Transmission resonances in above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential

TL;DR

This paper analytically investigates quantum above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential within the Gross-Pitaevskii framework, deriving accurate formulas for resonance positions including nonlinear effects.

Contribution

It introduces a modified perturbation theory approach to determine resonance conditions, revealing that nonlinear transmission resonances can occur for both potential wells and barriers, with an exact formula for the first resonance.

Findings

The first nonlinear resonance position is exactly determined by the derived formula.

Reflectionless transmission can occur for potential barriers, not just wells, in the nonlinear case.

The nonlinear shift of resonance positions is approximately linear with respect to resonance order.

Abstract

Quantum above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential is analytically considered within the mean field Gross-Pitaevskii approximation. Reformulating the problem of reflectionless transmission as a quasi-linear eigenvalue problem for the potential depth, an approximation for the specific height of the potential that supports reflectionless transmission of the incoming matter wave is derived via modification of the Rayleigh-Schroedinger time-independent perturbation theory. The approximation provides highly accurate description of the resonance position for all the resonance orders if the nonlinearity parameter is small compared with the incoming particles chemical potential. Notably, the result for the first transmission resonance turns out to be exact, i.e., the derived formula for the resonant potential height gives the exact value of the first nonlinear resonances position for all the allowed variation range of the involved parameters, the nonlinearity parameter and chemical potential. This has been shown by constructing the exact solution of the problem for the first resonance. Furthermore, the presented approximation reveals that, in contrast to the linear case, in the nonlinear case reflectionless transmission may occur not only for potential wells but also for potential barriers with positive potential height. It also shows that the nonlinear shift of the resonance position from the position of the corresponding linear resonance is approximately described as a linear function of the resonance order. Finally, a compact (yet, highly accurate) analytic formula for the n-th order resonance position is constructed via combination of analytical and numerical methods.