Multiple-scale analysis for resonance reflection by a one-dimensional rectangular barrier in the Gross-Pitaevskii problem

Abstract

We consider a quantum above-barrier reflection of a Bose-Einstein condensate by a one-dimensional rectangular potential barrier, or by a potential well, for nonlinear Schrödinger equation ͑Gross-Pitaevskii equation͒ with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is equal to zero for the linear Schrödinger equation. Then the reflection is determined only by small nonlinear term in the Gross-Pitaevskii equation. A simple analytic expression has been obtained for the reflection coefficient produced only by the nonlinearity. An analytical condition is found when common action of potential barrier and nonlinearity produces a zero reflection coefficient. The reflection coefficient is derived analytically in the vicinity of resonances which are shifted by nonlinearity.For studying quantum transmission and reflection, it is the most direct way to find exact solutions of the Schrödinger equation that dominates the dynamics of systems. However, only in a few cases with the simplest potentials, like rectangular well, the Schrödinger equation can be solved exactly. In most circumstances, exact solutions are difficult to obtain due to not only the effect of external field on particles, but also the interaction of particles. The most direct generalization of single-particle case is a tunneling of mean field through a barrier in the Gross-Pitaevskii, or nonlinear Schrödinger equation ͓1,2͔. We emphasize that this is a nonlinear tunneling problem in the mean-field approximation. There have been various theoretical studies. From the theoretical point of view, the main complication in description of a quasistationary scattering process of particles obviously comes from the presence of atom-atom interaction. In leading order, the effect of this interaction is included in a nonlinear term in the Schrödinger-like Gross-Pitaevskii equation for wave function, using the Hartree self-consistent approximation with zero range interaction potential between atoms. The dynamics of solutions of this equation is very complex and rich. The phenomena of instabilities, focusing, and blowup are all concepts related to the nonlinear nature of the systems. Low velocity quantum reflection of Bose-Einstein condensates ͑BEC͒ of ultracold 23 Na atoms from the attractive Casimir-Polder potential of silicon surface was observed experimentally in Refs. ͓3,4͔. The measured reflection probability is in agreement with the theoretical model. Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction was observed in Refs. ͓5,6͔. Their results verify the predicted nonlinear generalization of tunneling oscillations in superconducting and superfluid Josephson junctions for two weakly linked BoseEinstein condensates in a double-well potential. One of the first papers addressing nonlinear resonant tunneling of a BEC has been written by Paul et al. ͓7͔. The most promising results for tunneling experiments are obtained using atomchip-based waveguide interferometry wi...

Superfluid flow above the critical velocity

“…Our results are plotted in Fig. 6 together with a multiple-scale analytical derivation valid for small g , predicting values of k barrier (defined below) corresponding to a total transmission across the barrier 38 . These resonant momenta are given by k barrier d = nπ + δ , where (with velocity v = ħk / m and k incident momentum).…”

“…24 of ref. 38 by the factor (−1) n which is present there. The analytical predictions for small g match the numerical data well for the higher excitation-free points, but less so for the lower resonant velocities.…”

Higher order transmission resonances in above-barrier reflection of ultra-cold atoms

“…where ) (z u Ln is the solution of the linear problem for the n th-order linear resonance given by Eq. (5). It has been numerically proven that these formulas provide a highly accurate and start from the simplest zero-order initial approximation 1 0 = p , i.e., we put 1 0 = p in the right-hand-side of this equation to obtain the first approximation for the limit solution as…”

“…We will see that in the nonlinear case the situation is changed: the spectrum of the nonlinear resonances becomes a function of µ . [12] (we have applied this method to treat the first nonlinear resonance for the rectangular barrier [5] and the Rosen-Morse potential [9]) and the renormalization technique (this reveals that one should apply the linear solution using…”

Nonlinear scattering of a Bose-Einstein condensate on a rectangular barrier

“…The critical nonlinearity at which bound states appear was discovered. Third, Ishkhanyan and Krainov [27] considered the limit of very small nonlinearity. They used perturbation theory to expand the NLSE, and a multi-scale method to find the solutions.…”