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Physica Scripta

Resonant reflection of a Bose–Einstein condensate by a double barrier within the Gross–Pitaevskii equation

V. P. Kraynov, H. A. Ishkhanyan

Published Sep 1, 2010• 2 citations
Publisher

Abstract

Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrodinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.

DOI
10.1088/0031-8949/2010/t140/014052

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Resonant reflection of a Bose–Einstein condensate by a double delta-function barrier has been considered analytically using the Gross–Pitaevskii approximation for nonlinearity. The reflecting coefficient has been derived taking into account a weak nonlinearity of the Schrodinger equation produced by the interaction between cold alkali atoms. A nonlinear term is given in the Hartree approximation for short-range interaction between atoms. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflecting coefficient by a multiple-scale method in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without a nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflecting coefficient is derived in the first order on the nonlinearity parameter.
Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter.
Above-barrier reflection of cold alkali atoms by resonant laser light was considered analytically within the Gross-Pitaevskii approximation. Correction for the reflection coefficient because of a weak nonlinearity of the stationary Schrödinger equation has been derived using multiscale analysis as a form of perturbation theory. The nonlinearity adds spatial harmonics to linear incident and reflecting waves. It was shown that the role of nonlinearity increases when the kinetic energy of an atom is nearly to the height of the potential barrier. Results are compared to the known numerical derivations for wave functions of the Gross-Pitaevskii equation with the step potential.
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 Schr\"odinger equation (Gross-Pitaevskii equation) with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is equal to zero for the linear Schr\"odinger 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.
The reflectionless transmission resonances in above-barrier reflection of Bose-Einstein condensates by the Rosen-Morse potential are considered using the mean field Gross-Pitaevskii approach. Applying an exact third order nonlinear differential equation obeyed by the condensate's density, the exact solution of the problem for the first resonance is derived. It is shown that in the nonlinear case the total transmission is possible for positive potential heights, i.e., for potential barriers. Further, it is shown that an appropriate approximation for higher-order resonances can be constructed using a limit solution of the equation for the density written as a root of a polynomial equation of the third degree. Using this limit function and the solution for the first resonance, a simple approximation for the shift of the nonlinear resonance potential's depth from the corresponding linear resonance's position is constructed for higher order resonances. The result is written as a linear function of the resonance order. This behavior notably differs from the case of the rectangular barrier for which the nonlinear shift is approximately constant for all the resonance orders.
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.