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Physical review. B, Condensed matter

Density profile of the weakly interacting Bose gas confined in a potential well: Nonzero temperature

J. Oliva

Published Mar 1, 1989• 59 citations
Publisher

Abstract

A theory is developed for the density profile of a weakly interacting Bose gas confined within an arbitrary potential well at nonzero temperature. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (H\ensuremath{\downarrow}). The approach taken is conceptually simpler and of comparable accuracy as compared to previous approaches. It is based on the statement of constancy of the chemical potential in equilibrium. This in turn involves the internal chemical potential, a functional of the density, which we treat in a local-density approximation. For a given temperature we develop a form for the local internal chemical potential for a wide range of density using known low-density, intermediate-density (i.e., near the Bose-condensation density), and high-density analytic forms, with appropriate smooth interpolants connecting these three forms. Density profiles are evaluated within the Bogoliubov approximation for the local internal chemical potential for parameters appropriate to magnetically confined H\ensuremath{\downarrow} for both symmetric linear and quadratic one-dimensionally varying wells. Comparison is made to previous work. The nonanalytic region of the density profile near the Bose-condensation density is analyzed. We also calculate density profiles including the next-highest-order interaction correction (i.e., beyond Bogoliubov) in the internal chemical potential. For conditions relevant to hypothetical H\ensuremath{\downarrow} at moderate densities this leads to a small but discernable effect on the density profile.

