Geometric Origin of the Non-Adiabaticity Parameter and Self-Limiting Instability in Driven Nonlinear Systems

TL;DR

This paper reveals that the non-adiabaticity parameter has a geometric interpretation related to the evolution speed of quantum states, enabling continuous evaluation and control of instabilities in driven nonlinear systems.

Contribution

It introduces a local geometric framework for analyzing non-adiabatic instability and proposes a nonlinear regulator that suppresses instability in driven bosonic systems.

Findings

The non-adiabaticity parameter corresponds to the evolution speed in projective Hilbert space.

A nonlinear regulator effectively suppresses the geometric evolution speed.

A crossover parameter serves as a criterion for self-limited instability.

Abstract

We establish that the non-adiabaticity parameter has a direct geometric interpretation as the instantaneous evolution speed of a driven quantum state in projective Hilbert space under the Fubini Study metric. In contrast to conventional asymptotic approaches, the proposed framework provides a strictly local geometric criterion that allows non-adiabatic instability and its nonlinear suppression to be evaluated continuously at each stage of the driven evolution. We further show that an occupation-dependent nonlinear regulator Usuppresses the effective geometric evolution speed, leading to bounded low-occupancy dynamics. The resulting crossover parameter provides a compact criterion for self-limited non-adiabatic instability in driven nonlinear bosonic systems.