Bose-Einstein condensates under a spatially-modulated transverse confinement

TL;DR

This paper derives an effective 1D equation for Bose-Einstein condensates with spatially-modulated transverse confinement, analyzing localized and extended states, and comparing approximate solutions with full 3D results.

Contribution

The study introduces a nonpolynomial Schrödinger equation for BECs in modulated traps and validates it against full 3D simulations, providing insights into soliton formation and collapse thresholds.

Findings

Effective NPSE closely matches 3D solutions.

Variational approximation provides reasonable predictions.

Transition from delocalized states to gap solitons analyzed.

Abstract

We derive an effective nonpolynomial Schrodinger equation (NPSE) for self-repulsive or attractive BEC in the nearly-1D cigar-shaped trap, with the transverse confining frequency periodically modulated along the axial direction. Besides the usual linear cigar-shaped trap, where the periodic modulation emulates the action of an optical lattice (OL), the model may be also relevant to toroidal traps, where an ordinary OL cannot be created. For either sign of the nonlinearity, extended and localized states are found, in the numerical form (using both the effective NPSE and the full 3D Gross-Pitaevskii equation) and by means of the variational approximation (VA). The latter is applied to construct ground-state solitons and predict the collapse threshold in the case of self-attraction. It is shown that numerical solutions provided by the one-dimensional NPSE are always very close to full 3D solutions, and the VA yields quite reasonable results too. The transition from delocalized states to gap solitons, in the first finite bandgap of the linear spectrum, is examined in detail, for the repulsive and attractive nonlinearities alike.