Correlations in many-body quantum states lead to interesting phenomena while at the same time being enormously difficult to describe theoretically. Recent experiments conducted with ultracold atoms in optical lattices allow to explore quantum many-body correlations in a controlled and tunable way. The underlying theoretical model is the single-band Hubbard model. In this thesis we investigate the one-dimensional Hubbard model by applying the newly developed time-dependent two-particle reduced density matrix method. The method is based upon the assumption that three-particle correlations are negligible. We assess this approximation for equilibrium and non-equilibrium states. We show that the method is well suited for weakly interacting and weakly correlated systems.
Correlations in many-body quantum states lead to interesting phenomena while at the same time being enormously difficult to describe theoretically. Recent experiments conducted with ultracold atoms in optical lattices allow to explore quantum many-body correlations in a controlled and tunable way. The underlying theoretical model is the single-band Hubbard model. In this thesis we investigate the one-dimensional Hubbard model by applying the newly developed time-dependent two-particle reduced density matrix method. The method is based upon the assumption that three-particle correlations are negligible. We assess this approximation for equilibrium and non-equilibrium states. We show that the method is well suited for weakly interacting and weakly correlated systems.
We explore the nonequilibrium dynamics of a one-dimensional Fermi-Hubbard system as a sensitive testbed for the capabilities of the time-dependent two-particle reduced density matrix (TD2RDM) theory to accurately describe time-dependent correlated systems. We follow the time evolution of the out-of-equilibrium finite-size Fermi-Hubbard model initialized by a quench over extended periods of time. By comparison with exact calculations for small systems and with matrix product state calculations for larger systems but limited to short times, we demonstrate that the TD2RDM theory can accurately account for the nonequilibrium dynamics in the regime from weak to moderately strong interparticle correlations. We find that the quality of the approximate reconstruction of the three-particle cumulant (or correlation) required for the closure of the equations of motion for the reduced density matrix is key to the accuracy of the numerical TD2RDM results. We identify the size of the dynamically induced three-particle correlations and the amplitude of cross correlations between the two- and three-particle cumulants as critical parameters that control the accuracy of the TD2RDM theory when current state-of-the-art reconstruction functionals are employed.
Cutting-edge technology needs small, ultrafast devices, operational in a wide range of regimes. This calls for solids with novel, unconventional and tailorable properties. Great progress is expected from materials in which electron-electron and electron-phonon correlations strongly affect the dynamics ( i.e. "unforeseen" useful properties are expected to be most likely found in systems with complex behavior). In spite of the potentially huge technological pay-off, our understanding of these systems is rather incomplete, especially in non-equilibrium. Often, with knowledge of systems or phenomena at an early stage, it is rewarding to resort to model, simplified descriptions. This strategy is used in this thesis, where we study several models lattice systems via density-functional theory. The latter is a well established approach (in fact, it is the current method of choice) for investigations of real materials. In our research, we focussed on little understood properties of interacting many-particle systems, such as the time-dependent conduction properties of electronic devices in the presence of interactions, disorder, and lattice vibrations, or the expansion of ultracold fermion clouds in 3D optical lattices. All these systems were described in terms of Hubbard-type interactions for the electrons, and Holstein-type electron-phonon interactions. Our results show interesting features due to the interactions which depend on dimensionality; they also show a dynamical crossover for several properties, due to the competition between disorder and interaction. Finally, when lattice vibrations are included, we showed how it is possible to manipulate in a controlled way the nuclear dynamics of molecular device via fast electronic external fields, of potential interest for technologies employing nanomolecular motors.
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.
Experiments with ultracold fermionic atoms in optical lattices have attracted much interest in condensed matter physics as they serve as a model for electrons in solids. While experiments now allow the observation of nonequilibrium transport processes with single-site resolution, the theoretical description, especially for strongly correlated fermions, remains very challenging. Standard methods, such as the density matrix renormalization group (DMRG) and nonequilibrium Green functions (NEGF), have been primarily applied to one-dimensional systems and weak to moderate coupling, respectively. Here, the authors perform a detailed comparison of DMRG and NEGF and demonstrate that both can be benchmarked against each other. They observe complementary applicability ranges suggesting that a combination of both will allow to substantially expand the range and duration of first-principles quantum dynamics simulations for fermionic lattice systems.
Strongly correlated quantum systems display a myriad of complex, fascinating behaviour. The competition between charge, spin, orbital, or lattice degrees of freedom and their respective energy scales gives rise to rich and complex emergent phases. More recently, the non-equilibrium properties of correlated quantum many-body systems have come into the focus of active research, with remarkable advances. While the advent of tensor network based simulations on the theoretical side has allowed access to the full time-evolution of the quantum state as it traverses the Hilbert space, the experimental progress in the ultracold atom community has allowed physicists to enter the paradigm of analog quantum simulation. Their tunability and isolation from the environment makes them ideal platforms to emulate and study open problems in condensed matter physics and go beyond the simulational capabilities of state of the art classical simulations. In this thesis, we explore the dynamics of correlated Fermi gases. We develop and present different facets of applicability of radiofrequency modulation techniques to drive the system out of equilibrium. We begin by considering a homogeneous, three-dimensional Fermi gas in the BCS-BEC crossover and develop an off-resonant radiofrequency transfer scheme to excite the Higgs mode of the superfluid. In a subsequent study we investigate the tunability of the coherence between Cooper pairs by tuning the duration of interaction ramps of the internal interaction strength. These works highlight the possibility to externally tune and stabilise complex quantum many-body states away from equilibrium. Using quasi-exact matrix product state simulations, we study the response of an attractively interacting, one-dimensional Fermi-Hubbard model to weak radiofrequency perturbations. We reveal the emergence of two distinct dynamical regimes in the time evolution and are able to relate it back to the underlying excitation spectrum of the system. Finally, we explore the possibility to perform quantum quenches through radiofrequency π-pulses, achieving nearly complete population transfer between different interacting fermionic states. We reveal non-trivial dynamical effects in the pair correlation of the final state, as well as the excitation of a collective trap mode of the system.
Recently, many experiments with cold atomic gases have been conducted from interest in the non-equilibrium dynamics of correlated quantum systems. Of these experiments, the mixing dynamics of fermion clusters motivates us to research cluster-cluster collision dynamics in one-dimensional Fermi systems. We adopt the one-dimensional Fermi-Hubbard model and apply the time-dependent density matrix renormalization group method. We simulate collisions between two fermion clusters of spin-up and spin-down and calculate reflectance of the clusters R changing the particle number in each cluster and the interaction strength between two fermions with up and down spins. We also evaluate the quasi-classical (independent collision) reflectance R qc to compare it with R . The quasi-classical picture is quantitatively valid in the limit of weak interaction, but it is not valid when interaction is strong.