Commutability between Semiclassical Limit and Adiabatic Limit

TL;DR

This paper investigates the relationship between semiclassical and adiabatic limits in a two-mode bosonic system, revealing that their commutability depends on interaction strength and is linked to topological changes in energy bands.

Contribution

It demonstrates how the commutability of semiclassical and adiabatic limits depends on interaction strength, highlighting the impact of topological energy band changes.

Findings

Limits are commutable at weak interactions.

Limits become incommutable at strong interactions.

Topological changes in energy bands occur at critical interaction strength.

Abstract

We study the semiclassical limit and the adiabatic limit with a second-quantized two-mode model, which describes a many-boson interacting system. When its mean-field interaction is small, these two limits are commutable. However, when the interaction is strong and over a critical value, the two limits become incommutable. This change of commutability is associated with a topological change in the structure of the energy bands. These results reveal that nonlinear mean-field theories, such as Gross-Pitaevskii equations for Bose-Einstein condensates, can be invalid in the adiabatic limit.