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arXiv:cond-mat/0304546 · Physical Review Letters

Levinson-Like Theorem for Scattering from a Bose-Einstein Condensate

Joachim Brand, Ivo Häring, Jan Michael Rost

Published Aug 15, 2003• 6 citations
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Abstract

A relation between the number of bound elementary excitations of an atomic Bose-Einstein condensate and the phase shift of elastically scattered atoms is derived. Within the Bogoliubov model of a weakly interacting Bose gas this relation is exact and generalizes Levinson's theorem. Specific features of the Bogoliubov model such as complex energy and continuum bound states are discussed and a numerical example is given.

DOI
10.1103/physrevlett.91.070403

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A relation between the number of bound elementary excitations of an atomic Bose-Einstein condensate and the phase shift of elastically scattered atoms is derived. Within the Bogoliubov model of a weakly interacting Bose gas this relation is exact and generalizes Levinson's theorem. Specific features of the Bogoliubov model such as complex energy and continuum bound states are discussed and a numerical example is given.
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The Bogoliubov treatment of a weakly interacting Bose gas is extended by allowing macroscopic occupation of many single-particle quantum states. This generalization of having more than a single condensed mode gives rise to a nonuniform condensate and is being considered as a possible mechanism for the description of a dense strongly interacting Bose gas at low temperatures. The model with a repulsive $\ensuremath{\delta}$-function interparticle potential gives an energy spectrum of elementary excitations or quasiparticles with both phonon and roton features. The values obtained for the roton minimum energy $\ensuremath{\Delta}$ and the speed of (first) sound agree reasonably well with inelastic-neutron-scattering data.
The Bogoliubov theory of a uniform weakly-nonideal Bose-Einstein condensate leads to a divergent expression for the kinetic energy of atoms. However, the latter can be determined provided that the dependence of the scattering length on atomic mass is known. The explicit expressions are derived for the kinetic energy through parameters that specify interatomic interaction. The kinetic energy of condensate atoms noticeably exceeds the total energy. Numerical data are presented for Rb-87 and Na-23 condensates.
The exactness of the Bogoliubov approximation is discussed from the point of view of the condensation properties of the Bose gas. For the imperfect Bose gas, the authors find that the approximation yields the correct condensate density when the thermodynamic limit is taken by isotropic dilation; this is not the case in general for other ways of going to the infinite volume limit. They also prove the existence of Bose-Einstein condensation for a class of Bose gases with weak interactions and an energy gap.