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...
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-Schrödinger 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 particle's 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 resonance's 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.Bose-Einstein condensates of ultracold gases [1,2] provide an ideal ground for testing of many important nonlinear phenomena occurring in many-body quantum systems. An increasing number of such phenomena, e.g., the creation of topological structures such as vortices [3], the generation of bright [4] and dark [5] solitons, the self-trapping effect [6], etc., has been recently extensively studied both theoretically and experimentally. One of such important phenomena offered by the Bose-condensates is the macroscopic quantum tunneling through and reflection from a potential barrier (well) of a many-body wave function. This is because the basic concepts of tunneling through a barrier and above-barrier reflection of a particle are fundamental effects in quantum mechanics not present in classical physics [7].Since the many-body macroscopic tunneling and reflection are essentially nonlinear
Abstract-In the framework of a basic semiclassical time dependent nonlinear two state problem, we study the weak coupling limit of the nonlinear Landau-Zener transition at coherent photo and magneto associ ation of an atomic Bose-Einstein condensate. Using an exact third order nonlinear differential equation for the molecular state probability, we develop a variational approach which enables us to construct an accurate analytic approximation describing time dynamics of the coupled atom molecular system for the case of weak coupling. The approximation is written in terms of the solution to an auxiliary linear Landau-Zener problem with some effective Landau-Zener parameter. The dependence of this effective parameter on the input Lan dau-Zener parameter is found to be unexpected: as the generic Landau-Zener parameter increases, the effective Landau-Zener parameter first monotonically increases (starting from zero), reaches its maximal value and then monotonically decreases again reaching zero at some point. The constructed approximation quantitatively well describes many characteristics of the time dynamics of the system, in particular, it provides a highly accurate formula for the final transition probability to the molecular state. The present result for the final transition probability improves the accuracy of the previous approximation by Ishkhanyan et al.
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