DOI
10.1103/physrevb.39.4197

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A theory is developed for the density profile of a weakly interacting Bose gas confined within an arbitrary potential well at nonzero temperature. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (H\ensuremath{\downarrow}). The approach taken is conceptually simpler and of comparable accuracy as compared to previous approaches. It is based on the statement of constancy of the chemical potential in equilibrium. This in turn involves the internal chemical potential, a functional of the density, which we treat in a local-density approximation. For a given temperature we develop a form for the local internal chemical potential for a wide range of density using known low-density, intermediate-density (i.e., near the Bose-condensation density), and high-density analytic forms, with appropriate smooth interpolants connecting these three forms. Density profiles are evaluated within the Bogoliubov approximation for the local internal chemical potential for parameters appropriate to magnetically confined H\ensuremath{\downarrow} for both symmetric linear and quadratic one-dimensionally varying wells. Comparison is made to previous work. The nonanalytic region of the density profile near the Bose-condensation density is analyzed. We also calculate density profiles including the next-highest-order interaction correction (i.e., beyond Bogoliubov) in the internal chemical potential. For conditions relevant to hypothetical H\ensuremath{\downarrow} at moderate densities this leads to a small but discernable effect on the density profile.
A theory is developed for the low-temperature density profile of a weakly interacting normal Fermi gas confined within an arbitrary potential well. This problem is of interest in connection with magnetically confined spin-aligned atomic deuterium (${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$). The approach used is based on the statement of constancy of the chemical potential in equilibrium. This involves the internal chemical potential, a functional of the density, which we treat within the local-density approximation. We develop an accurate approximate form for the local internal chemical potential: For a given low temperature, it is taken equal to the known forms at low density (classical) and higher density (quantum); for intermediate density we use an appropriate interpolant which smoothly connects the low- and high-density forms. Finite-temperature density profiles are evaluated for the ideal Fermi gas for parameters appropriate to magnetically confined ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ for both symmetric linear and quadratic one-dimensionally varying wells. We also evaluate density profiles including the leading-order interparticle interaction effect. For conditions relevant to hypothetical ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ at moderate densities this effect is found to be small but measurable.
A theory is developed for the nonuniform ground-state density profile of weakly interacting quantum gases confined within an arbitrary potential well. Both Bose-Einstein and Fermi-Dirac cases are considered. This problem is of interest in connection with magnetically confined spin-aligned atomic hydrogen (${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$) and deuterium (${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$). The approach is based on a macroscopic formulation involving the principle of constancy of the chemical potential throughout the system. The main input is the internal chemical potential for which we use a local-density approximation (LDA). In turn we use for the LDA internal chemical potential standard results for the uniform interacting quantum gases. It is shown that generally, the systems mentioned develop macroscopically sharp surfaces (again, while in the gas phase). Density profiles are analytically calculated for the ideal Fermi gas and the leading-order weakly interacting Bose gas for the case of an arbitrary, symmetric power-law potential well. We also calculate the density profile including the next-highest-order correction in the interparticle interaction in the Bose case and the leading-order correction in the interparticle interaction in the Fermi case. The accuracy of the LDA density profiles is assessed. Numerical results are given appropriate for ${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$ and ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$. For weakly interacting ${\mathrm{D}}_{\mathrm{\ensuremath{\downarrow}}}$ a notable contrast between the case of single hyperfine level occupation and the case of occupation of the lowest three hyperfine levels is pointed out.
An analytical description of the low temperature behaviour of a trapped interacting Bose gas is presented by using a simple approach that is based on the principle of the constancy of chemical potentials in equilibrium and the local-density approximation. Several thermodynamic quantities, which include the ground-state fraction, chemical potential, total energy, entropy and heat capacity, are derived analytically. It is shown that the results obtained here are in excellent agreement with the experimental data and the theoretical predictions based on the numerical calculation. Meanwhile, by selecting a suitable variable, the divergent problem existing in some papers is solved.
Abstract We derive and analyze the coupled equations of quadratic approximation of the Bogoliubov model for a weakly interacting Bose gas. The first equation determines the condensate density as a variational parameter and ensures the minimum of the grand thermodynamic potential. The second one provides a relation between the total number of particles and chemical potential. Their consistent theoretical analysis is performed for a number of model interaction potentials including contact (local) and nonlocal interactions, where the latter provide nontrivial dependencies in momentum space. We demonstrate that the derived equations have no solutions for the local potential, although they formally reproduce the well-known results of the Bogoliubov approach. At the same time, it is shown that these equations have the solutions for physically relevant nonlocal potentials. We show that in the regimes close to experimental realizations with ultracold atoms, the contribution of the terms originating from the quadratic part of the truncated Hamiltonian to the chemical potential can be of the same order of magnitude as from its c -number part. Due to this fact, in particular, the spectrum of single-particle excitations in the quadratic approximation acquires a gap. The issue of the gap is also discussed.
The low-temperature properties of a weakly interacting Bose gas in weak magnetic field were studied by using the conclusions of local-density approximation method and invariability principle of chemical potential in inhomogenous system.The analytical expressions of the ground-state fraction,boundary magnetic potential in condensation region,chemical potential,total energy and heat capacity were derived.The effects of magnetic field and interparticle interactions on properties of the system were discussed.It is shown that when repulsive interactions are kept constant,the effects of variational magnetic filed on condensation volume,total energy and heat capacity of the system have boundary temperature respectively.The repulsive interactions make the chemical potential of the system always positive and increase with the decreasing temperature,the total energy of the system is lifted a platform and a maximum is obtained near condensation temperature;the heat capacity of the system have the reversal point and the maximum point with the changing temperature;and the total energy and heat capacity decrease rapidly from their own maximums with the temperature approaching to condensation temperature.
The thermodynamical properties of interacting Bose atoms in a harmonic potential are studied within the mean-field approximation. For weak interactions, the quantum statistics is equivalent to an ideal gas in an effective mean-field potential. The eigenvalue of the Gross–Pitaevskii equation is identified as the chemical potential of the ideal gas. The condensation temperature and density profile of atoms are calculated. It is found that the critical temperature Tc decreases as the interactions increase. Below the critical point, the condensation fraction exhibits a universal relation of N0/N = 1-(T/Tc)γ, with the index γ ≈ 2.3 independent of the interaction strength, the chemical potential, as well as the frequency of the confining potential.