$N$-coherence vs. $t$-coherence: An alternative route to the Gross-Pitaevskii equation

TL;DR

This paper presents a method to construct a mean-field wave function from an exact N-Boson system without breaking particle number symmetry, showing it satisfies the Gross-Pitaevskii equation under certain conditions.

Contribution

The authors introduce a new approach to derive the Gross-Pitaevskii equation from exact many-body dynamics without symmetry breaking, using the concept of $t$-coherence.

Findings

Macroscopic wave functions can persist under strong forcing.

Chaotic dynamics rapidly destroy macroscopic coherence.

Numerical simulations validate the theoretical framework.

Abstract

We show how a candidate mean-field amplitude can be constructed from the exact wave function of an externally forced $N$-Boson system. The construction makes use of subsidiary $(N-1)$-particle states which are propagated in time in addition to the true $N$-particle state, but does not involve spontaneous breaking of the $U(1)$ symmetry associated with particle number conservation. Provided the flow in Fock space possesses a property which we call maximum stiffness, or $t$-coherence, the candidate amplitude actually satisfies the time-dependent Gross-Pitaevskii equation, and then serves as macroscopic wave function of the forced $N$-particle system. The general procedure is illustrated in detail by numerical calculations performed for the model of a driven bosonic Josephson junction, which allows one to keep track of all contributions which usually are subject to uncontrolled assumptions. These calculations indicate that macroscopic wave functions can persist even under conditions of strong forcing, but are rapidly destroyed upon entering a regime of chaotic dynamics. Our results provide a foundation for future attempts to manipulate, and actively control, macroscopic wave functions by means of purposefully designed force protocols.