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arXiv:cond-mat/0008025 · Physical Review A

Bose-Einstein condensation in shallow traps

J.-P. Martikainen

Published Mar 6, 2001• 21 citations
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Abstract

In this paper we study the properties of Bose-Einstein condensates in shallow traps. We discuss the case of a Gaussian potential, but many of our results also apply to the traps having a small quadratic anharmonicity. We show the errors introduced when a Gaussian potential is approximated with a parabolic potential; these errors can be quite large for realistic optical trap parameter values. We study the behavior of the condensate fraction as a function of trap depth and temperature and calculate the chemical potential of the condensate in a Gaussian trap. Finally we calculate the frequencies of the collective excitations in shallow spherically symmetric and one-dimensional traps.

DOI
10.1103/physreva.63.043602

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In this paper we study the properties of Bose-Einstein condensates in shallow traps. We discuss the case of a Gaussian potential, but many of our results also apply to the traps having a small quadratic anharmonicity. We show the errors introduced when a Gaussian potential is approximated with a parabolic potential; these errors can be quite large for realistic optical trap parameter values. We study the behavior of the condensate fraction as a function of trap depth and temperature and calculate the chemical potential of the condensate in a Gaussian trap. Finally we calculate the frequencies of the collective excitations in shallow spherically symmetric and one-dimensional traps.
The stability of Bose–Einstein condensates (BECs) loaded into a two-dimensional shallow harmonic potential well is studied. By using the variational method, the ground state properties for interacting BECs in the shallow trap are discussed. It is shown that the possible stable bound state can exist. The depth of the shallow well plays an important role in stabilizing the BECs. The stability of BECs in the shallow trap with the periodic modulating of atom interaction by using the Feshbach resonance is also discussed. The results show that the collapse and diffusion of BECs in a shallow trap can be controlled by the temporal modulation of the scattering length.
We study Bose—Einstein condensation in a linear trap with a dimple potential where we model dimple potentials by Dirac δ function. Attractive and repulsive dimple potentials are taken into account. This model allows simple, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schrödinger equation is used instead of the Gross—Pitaevski equation. We calculate the atomic density, the chemical potential, the critical temperature and the condensate fraction. The role of the relative depth of the dimple potential with respect to the linear trap in large condensate formation at enhanced temperatures is clearly revealed. Moreover, we also present a semi-classical method for calculating various quantities such as entropy analytically. Moreover, we compare the results of this paper with the results of a previous paper in which the harmonic trap with a dimple potential in 1D is investigated.
We study the instability of collective excitations of a three-dimensional Bose-Einstein condensate with repulsive and attractive interactions in a shallow trap designed as a quadratic plus a quartic potential. By using a correlated many-body theory, we determine the excitation modes and probe the critical behavior of collective modes, having a crucial dependence on the anharmonic parameter. We examine the power-law behavior of monopole frequency near criticality. In Gross-Pitaevskii variational treatment [Phys. Rev. Lett. 80, 1576 (1998)] the power-law exponent is determined as one-fourth power of $(1\ensuremath{-}\frac{A}{{A}_{\mathit{cr}}})$, AAA is the number of condensate atoms and Acr{A}_{\mathit{cr}}Acr​ is the critical number near collapse. We observe that the power-law exponent becomes 16\frac{1}{6}61​ in our calculation for the pure harmonic trap and it becomes 17\frac{1}{7}71​, for traps with a small anharmonic distortion. However for large anharmonicity the power law breaks down.
Abstract In two-dimensional traps, since the theoretical study of Bose–Einstein condensation will encounter the problem of divergence, the actual contribution of the divergent terms is often estimated in some indirect ways with the accuracy to the leading order. In this paper, by using an analytical continuation method to solve the divergence problem, we obtain the analytical expressions of critical temperature and condensate fraction for Bose gases in a 2D anisotropic box and harmonic trap, respectively. They are consistent with or better than previous studies. Then, we further consider the nonvanishing chemical potential, and obtain the expressions of chemical potential and more precise condensate fraction. These results agree with the numerical calculation well, especially for the case of harmonic traps. The comparison between the grand canonical and canonical ensembles shows that our calculation in the grand canonical ensemble is reliable.
We study Bose-Einstein condensation in a harmonic trap with a dimple potential. We specifically consider the case of a tight and deep dimple potential which is modeled by a Dirac $\ensuremath{\delta}$ function. This allows for simpler, explicit numerical and analytical investigations of noninteracting gases. Thus, the Schr\"odinger equation is used instead of the Gross-Pitaevski equation. Calculating the atomic density, chemical potential, critical temperature, and condensate fraction, the role of the relative depth of the dimple potential with respect to the harmonic trap in large condensate formation at enhanced temperatures is clearly revealed.
We study the thermodynamic behavior of an ideal gas of bosons trapped in a three-dimensional anisotropic harmonic-oscillator potential. The condensate fraction as well as the specific heat is calculated using the Euler-Maclaurin approximation. For a finite number of particles there is no phase transition, but there is a well-defined temperature at which the condensation starts. We also consider condensation in lower dimensions, and, for one-dimensional systems, we discuss the dependence of the condensate fraction and heat capacity on the ensemble used.