatz parameters. On its basis we precisely isolate the origin of the issues related to the multi Davydov-Ansatz and solve the long-standing convergence problem of the method by a regularization termed apoptosis. We show exemplary for the ohmic and sub-ohmic Spin-Boson model that apoptosis renders the multi Davydov-Ansatz a highly stable method with an outstanding speed of convergence, suited to numerically exactly reproduce the dynamics of the model at surprisingly humble numerical effort even for strong coupling strengths. Furthermore, since they are not suited to efficiently reproduce Fock number states in many-body systems, we shed some light on possible extensions of the Gaussian basis functions in the multi Davydov-Ansatz in terms of displaced number states and in terms of squeezed states. In particular we argue that due to the emergence of an inappropriate number of equations of motion, there is no straightforward generalization of the multi Davydov-Ansatz by displaced number states. For the purpose of further optimization of the multi Davydov-Ansatz, we investigate in detail the impact on the numerical effort of different representations of an open system's environment. In particular, different frequency discretizations for given continuous spectral densities are examined with respect to the speed of convergence of the system dynamics to the continuum limit. We utilize a Windowed Fourier Transform as an a priori measure for the quality of the discretized representation of bath correlations. Furthermore, efficient representations of the environment for shifted initial conditions in general and non-zero temperature in particular are found systematically. As an alternative representation of an environment of mutually uncoupled harmonic oscillators, we investigate an environment represented in terms of a linear chain of effective modes. In this context we detail how to consistently reformulate the effective mode representation in second quantization, removing inadvertent double excitations introduced by the original formulation. We show that the alternative representation is beneficial in cases where the bath spectral density is highly structured, while for the ohmic and sub-ohmic spectral density of the Spin-Boson model it is of no advantage. Once we have identified the numerically most efficient representation of the environment, we apply the multi Davydov-Ansatz in order to illuminate several aspects of open quantum system dynamics whose investigation has previously remained occlusive. In particular, the access to the exact dynamics of the environmental degrees of freedom allows to shed light on the question for the channels through which energy can be interchanged between system and environment in the considered systems. Firstly, in a system-bath setup we survey the vibrational relaxation dynamics of deuterium dimers at a silicon surface. The investigation of the relaxation dynamics requires the quantum mechanical treatment of multiple system levels, which in turn prohibits a treatment of the environmental dynamics on a perturbative level. We demonstrate that the multi Davydov-Ansatz allows for a numerically exact calculation of the system dynamics with multiple system levels and a huge number of surface vibrations explicitly taken into account. Furthermore, due to the structure of the spectral density of the environment, the effective mode representation allows for this system to dramatically reduce the numerical effort. Secondly we shall investigate in detail the relaxation dynamics of an exciton in a one-dimensional molecular crystal. Since the strong coupling regime renders highly complicated the phonon dynamics, apoptosis turns out to be inevitably required in order to reliably converge the system dynamics. We show that the multi Davydov-Ansatz equipped with apoptosis allows for an extremely efficient calculation of the exciton and phonon dynamics, for both large hopping integrals and large molecular crystals. Furthermore we illuminate diverse aspects of laser-driven molecular dynamics in a dissipative environment. By restriction to two electronic energy levels we determine the channels through which system and environment interchange energy in the vicinity of an avoided crossing in a dissipative Landau-Zener model. In particular, we reveal that the final transition probability can be tuned by coupling to the environment for both diagonal and off-diagonal coupling. By appropriately adjusting the initial excitation of the system, the final transition probability is shown to converge to a fixed value for increasing coupling. Finally, we investigate in detail laser-induced population transfer by rapid adiabatic passage in a dissipative environment. By application of the multi Davydov-Ansatz it is shown for zero as well as for non-zero temperature that strongly coupling the system to an environment can serve as a resource for the population inversion. In particular, we shall examine how the coupling to the environment compensates for the decay channels in the system even if the laser pulse is only weakly chirped.:1. Introduction 2. Prerequisites 2.1. Harmonic