Effective self-similar expansion for the Gross-Pitaevskii equation

Abstract

We consider an effective scaling approach for the free expansion of a one-dimensional quantum wave packet, consisting in a self-similar evolution to be satisfied on average, i.e. by integrating over the coordinates. A direct comparison with the solution of the Gross-Pitaevskii equation shows that the effective scaling reproduces with great accuracy the exact evolution -the actual wave function is reproduced with a fidelity close to one -for arbitrary values of the interactions. This result represents a proof-of-concept of the effectiveness of the scaling ansatz, which has been used in different forms in the literature but never compared against the exact evolution.

Effective self-similar expansion of a Bose-Einstein condensate: Free space versus confined geometries

“…In case of the expansion in free space, we find that the effective scaling approach for the expansion of a spherically symmetric condensate is rather accurate even for intermediate values of the interaction-between the noninteracting and hydrodynamic limits where the scaling is exact-similarly to what found for the quasi-1D case [20]. Deviations from this behavior are observed instead for prolate and especially oblate condensates, signaling that the scaling approach becomes less accurate when the expansion along certain directions is faster than along the others, causing local variations of the density that do not conform to the hypothesis of self-similarity.…”

“…Recently, this effective approach was tested for the case of a freely expanding quasi-one-dimensional Bose-Einstein condensate (BEC) by comparing the approximate solution with the exact evolution of the system as obtained from the solution of the Gross-Pitaevskii (GP) equation. Remarkably, in this case it was found that the effective scaling approach is indeed very accurate in reproducing the exact evolution for arbitrary values of the interactions [20].…”

“…Contrarily, when such a factorization is not possible, one may follow a different approach by requiring the self-similarity to be satisfied on average, as discussed in Refs. [17][18][19][20]. First, it is convenient to transform the expression in Eq.…”

“…and we get rid of the coordinate dependence by integrating over the volume. This sort of averaging procedure-that constitutes the essence of the effective scaling approach [17,20]-yields an equation for the scaling parameters depending only on the time variable. Specific expressions for each case considered in this paper will be discussed in the following sections.…”

“…In the present paper we extend this analysis to the case of a three-dimensional (3D) BEC [21] for which we consider the expansion in free space and in trapped geometries (namely in cylindrical and planar waveguides). The aim of this work to understand in a quantitative framework how nonlinearity affects self-similar dynamics, in the context of the GP equation, for which scaling approaches have been widely employed [1][2][3][4][5][6][7]9,20]. This is particularly relevant in 3D-where the interplay between spatial anisotropies and interactions may not be trivial-and where there are not actual proofs that justify the usage of an effective scaling approach.…”