Tunable tunneling: An application of stationary states of Bose-Einstein condensates in traps of finite depth

Abstract

The fundamental question of how Bose-Einstein condensates tunnel into a barrier is addressed. The cubic nonlinear Schrödinger equation with a finite square well potential, which models a Bose-Einstein condensate in a quasi-one-dimensional 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 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 wavefunction. We find both linear-type localized solutions and uniquely nonlinear partially localized solutions where the tails of the wavefunction become nonzero at infinity when the nonlinearity increases. The tunneling of the wavefunction into the nonclassical regime and thus its localization therefore becomes an external experimentally controllable parameter.

Deformation of dark solitons in inhomogeneous Bose–Einstein condensates

“…Note that such density profiles can also be obtained by matching analytically known solutions of the Gross-Pitaevskii equation in finite and semi-infinite intervals (see, e.g. [42,43]). Fig.…”

Nonlinear tunneling in a fiber guide array resonator

“…The nonlinearity is managed such as to be repulsive inside the well and attractive outside. The nonlinear states for NLS with a definite sign of the nonlinearity has been derived in [1] with interesting solutions having no counterparts in the linear limit. Using nonlinearity management as in (1) actually allows to obtain a system whose nonlinear states solutions possess remarkable properties: i) they uniformly tend to the linear eigenstates in the small amplitude limit, ii) they are stable solutions for amplitudes below an explicit threshold, iii) above the threshold an instability generates gap solitons, emitted outside the well, hence realizing a classical nonlinear tunneling process.…”

“…As learned from [1], the main tool to derive the nonlinear states is to connect a periodic solution inside the well to a static one-soliton tail outside. We make use of the fundamental solutions of NLS in terms of Jacobi elliptic functions for both symmetric (even) and antisymmetric (odd) cases [9].…”

Qualitative and quantitative analysis of stability and instability dynamics of positive lattice solitons

“…In recent years there has been a considerable interest in the study of solitons in lattice-type systems. Such solitons have been observed in optics using waveguide arrays, photo-refractive materials, photonic crystal fibers, etc., in both one-dimensional and multidimensional lattices, mostly periodic sinusoidal square lattices [1,2,3,4,5,6,7,8] or single waveguide potentials [9,10,11], but also in discontinuous lattices (surface solitons) [12], radially-symmetric Bessel lattices [13], lattices with triangular or hexagonal symmetry [14,15], lattices with defects [16,17,18,19,20,21,22], with quasicrystal structures [16,23,24,25,26,27,28] or with random potentials [29,30]. Solitons have also been observed in the context of Bose-Einstein Condensates (BEC) [31,32], where lattices have been induced using a variety of techniques.…”