oscillator basics 2.2. Canonical coherent states of the harmonic oscillator 2.3. Overcompleteness of CS and the Segal-Bargmann transformation 2.4. Density operator representation in terms of CS 2.5. Ideal squeezed states 2.6. Displaced number states 2.7. On the variational principle 3. Real time propagation with CS 3.1. Variational principle with CS 3.1.1. Gauge freedom in the vMCG Ansatz 3.1.2. Equations of motion for the vMCG Ansatz 3.2. Standard form of the linear system 3.3. Regularity of the coefficient matrix 3.3.1. Regularization in the case of vanishing coefficients 3.3.2. Apoptosis of CS 3.4. The route to Semiclassics 3.5. Variational principle with DNS and squeezed states 3.6. The multi Davydov-Ansatz 3.7. The multi Davydov-Ansatz at non-zero temperature 4. Open Quantum Systems 4.1. System-Bath Hamiltonian 4.2. The road to classical dissipation 4.3. The impact of apoptosis and regularization of the 𝜌-matrix 4.3.1. Multi Davydov-Ansatz for the Quantum Rabi model 4.3.2. Multi Davydov-Ansatz and the Spin-Boson model 4.3.2.1. Spin-Boson model in the ohmic regime 4.3.2.2. Spin-Boson model in the sub-ohmic regime 4.4. The Windowed Fourier Transform 4.5. The sub-ohmic case and the problem of oversampling 4.5.1. On the polarized initial condition 4.5.2. On the treatment of non-zero temperature 4.6. The Effective Mode Representation 5. Applications 5.1. Vibrational relaxation dynamics at surfaces 5.2. Relaxation dynamics of the Holstein polaron 5.3. The dissipative Landau Zener Model 5.3.1. Coupling to a single environmental mode 5.3.2. Coupling to multiple environmental modes 5.4. Rapid Adiabatic Passage with a dissipative environment 6. Summary And Outlook List of abbreviations Appendix A. Closure relation of displaced number states B. Hamilton equations: classical vs. CCS for a Morse oscillator C. Equations of motion for the multi Davydov-Ansatz C.1. D2-Ansatz C.2. D1-Ansatz D. Details of implementation E. Calculation of the BCF F. Calculation of the polarized initial condition for 𝑇 = 0 Bibliography List of publications

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第18774号 / 理博第4032号 / 新制||理||1581(附属図書館) / 31725 / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 川上 則雄, 教授 佐々 真一, 教授 高橋 義朗 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2¿,Z(p)) in ¿-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x¿p¿X¿P where X, P ¿ Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained. / Ministry of Higher Education, Sultanate of Oman

No description available.

Thermalization, the evolution of an interacting many-body system towards a thermal Gibbs ensemble after initialization in an arbitrary non-equilibrium state, is currently a phenomenon of great interest, both in theory and experiment. As the time evolution of a quantum system is unitary, the proposed mechanism of thermalization in quantum many-body systems corresponds to the so-called eigenstate thermalization hypothesis (ETH) and the typicality of eigenstates. Although this formally solves the contradiction of thermalizing but unitary dynamics in a closed quantum many-body system, it does neither make any statement on the dynamical process of thermalization itself nor in which way the coupling of the system to an environment can hinder or modify the relaxation dynamics. In this thesis, we address both the question whether or not a quantum system driven away from equilibrium is able to relax to a thermal state, which fulfills detailed balance, and if one can identify universal behavior in the non-equilibrium relaxation dynamics. As a first realization of driven quantum systems out of equilibrium, we investigate a system of Ising spins, interacting with the quantized radiation field in an optical cavity. For multiple cavity modes, this system forms a highly entangled and frustrated state with infinite correlation times, known as a quantum spin glass. In the presence of drive and dissipation, introduced by coupling the intra-cavity radiation field to the photon vacuum outside the cavity via lossy mirrors, the quantum glass state is modified in a universal manner. For frequencies below the photon loss rate, the dissipation takes over and the system shows the universal behavior of a dissipative spin glass, with a characteristic spectral density $\\mathcal{A}(\\omega)\\sim\\sqrt{\\omega}$. On the other hand, for frequencies above the loss rate, the system retains the universal behavior of a zero temperature, quantum spin glass. Remarkably, at the glass transition, the two subsystems of spins and photons thermalize to a joint effective temperature, even in the presence of photon loss. This thermalization is a consequence of the strong spin-photon interactions, which favor detailed balance in the system and detain photons from escaping the cavity. In the thermalized system, the features of the spin glass are mirrored onto the photon degrees of freedom, leading to an emergent photon glass phase. Exploiting the inherent photon loss of the cavity, we make predictions of possible measurements on the escaping photons, which contain detailed information of the state inside the cavity and allow for