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Physical Review A

Effective potentials for dilute Bose-Einstein condensates

John L. Bohn, Brett D. Esry, Chris H. Greene

Published Jul 1, 1998• 77 citations
Publisher

Abstract

We present a theoretical formulation of trapped, dilute Bose-Einstein condensates (BEC's) at zero temperature based on ordinary Schr\"odinger quantum mechanics. By a judicious choice of coordinates and of a variational trial wave function we reduce the many-atom problem to a linear Schr\"odinger equation that is easier to handle and interpret than the usual nonlinear Schr\"odinger equation of BEC theory. Ordinary quantum mechanics then reproduces, semiquantitatively, many of the main features of zero-temperature BEC, including the critical number of atoms in a condensate with negative scattering length. The procedure is similar in results, but completely different in spirit, to recent variational approaches to solving the nonlinear Schr\"odinger equation. Moreover, the present procedure represents a step in a systematic alternative method for computing quantitatively accurate wave functions for trapped bosonic atoms.

DOI
10.1103/physreva.58.584

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We present a theoretical formulation of trapped, dilute Bose-Einstein condensates (BEC's) at zero temperature based on ordinary Schr"odinger quantum mechanics. By a judicious choice of coordinates and of a variational trial wave function we reduce the many-atom problem to a linear Schr"odinger equation that is easier to handle and interpret than the usual nonlinear Schr"odinger equation of BEC theory. Ordinary quantum mechanics then reproduces, semiquantitatively, many of the main features of zero-temperature BEC, including the critical number of atoms in a condensate with negative scattering length. The procedure is similar in results, but completely different in spirit, to recent variational approaches to solving the nonlinear Schr"odinger equation. Moreover, the present procedure represents a step in a systematic alternative method for computing quantitatively accurate wave functions for trapped bosonic atoms.
For a trapped, dilute atomic gas of short-range, repulsive interactions at extremely low temperatures, when Bose–Einstein condensation is nearly complete, some special forms of the time-dependent condensate wave function and the pair-excitation function, the latter being responsible for phonon creation, are investigated. Specifically, (i) a class of external potentials Ve(r,t) that allow for localized, shape-preserving solutions to the nonlinear Schrödinger equation for the condensate wave function, each recognized as a solitary wave moving along an arbitrary trajectory, is derived and analyzed in any number of space dimensions; and (ii) for any such external potential and condensate wave function, the nonlinear integro-differential equation for the pair-excitation function is shown to admit solutions of the same nature. Approximate analytical results are presented for a sufficiently slowly varying trapping potential. Numerical results are obtained for the condensate wave function when Ve is a time-independent, spherically symmetric harmonic potential.
We evaluate the zero temperature properties and Bose-Einstein condensation (BEC) of hard sphere bosons in a harmonic trap using Quantum Monte Carlo methods. The chief purpose of this work is to go beyond the dilute limit, to test the limits of the Gross-Pitaevskii equation and related mean field approximations and to explore the role of interactions in determining the zero temperature properties of the trapped Bose gas. The density is increased by adjusting both the number of trapped bosons, N , and the s-wave scattering length, a, to study systems from the highly dilute region corresponding to early experiments in trapped BEC [2, 3] up to liquid He densities and beyond. Rather than relying on the perturbative Bogoliubov approximation (which assumes a large condensate fraction) to describe the condensate, we use the one-body density matrix (OBDM) formulation of BEC so that the properties of the condensate for systems with arbitrarily large interactions may be studied. In this formulation of BEC, condensate properties are obtained by diagonalizing the OBDM and obtaining the corresponding single particle “natural orbitals” and their occupation numbers for the system. The condensate wave function and condensate fraction are then obtained from the single particle orbital(s) with macroscopic occupation (N0 >> 1). Within this framework, we calculate the effects of interactions and increased density on the ground state energy, the density profile, the momentum distribution, the condensate fraction and condensate “wave-function” and several other properties.
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Bose-Einstein condensation is described in terms of the condensate wave function and the pair-excitation function, the latter being responsible for the existence of phonons. This minimal description in terms of these two functions is generalized to the case with an external potential. For a dilute gas with short-range pairwise repulsive interaction and at very low temperatures when the Bose-Einstein condensation is nearly complete, a partial differential equation is obtained for the condensate wave function and an integro-differential equation for the pair excitation. Experimentally, the external potential is used to trap the atoms, i.e., to keep them together. Since the trap is of macroscopic dimensions, the resulting external potential is often slowly varying. In these cases and when the condensate is in the lowest state, the partial differential equation for the condensate wave function and the integro-differential equation for the pair excitation are solved approximately for the case of a time-independent trap.
Bose–Einstein condensation has been realized as dilute atomic vapors. This achievement has generated immense interest in this field. This article review of recent theoretical research into the properties of trapped dilute-gas Bose–Einstein condensates. Among these properties, stability of Bose–Einstein condensates confined in traps is mainly discussed. Static properties of the ground state are investigated by using the variational method. The analysis is extended to the stability of two-component condensates. Time-development of the condensate is well-described by the Gross–Pitaevskii equation which is known in nonlinear physics as the no nlinear Schrödinger equation. For the case that the inter-atomic potential is effectively attractive, a singularity of the solution emerges in a finite time. This phenomenon which we call collapse explains the upper bound for the number of atoms in such condensates under traps.