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arXiv:cond-mat/0204611 · Physical Review Letters

Dynamical Transition from a Quasi-One-Dimensional Bose-Einstein Condensate to a Tonks-Girardeau Gas

Patrik Öhberg, Luis Santos

Published Nov 21, 2002• 82 citations
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Abstract

We analyze in detail the expansion of a 1D Bose gas after removing the axial confinement. We show that during its one-dimensional expansion the density of the Bose gas does not follow a self-similar solution. Our analysis is based on a nonlinear Schrödinger equation with variable nonlinearity whose validity is discussed for the expansion problem, by comparing with an exact Bose-Fermi mapping for the case of an initial Tonks-Girardeau gas. For this case, the gas is shown to expand self-similarly, with a different scaling law compared to the one-dimensional Thomas-Fermi condensate.

DOI
10.1103/physrevlett.89.240402

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We analyze in detail the expansion of a 1D Bose gas after removing the axial confinement. We show that during its one-dimensional expansion the density of the Bose gas does not follow a self-similar solution. Our analysis is based on a nonlinear Schrödinger equation with variable nonlinearity whose validity is discussed for the expansion problem, by comparing with an exact Bose-Fermi mapping for the case of an initial Tonks-Girardeau gas. For this case, the gas is shown to expand self-similarly, with a different scaling law compared to the one-dimensional Thomas-Fermi condensate.
The expansion of a one-dimensional Bose gas after releasing its initial harmonic confinement is investigated employing the Lieb-Liniger equation of state within the local-density approximation. We show that during the expansion the density profile of the gas does not follow a self-similar solution, as one would expect from a simple scaling ansatz. We carry out a variational calculation, which recovers the numerical results for the expansion, the equilibrium properties of the density profile, and the frequency of the lowest compressional mode. The variational approach allows for the analysis of the expansion in all interaction regimes between the mean-field and the Tonks-Girardeau limits, and in particular shows the range of parameters for which the expansion violates self-similarity.
We study the free expansion of Tonks‐Girardeau gases on one‐dimensional lattices after removing the confining potential. We find that for low initial densities the momentum distribution of the expanding gas rapidly approaches the one of noninteracting fermions. In the opposite limit, i.e, when the initial system is a pure Mott insulator with one particle per lattice site, the expansion leads to the emergence of quasi‐condensates at finite momentum. These traveling quasi‐condensates develop quasi‐long range one‐particle correlations similar to the ones shown to be universal in the equilibrium case.
We investigate the expansion dynamics of a Bose–Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the nonuniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, which reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas–Fermi approximation. We have obtained self-similar solutions of these differential equations for several important special cases and write out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms may be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves is considered. We compare the numerical solutions of the Gross–Pitaevskii equations with their approximate analytical solutions and numerically study the situations where the analytical method being used admits no exact solutions.
One-dimensional Bose gases are considered, interacting either through the hard-core potentials or through the contact delta potentials. Interest in these gases gained momentum because of the recent experimental realization of quasi-one-dimensional Bose gases in traps with tightly confined radial motion, achieving the Tonks-Girardeau (TG) regime of strongly interacting atoms. For such gases the Fermi-Bose mapping of wavefunctions is applicable. The aim of the present communication is to give a brief survey of the problem and to demonstrate the generality of this mapping by emphasizing that: (i) It is valid for nonequilibrium wavefunctions, described by the time-dependent Schrödinger equation, not merely for stationary wavefunctions. (ii) It gives the whole spectrum of all excited states, not merely the ground state. (iii) It applies to the Lieb-Liniger gas with the contact interaction, not merely to the TG gas of impenetrable bosons.
We study ground state properties of spinless, quasi one-dimensional bosons which are confined in a harmonic trap and interact via repulsive delta-potentials. We use the exact diagonalization method to analyze the pair correlation function, as well as the density, the momentum distribution, different contributions to the energy and the population of single-particle orbitals in the whole interaction regime. In particular, we are able to trace the fascinating transition from bosonic to fermi-like behavior in characteristic features of the momentum distribution which is accessible to experiments. Our calculations yield quantitative measures for the interaction strength limiting the mean-field regime on one side and the Tonks-Girardeau regime on the other side of an intermediate regime.