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arXiv:0811.1532 · New Journal of Physics

The granularity of weakly occupied bosonic fields beyond the local density approximation

M. Eckart, R. Walser, W. P. Schleich, Sascha Zöllner, Peter Schmelcher

Published Feb 4, 2009• 3 citations
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Abstract

We examine ground state correlations for repulsive, quasi one-dimensional bosons in a harmonic trap. In particular, we focus on the few particle limit N =2, 3, 4, ..., where exact numerical solutions of the many particle Schrödinger equation are available, by employing the multi-configuration time-dependent Hartree method. Our numerical results for the inhomogeneous system are modeled with the analytical solution of the homogeneous problem using the Bethe ansatz and the local density approximation. Tuning the interaction strength from the weakly correlated Gross–Pitaevskii to the strongly correlated Tonks–Girardeau regime reveals finite particle number effects in the second-order correlation function beyond the local density approximation.

DOI
10.1088/1367-2630/11/2/023010

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We examine ground state correlations for repulsive, quasi one-dimensional bosons in a harmonic trap. In particular, we focus on the few particle limit N =2, 3, 4, ..., where exact numerical solutions of the many particle Schrödinger equation are available, by employing the multi-configuration time-dependent Hartree method. Our numerical results for the inhomogeneous system are modeled with the analytical solution of the homogeneous problem using the Bethe ansatz and the local density approximation. Tuning the interaction strength from the weakly correlated Gross–Pitaevskii to the strongly correlated Tonks–Girardeau regime reveals finite particle number effects in the second-order correlation function beyond the local density approximation.
Many-body densities and correlation functions are of paramount importance for understanding quantum many-body physics. Here, we present a method to compute them; our approach is general and based on the action of bosonic or fermionic annihilation field operators on the many-body wavefunction. We analyze N=6N = 6N=6 quasi-one-dimensional harmonically-trapped bosons with weak to strong contact interaction strength up to the Tonks-Girardeau limit with infinite repulsion using the MultiConfigurational Time-Dependent Hartree method for indistinguishable particles (MCTDH-X). We compare our MCTDH-X solutions to the analytical ones in the infinite repulsion regime as well as to the so-called correlated pair wavefunction approach and find a good agreement. Since numerical approximations are not bound to the cases where analytical solutions are known, we thus demonstrate a general method to investigate high-order reduced density matrices and correlation functions in systems for which analytical solutions are unknown. We trace the build-up of correlation features in the crossover from weak interactions to the Tonks-Girardeau limit and find that the higher-order correlation functions and densities resemble those in the Tonks-Girardeau limit for way smaller interactions than anticipated from just the one-body density.
We calculate the density profiles and density correlation functions of the one-dimensional Bose gas in a harmonic trap, using the exact finite-temperature solutions for the uniform case, and applying a local density approximation. The results are valid for a trapping potential that is slowly varying relative to a correlation length. They allow a direct experimental test of the transition from the weak-coupling Gross-Pitaevskii regime to the strong-coupling, ``fermionic'' Tonks-Girardeau regime. We also calculate the average two-particle correlation which characterizes the bulk properties of the sample, and find that it can be well approximated by the value of the local pair correlation in the trap center.
We investigate few-boson systems in finite one-dimensional multiwell traps covering the full interaction crossover from uncorrelated to fermionized particles. Our treatment of the ground-state properties is based on the numerically exact multiconfigurational time-dependent Hartree method. For commensurate filling, we trace the fingerprints of localization as the interaction strength increases, in several observables like reduced-density matrices, fluctuations, and momentum distribution. For a filling factor larger than 1 we observe on-site repulsion effects in the densities and fragmentation of particles beyond the validity of the Bose-Hubbard model upon approaching the Tonks-Girardeau limit. The presence of an incommensurate fraction of particles induces incomplete localization and spatial modulations of the density profiles, taking into account the finite size of the system.
We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.
We calculate the density profiles and density correlation functions of the one-dimensional Bose gas in a harmonic trap, using the exact finite-temperature solutions for the uniform case, and applying a local density approximation. The results are valid for a trapping potential which is slowly varying relative to a correlation length. They allow a direct experimental test of the transition from the weak coupling Gross-Pitaevskii regime to the strong coupling, \'fermionic\' Tonks-Girardeau regime. We also calculate the average two-particle correlation which characterizes the bulk properties of the sample, and find that it can be well approximated by the value of the local pair correlation in the trap center.
The emergence of a Bose-glass region in a quasi one-dimensional Bose–Einstein-condensed gas in a harmonic trapping potential with an additional delta-correlated disorder potential at zero temperature is studied using three approaches. At first, the corresponding time-independent Gross–Pitaevskii equation is numerically solved for the condensate wave function, and disorder ensemble averages are evaluated. In particular, we analyse quantitatively the emergence of mini-condensates in the local minima of the random potential, which occurs for weak disorder preferentially at the border of the condensate, while for intermediate disorder strength this happens in the trap centre. Second, in view of a more detailed physical understanding of this phenomenon, we extend a quite recent non-perturbative approach towards the weakly interacting dirty boson problem, which relies on the Hartree–Fock theory and is worked out on the basis of the replica method, from the homogeneous case to a harmonic confinement. Finally, in the weak disorder regime we also apply the Thomas–Fermi approximation, while in the intermediate disorder regime we additionally use a variational ansatz in order to describe analytically the numerically observed redistribution of the fragmented mini-condensates with increasing disorder strength.