a precise, non-destructive measurement of the glass state. As a further set of non-equilibrium systems, we consider one-dimensional quantum fluids driven out of equilibrium, whose universal low energy theory is formed by the so-called Luttinger Liquid description, which, due to its large degree of universality, is of intense theoretical and experimental interest. A set of recent experiments in research groups in Vienna, Innsbruck and Munich have probed the non-equilibrium time-evolution of one-dimensional quantum fluids for different experimental realizations and are pushing into a time regime, where thermalization is expected. From a theoretical point of view, one-dimensional quantum fluids are particular interesting, as Luttinger Liquids are integrable and therefore, due to an infinite number of constants of motion, do not thermalize. The leading order correction to the quadratic theory is irrelevant in the sense of the renormalization group and does therefore not modify static correlation functions, however, it breaks integrability and will therefore, even if irrelevant, induce a completely different non-equilibrium dynamics as the quadratic Luttinger theory alone. In this thesis, we derive for the first time a kinetic equation for interacting Luttinger Liquids, which describes the time evolution of the excitation densities for arbitrary initial states. The resonant character of the interaction makes a straightforward derivation of the kinetic equation, using Fermi\'s golden rule, impossible and we have to develop non-perturbative techniques in the Keldysh framework. We derive a closed expression for the time evolution of the excitation densities in terms of self-energies and vertex corrections. Close to equilibrium, the kinetic equation describes the exponential decay of excitations, with a decay rate $\\sigma^R=\\mbox\\Sigma^R$, determined by the self-energy at equilibrium. However, for long times $\\tau$, it also reveals the presence of dynamical slow modes, which are the consequence of exactly energy conserving dynamics and lead to an algebraic decay $\\sim\\tau^$ with $\\eta_D=0.58$. The presence of these dynamical slow modes is not contained in the equilibrium Matsubara formalism, while they emerge naturally in the non-equilibrium formalism developed in this thesis. In order to initialize a one-dimensional quantum fluid out of equilibrium, we consider an interaction quench in a model of interacting, dispersive fermions in Chap.~\\ref. In this scenario, the fermionic interaction is suddenly changed at time $t=0$, such that for $t>0$ the system is not in an eigenstate and therefore undergoes a non-trivial time evolution. For the quadratic theory, the stationary state in the limit $t\\rightarrow\\infty$ is a non-thermal, or prethermal, state, described by a generalized Gibbs ensemble (GGE). The GGE takes into account for the conservation of all integrals of motion, formed by the eigenmodes of the Hamiltonian. On the other hand, in the presence of non-linearities, the final state for $t\\rightarrow\\infty$ is a thermal state with a finite temperature $T>0$. . The spatio-temporal, dynamical thermalization process can be decomposed into three regimes: A prequench regime on the largest distances, which is determined by the initial state, a prethermal plateau for intermediate distances, which is determined by the metastable fixed point of the quadratic theory and a thermal region on the shortest distances. The latter spreads sub-ballistically $\\sim t^$ in space with $0<\\alpha<1$ depending on the quench. Until complete thermalization (i.e. for times $t<\\infty$), the thermal region contains more energy than the prethermal and prequench region, which is expressed in a larger temperature $T_{t}>T_$, decreasing towards its final value $T_$. As the system has achieved local detailed balance in the thermalized region, energy transport to the non-thermal region can only be performed by the macroscopic dynamical slow modes and the decay of the temperature $T_{t}-T_\\sim t^$ again witnesses the presence of these slow modes. The very slow spreading of thermalization is consistent with recent experiments performed in Vienna, which observe a metastable, prethermal state after a quench and only observe the onset of thermalization on much larger time scales. As an immediate indication of thermalization, we determine the time evolution of the fermionic momentum distribution after a quench from non-interacting to interacting fermions. For this quench scenario, the step in the Fermi distribution at the Fermi momentum $k\\sub$ decays to zero algebraically in the absence of a non-linearity but as a stretched exponential (the exponent being proportional to the non-linearity) in the presence of a finite non-linearity. This can serve as a proof for the presence or absence of the non-linearity even on time-scales for which thermalization can not yet be observed. Finally, we consider a bosonic quantum fluid, which is driven away from equilibrium by permanent heating. The origin of the heating is atomic spontaneous emission of laser photons, which are used to create a coherent lattice