We study the occupation numbers and number fluctuations of ultracold atoms in deep optical lattices for finite-temperatures within the Bose-Hubbard model. Simple analytical expressions for the mean occupation number and number fluctuations are obtained in the weak-hopping regime using an interpolation between results from different perturbation approaches in the Mott-insulator and superfluid phases. With this approach the magnitude of number fluctuations under a wide range of experimental conditions can be estimated and the properties of the finite-temperature phase diagram can be studied. These analytical results are compared to exact one-dimensional numerical calculations using a finite temperature variant of the density-matrix renormalization group (DMRG) method and found to have a high degree of accuracy. We find very good agreement, also in the crossover ``thermal'' region. We also analyze the influence of finite temperature on the behavior of the system in the vicinity of the zero-temperature phase transition, in one, two, and three dimensions.
We study the occupation numbers and number fluctuations of ultracold atoms in deep optical lattices for finite-temperatures within the Bose-Hubbard model. Simple analytical expressions for the mean occupation number and number fluctuations are obtained in the weak-hopping regime using an interpolation between results from different perturbation approaches in the Mott-insulator and superfluid phases. With this approach the magnitude of number fluctuations under a wide range of experimental conditions can be estimated and the properties of the finite-temperature phase diagram can be studied. These analytical results are compared to exact one-dimensional numerical calculations using a finite temperature variant of the density-matrix renormalization group (DMRG) method and found to have a high degree of accuracy. We find very good agreement, also in the crossover ``thermal'' region. We also analyze the influence of finite temperature on the behavior of the system in the vicinity of the zero-temperature phase transition, in one, two, and three dimensions.
We study the thermodynamics of ultracold Bose atoms in optical lattices by numerically diagonalizing the mean-field Hamiltonian of the Bose-Hubbard model. This method well describes the behavior of long-range correlations and therefore is valid deep in the superfluid phase. For the homogeneous Bose-Hubbard model, we draw the finite-temperature phase diagram and calculate the superfluid density at unity filling. We evaluate the finite-temperature effects in a recent experiment probing number fluctuation [Phys. Rev. Lett. 96 090401 (2006)], and find that our finite-temperature curves give a better fitting to the experimental data, implying non-negligible temperature effects in this experiment.
For systems of interacting, ultracold spin-zero neutral bosonic atoms, harmonically trapped and subject to an optical lattice potential, we derive an Extended Bose Hubbard (EBH) model by developing a systematic expansion for the Hamiltonian of the system in powers of the lattice parameters and of a scale parameter, the lattice attenuation factor. We identify the dominant terms that need to be retained in realistic experimental conditions, up to nearest-neighbor interactions and nearest-neighbor hoppings conditioned by the on-site occupation numbers. In the mean field approximation, we determine the free energy of the system and study the phase diagram both at zero and at finite temperature. At variance with the standard on site Bose Hubbard model, the zero-temperature phase diagram of the EBH model possesses a dual structure in the Mott insulating regime. Namely, for specific ranges of the lattice parameters, a density wave phase characterizes the system at integer fillings, with domains of alternating mean occupation numbers that are the atomic counterparts of the domains of staggered magnetizations in an antiferromagnetic phase. We show as well that in the EBH model, a zero-temperature quantum phase transition to pair superfluidity is, in principle, possible, but completely suppressed at the lowest order in the lattice attenuation factor. Finally, we determine the possible occurrence of the different phases as a function of the experimentally controllable lattice parameters.
We use the density-matrix renormalization group method to investigate ground-state and dynamic properties of the one-dimensional Bose-Hubbard model, the effective model of ultracold bosonic atoms in an optical lattice. For fixed maximum site occupancy n b = 5, we calculate the phase boundaries between the Mott insulator and the "superfluid" phase for the lowest two Mott lobes. We extract the Tomonaga-Luttinger parameter from the density-density correlation function and determine accurately the critical interaction strength for the Mott transition. For both phases, we study the momentum distribution function in the homogeneous system, and the particle distribution and quasi-momentum distribution functions in a parabolic trap. With our zero-temperature method we determine the photoemission spectra in the Mott insulator and in the "superfluid" phase of the one-dimensional Bose-Hubbard model. In the insulator, the Mott gap separates the quasi-particle and quasi-hole dispersions. In the "superfluid" phase the spectral weight is concentrated around zero momentum.
We study on-site occupation number fluctuations in a system of interacting bosons in an optical lattice. The ground-state distribution is obtained analytically in the limiting cases of strong and weak interaction, and by means of exact Monte Carlo simulations in the strongly correlated regime. As the interaction is increased, the distribution evolves from Poissonian in the noninteracting gas to a sharply peaked distribution in the Mott-insulator (MI) regime. In the special case of large occupation numbers, we demonstrate analytically and check numerically that there exists a wide interval of interaction strength, in which the on-site number fluctuations remain Gaussian and are gradually squeezed until they are of order unity near the superfluid (SF)-MI transition. Recently, the on-site number statistics were studied experimentally in a wide range of lattice potential depths [Phys. Rev. Lett. 96, 090401 (2006)]. In our simulations, we are able to directly reproduce experimental conditions using temperature as the only free parameter. Pronounced temperature dependence suggests that measurements of on-site atom number fluctuations can be employed as a reliable method of thermometry in both SF and MI regimes.
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 . . . . . .
We investigate the ground-state and finite-temperature properties of the Bose-Hubbard model using the standard basis operator method applied to the equation of motion for Green's functions. We show that by introducing the self-consistent approach, the theory goes significantly beyond the mean-field approximation, allowing more-precise description of quantum and thermal fluctuations. For optical lattice systems with ultracold bosons, we find relevant quantitative agreement with numerical and experimental data. In particular, the presented method reveals the importance of thermal fluctuations in time-of-flight experiments. For the most reliable fitting of the visibility analytical curve to the experimental data, the value of temperature is determined and proximity to the quantum regime is discussed. The problem of identification of the superfluid--Mott-insulator phase boundary from time-of-flight experiments is also considered.