We investigate the dynamic structure factor of atomic Bose and Fermi gases in one-dimensional optical lattices at zero temperature. The focus is on the generic behaviour of S ( k , ω) as a function of filling and interaction strength with the aim of identifying possible experimental signatures for the different quantum phase transitions. We employ the Hubbard or Bose–Hubbard model and solve the eigenvalue problem of the Hamiltonian exactly for moderate lattice sizes. This allows us to determine the dynamic structure factor and other observables directly in the phase transition regime, where approximation schemes are generally not applicable. We discuss the characteristic signatures of the various quantum phases appearing in the dynamic structure factor and illustrate that the centroid of the strength distribution can be used to estimate the relevant excitation gaps. Employing sum rules, these quantities can be evaluated using ground-state expectation values only. Important differences between bosonic and fermionic systems are observed, e.g., regarding the origin of the excitation gap in the Mott-insulator phase.
We investigate the dynamic structure factor of atomic Bose and Fermi gases in one-dimensional optical lattices at zero temperature. The focus is on the generic behaviour of S ( k , ω) as a function of filling and interaction strength with the aim of identifying possible experimental signatures for the different quantum phase transitions. We employ the Hubbard or Bose–Hubbard model and solve the eigenvalue problem of the Hamiltonian exactly for moderate lattice sizes. This allows us to determine the dynamic structure factor and other observables directly in the phase transition regime, where approximation schemes are generally not applicable. We discuss the characteristic signatures of the various quantum phases appearing in the dynamic structure factor and illustrate that the centroid of the strength distribution can be used to estimate the relevant excitation gaps. Employing sum rules, these quantities can be evaluated using ground-state expectation values only. Important differences between bosonic and fermionic systems are observed, e.g., regarding the origin of the excitation gap in the Mott-insulator phase.
In order to identify possible experimental signatures of the superfluid to Mott-insulator quantum phase transition we calculate the charge structure factor S ( k , ω ) for the one-dimensional Bose-Hubbard model using the dynamical density-matrix renormalisation group (DDMRG) technique. Particularly we analyse the behaviour of S ( k , ω ) by varying – at zero temperature–the Coulomb interaction strength within the first Mott lobe. For strong interactions, in the Mott-insulator phase, we demonstrate that the DDMRG results are well reproduced by a strong-coupling expansion, just as the quasi-particle dispersion. In the super-fluid phase we determine the linear excitation spectrum near k = 0. In one dimension, the amplitude mode is absent which mean-field theory suggests for higher dimensions.
Ultracold atomic gases in optical lattices provide an unique framework to study quantum phenomena in strongly correlated systems. In addition to the precise control over all relevant parameters in the experiment, these experiments can be mapped to a fundamental model from solid-state physics. For the bosonic version of this model, the so-called Bose-Hubbard model, a phase transition from a superfluid to a Mott insulator was theoretically predicted and later experimentally observed in an ultracold gas of Rb-87 atoms in three-dimensional as well as in one-dimensional optical lattices. Apart from homogeneous optical lattices one can introduce more complex lattice topologies such as two-color superlattices which give rise to a rich phase diagram including more exotic phases like the Bose-glass. We employ various powerful many-body techniques like exact diagonalization in complete and truncated Hilbert spaces and the Density-Matrix Renormalization Group (DMRG) algorithm to study the phase diagrams of the one-dimensional Bose-Hubbard and the one-dimensional Bose-Fermi-Hubbard Hamiltonian. Most theoretical studies of these systems discuss the phase diagrams with respect to the generic parameters of the Hubbard model. These Hubbard parameters, however, depend non-trivially on the control parameters used in experiments. The focus of this work is on the ab-initio calculation of the phase diagram of ultracold Rb-87 in one-dimensional optical superlattices starting directly from the experimental setup. To this end, we first employ band-structure calculations to extract the Hubbard parameters from the experimental parameters. Then, we use state-of-the-art DMRG techniques to solve the many-body problem for realistic particle numbers and lattice sizes that occur in experiments. Our results show that by using the intensities of the two laser fields forming the two-color superlattice as control parameters while keeping all other experimental parameters fixed, it is possible to access all relevant quantum phases of the system. Furthermore, we found out that the critical values of the laser intensities for the different phase transitions depend strongly on a third parameter that has to be included for a realistic description of the experiment. This third parameter is the strength of a harmonic trapping potential which accounts for the Gaussian shape of the laser fields and an additional magnetic potential used to confine the atoms in the center of the trap.
