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Physical Review Letters

Bose Condensation of Idealized Spin-Polarized Atomic Hydrogen in Equilibrium

William J. Mullin

Published May 26, 1980• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.44.1420

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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 calculation is presented of the relaxation rate T 1 -1 between the lower two branches of the atomic hyperfine states in spin-polarized atomic hydrogen H↓ below the Bose-Einstein condensation temperature T 0 . We find that T 1 -1 has a maximum between T =0 and T 0 , which is due to appearance of additional relaxation processes mediated by atoms in the condensed state. Theoretical estimation of T 1 is of the order of a minute with a magnetic field ∼100 kG and with density of H ↓∼10 18 atoms/cm 3 .
A study is made of the quantum mechanics of a gas of spin-polarized (b-state) atomic hydrogen confined in a three-dimensional dynamic (ac) magnetic trap. A previous classical treatment indicated that evaporative cooling from such a trap could be used to attain conditions for Bose-Einstein condensation (BEC). An approximate solution is developed for the quantum mechanics of a single atom in an ac trap. This solution is then used to treat the statistical mechanics of the BEC in an ac trap. The ideal-gas limit is briefly considered, and this is compared with the more realistic approximation of a nonideal gas described by the finite-temperature Hartree-Fock equations with an s-wave scattering-length interaction. The threshold temperature for BEC differs only slightly from the value for an ideal gas. However, the nature of the condensate is strongly dependent on the scattering length.
Spin-polarized, atomic hydrogen at temperatures below 1K shows promise of becoming the first physically realizable example of a degenerate, weakly interacting Bose gas. Because of the hyperfine structure of the hydrogen atom the Bose condensed state is expected to exhibit novel magnetic properties such as spin wave excitations of the superfluid and possibly a coherent precessing, spontaneous magnetization. This paper discusses a variety of theoretical problems which arise in the study of atomic hydrogen gas. Theories are presented for spin relaxation of the hyperfine levels and for the recombination of H-atoms into H2 molecules. The problem of the binding of H atoms to a surface of superfluid 4He is reviewed together with the theory of surface superfluidity in the hydrogen film proposed by Edwards and Mantz. Finally we consider the problem of superfluidity in the bulk atomic hydrogen gas. We describe the theory of Siggia and Ruckenstein of double Bose condensation into the two lowest hyperfine states and speculate about the possible observable consequences of this phenomenon.
We investigate the thermodynamics of a spin-1 Bose gas with fixed magnetization including the quadratic Zeeman energy shift. Our calculations are based on the grand canonical description for the ideal gas and the classical field approximation for atoms with ferromagnetic and antiferromagnetic interactions. We confirm the occurrence of a double phase transition in the system that takes place due to two global constraints. We show analytically for the ideal gas how critical temperatures and condensed fractions are changed by a nonzero magnetic field. The interaction strongly affects the condensate scenario below the second critical temperature. The effect imposed by interaction energies becomes diminished in high magnetic fields where condensation of both ferromagnetic and antiferromagnetic atoms agrees with the ideal gas results.
Spin-polarized atomic hydrogen continues to be one of the most promising candidates for Bose condensation of an atomic system. In contrast to liquid helium, hydrogen is gaseous and therefore its density can be changed to vary its behavior from the weakly to the strongly interacting Bose gas. Until now, efforts to Bose condense hydrogen have been thwarted by recombination and relaxation phenomena. After a long introduction to the subject, two promising approaches to observe quantum degenerate behavior are discussed: the microwave trap and a two-dimensional gas of hydrogen. Introduction Since the stabilization of atomic hydrogen as a spin-polarized gas (H↓), reported in 1980 [1], there has been a continuous effort to observe Bose–Einstein condensation (BEC) or other effects of quantum degeneracy in this Bose gas. This challenge has not yet been realized due to the instability of H↓ towards recombination to H 2 or relaxation among hyperfine states as the conditions for BEC are approached. Although this difficulty is formidable, hydrogen presents a unique opportunity to study BEC and the related superfluidity, as it remains a gas to T = 0 [2]. By comparison, 4 He, the only experimentally observed boson fluid, is a strongly interacting superfluid Bose liquid, with little flexibility for varying its density. As a result of the gaseous nature of hydrogen, its density, n , can be varied over several orders of magnitude with the possibility of studying BEC and its relationship to superfluidity, ranging from the weakly to the strongly interacting boson gas.
The static fluctuation approximation (SFA) is used to study the condensate fraction and the specific heat capacity of finite two-dimensional spin-polarized atomic hydrogen. It is found that Bose-Einstein condensation occurs in this system. The transition temperature at different densities decreases as the number of particles of the system increases. At low density, a sharp peak in the specific heat capacity is observed at the transition temperature. On the other hand, as the density of the system increases, the transition temperature becomes no longer well-defined, and a hump is observed in the specific heat capacity around the transition temperature. A qualitative comparison of our results to published results for finite Bose systems shows good agreement.