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Physical Review A

Bose-Einstein condensation in low-dimensional traps

Vanderlei Salvador Bagnato, Daniel Kleppner

Published Dec 1, 1991• 385 citations
Publisher

Abstract

We demonstrate the possibility of Bose-Einstein condensation (BEC) of an ideal Bose gas confined by one- and two-dimensional power-law traps. One-dimensional systems display BEC in traps that are more confining than parabolic: U(x)\ensuremath{\sim}${\mathit{x}}^{\mathrm{\ensuremath{\eta}}}$,\ensuremath{\eta}2. Two-dimensional systems display BEC for any finite value of \ensuremath{\eta}. A possible experimental configuration for a two-dimensional trap is described.

DOI
10.1103/physreva.44.7439

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We show that the critical temperature of a one-dimensional gas confined by a power-law potential should be lower than that in the paper of Bagnato and Kleppner [Phys. Rev. A 44, 7439 (1991)]. Moreover, a sketch of the critical temperature is given in greater detail.

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We demonstrate the possibility of Bose-Einstein condensation (BEC) of an ideal Bose gas confined by one- and two-dimensional power-law traps. One-dimensional systems display BEC in traps that are more confining than parabolic: U(x)\ensuremath{\sim}x\ensuremathη{\mathit{x}}^{\mathrm{\ensuremath{\eta}}}x\ensuremathη,\ensuremath{\eta}2. Two-dimensional systems display BEC for any finite value of \ensuremath{\eta}. A possible experimental configuration for a two-dimensional trap is described.
Bose-Einstein condensation (BEC) of an ideal gas is investigated for a finite number of particles. In three dimensions, we find a transition temperature which is lower than in the thermodynamic limit. Lowering the dimension increases the transition temperature and is therefore favorable for BEC. This is in contrast to the standard result obtained in the thermodynamic limit which states that BEC is not possible in, e.g., a one-dimensional (1D) harmonic potential. As a result, 1D atom traps, such as radially tightly confining magnetic traps or optical dipole traps, are promising for studying BEC. \textcopyright{} 1996 The American Physical Society.
We examine the possibility of Bose-Einstein condensation in one-dimensional interacting Bose gas subjected to confining potentials of the form ${V}_{\mathrm{ext}}{(x)=V}_{0}(|x|{/a)}^{\ensuremath{\gamma}},$ in which $\ensuremath{\gamma}<2,$ by solving the Gross-Pitaevskii equation within the semiclassical two-fluid model. The condensate fraction, chemical potential, ground state energy, and specific heat of the system are calculated for various values of interaction strengths. Our results show that a significant fraction of the particles is in the lowest energy state for a finite number of particles at low temperature, indicating a phase transition for weakly interacting systems.
Bose-Einstein condensation is known to occur in one-dimensional power-law potentials $V(x)\ensuremath{\propto}|x{|}^{\ensuremath{\eta}}$ as a true phase transition for \ensuremath{\eta}2, but only if there are no interactions between the particles. We show, via a path-integral quantum Monte Carlo scheme, that for both \ensuremath{\eta}2 and \ensuremath{\eta}=2 the spatial distribution of finite numbers of hard-core bosons suddenly becomes bimodal below a certain temperature, with a central condensate of particles distributed according to the lowest single-particle eigenstate. At still lower temperatures, the hard-core interactions cause the single-particle ground state to become a poor description for the interacting gas. If \ensuremath{\eta}2, the energy per particle undergoes a sudden decrease near the same temperature at which the bimodal distribution appears. It is found that this drop in energy disappears if the range of the interaction potential is sufficiently large.
The critical temperature for Bose-Einstein condensation, ground-state fraction, and heat capacity of an ideal gas of Bose particles that are confined by an n-dimensional generic power-law potential trap are derived. The conditions for Bose-Einstein condensation and the discontinuous conditions of the heat capacity at critical temperature are obtained. All these quantities and conditions are found to vary markedly with the dimensionality of space, characteristics of particles, and shape of the external potential. The results obtained here are universal.