This thesis deals with the determination of the eigenstates of the Hamiltonian and quantum dynamics in many-body interacting systems. As an example, the Bose- Hubbard model is used. Methods of generating the basis of system state space and calculation of Hamiltonian matrix in selected representations are studied. After the complete specification of the system, the energy spectrum and the eigenstates are determined. By using the appropriate techniques, the mapping of the vector index and quantum numbers is established for efficient access and manipulation for physical quantities evaluations and simulating dynamics in quantum interacting systems. Furthermore, the expected distribution of particles and momenta are examined, studying the behaviour of characteristic quantities such as energy gaps, the excitation spectra of such a system as a function of the Hamiltonian parameters is determined for a couple of cases. An overview of the formalism of second quantization is given in the paper as well as the derivation of the Bose-Hubbard model in this formalism. Also, the limit of infinitely strong repulsive interactions between the bosons is investigated when the system shows certain similarities with the system of non-interacting fermions onto which it is mapped. In the end, the time evolution of the system of bosons in an optical lattice is investigated in the presence of an external potential in the case of a constant and a time-varying potential.
This thesis deals with the determination of the eigenstates of the Hamiltonian and quantum dynamics in many-body interacting systems. As an example, the Bose- Hubbard model is used. Methods of generating the basis of system state space and calculation of Hamiltonian matrix in selected representations are studied. After the complete specification of the system, the energy spectrum and the eigenstates are determined. By using the appropriate techniques, the mapping of the vector index and quantum numbers is established for efficient access and manipulation for physical quantities evaluations and simulating dynamics in quantum interacting systems. Furthermore, the expected distribution of particles and momenta are examined, studying the behaviour of characteristic quantities such as energy gaps, the excitation spectra of such a system as a function of the Hamiltonian parameters is determined for a couple of cases. An overview of the formalism of second quantization is given in the paper as well as the derivation of the Bose-Hubbard model in this formalism. Also, the limit of infinitely strong repulsive interactions between the bosons is investigated when the system shows certain similarities with the system of non-interacting fermions onto which it is mapped. In the end, the time evolution of the system of bosons in an optical lattice is investigated in the presence of an external potential in the case of a constant and a time-varying potential.
In this thesis, strongly correlated quantum many-body systems in equilibrium and in out-of-equilibrium situations are investigated. This is done by applying and developing well established numerical methods. The focus of the thesis lies in the development and application of the density matrix renormalization group method (DMRG) to quantum many-body systems out of equilbrium. In this thesis, in addition to the DMRG, methods for the exact diagonalization of the Hamiltonian of the system, like the Lanczos- or the Jacobi-Davidson method, are treated. An extension of the Lanczos method makes it possible to treat the time evolution of strongly correlated quantum systems with an accuracy comparable to machine precision. This method is the basis for a possible extension of the DMRG for the treatment of systems out of equilibrium, the so-called adaptive time-dependent DMRG ("adaptive t-DMRG"). A second variant of the adaptive t-DMRG uses the Suzuki-Trotter decomposition of the time-evolution operator. An error analysis demonstrates that both methods, for a suitable choice of control parameters, have errors < 1% at the end of the time evolution in complicated quantities like, e.g., the momentum distribution function. We apply these numerical methods to investigate the quantum critical behavior of a variant of the Hubbard model and to treat two non-equilibrium situations of current interest. Extensive use of the DMRG makes it possible to clarify the quantum-critical behavior of the ionic Hubbard model; in particular, the numerical results demonstrate quite clearly that the correct scenario has two critical points; at the first critical point, only the charge degrees of freedom are critical, while at the second one only the spin degrees of freedom are critical. Quite surprisingly, the Mott-insulator phase in the strong-coupling regime shows a divergent electrical susceptibility. Next, the dynamics of a system of so-called soft-core bosons is treated. The particles initially are trapped in a deep box-potential and the dynamics is investigated after releasing them from this trapping potential. Similar to exact results obtained for so-called hard-core bosons, (quasi-)coherent matter waves emerge, demonstrating the possibility of realizing an atom laser in experiments on optical lattices. We find that the wave vector of the emerging matter wave can be tuned by changing the strength of the interaction between the particles. The second non-equilibrium situation is a so-called "quantum-quench" of a system of strongly correlated fermions, i.e., the evolution of the system after suddenly changing an intrinsic parameter like, e.g., the interaction strength between the particles, is investigated. In this case, we are mainly interested in the long-time behavior; in particular, the question arises whether, relying on Boltzmann's ergodic hypothesis, relaxation to a thermal state is obtained. We find that, in general, non-thermal final states are reached, which can be described by generalized Gibbs-Boltzmann ensembles.
