We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas-Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.
We study the fate of algebraic decay of correlations in a harmonically trapped two-dimensional degenerate Bose gas. The analysis is inspired by recent experiments on ultracold atoms where power-law correlations have been observed despite the presence of the external potential. We generalize the spin wave description of phase fluctuations to the trapped case and obtain an analytical expression for the one-body density matrix within this approximation. We show that algebraic decay of the central correlation function persists to lengths of about 20% of the Thomas-Fermi radius. We establish that the trap-averaged correlation function decays algebraically with a strictly larger exponent weakly changing with trap size and find indications that the recently observed enhanced scaling exponents receive significant contributions from the normal component of the gas. We discuss radial and angular correlations and propose a local correlation approximation which captures the correlations very well. Our analysis goes beyond the usual local density approximation and the developed summation techniques constitute a powerful tool to investigate correlations in inhomogeneous systems.
We analyze the ground-state and low-temperature properties of a one-dimensional Bose gas in a harmonic trapping potential using the numerical density-matrix renormalization group. Calculations cover the whole range from the Bogoliubov limit of weak interactions to the Tonks-Girardeau limit. Local quantities such as density and local three-body correlations are calculated and shown to agree very well with analytic predictions within a local-density approximation. The transition between temperature-dominated to quantum-dominated correlation is determined. It is shown that despite the presence of the harmonic trapping potential, first-order correlations display, over a large range, the algebraic decay of a harmonic fluid with a Luttinger parameter determined by the density at the trap center.
We study various properties of an ultracold two-dimensional (2D) Bose gas that are beyond a mean-field description. We first derive the effective interaction for such a system as realized in current experiments, which requires the use of an energy dependent T matrix. Using this result, we then solve the mean-field equation of state of the modified Popov theory, and compare it with the usual Hartree-Fock theory. We show that even though the former theory does not suffer from infrared divergences in both the normal and superfluid phases, there is an unphysical density discontinuity close to the Berezinskii-Kosterlitz-Thouless transition. We then improve upon the mean-field description by using a renormalization group approach and show how the density discontinuity is resolved. The flow equations in two dimensions, in particular, of the symmetry-broken phase, already contain some unique features pertinent to the 2D XY model, even though vortices have not been included explicitly. We also compute various many-body correlators, and show that correlation effects beyond the Hartree-Fock theory are important already in the normal phase as criticality is approached. We finally extend our results to the inhomogeneous case of a trapped Bose gas using the local-density approximation and show that close to criticality, the renormalization group approach is required for the accurate determination of the density profile.
We study the quasi-two-dimensional Bose gas in harmonic traps at temperatures above the Kosterlitz-Thouless transition, where the gas is in the normal phase. We show that mean-field theory takes into account the dominant interaction effects for experimentally relevant trap geometries. Comparing with quantum Monte Carlo calculations, we quantify the onset of the fluctuation regime, where correlations beyond mean-field become important. Although the density profile depends on the microscopic parameters of the system, we show that the correlation density (the difference between the exact and the mean-field density) is accurately described by a universal expression, obtained from classical-field calculations of the homogeneous strictly two-dimensional gas. Deviations from universality, due to the finite value of the interaction or to the trap geometry, are shown to be small for current experiments. We further study coherence and pair correlations on a microscopic scale. Finite-size effects in the off-diagonal density matrix allow us to characterize the crossover from Kosterlitz-Thouless to Bose-Einstein behavior for small particle numbers. Bose-Einstein condensation occurs below a characteristic number of particles which rapidly diverges with vanishing interactions.
Interacting quantum spin models are remarkably useful for describing different types of physical, chemical, and biological systems.Significant understanding of their equilibrium properties has been achieved to date, especially for the case of spin models with short-range couplings.However, progress toward the development of a comparable understanding in long-range interacting models, in particular out-of-equilibrium, remains limited.In a recent work, we proposed a semiclassical numerical method to study spin models, the discrete truncated Wigner approximation (DTWA), and demonstrated its capability to correctly capture the dynamics of one-and two-point correlations in one-dimensional (1D) systems.Here we go one step forward and use the DTWA method to study the dynamics of correlations in two-dimensional (2D) systems with many spins and different types of long-range couplings, in regimes where other numerical methods are generally unreliable.We compute spatial and time-dependent correlations for spin-couplings that decay with distance as a power-law and determine the velocity at which correlations propagate through the system.Sharp changes in the behavior of those velocities are found as a function of the power-law decay exponent.Our predictions are relevant for a broad range of systems including solid state materials, atom-photon systems and ultracold gases of polar molecules, trapped ions, Rydberg, and magnetic atoms.We validate the DTWA predictions for small 2D systems and 1D systems, but ultimately, in the spirt of quantum simulation, experiments will be needed to confirm our predictions for large 2D systems.
In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.
We simulate a trapped quasi-two-dimensional Bose gas using a classical field method. To interpret our results we identify the uniform Berezinskii-Kosterlitz-Thouless (BKT) temperature TBKT as where the system phase-space density satisfies a critical value. We observe that density fluctuations are suppressed in the system well above TBKT when a quasicondensate forms as the first occurrence of degeneracy. At lower temperatures, but still above TBKT, we observe the development of appreciable coherence as a prominent finite-size effect, which manifests as bimodality in the momentum distribution of the system. At TBKT algebraic decay of off-diagonal correlations occurs near the trap center with an exponent of 0.25, as expected for the uniform system. Our results characterize the low-temperature phase diagram for a trapped quasi-two-dimensional Bose gas and are consistent with observations made in recent experiments.