We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
We discuss interacting and noninteracting one dimensional atomic systems trapped in an optical lattice plus a parabolic potential. We show that, in the tight-binding approximation, the noninteracting problem is exactly solvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations of ideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat the interacting boson problem by numerical diagonalization of the Bose-Hubbard Hamiltonian. From analysis of the dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we consider the problems of ``fermionization'' of a Bose gas, and the superfluid-Mott insulator transition. The spectrum of the noninteracting system turns out to provide a useful guide to understanding the collective oscillations of the interacting system, throughout a large and experimentally relevant parameter regime.
This dissertation presents my research covering the field of ultracold atoms loaded in optical lattices. The static and dynamical properties of atoms in combined periodic and parabolic potentials are studied, with a focus on the strongly interacting regimes. Because parabolic magnetic and optical potentials are routinely used to confine atoms, the results of this research are directly relevant to ongoing experimental endeavours in atomic physics. After a review of the basic theory of atoms in homogeneous periodic potentials, the equilibrium and non-equilibrium properties of non-interacting and interacting atoms in periodic plus parabolic potentials are studied. The problem of the localization of the many-body wavefunction for systems with arbitrary peak onsite density is presented in Chapters 3 and 4. The physics pertaining to the experimental realization of Mott insulator states with one or more atoms per sites in inhomogeneous lattices is elucidated by introducing an intuitive model for strongly interacting bosons in one dimension. This model is then utilized to study the decay of the dipole oscillations of atomic ensembles subject to a small displacement of the parabolic potential. Good agreement is found with results of recent experiments. Chapters 5 and 6 are dedicated to the characterization of the Mott insulator state with unit filling, which plays a central role in proposed schemes for neutral atom quantum computation. The usefulness of Bragg spectroscopy to probe the excitation spectrum of the Mott state in homogeneous lattices is analyzed in Chapter 5, where the limits of validity of linear response theory in this strongly correlated regime are delimited. In Chapter 6 the effects of finite temperature on the confined Mott insulator state are studied, and a scheme is devised for possibly estimating the system's temperature, at energies of the order of the inter-particle interaction energy. Finally, in Chapter 7, a proposal is introduced to utilize the Mott state as a robust register for neutral atom quantum computation. Unwanted residual quantum coherences inherent to the Mott insulator ground state are eliminated by a judicious choice of the trapping potentials and a selective measurement on a molecular photo-associative transition.
Motivated by the recent rapid development of the field of quantum gases in optical lattices, we present a comprehensive study of the spectrum of ultracold atoms in a one-dimensional optical lattice subjected to a periodic lattice modulation. Using the time-dependent density matrix renormalization group method, we study the dynamical response due to lattice modulations in different quantum phases of the system with varying density. For the Mott-insulating state, we identify several excitation processes, which provide important information about the density profile of the gases. For the superfluid, the dynamical response can be well described in a local density approximation. This simplification can be valuable in understanding the strong-correlated superfluid in a slow-varying harmonic potential. All these spectroscopic features of an inhomogeneous system can be used as a test for the validity of the Bose-Hubbard model in a parabolic trapping potential.
Bosonic atoms confined in optical lattices are described by the Bose-Hubbard model and can exist in two different phases, Mott insulator or superfluid, depending on the strength of the system parameters. In the vicinity of the phase boundary, there are degeneracies that occur between every two adjacent Mott lobes. Because of this, nondegenerate perturbation theory fails to give meaningful results for the condensate density: it predicts a phase transition in a point of the phase diagram where no transition occurs. Motivated by this, we develop two different degenerate perturbative methods to solve the degeneracy-related problems. Moreover, we study the one-dimensional repulsively interacting Bose gas under harmonic confinement, with special attention to the asymptotic behavior of the momentum distribution, which is a universal k−4 decay characterized by the Tan's contact. The latter constitutes a direct signature of the short-range correlations in such an interacting system and provides valuable insights about the role of the interparticle interactions. We investigate the system constituted of N interacting particles in the strongly interacting limit. In such a regime, the strong interparticle interaction makes the bosons behave similarly to the ideal Fermi gas. Because of the difficulty in analytically solving the system for N particles at finite interaction, the Tonks-Girardeau regime provides a favorable scenario to probe the contact. Therefore, we are able to provide an analytical formula for the Tan's contact. Furthermore, we analyze the scaling properties of the Tan's contact in terms of N in the high-temperature regime as well as in the strongly interacting regime. Finally, we compare our analytical calculations of the Tan's contact to quantum Monte Carlo simulations and discuss some fundamental differences between the canonical and the grand-canonical ensembles.
We present a model that generalizes the Bose–Fermi mapping for strongly correlated one-dimensional (1D) bosons trapped in combined optical lattice plus parabolic potentials, to cases in which the average number of atoms per site is larger than one. Using a decomposition in arrays of disjoint strongly interacting gases, this model gives an accurate account of equilibrium properties of such systems, in parameter regimes relevant to current experiments. The application of this model to non-equilibrium phenomena is explored by a study of the dynamics of an atom cloud subject to a sudden displacement of the confining potential. Excellent agreement is found with results of recent experiments, without the use of any adjustable parameters. The simplicity and intuitive appeal of this model make it attractive as a general tool for understanding bosonic systems in the strongly correlated regime.
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.
Ultracold atoms have emerged as an indispensable setting to study quantum many-body systems. Recent experimental and theoretical work has explored the curious phases and novel properties of Bose-Einstein Condensate with optical lattices, and Bose-Einstein Condensate with light-induced artificial spin-orbit coupling. In this thesis, we report our research on these two types of boson systems. In the first topic, in contrast with calculations of bosons in optical lattices that focus on the tight-binding regime, we note that the single-particle states of bosons in a periodic potential generally satisfy the Mathieu equation, and have developed a formalism for studying bosons in an optical lattice using the Mathieu equation. Moreover, based on this formalism, we have proposed a self-consistent scheme for describing interacting bosons in an optical lattice using Hartree Fock approximation. We apply this scheme to quantify the effects of inter-atomic interactions on the properties of bosons in an optical lattice, as exhibited in the comparison between observables of non-interacting and interacting systems, such as the superfluid transition temperature and momentum distribution as probed in time-of-flight expansion. In the second topic, the phases of a Bose-Einstein condensate with light-induced spin-orbit coupling are studied within the mean-field approximation. We obtain the phase diagram at fixed chemical potential and at fixed density for bosons with spin-orbit coupling, finding a regime of phase separation and a regime in which the bosons condensed into a mixed phase. We determine how this phase diagram evolves as a function of the atom interaction parameters and as a function of the strength of light-atom coupling. The mixed phase is found to be stable for sufficiently small light-atom coupling. Specifically, we show that the structure of the phase diagram at fixed chemical potential suggests an unusual density dependence for the mixed phase in a harmonic trapping potential, in which the density of one spin increases with increasing radius, suggesting a unique experimental signature of this state. The collective Bogoliubov sound mode is shown to also provide a signature of the mixed phase, vanishing as the boundary to the regime of phase separation is approached. Together, in these two topics we address the need to enhance the understanding of the unconventional physical properties of Bose-Einstein Condensate in a controlled electromagnetic environment (optical lattices, Raman lasers, etc.), and provide predictions for possible experimental findings.