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arXiv:quant-ph/0108027

Atomic scattering from Bose-Einstein condensates

Ivo Haering, Jan Michael Rost

Published Aug 7, 2001• 0 citations
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Abstract

Elastic scattering probes directly the interaction potential. For weakly interacting condensates this potential is given by the condensate density. We investigate how the differential and total cross sections reflect the density. In particular, we have determined which signatures the Thomas Fermi approximation leaves in contrast to an exact solution for the condensate wave function within the Gross-Pitaevskii theory.

DOI
10.48550/arxiv.quant-ph/0108027

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Elastic scattering probes directly the interaction potential. For weakly interacting condensates this potential is given by the condensate density. We investigate how the differential and total cross sections reflect the density. In particular, we have determined which signatures the Thomas Fermi approximation leaves in contrast to an exact solution for the condensate wave function within the Gross-Pitaevskii theory.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas–Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. In Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. Finally, in Sec. 6.5 we show how the magnetic dipole–dipole interaction, which is long-ranged and anisotropic, may be included in the Gross–Pitaevskii equation and determine within the Thomas–Fermi approximation its effect on the density distribution. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4π ħ 2 a/m .
Bose-Einstein condensates of 10485Rb{10}^{4}{}^{85}\mathrm{Rb}10485Rb atoms in a cylindrical trap are studied using a recently proposed approach based on a local-density approximation. Since the existence of a Feshbach resonance allows for widely tuning the scattering length of the atoms, values of the peak gas parameter xpk{x}_{\mathrm{pk}}xpk​ of the order of ${10}^{\ensuremath{-}2}$ may be attained and the standard analysis based on Gross-Pitaevskii and/or Thomas-Fermi equations may result in being questionable. Energy functionals derived from the correlated basis functions theory and the low-density expansion for a homogeneous hard-spheres gas are used to estimate corrections to the Gross-Pitaevskii equation. The two functionals give similar results, both showing large differences with respect to the Thomas-Fermi and Gross-Pitaevskii ones. The column densities at z=0z=0z=0 may differ by as much as $\ensuremath{\sim}30%$ and the half maximum radius by $\ensuremath{\sim}20%.$ The scattering lengths estimated by fitting the half maximum radius may differ by $\ensuremath{\sim}40%$ from those given by a Thomas-Fermi-based analysis of the experimental data.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
In the present chapter we consider the structure of the Bose–Einstein condensed state in the presence of interactions. Our discussion is based on the Gross–Pitaevskii equation, which describes the zero-temperature properties of the non-uniform Bose gas when the scattering length a is much less than the mean interparticle spacing. We shall first derive the Gross–Pitaevskii equation at zero temperature by treating the interaction between particles in a mean-field approximation (Sec. 6.1). Following that, in Sec. 6.2 we discuss the ground state of atomic clouds in a harmonic-oscillator potential. We compare results obtained by variational methods with those derived in the Thomas-Fermi approximation, in which the kinetic energy operator is neglected in the Gross–Pitaevskii equation. The Thomas–Fermi approximation fails near the surface of a cloud, and in Sec. 6.3 we calculate the surface structure using the Gross–Pitaevskii equation. Finally, in Sec. 6.4 we determine how the condensate wave function ‘heals’ when subjected to a localized disturbance. The Gross–Pitaevskii equation In the previous chapter we have shown that the effective interaction between two particles at low energies is a constant in the momentum representation, U 0 = 4πħ 2 a / m . In coordinate space this corresponds to a contact interaction U 0 δ( r – r ′), where r and r ′ are the positions of the two particles. To investigate the energy of man y-body states we adopt a Hartree or mean-field approach, and assume that the wave function is a symmetrized product of single-particle wave functions.