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Journal of Physics A Mathematical and General

Bose–Einstein condensation of atomic hydrogen Feynman path integral approach

Wattana Lim, Virulh Sa‐Yakanit

Published Apr 28, 2004• 1 citation
Publisher

Abstract

We study the ground state properties of Bose–Einstein condensation of atomic hydrogen in the Ioffe–Pritchard magnetic trap using many-body Feynman path integral theory, which leads to the calculation of the ground state energy and the wavefunction. We also calculate the size, peak condensate density and the value of the ground state energy, which are in good agreement with the experimental results obtained by laser spectroscopy of the 1S–2S transition.

DOI
10.1088/0305-4470/37/19/012

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We study the ground state properties of Bose–Einstein condensation of atomic hydrogen in the Ioffe–Pritchard magnetic trap using many-body Feynman path integral theory, which leads to the calculation of the ground state energy and the wavefunction. We also calculate the size, peak condensate density and the value of the ground state energy, which are in good agreement with the experimental results obtained by laser spectroscopy of the 1S–2S transition.
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A Bose–Einstein condensate in a double-well potential is considered by using the Feynman path integral theory combined with a variational approach. The system consists of N interacting bosons confined in the double well which is taken as a harmonic potential with a Gaussian barrier. The calculation is carried out within the first cumulant approximation measured with respect to the harmonic action containing variational parameters. Assuming a separable expression for the trial action corresponds to the usual mean field approximation. Performing the variational calculations, we obtain analytical results for the ground state energy and its condensate wavefunction. Using a projection onto the odd contributions, the first excited state energy and condensate wavefunction are determined, too. We find very good agreement with a full numerical solution of the Gross–Pitaevskii equation over the full range of potential parameters.
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We report observation of Bose-Einstein condensation (BEC) of a trapped, dilute gas of atomic hydrogen. The condensate and normal gas are studied by two-photon spectroscopy of the 1S1S1S- 2S2S2S transition. Interactions among the atoms produce a shift of the resonance frequency proportional to density. The condensate is clearly distinguished by its large frequency shift. The peak condensate density is $4.8\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\times\else\texttimes\fi{}{10}^{15}{\mathrm{cm}}^{\ensuremath{-}3}$, corresponding to a condensate population of 109{10}^{9}109 atoms. The BEC transition occurs at about $T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}50\ensuremath{\mu}\mathrm{K}$ and $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1.8\ifmmode\times\else\texttimes\fi{}{10}^{14}{\mathrm{cm}}^{\ensuremath{-}3}$.
The dilute weakly interacting alkali gas was shown to exhibit the phase transition called Bose-Einstein condensation for the first time in the experiment of Wieman's group in 1995. in this thesis, the ground state properties of the alkali gas trapped in an anisotropic magnetic field are investigated by using the variational method in path integral formalism. The interatomic interaction is approximated by the delta function which is short-range potential. The approximated density matrix is derived. Consequently the ground state energy and wavefunction are obtained. The ground state energy is then compared to the result obtained by the mean field approach both analytically and numerically. The result is shown to equivalent to the variational mean field approach. In addition, a comparison between path integral approach and the numerical mean field approach is shown to give no significant difference. The ground state wavefunction is extracted from the weighting function. Substitute the variational parameters into the expression of wavefunction one can see that the result is close to the numerical method.