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Physics and Chemistry of Liquids

Momentum Distribution for an Inhomogeneous Bose Fluid atT= 0 with Both Confinement and Particle Interactions Modelled as Harmonic

Norman Henry March, M. P. Tosi

Published Mar 1, 2001• 1 citation
Publisher

Abstract

Because of experimental interest in the effects of particle interactions in Bose condensates, such as a dilute atomic vapour of 87Rb, we have worked out the momentum distribution in an N-Boson many-body assembly at T = 0, for which both confinement and interbosonic interactions are modelled as purely harmonic. The half-width of the Gaussian momentum distribution is displayed as a function of N and of the strength and sign of the harmonic interactions. The bosonic kinetic energy is finally treated.

DOI
10.1080/00319100108030338

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Because of experimental interest in the effects of particle interactions in Bose condensates, such as a dilute atomic vapour of 87Rb, we have worked out the momentum distribution in an N-Boson many-body assembly at T = 0, for which both confinement and interbosonic interactions are modelled as purely harmonic. The half-width of the Gaussian momentum distribution is displayed as a function of N and of the strength and sign of the harmonic interactions. The bosonic kinetic energy is finally treated.
Abstract In earlier work, the momentum distribution of harmonically confined bosons has been calculated for interparticle interactions which are also harmonic for an arbitrary number N of repelling Bose particles. After a brief discussion of the pair function in this model, attention is also given, but now for N = 2 , to the single-particle density for the repulsive interaction potential λ / r 12 2 , where r 12 is the interboson distance and λ the repulsive coupling strength. The single-particle density is displayed for specific values of λ , with attention focusing on the limit of strong repulsive coupling. Throughout, the focus is on the fingerprints of Wigner bosonic molecules.
We use a semiclassical two-fluid model to study the momentum distribution of a Bose-Einstein condensed gas with repulsive interactions inside a harmonic trap at finite temperature, with specific focus on atomic hydrogen. We give particular attention to the average kinetic energy, which is almost entirely associated with the thermal cloud. A nonlinear dependence of the kinetic energy on temperature is displayed, affording a precise way to assess the temperature of the gas. We also show that the kinetic energy increases with the strength of the interactions, reflecting an enhanced rate of depletion of the condensate with increasing temperature.
We investigate the low energy excitations of a dilute atomic Bose gas confined in a harmonic trap of frequency ${\ensuremath{\omega}}_{0}$ and interacting with repulsive forces. The dispersion law $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}({2n}^{2}+2n\ensuremath{\ell}+3n+\ensuremath{\ell}{)}^{1/2}$ for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}(2n+\ensuremath{\ell})$ of the noninteracting harmonic oscillator model. Here nnn is the number of radial nodes and $\ensuremath{\ell}$ is the orbital angular momentum. The effects of the kinetic energy pressure are estimated using a sum rule approach. Results are also presented for deformed traps and attractive forces.
The momentum distribution function of a parabolically confined gas of bosons with harmonic interparticle interactions is derived. In the Bose-Einstein condensation region, this momentum distribution substantially deviates from a Maxwell-Boltzmann distribution. It is argued that the determination of the temperature of the boson gas from the Bose-Einstein momentum distribution function is more appropriate than the currently used fitting to the high-momentum tail of the Maxwell-Boltzmann distribution.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
In the Thomas–Fermi (TF) regime, S. Stringari [Phys. Rev. Lett., 77, 2360 (1996)] established the hydrodynamic normal modes of an interacting dilute Bose-condensed fluid confined in an isotropic harmonic trap. Here, we extend this treatment beyond the TF condensate using the work of B. Hu, G. Huang and Y. Ma [Phys. Rev. A, 69, 063608 (2004)] as a starting point. Both numerical results for low-lying eigenfrequencies, for various angular momentum quantum numbers ℓ and n = 0, and also samples of corresponding eigenfunctions, are presented. Finally, eigenfrequencies are also given for some collective excitations in an axially symmetrical trap.