We study dynamically coupled one-dimensional Bose-Hubbard models and solve for the wave functions and energies of two-particle eigenstates. Even though the wave functions do not directly follow the form of a Bethe ansatz, we describe an intuitive construction to express them as combinations of Choy-Haldane states for models with intra- and interspecies interactions. We find that the two-particle spectrum of the system with generic interactions comprises in general four different continua and three doublon dispersions. The existence of doublons depends on the coupling strength $\mathrm{\ensuremath{\Omega}}$ between two species of bosons, and their energies vary with $\mathrm{\ensuremath{\Omega}}$ and interaction strengths. We give details on one specific limit, i.e., with infinite interaction, and derive the spectrum for all types of two-particle states and their spatial and entanglement properties. We demonstrate the difference in time evolution under different coupling strengths and examine the relation between the long-time behavior of the system and the doublon dispersion. These dynamics can in principle be observed in cold atoms and might also be simulated by digital quantum computers.
In this thesis, we study interacting bosons on lattices consisting of a few sites. The main focus is on their response to external driving fields, their transport and decay properties. We address fundamental questions of quantum classical correspondence and investigate the implications of interactions in the transport properties of (ultra-)cold atom devices. Quantum mechanically, we describe bosons on lattices using a parametric Bose-Hubbard Hamiltonian while the corresponding classical limit is described by the discrete nonlinear Schrödinger equation.In the first part of this thesis we focus on the chaotic regime of the Bose-Hubbard Hamiltonian. We investigate the parametric change of eigenstates as the coupling strength between neighboring sites is varied, using perturbative and semiclassical methods. We then turn to the quantum evolution of driven interacting bosonic systems and study the resulting quantum dynamics and the irreversibility (fidelity) of the quantum motion.In the second part we initiate the study of quantum pumping/stirring in Bose-Einstein condensates with the aim to identify optimal pumping cycles and propose such a device in order to probe the interatomic interactions. Finally, in the last part we focus on the interplay of intrinsic dynamics with coupling to the continuum and investigate the structure of the resonance widths of a Bose-Hubbard dimer which is coupled to the continuum at one of the sites using an effective non-Hermitian Hamiltonian formalism.
Experimental advances have made ultracold atoms in optical lattices a favourable setting to study out-of-equilibrium phenomena and attracted considerable attention in recent years.These systems are highly versatile in that experimental parameters can be tuned over a wide range of values in real time.This facilitates the study of quantum quenches, in which parameters in the corresponding Hamiltonian are varied in time faster than the system can respond adiabatically.Such protocols open the door to a rich range of many-body physics and have been studied intensely both theoretically and experimentally.The Bose-Hubbard model (BHM) has been shown to describe interacting ultracold bosons in an optical lattice, allowing the opportunity for experiments to probe the out-of-equilibrium dynamics of the model.The BHM is a particularly convenient context for studying quantum quenches as it displays a quantum phase transition between superfluid and Mott-insulator phases.In this thesis, we develop a strong-coupling approach that allows the study of correlations in space and time in both the superfluid and Mott-insulating phases of the BHM.Specifically, we obtain a two-particle irreducible (2PI) effective action within the contourtime formalism that allows for a description of both equilibrium and out-of-equilibrium phenomena.We derive equations of motion for both the superfluid order parameter and the single-particle many-body Green's functions.First, we assess the accuracy of our formalism by studying the equilibrium solution for the homogeneous BHM and comparing our results to existing strong-coupling methods as well as to exact methods where possible.We then consider homogeneous systems that are initially thermalized in the Mott phase, and which are then subjected to quenches.We solve numerically the equations of motion for this scenario and calculate the single-particle density matrix.We demonstrate a Lieb-Robinson-like maximal propagation velocity for the spreading of single-particle correlations in one, two, and three dimensions.We study the dependence of the maximal propagation velocity on the quench protocol, chemical potential, temperature, and dimensionality.We compare our results to exact methods, existing strong-coupling approaches, and experiments where possible.Lastly, we extend our strong-coupling approach to the disordered BHM and derive equations of motion for the disorder-averaged single-particle Green's function.We discuss how these equations of motion can be used to study the phase stability of many-body localization in the disordered BHM for dimensions higher than one.
