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Physical Review A

Theory of Bose-Einstein condensation of atomic hydrogen in a dynamic trap

R. V. E. Lovelace, T. J. Tommila

Published May 1, 1987• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.35.3597

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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.
We report experiments on evaporative cooling of spin-polarized atomic hydrogen gas confined in a cryogenic magnetic trap, and present a model of the trapped gas based on a generalized truncated Boltzmann approximation for the phase-space distribution function. The model takes into account the dimension of the evaporation [three-dimensional (3D) or 1D] and the time dependence of both the depth and the shape of the confining potential. Our observations are consistent with 1D evaporative cooling. To attain maximal phase-space density, we used the model assuming 1D evaporative cooling to optimize the evaporation procedure. With this work, the low dimension of evaporation was identified as the bottleneck thus far preventing us from achieving Bose-Einstein condensation in trapped atomic hydrogen.
Experimental efforts to produce a Bose Einstein condensed (BEC) gas of spin-polarized atomic hydrogen have been intensely pursued since 1980. Initial efforts studied hydrogen in contact with superfluid helium surfaces. In order to avoid the limiting wall recombination a static magnetic trap was developed which utilizes magnetic barriers to isolate atoms from the van der Wals walls. In this approach traps can be filled to about 10{sup 14} atoms/cm{sup 3} for which the critical temperature is in the region of 20-30 {mu}K. However, approaching BEC conditions, a new instability dominates the behavior. The authors have developed a new type of electromagnetic trap, a microwave (mw) trap in which the trapped atoms are in the ground spin-state so that at low temperature relaxation to a barrier state is strongly suppressed and BEC should be attainable. The shallow mw trap has recently been demonstrated experimentally for the cesium atom. Because the mw trap is so shallow it is necessary to precool the atoms for loading. The authors are constructing a hybrid trap in which the atoms are cooled in a static trap and then transferred to a mw trap where they can be further evaporatively cooled in an effort to achieve BEC.more » This trap should have density limits in the region of 10{sup 16} to 10{sup 17}/cm{sup 3} due to three-body recombination. A second, promising approach is in a 2-D geometry for which in finite sized systems BEC is allowed. Hydrogen atoms are confined to a superfluid helium surface by the absorption potential to form a 2-D gas. Although recombination is an important process on helium surfaces, recently the authors have shown that only a very small fraction of the recombination energy is coupled into the surface. With an appropriate geometry the recombination energy can be removed from the sample region to suppress heating to a negligible level. Furthermore in 2-D a magnetic field can have a maximum so that atoms can be confined to a very small area.« less
We study the evaporative cooling of magnetically trapped atomic hydrogen on the basis of the kinetic theory of a Bose gas. The dynamics of trapped atoms is described by the coupled differential equations, considering both the evaporation and dipolar spin relaxation processes. The numerical time-evolution calculations quantitatively agree with the recent experiment of Bose-Einstein condensation with atomic hydrogen. It is demonstrated that the balance between evaporative cooling and heating due to dipolar relaxation limits the number of condensates to $9\ifmmode\times\else\texttimes\fi{}{10}^{8}$ and the corresponding condensate fraction to a small value of 4% as observed experimentally.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
A simple picture describes the results of recent treatments of partially--condensed, dilute, trapped Bose gases at temperature T > 0. The condensate wave function is nearly identical to that of a T = 0 condensate with the same number of condensate atoms, Ν0.The cloud of non-condensed atoms is described by the statistical mechanics of an ideal Bose gas in the com- bined potentials of the magnetic trap and the cloud-condensate interaction.We provide a physical motivation for this result, show how it emerges in the Hartree-Fock-Bogo liubov-Popov approximation, and explore some of its implications for future experiments.
A gas of spin-polarized atomic hydrogen can be prepared in the upper hyperfine states and loaded into a static magnetic trap. Evaporative cooling and magnetic compression of such a gas can produce temperatures of 30 \ensuremath{\mu}K and densities of 1014{10}^{14}1014 ${\mathrm{cm}}^{\ensuremath{-}3}$. Under these conditions a Bose-Einstein condensate may form.