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Physical review. B, Condensed matter

Atomic hydrogen in an inhomogeneous magnetic field: Density profile and Bose-Einstein condensation

Victor V. Goldman, Isaac F. Silvera, Anthony J. Leggett

Published Sep 1, 1981• 175 citations
Publisher

Abstract

The density profiles for an interacting gas of atomic hydrogen in an inhomogeneous axial magnetic field are determined by extending the Gross-Pitaevskii theory to finite temperature. The profiles can be used as an identifying feature of Bose-Einstein condensation for all $T$. Radial inhomogeneities in the field are also treated.

DOI
10.1103/physrevb.24.2870

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Physical review. B, Condensed matter•Mar 1, 1989•59 citations
The density profiles for an interacting gas of atomic hydrogen in an inhomogeneous axial magnetic field are determined by extending the Gross-Pitaevskii theory to finite temperature. The profiles can be used as an identifying feature of Bose-Einstein condensation for all TTT. Radial inhomogeneities in the field are also treated.
Atomic hydrogen (H) has recently been created in a long-lived state referred to as spin-polarized atomic hydrogen (H↓). A moderate density gas (n≃1017 atoms/ cc) has been produced at temperatures T≃400 mK. H↓ is predicted to be the only atomic substance that remains gaseous to T = 0 K; current experiments are consistent with this picture. Since the H↓ atoms behave as composite bosons the gas is expected to have a Bose-Einstein condensation (BEC) and related superfluidity at a sufficiently high density or low temperature. H↓ is expected to display spectacular static and dynamic properties. Since the gas is polarized, the dominant electronic magnetization, M⃗e, is proportional to the atomic density. In an inhomogeneous magnetic field M⃗e will also be highly inhomogeneous. BEC will be spatially localized and the static M⃗e should be an identifying feature of this state. The low-lying excitation spectrum should not only feature electronic and nuclear spin-transitions, but should also have magnetic translational modes of the gas. The experimental technique for stabilization of H↓ as well as many of the predicted magnetic properties are discussed.
After a brief historical introduction to Bose-Einstein condensation and Fermi degeneracy, we discuss theoretical results we have recentely obtained for trapped degenerate quantum gases by means of a thermal field theory approach. In particular, by using Gross-Pitaevskii and Bogoliubov-Popov equations, we consider thermodynamical properties of two Bosonic systems: a gas of Lithium atoms and a gas of Hydrogen atoms. Finally, we investigate finite-temperature density profiles of a dilute Fermi gas of Potassium atoms confined in a harmonic potential.
The energy of interacting spin-polarized atomic hydrogen (${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$) on a helium surface and the density dependence of the adsorption energy have been calculated, permitting a realistic determination of the adsorption isotherms. Surface interactions are shown to allow Bose-Einstein condensation in the gas. At the Bose-Einstein condensation the surface coverage attains a maximum which is essentially independent of the isotherm temperature. Also calculated is the stability condition for surfaces in magnetic fields. Easily met requirements are found.
A model of spin-polarized hydrogen (H\ensuremath{\uparrow}) is treated in which interactions between atoms are neglected while the single-atom Zeeman and hyperfine interactions are treated exactly. These magnetic terms in the Hamiltonian are found to affect substantially the Bose-Einstein condensation and the various thermodynamic variables. Computations are discussed of the condensation temperature, condensate density, and specific heat in order to indicate how changes in magnetic field strength might be expected to affect future measurements on this quantum system.
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.