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Physical Review A

Finite number of trapped bosons interacting through the harmonic Calogero interaction in one dimension

Barnali Chakrabarti, Tapan Kumar Das

Published Jan 27, 2010• 4 citations
Publisher

Abstract

We consider a finite number of trapped bosons in one dimension (1D) interacting through the harmonic Calogero interaction. Using an exact analytic solution for the energy eigenvalues we calculate the condensate fraction, ground-state energy, and specific heat of the system. For weak interaction the behavior of thermodynamic properties is close to that of noninteracting trapped bosons.

DOI
10.1103/physreva.81.015601

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We consider a finite number of trapped bosons in one dimension (1D) interacting through the harmonic Calogero interaction. Using an exact analytic solution for the energy eigenvalues we calculate the condensate fraction, ground-state energy, and specific heat of the system. For weak interaction the behavior of thermodynamic properties is close to that of noninteracting trapped bosons.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.
We consider systems of a small number of interacting bosons confined to harmonic potentials in one and two dimensions. By exact numerical diagonalization of the many-body Hamiltonian we determine the low-lying excitation energies and the ground-state energy and density profile. We discuss the dependence of these quantities on both interaction strength ggg and particle number NNN. The ground-state properties are compared to the predictions of the Gross-Pitaevskii equation, and the agreement is surprisingly good even for relatively low particle numbers. We also calculate the specific heat based on the obtained energy spectra.
We study the Bose-Einstein condensation phenomenon in a two-dimensional (2D) system of bosons subjected to a harmonic-oscillator-type confining potential. The interaction among the 2D bosons is described by a \ensuremath{\delta} function in configuration space. Solving the Gross-Pitaevskii equation within the two-fluid model we calculate the condensate fraction, ground-state energy, and specific heat of the system. Our results indicate that interacting bosons have similar behavior to those of an ideal system for weak interactions.
We consider the energetics and structural properties of a many-particle system in one dimension with pairwise contact interactions confined in a parabolic external potential. To render the problem analytically solvable, we use the harmonic approximation scheme at the level of the Hamiltonian. We investigate the scaling with particle number of the ground-state energies for systems consisting of identical bosons or fermions. We then proceed to focus on bosonic systems and make a detailed comparison with known exact results in the absence of the parabolic external trap for three-body systems. We also consider the thermodynamics of the harmonic model, which turns out to be similar for bosons and fermions due to the lack of degeneracy in one dimension.
Exact solutions to many-body interacting systems of both bosonic and fermionic particles confined to harmonic potential in an arbitrary number of dimensions are given. Energy levels and their degeneracies for trapped identical particles interacting via harmonic forces are calculated. This specific form of the interaction allows for analytical solutions. The mutual interaction, attractive or repulsive, modifies significantly the properties of the considered system. For a large number of particles the interaction essentially results in a frequency shift. Statistical properties (e.g., microcanonical and grand canonical partition functions) as well as some illustrative, physically relevant examples are discussed. Our results give an unusual opportunity for further studies of interacting systems in the framework of the exactly soluble model.