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arXiv:cond-mat/0304071 · Physical Review Letters

One-Dimensional Bosons in Three-Dimensional Traps

Élliott H. Lieb, Robert Seiringer, Jakob Yngvason

Published Oct 7, 2003• 159 citations
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Abstract

Recent experimental and theoretical work has indicated conditions in which a trapped, low density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up until now, the theoretical arguments have been based on variational/perturbative ideas or numerical investigations. There are four parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate five parameter regions in which various types of 1D or 3D behavior occur in the ground state. Our treatment is based on a rigorous analysis of the many-body Schrödinger equation.

DOI
10.1103/physrevlett.91.150401

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Recent experimental and theoretical work has indicated conditions in which a trapped, low density Bose gas ought to behave like the 1D delta-function Bose gas solved by Lieb and Liniger. Up until now, the theoretical arguments have been based on variational/perturbative ideas or numerical investigations. There are four parameters: density, transverse and longitudinal dimensions, and scattering length. In this paper we explicate five parameter regions in which various types of 1D or 3D behavior occur in the ground state. Our treatment is based on a rigorous analysis of the many-body Schrödinger equation.
We show that the Lieb-Liniger model for one-dimensional bosons with repulsive δδδ-function interaction can be rigorously derived via a scaling limit from a dilute three-dimensional Bose gas with arbitrary repulsive interaction potential of finite scattering length. For this purpose, we prove bounds on both the eigenvalues and corresponding eigenfunctions of three-dimensional bosons in strongly elongated traps and relate them to the corresponding quantities in the Lieb-Liniger model. In particular, if both the scattering length aaa and the radius rrr of the cylindrical trap go to zero, the Lieb-Liniger model with coupling constant g∼a/r2g \sim a/r^2g∼a/r2 is derived. Our bounds are uniform in ggg in the whole parameter range 0≤g≤∞0\leq g\leq \infty0≤g≤∞, and apply to the Hamiltonian for three-dimensional bosons in a spectral window of size ∼r−2\sim r^{-2}∼r−2 above the ground state energy.
We consider two regimes where a trapped Bose gas behaves as a one-dimensional (1D) system. In the first one the Bose gas is microscopically described by 3D mean-field theory, but the trap is so elongated that it behaves as a 1D gas with respect to low-frequency collective modes. In the second regime we assume that the 1D gas is truly 1D and that it is properly described by the Lieb-Liniger model. In both regimes we find the frequency of the lowest compressional mode by solving the hydrodynamic equations. This is done by making use of a method which allows us to find analytical or quasianalytical solutions of these equations for a large class of models approaching very closely the actual equation of state of the Bose gas. We find an excellent agreement with the recent results of Menotti and Stringari obtained from a sum-rule approach.
Abstract The coordinate Bethe ansatz can be extended to a model, the Lieb–Liniger model, of a one-dimensional gas of Bosons interacting with repulsive δ-function potentials. It has attracted attention due to its relevance for experimental developments in the fields of ultracold gases and optical lattices. This chapter provides an exposition of the related classical nonlinear Schrödinger equation, followed by its generalization to the quantum model. It explores a limiting case, the Tonks-Girardeau gas. The δ-function potentials supply a kind of boundary condition on the wave functions allowing us to analyze the eigenfunctions of the Bethe ansatz, which are examined on the infinite line and for periodic boundary conditions. The latter leads to the Bethe ansatz equations. The solution of these equations is achieved in the thermodynamic limit for the ground state and for low-lying excited states.
We consider a trapped repulsive one-dimensional (1D) Bose gas at very low temperature. In order to study the collective modes of this strongly interacting system, we use a hydrodynamic approach, where the gas is locally described by the Lieb-Liniger model of bosons interacting via a repulsive delta potential. Solving the corresponding linearized hydrodynamic equations, we obtain the collective modes and concentrate more specifically on the lowest compressional mode. This is done by finding models, approaching very closely the exact equation of stae of the gas, for which the linearized hydrodynamic equations are exactly solvable. Results are in excellent agreement with those of the sum rule approach of Menotti and Stringari.
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.