Effective self-similar expansion for the Gross-Pitaevskii equation

TL;DR

This paper introduces an effective self-similar scaling approach for the free expansion of a one-dimensional quantum wave packet, accurately approximating the Gross-Pitaevskii equation's exact evolution across various interaction strengths.

Contribution

It demonstrates that the self-similar scaling ansatz can reliably reproduce the exact wave function evolution, validating its effectiveness for arbitrary interactions.

Findings

High fidelity between the scaled and exact wave functions

Effective for arbitrary interaction strengths

Validates the scaling approach as a reliable approximation

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 unity - 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 with the exact evolution.