Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates

Abstract

By applying the standard adiabatic approximation and using the accurate analytical expression for the corresponding local chemical potential obtained in our previous work [Phys. Rev. A 75, 063610 (2007)] we derive an effective 1D equation that governs the axial dynamics of mean-field cigar-shaped condensates with repulsive interatomic interactions, accounting accurately for the contribution from the transverse degrees of freedom. This equation, which is more simple than previous proposals, is also more accurate. Moreover, it allows treating condensates containing an axisymmetric vortex with no additional cost. Our effective equation also has the correct limit in both the quasi-1D mean-field regime and the Thomas-Fermi regime and permits one to derive fully analytical expressions for ground-state properties such as the chemical potential, axial length, axial density profile, and local sound velocity. These analytical expressions remain valid and accurate in between the above two extreme regimes. Following the same procedure we also derive an effective 2D equation that governs the transverse dynamics of mean-field disk-shaped condensates. This equation, which also has the correct limit in both the quasi-2D and the Thomas-Fermi regime, is again more simple and accurate than previous proposals. We have checked the validity of our equations by numerically solving the full 3D Gross-Pitaevskii equation.

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

“…This way, the same authors presented in the recent work of Ref. [119] the effective 1D NLS Eq. (33), but with the nonlinearity function given by…”

“…Following Refs. [115][116][117][118][119], one may factorize the wavefunction as per Eq. (20), but with the transverse wavefunction Φ depending also on the longitudinal variable z.…”

“…[118][119][120]) to NLS-type equations with generalized nonlinearities [121][122][123][124][125]. Among these models, the so-called non-polynomial Schrödinger equation (NPSE) has attracted considerable attention.…”

Self-trapping of Fermi and Bose gases under spatially modulated repulsive nonlinearity and transverse confinement

“…Then, in Secs. IV and V, following previous results obtained in the framework of the mean-field description of BEC [23][24][25], we consider Fermi gases without intrinsic interactions (in particular, it may be a spin-polarized, i.e., single-spin-component, gas) and demonstrate that they give rise to localized ground states under the action of the 2D or 1D HO confinement, if its tightness, i.e., the corresponding confinement frequency, grows in the transverse directions faster than √ |z| or r 4 , respectively. In addition, we demonstrate that the same confinements with the spatially growing tightness support, in a similar way, localized ground states in BEC with the self-repulsive nonlinearity.…”

“…The respective effective 1D equation for the mean-field wave function (z,t) was derived in Ref. [25], starting from the 3D Gross-Pitaevskii equation…”