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

Abstract

By exact numerical solutions of the Gross-Pitaevskii (GP) equation in 3D, we assess the validity of 1D and 2D approximations in the study of Bose-Einstein condensates confined in harmonic trap potentials. Typically, these approximations are performed when one or more of the harmonic frequencies are much greater than the remaining ones, using arguments based on the adiabatic evolution of the initial approximated state. Deviations from the 3D solution are evaluated as a function of both the effective interaction strength and the ratio between the trap frequencies that define the reduced dimension where the condensate is confined. The observables analyzed are both of stationary and dynamical character, namely, the chemical potential, the wave function profiles, and the time evolution of the approximated 1D and 2D stationary states, considered as initial states in the 3D GP equation. Our study, besides setting quantitative limits on approximations previously developed, should be useful in actual experimental studies where quasi-1D and quasi-2D conditions are assumed. From a qualitative perspective, 1D and 2D approximations certainly become valid when the anisotropy is large, but in addition the interaction strength needs to be above a certain threshold.

Intrinsic Decoherence and Recurrences in a Large Ferromagnetic F = 1 Spinor Bose–Einstein Condensate

“…Therefore, if we set Ψ 0α (r, t) = ψ 0 ( r)ζ α (t) with ζ α (t) a simple threecomponent spinor, we can integrate out the spatial part and find very simple dynamical equations, called SMA-GP (not shown here), for the time evolution of ζ(t). The result of those is shown also in Figure 1 where we plot the time evolution of the magnetization f (t) = drΨ * 0α (r, t)F αβ Ψ * 0α (r, t) with full 3D GP (see [39] for the details of our methods), and…”

Relaxation of ferromagnetic domains in a disordered lattice in 2D

“…The values of the effective interaction couplings g σσ with σ, σ = {↑, ↓} are written in terms of the s-wave scattering length a σ,σ as, g σσ = 4πN 2 a σσ /m, being N the number of particles in the condensate. We should point here that these interaction coefficients must be substituted by effective interaction couplings that take into account that the atom collision processes occur in 2D [25][26][27][28][29][30][31]. The effective scattering length in the plane x − y becomes a σσ → a σσ / √ 2πl z , with l z = /mω z , being ω z a typical frequency of condensates confined in 2D [32,33].…”

Decoherence and Recurrences in a Large Ferromagnetic $F = 1$ Spinor Bose-Einstein Condensate

“…Although not shown here for brevity, we have performed full evolution of an F = 1 Gross-Pitaevskii equation, in the ferromagnetic regime, and found that SMA holds perfectly well for uniform magnetic fields, see Ref. [30] for details of our method.…”