In the last years the impressive progress on the experimental side led to a variety of new experiments allowing to address systems out of equilibrium.In this way the behavior of such systems far from equilibrium is no longer a purely theoretical issue but indeed observable.New experimental techniques, like particles trapped in optical lattices, render a realization of quantum systems with nearly arbitrary system parameters possible and provide a possibility to study their time evolution.Systems out of equilibrium are characterized by the fact, that these systems are in highly excited states giving rise to totally new fascinating properties.In the present thesis one-and two-dimensional fermionic Hubbard models out of equilibrium are discussed.The system is taken out of equilibrium by a so-called interaction quench.At the beginning the system is prepared in the groundstate of the non-interacting Hamiltonian.At a time t the interaction between the fermions is suddenly turned on so that the time evolution is governed by the whole, interacting Hamiltonian.Hence the system is prepared in the groundstate of one Hamiltonian but evolves according to a di_erent Hamiltonian.Consequently the system ends up in a highly excited state.To describe such a system a method based on an expansion of the Heisenberg equations of motion to highest order possible is developed in this thesis.This method provides an exact description of the time evolution on short and intermediate time scales after the quench.As the method reveal exact results and does not rely on any perturbative assumption, a study of arbitrarily large interaction strengths is possible.Besides, the method is one of the few methods capable of two-dimensional systems.In the following the method used in this thesis is explained and advantages and disadvantages of the approach are thematized.For this purpose the results of the developed iterated equation of motion approach are compared to results obtained in calculation up to second order in the interaction.Then the dynamics of the onedimensional model is discussed with a focus on the relation of the results derived by the iterated equations of motion approach to results obtained by bosonization theory, the behavior for strong interactions and the dynamical transition from the weak to the strong quench regime.Furthermore, the Hubbard model is studied on a two-dimensional square lattice.This model is fundamentally di_erent from the one-dimensional model: In contrast to the one-dimensional model the two-dimensional model is not integrable allowing a true relaxation of the system.For this system a calculation up to second order in the interaction is performed and compared to the results of the iterated equation of motion approach.Besides, the time evolution of the momentum distribution and the influence of doping on the dynamics is studied.Moreover, a first estimate for relaxation times is provided without relying on the assumption of a mixed state.
In this thesis, we study the physical properties of several ultracold few-boson systems depending on the interactions between their constituents. Nowadays, experimentally, it is possible to have great control with high precision over the geometry and the interactions between the particles, making them an excellent setup to test directly the principles of quantum mechanics. A very interesting point is to study the evolution of their properties with the number of particles. The theoretical study of these systems pretends to microscopically understand the current experimental results and give support to new experimental developments.\n\n\t\t\t\t The method that will be used is the exact diagonalization of the Hamiltonian of the system. As we will see, in spite of the attempts to improve it, the method is limited by the fact that, in practice, it is only useful to study few-particle systems. The method has several advantages. First of all, one has access to both the ground and the excited states. In second place, the method is variational and converges to the exact solution as long as the Hilbert space in which we diagonalize is enlarged.\n\t\t\t\t Moreover, since we have access to the states of the system, it is possible to calculate any observable quantity of interest.\n\n\t\t\t\t First, we will study a system of spinless bosons trapped in a two-dimensional harmonic potential. The effect of the trap is to keep the system bound. It will be seen how the presence of a repulsive interaction changes the energy spectrum and other properties of the system. For instance, the density profile, which is usually measurable, and also the two-body distribution function, which is intimately related to the existence of correlations.\n\n\t\t\t\t Afterwards, the focus will be on the particular case of having only two bosons in the system interacting through a strong repulsive force. Inspired by the one-dimensional case where the fermionization phenomenon takes place in the strongly-interacting limit, we will study whether in two dimensions there is a resembling reminiscent effect. In other words, we will analyze if there are properties of the two strongly-interacting bosons in two dimensions that are like the ones of two noninteracting fermions.\n\n\t\t\t\t After that, we will tackle the localization phenomenon in a one-dimensional system that is caused by an external speckle potential that introduces disorder in the system. We will show that the localization is a robust phenomenon against repulsive contact interactions.\n\n\t\t\t\t Finally, we will study the influence of the spin-orbit coupling in a system of bosons with two possible pseudospin components, associated, for instance, to two hyperfine levels, confined in a two-dimensional harmonic trap. We will present an exhaustive analysis of the combined effects of the interaction and the spin-orbit coupling in the spectrum and the properties of the system. In particular we show the existence of a crossover in the ground state of the system susceptible to be experimentally identified.