potential in optical lattice experiments. This process preserves the system\'s $U(1)$-invariance, i.e. conserves the global particle number, and the corresponding long-wavelength description is a heated, interacting Luttinger Liquid, for which phonon modes are continuously populated with a momentum dependent rate $\\partial_tn_q\\sim\\gamma |q|$. In the dynamics, we identify a quasi-thermal regime for large momenta, featuring an increasing time-dependent effective temperature. In this regime, due to fast phonon-phonon scattering, detailed balance has been achieved and is expressed by a time-local, increasing temperature. The thermal region emerges locally and spreads in space sub-ballistically according to $x_t\\sim t^{4/5}$. For larger distances, the system is described by an non-equilibrium phonon distribution $n_q\\sim |q|$, which leads to a new, non-equilibrium behavior of large distance observables. For instance, the phonon decay rate scales universally as $\\gamma_q\\sim |q|^{5/3}$, with a new non-equilibrium exponent $\\eta=5/3$, which differs from equilibrium. This new, universal behavior is guaranteed by the $U(1)$ invariant dynamics of the system and is insensitive to further subleading perturbations. The non-equilibrium long-distance behavior can be determined experimentally by measuring the static and dynamic structure factor, both of which clearly indicate the exponents for phonon decay, $\\eta=5/3$ and for the spreading of thermalization $\\eta_T=4/5$. Remarkably, even in the presence of this strong external drive, the interactions and their aim to achieve detailed balance are strong enough to establish a locally emerging and spatially spreading thermal region. The physical setups in this thesis do not only reveal interesting and new dynamical features in the out-of-equilibrium time evolution of interacting systems, but they also strongly underline the high degree of universality of thermalization for the classes of models studied here. May it be a system of coupled spins and photons, where the photons are pulled away from a thermal state by Markovian photon decay caused by a leaky cavity, a one-dimensional fermionic quantum fluid, which has been initialized in an out-of-equilibrium state by a quantum quench or a one-dimensional bosonic quantum fluid, which is driven away from equilibrium by continuous, external heating, all of these systems at the end establish a local thermal equilibrium, which spreads in space and leads to global thermalization for $t\\rightarrow\\infty$. This underpins the importance of thermalizing collisions and endorses the standard approach of equilibrium statistical mechanics, describing a physical system in its steady state by a thermal Gibbs ensemble.

We investigate the simulations of the the Schrödinger equation using the onedimensional quantum lattice Boltzmann (QLB) scheme and the irregular behaviour of solution. We isolate error due to approximation of the Schrödinger solution with the non-relativistic limit of the Dirac equation and numerical error in solving the Dirac equation. Detailed analysis of the original scheme showed it to be first order accurate. By discretizing the Dirac equation consistently on both sides we derive a second order accurate QLB scheme with the same evolution algorithm as the original and requiring only a one-time unitary transformation of the initial conditions and final output. We show that initializing the scheme in a way that is consistent with the non-relativistic limit supresses the oscillations around the Schrödinger solution. However, we find the QLB scheme better suited to simulation of relativistic quantum systems governed by the Dirac equation and apply it to the Klein paradox. We reproduce the quantum tunnelling results of previous research and show second order convergence to the theoretical wave packet transmission probability. After identifying and correcting the error in the multidimensional extension of the original QLB scheme that produced asymmetric solutions, we expand our second order QLB scheme to multiple dimensions. Next we use the QLB scheme to simulate Klein tunnelling of massless charge carriers in graphene, compare with theoretical solutions and study the dependence of charge transmission on the incidence angle, wave packet and potential barrier shape. To do this we derive a representation of the Dirac-like equation governing charge carriers in graphene for the one-dimensional QLB scheme, and derive a two-dimensional second order graphene QLB scheme for more accurate simulation of wave packets. We demonstrate charge confinement in a graphene device using a configuration of multiple smooth potential barriers, thereby achieving a high ratio of on/off current with potential application in graphene field effect transistors for logic devices. To allow simulation in magnetic or pseudo-magnetic fields created by deformation of graphene, we expand the scheme to include vector potentials. In addition, we derive QLB schemes for bilayer graphene and the non-linear Dirac equation governing Bose-Einstein condensates in hexagonal optical lattices.

No description available.