Ultracold bosonic gases in optical lattices are strongly correlated quantum systems similar to solids. The strong correlation between the electrons in a solid on the one hand, and the bosonic atoms in optical lattice on the other, exhibit various quantum phenomena like insulation, conductivity, localization of electrons and atoms, respectively. Controlled by the intensity of the lattice laser, the ultracold bosonic gas can be transferred from a regime with superfluid character for shallow lattices into a regime of strong correlations, the Mott insulator. As an additional external parameter besides the lattice depth, one can generate spatial inhomogeneities by superimposing an additional standing wave (so-called two-color superlattices), which gives rise to localization effects or the formation of a Bose-glass phase. In the present work, numerical simulations are employed in order to investigate characteristic signatures of the quantum phases in the low-energy excitation spectrum of one-dimensional systems. We simulate temporal small amplitude modulations of the optical lattice in analogy to experiments, and evaluate the response of the system from the time-evolved initial state. The lattice systems are described in the framework of the Bose-Hubbard model. For the evaluation of the time-evolved state, we employ several numerical methods. We analyze systems of small size (6 particles on 6 sites) using an exact time-evolution by integration of the time-dependent Schrödinger equation. The formulation of an importance truncation scheme enables us to retain only the relevant components of the model space in the strongly correlated regime and, thus, allows for the investigation of systems with 10 particles on 10 sites using exact time-evolution. Based on this method, we present results of the Mott-insulating regime as well as for the Bose-glass phase. Furthermore, we employ particle-hole methods, which allow for the treatment of systems with experimental lattice sizes and particle numbers. Starting from the equation of motion method we adapt the Tamm-Dancoff approximation as well as the random-phase approximation for the occupation number representation of the Bose-Hubbard model. We present results of simulations of up to 50 particles on 50 sites and discuss the impact of the lattice depth on the low-energy excitations (U-resonance). Moreover, the impact of a two-color superlattice and the variation of its amplitude is investigated.
We study the dynamic structure factor of a one-dimensional Bose gas confined in an optical lattice and modeled by the Bose–Hubbard Hamiltonian, using a variety of numerical and analytical approaches. The dynamic structure factor, experimentally measurable by Bragg spectroscopy, is studied in three relevant cases: in the clean regime, featuring either a superfluid or a Mott phase; and in the presence of two types of (quasi-)disordered external potentials: a quasi-periodic potential obtained from a bichromatic superlattice and a random box disorder—both featuring a Bose-glass phase. In the clean case, we show the emergence of a gapped doublon mode (corresponding to a repulsively bound state) for incommensurate filling, well separated from the low-energy acoustic mode. In the disordered case, we show that the dynamic structure factor provides direct insight into the spatial structure of the excitations, unveiling their localized nature, which represents a fundamental signature of the Bose-glass phase. Furthermore, it provides a clear fingerprint of the very nature of the localization mechanism which differs for the two kinds of disorder potentials we consider. In special cases, the dynamic structure factor may provide an estimate of the position of the localization transition from superfluid to Bose glass, in a complementary manner to the information deduced from the momentum distribution.
The dynamic response of ultracold Bose gases in one-dimensional optical lattices and superlattices is investigated based on exact numerical time evolutions in the framework of the Bose–Hubbard model. The system is excited by a temporal amplitude modulation of the lattice potential, as it was done in recent experiments. For regular lattice potentials, the dynamic signatures of the superfluid to Mott-insulator transition are studied and the position and the fine-structure of the resonances is explained by a linear response analysis. Using direct simulations and the perturbative analysis it is shown that in the presence of a two-colour superlattice the excitation spectrum changes significantly when going from the homogeneous Mott-insulator to the quasi-Bose-glass phase. A characteristic and experimentally accessible signature for the quasi-Bose-glass is the appearance of low-lying resonances and a suppression of the dominant resonance of the Mott-insulator phase.
Ultracold atomic gases may be the ultimate quantum simulator.These isolated systems have the lowest temperatures in the observable universe, and their properties and interactions can be precisely and accurately tuned across a full spectrum of behaviors, from few-body physics to highly-correlated many-body effects.The ability to impose potentials on and tune interactions within ultracold gases to mimic complex systems mean they could become a theorist's playground.One of their great strengths, however, is also one of the largest obstacles to this dream: isolation.This thesis touches on both of these themes.First, methods to characterize phases and quantum critical points, and to construct finite temperature phase diagrams using experimentally accessible observables in the Bose Hubbard model are discussed.Then, the transition from a weakly to a strongly interacting Bose-Fermi mixture in the continuum is analyzed using zero temperature numerical techniques.Real materials can be emulated by ultracold atomic gases loaded into optical lattice potentials.We discuss the characteristics of a single boson species trapped in an optical lattice (described by the Bose Hubbard model) and the hallmarks of the quantum critical region that separates the superfluid and the Mott insulator ground states.We propose a method to map the quantum critical region using the single, experimentally accessible, local quantity R, the ratio of compressibility to local number fluctuations.The procedure to map a phase diagram with R is easily generalized to inhomogeneous systems and generic many-body Hamiltonians.We illustrate it here using quantum Monte Carlo simulations of the 2D Bose Hubbard model.Secondly, we investigate the transition from a degenerate Fermi gas weakly coupled List of Tables Table Page 2.1 Techniques to Probe Atoms in Optical Lattices . . . . . .