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Bulletin of the American Physical Society

Nonlinear Scattering of Bose-Einstein Condensates on a Finite Barrier

Rachel Miller, Matthew S. Heller, Daniel R. Bolton, Lincoln D. Carr

Published Mar 6, 2007• 0 citations

Abstract

We seek stationary state solutions to the nonlinear Schrodinger equation (NLS) for piecewise-constant potentials V(x). In this case, we are able to find complete stationary solutions, i.e. exact solutions which are separable in time and space, in terms of Jacobi elliptic functions. These special functions can be defined as an inverse integral and are an interpolation between the circular and hyperbolic trigonometric functions. As an important mathematical application of our exact solutions, we will seek a nonlinear generalization of the transfer matrix. When this matrix, which is commonly used in electromagnetic transmission problems, is applied to the coefficient vector characterizing the solution in the region to the left of the first boundary, it yields the coefficient vector for any region to the right. To solve this general problem, we begin with the nontrivial case of two boundaries, which corresponds to a potential step or barrier. This has a direct physical application to an atom laser incident on such a potential, as occurs in Bose-Einstein condensates on a chip. Our work provides the first step towards formulating a completely general nonlinear scattering theory.

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We seek stationary state solutions to the nonlinear Schrodinger equation (NLS) for piecewise-constant potentials V(x). In this case, we are able to find complete stationary solutions, i.e. exact solutions which are separable in time and space, in terms of Jacobi elliptic functions. These special functions can be defined as an inverse integral and are an interpolation between the circular and hyperbolic trigonometric functions. As an important mathematical application of our exact solutions, we will seek a nonlinear generalization of the transfer matrix. When this matrix, which is commonly used in electromagnetic transmission problems, is applied to the coefficient vector characterizing the solution in the region to the left of the first boundary, it yields the coefficient vector for any region to the right. To solve this general problem, we begin with the nontrivial case of two boundaries, which corresponds to a potential step or barrier. This has a direct physical application to an atom laser incident on such a potential, as occurs in Bose-Einstein condensates on a chip. Our work provides the first step towards formulating a completely general nonlinear scattering theory.

The fundamental question of how Bose-Einstein condensates tunnel into a barrier is addressed. The cubic nonlinear Schr\"odinger equation with a finite square well potential, which models a Bose-Einstein condensate in a trap of finite depth, is solved for the complete set of localized and partially localized stationary states, which the former evolve into when the nonlinearity is increased. An immediate application of these different solution types is tunable tunneling. Magnetically tunable Feshbach resonances can change the scattering length of certain Bose-condensed atoms, such as

{}^{85}\mathrm{Rb},

by several orders of magnitude, including the sign, and thereby also change the mean field nonlinearity term of the equation and the tunneling of the wave function. We find both linear-type localized solutions and uniquely nonlinear partially localized solutions where the tails of the wave function become nonzero at infinity when the nonlinearity increases. The tunneling of the wave function into the nonclassical regime (and thus its localization) therefore becomes an external experimentally controllable parameter.

We consider the nonlinear scattering and transmission of an atom laser or Bose-Einstein condensate (BEC) on a finite rectangular potential barrier. The nonlinearity inherent in this problem leads to several physical features beyond the well-known picture from single-particle quantum mechanics. We find numerical evidence for a denumerably infinite string of bifurcations in the transmission resonances as a function of nonlinearity and chemical potential, when the potential barrier is wide compared to the wavelength of oscillations in the condensate. Near the bifurcations, we observe extended regions of near-perfect resonance, in which the barrier is effectively invisible to the BEC. Unlike in the linear case, it is mainly the barrier width, not the height, that controls the transmission behavior. We show that the potential barrier can be used to create and localize a dark soliton or dark-soliton train from a phononlike standing wave.

The cubic vector nonlinear Schrödinger equation with an external trigonometric potential models a quasi-one-dimensional multi-component Bose–Einstein condensate trapped in a standing light wave. We construct families of exact stationary solutions for the more general case of an elliptic function potential. Some of these solutions degenerate to zero as the effect of the external potential disappears, whereas others limit to solutions of the free vector nonlinear Schrödinger equation. The stability of these solutions is examined both analytically and numerically. The stability results depend on the nature of the atomic interactions not only within the components but also between components. As in the scalar case (one component) with repulsive interaction, all linearly stable solutions are deformations of the ground state of the linear Schrödinger equation. Unlike the scalar case with attractive interaction, no solutions are found to be stable if there is any attractive interaction present.

A simple yet exact method of solving the Schrodinger equation across an arbitrary one-dimensional piecewise-linear potential is described. It is based on the analytical solution of the Schrodinger equation across a linear potential, where the solution can be expressed as a linear combination of the Airy functions. Proper boundary conditions are imposed at the interface between adjacent linear intervals of the potential, and a transfer-matrix procedure is utilized in the derivation. The quantum mechanical transmission coefficient across a potential barrier and eigenenergies of a potential well can also be easily calculated using this method.

Nonlinear effects in the interference of Bose-Einstein condensates are studied using exact solutions of the one-dimensional nonlinear Schrodinger equation, which is applicable when the lateral motion is confined or negligible. With the inverse scattering method, the interference pattern is studied as a scattering problem with the linear Schrodinger equation, whose potential is profiled by the initial density distribution of the condensates. Our theory not only provides an analytical framework for quantitative predictions for the one-dimensional case, it also gives an intuitive understanding of some mysterious features of the interference patterns observed in experiments and numerical simulations.