Bogoliubov excitation spectrum of an elongated condensate throughout a transition from quasi-one-dimensional to three-dimensional

Abstract

The quasiparticle excitation spectra of a Bose gas trapped in a highly anisotropic trap is studied with respect to varying total number of particles by numerically solving the effective one-dimensional (1D) Gross-Pitaevskii (GP) equation proposed recently by Mateo et al.. We obtain the static properties and Bogoliubov spectra of the system in the high energy domain. This method is computationally efficient and highly accurate for a condensate system undergoing a 1D to threedimensional (3D) cigar-shaped transition, as is shown through a comparison our results with both those calculated by the 3D-GP equation and analytical results obtained in limiting cases. We identify the applicable parameter space for the effective 1D-GP equation and find that this equation fails to describe a system with large number of atoms. We also identify that the description of the transition from 1D Bose-Einstein condensate (BEC) to 3D cigar-shaped BEC using this equation is not smooth, which highlights the fact that for a finite value of a ⊥ /a s the junction between the 1D and 3D crossover is not perfect.

Validity of Gross–Pitaevskii solutions of harmonically confined BEC gases in reduced dimensions

“…What we find, however, with our exact 3D calculations, is that the strong interaction regime may apply when the anisotropy is very small only, namely, when γ∼1, since it is in this limit when the TF approximation for the transverse part is approached. Although those approximations [29][30][31] may be useful in some particular cases, it is our conclusion that for condensates nearly isotropic one should remain within a numerical solution of the full 3D GP equation. The third and fourth columns in figure 1 show the fidelity of the stationary states, for α=2 and α=3, when evolved with the full time-dependent 3D GP equation, equation (1).…”

“…However, for small γ the approach to the HO profile is too slow, deteriorating even more for larger values of g. The overall conclusion is that as the interaction is large enough, g60, the 2D harmonic oscillator wave function is a reasonable good approximation for γ sufficiently small, namely, for g=60, γ0.1 and for g=600, γ0.001, and so on. At this stage it is worth mentioning that there are two other variational approaches that lead to approximated 1D and 2D GP-like equations [29][30][31] which show two different regimes, a weakly and a strong interacting ones. The former coincides with the 1D and 2D approaches analyzed here but the latter seemingly describes a different regime where interactions are more relevant.…”

“…This procedure, however, requires a theoretical justification that both yields the correct equation in the reduced dimension and sets their bounds of validity. This theoretical procedure, of how to conduct such a dimension reduction, has been the subject of several studies [28][29][30][31][32][33][34][35][36][37][38], with the most important result that the atomic interaction strength becomes modified in 1D and 2D with respect to the original interaction in 3D, usually being scaled by the length of the oscillator in the constrained direction or directions. In some of those cases [29][30][31], however, the reduced equations show a non-linearity different from the usual one in GP equations.…”

Accurate one-dimensional effective description of realistic matter-wave gap solitons

“…The equations of Refs. [32] and [33] have also been used to study BECs in nonlinear lattices [45], bright solitons in condensates with inhomogeneous defocusing nonlinearities [46], or the Bogoliubov spectrum of elongated condensates [47]. Since these effective equations are extensions of the standard 1D GPE (to which they reduce in the proper limit) and can be numerically solved with similar computational effort, they may offer a clear advantage over the latter.…”

Effective Equations for Repulsive Quasi‐One Dimensional Bose–Einstein Condensates Trapped with Anharmonic Transverse Potentials

“…[20][21][22][23][24][25][26][27][28][29] and via standard adiabatic approximations in Refs. [30][31][32][33][34][35][36][37][38][39]. However, in all these papers the effective equations are obtained based on the assumption that the transverse potential is quadratic.…”