Entanglement-based perturbation theory for highly anisotropic Bose-Einstein condensates

摘要

We investigate the emergence of three-dimensional behavior in areduced-dimension Bose-Einstein condensate trapped by a highly anisotropicpotential. We handle the problem analytically by performing a perturbativeSchmidt decomposition of the condensate wave function between the tightlyconfined (transverse) direction(s) and the loosely confined (longitudinal)direction(s). The perturbation theory is valid when the nonlinear scatteringenergy is small compared to the transverse energy scales. Our approach providesa straightforward way, first, to derive corrections to the transverse andlongitudinal wave functions of the reduced-dimension approximation and, second,to calculate the amount of entanglement that arises between the transverse andlongitudinal spatial directions. Numerical integration of the three-dimensionalGross-Pitaevskii equation for different cigar-shaped potentials andexperimentally accessible parameters reveals good agreement with our analyticalmodel even for relatively high nonlinearities. In particular, we show that evenfor such stronger nonlinearities the entanglement remains remarkably small,which allows the condensate to be well described by a product wave functionthat corresponds to a single Schmidt term.