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arXiv:quant-ph/0603134 · Physics Letters A

Quantum particles trapped in a position-dependent mass barrier; a d-dimensional recipe

Omar Mustafa, S. Habib Mazharimousavi

Published Jun 11, 2006• 75 citations
arXivHTMLPDF
DOI
10.1016/j.physleta.2006.05.037

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Abstract Using the time-dependent variational approach and the numerical solution of the Gross–Pitaevskii (GP) equation, the dynamics of a matter-wave soliton are investigated in a two-dimensional constant external force field. Firstly, the time-dependent variational approach is used to study the movements of the center-of-mass position in a simple harmonic trap and a two-dimensional weakly transmissive reflective barrier. Because the mathematical model for describing soliton dynamics in simple harmonic traps is simple, the analytic dynamical results of a soliton wave packet center-of-mass position is given directly. Since the model describing the weakly transmissive reflective barrier is difficult to solve analytically, we give the Newton’s equation of the center-of-mass position of motion, and then use the numerical solution method to study the movement of the center-of-mass in the case of several parameters. To study the motion of the soliton in the harmonic potential trap in the two-dimensional external force field under rotation , we use the split-step finite difference method to solve the two-dimensional GP equation; the motion of the center-of-mass position is discussed under the conditions of several parameter combinations. The results show that the rotation has a very important influence on the movement of the center-of-mass. When there is no rotation effect, the center-of-mass position acts as a simple harmonic motion under the combined action of the external force field and the potential well. When there is a rotation effect, the center-of-mass position is almost static.
An algebraic treatment of shape-invariant quantum-mechanical position-dependent effective mass systems is discussed. Using shape invariance, a general recipe for construction of ladder operators and associated algebraic structure of the pertaining system, is obtained. These operators are used to find exact solutions of general one-dimensional systems with spatially varying mass. We apply our formalism to specific translationally shape-invariant potentials having position-dependent effective mass.
We explore the impact of dimensionality on the scattering of a small bosonic ensemble in an elongated harmonic trap off a centered repulsive barrier, thereby taking particle correlations into account. The loss of coherence as well as the oscillation of the center of mass are studied and we analyze the influence of both particle and spatial correlations. Two different mechanisms of coherence losses in dependence of the aspect ratio are found. For small aspect ratios, loss of coherence between the region close to the barrier and outer regions occurs, due to spatial correlations, and for large aspect ratios, incoherence between the two density fragments of the left and right side of the barrier arises due to particle correlations. Apart from the decay of the center-of-mass motion induced by the reflection and transmission, further effects due to the particle and spatial correlations are explored. For tight transversal traps, the amplitude of the center-of-mass oscillation experiences a weaker damping, which can be traced back to the population of a second natural orbital, and for a weaker transversal confinement, we detect a strong decay due to the possibility of transferring energy to transversal excited modes. These effects are enhanced if the aspect ratio is integer valued.