Effective one-dimensional dynamics of elongated Bose-Einstein condensates

TL;DR

This paper derives a new effective 1D equation for elongated Bose-Einstein condensates using a variational and adiabatic approach, providing accurate analytical formulas for axial dynamics and validating them with 3D simulations.

Contribution

It introduces a unified derivation of effective 1D equations for elongated condensates, including a new equation valid for vorticity |q|=0 or 1, and offers precise analytical formulas for axial breathing modes.

Findings

New effective 1D equation derived for axial dynamics.

Analytical formulas for axial breathing mode frequency with <1% error.

Validation of formulas through numerical solutions of the 3D Gross-Pitaevskii equation.

Abstract

By using a variational approach in combination with the adiabatic approximation we derive a new effective 1D equation of motion for the axial dynamics of elongated condensates. For condensates with vorticity $|q|=0$ or 1, this equation coincides with our previous proposal [A. Mu\~noz Mateo and V. Delgado, Phys. Rev. A \textbf{77}, 013617 (2008)]. We also rederive the nonpolynomial Schr\"{o}dinger equation (NPSE) in terms of the adiabatic approximation. This provides a unified treatment for obtaining the different effective equations and allows appreciating clearly the differences and similarities between the various proposals. We also obtain an expression for the axial healing length of cigar-shaped condensates and show that, in the local density approximation and in units of the axial oscillator length, it coincides with the inverse of the condensate axial half-length. From this result it immediately follows the necessary condition for the validity of the local density approximation. Finally, we obtain analytical formulas that give the frequency of the axial breathing mode with accuracy better than 1%. These formulas can be relevant from an experimental point of view since they can be expressed in terms only of the axial half-length and remain valid in the crossover between the Thomas-Fermi and the quasi-1D mean-field regimes. We have corroborated the validity of our results by numerically solving the full 3D Gross-Pitaevskii equation.