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Physical Review A

Configuration-interaction studies of Bose-Einstein condensates

Tommy Vänskä, Dage Sundholm, Markus Lindberg

Published Feb 26, 2007• 3 citations
Publisher

Abstract

A direct configuration-interaction (CI) approach for studies of Bose-Einstein condensates has been developed and implemented. The reference state is obtained by solving the Hartree-Fock equation in a nonorthogonal Gaussian basis. The CI problem is solved by diagonalizing the Hamiltonian constructed in a many-body basis generated using the orthogonal canonical orbital basis obtained in a Hartree-Fock calculation. For Hamiltonian matrices, which are too large for full diagonalization, a direct iterative diagonalization scheme is employed for obtaining the lowest-lying eigenstates. The computational methods were tested by applications to Bose-Einstein condensates using various isotropic and anisotropic trap geometries.

DOI
10.1103/physreva.75.023621

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A direct configuration-interaction (CI) approach for studies of Bose-Einstein condensates has been developed and implemented. The reference state is obtained by solving the Hartree-Fock equation in a nonorthogonal Gaussian basis. The CI problem is solved by diagonalizing the Hamiltonian constructed in a many-body basis generated using the orthogonal canonical orbital basis obtained in a Hartree-Fock calculation. For Hamiltonian matrices, which are too large for full diagonalization, a direct iterative diagonalization scheme is employed for obtaining the lowest-lying eigenstates. The computational methods were tested by applications to Bose-Einstein condensates using various isotropic and anisotropic trap geometries.
I specialize the Hartree-Fock, random-phase, and configuration-interaction approximations to a system of interacting bosons in an external potential. This approach affords a more direct connection to the methods and language of atomic-structure calculations. The derivations and equations parallel those found in atomic structure and are contrasted with the more commonly used Bogoliubov approach for bosons. I present a numerical method for solving the Hartree-Fock and random-phase equations to obtain the low-lying excitation energies for trapped alkali-metal atoms.
A self-consistent mean-field (Hartree-Fock) theory for bosons for temperature T>0 is used to compute the spatial distribution of 7Li^{7}\mathrm{Li}7Li atoms in a harmonic trap. In light of recent experimental observations at Rice University, the goal is to establish the extent to which Bose-Einstein condensation (BEC) is possible with 7Li^{7}\mathrm{Li}7Li atoms, and to deduce the most effective signature of BEC from observations of the spatial profile. Our Hamiltonian, which includes the anisotropic external trap field and atom-atom interactions, is equivalent to that used previously for trapped atomic hydrogen. However, the attractive atom-atom interaction creates a localized well near the origin and precludes use of the usual local-density approximation. Eigenfunctions for the lowest orbitals are obtained by iteration. The total density is obtained from a sum with Bose population factors for these explicitly computed orbitals, together with analytic sums for unperturbed (ideal gas) orbitals for higher-lying states. For a small range of temperatures near Tc{\mathrm{T}}_{\mathrm{c}}Tc​, it is found that the atomic cloud exhibits a spatially localized ``condensate peak'' composed primarily of atoms in the ground state. For lower T, such that more than 800--1200 atoms are in the ground state, the calculations become numerically unstable as found previously for T=0. Computational results for the collisional spin-flip ejection rate for T near Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and an estimate of the fluctuation in the number of atoms in the condensate are presented.
A rotating bosonic many-body system in a harmonic trap is studied with the three-dimensional cranked Hartree-Fock-Bogoliubov method at zero temperature, which has been applied to nuclear many-body systems at high spin. This method is a variational method extended from Hartree-Fock theory, which can treat the pairing correlations in a self-consistent manner. An advantage of this method is that a finite-range interaction between constituent particles can be used in the calculation, unlike the original Gross-Pitaevskii approach. To demonstrate the validity of our method, we present a calculation for a toy model---that is, a rotating system of ten bosonic particles interacting through the repulsive quadrupole-quadrupole interaction in a harmonic trap. It is found that the yrast states, the lowest-energy states for the given total angular momentum, do not correspond to the Bose-Einstein condensate, except for a few special cases. One such case is a vortex state, which appears when the total angular momentum LLL is twice the particle number NNN (i.e., L=2NL=2NL=2N).
We consider two models of interacting Bose gases: a gas of spin one particles in the ground state of a cubic box and a one-dimension Bose gas with contact interactions. We show how to calculate exact eigenstates of the corresponding N-body Hamiltonians. Both models share the property of not leading to the formation of a Bose-Einstein condensate, even at zero temperature, in the strict sense of the existence of a single one-particle state with a macroscopic population. We show that a lot of physical insight can be gained on these two model systems by using the usual Hartree-Fock mean field approach: in this approximation, that we test against the exact result, everything happens as if a single realization of the system was a Bose-Einstein condensate in a state phi breaking the rotational or translational symmetry, and varying in a random way for any new experimental realization.
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.
The mean-field properties of trapped, two-dimensional interacting Bose gases of finite size are surveyed numerically. The coupled nonlinear equations of the Gross-Pitaevskii (GP) equation and the Hartree-Fock equation for the condensate and the thermal component, respectively, are calculated by the hybrid method, in which the GP equation and the low-lying excitations less than cut-off energy are calculated quantum mechanically using the basis representation, while the high-lying states are obtained within the local density approximation. Consistent solutions are obtained for the whole range of temperatures including the critical temperature by determining the thermodynamic chemical potential µ from the condition that the total atomic number must be the desired value, instead of identifying µ with the eigenvalue of the GP equation. Comparisons with the results of the semiclassical two-fluid model are presented.