Effect of a potential step or impurity on the Bose-Einstein condensate mean field

Abstract

The full set of stationary states of the mean field of a Bose-Einstein condensate in the presence of a potential step or point-like impurity are presented in closed analytic form. The nonlinear Schrödinger equation in one dimension is taken as a model. The nonlinear analogs of the continuum of stationary scattering states, as well as evanescent waves, are discussed. The solutions include asymmetric soliton trains and other wavefunctions of novel form, such as a pair of dark solitons bound by an impurity.

Instability of bound states of a nonlinear Schrödinger equation with a Dirac potential

“…In BEC, this equation models the dynamics of a condensate in the presence of an impurity of a length-scale much smaller than the healing length. Such an impurity can be realized by a tightly focused beam, by another spin state of the same atom or by another alkali atom confined in an optical trap [41]. In contrast to wide solitons in a periodic potential, in Eq.…”

Nonlinear bifurcation diagrams of the current states of microcavity exciton-polaritons in the lattice

“…With the further decrease of the potential depth the edges of this knot stretch towards the second band forming a typical loop structure shown in panel (d) of figure 2. The shape of this loop resembles the "swallow tail" structure [13][14][15][16][17][18] but with the twisted upper edge.…”

“…However the investigated solutions satisfy the nonlinear problem and possess several distinctive features which will be discussed latter. That is why we address these states as the nonlinear Bloch states [13]. In the case of weak nonlinearity the problem can be approached perturbatively when the solution is sought in the form of the expansion over the basis of the Bloch modes calculated for the corresponding linear problem and treating the nonlinearity as a perturbation.…”

“…The existence of such nonlinear current-currying states was interpreted as a consequence of superfluid behaviour of a condensate in periodic potential [10]. The linear stability properties of these nonlinear Bloch waves have been intensively studied [12][13][14]. The regions of both the energetic and dynamical instabilities have been identified for such essentially conservative systems.…”

“…In such class of problems analytical methods are suitable only for special kinds of a potential such as elliptic [12] or Kronig-Penney [13,14] potentials. That is why in this paper we use the numerical methods of finding nonlinear Bloch waves which are appropriate for the strongly nonlinear case when perturbative approaches fail.…”