On separation of quantum time evolution into holonomic and dynamical parts

TL;DR

This paper investigates the conditions under which quantum time evolution can be decomposed into holonomic and dynamical components in non-adiabatic, non-Abelian systems, highlighting gauge invariance and illustrating with a three-level system.

Contribution

It identifies all cases where such a separation is possible and proves that separability is a gauge-invariant property of quantum evolution.

Findings

Separation is only possible in specific cases.

Separable evolution is gauge invariant.

Typical non-separable cases involve non-commuting Hamiltonians.

Abstract

The issue of separating Schr\"odinger-type quantum time evolution into a product of holonomic and dynamical parts in the non-adiabatic non-Abelian case is examined. We identify all special cases in which this kind of separation is possible, and we prove that separability is a gauge invariant property of quantum time evolution. The general analysis is implemented in a three-level system with $\Lambda$-type coupling structures. The typical case where the holonomic and dynamical parts do not separate is illustrated by means of two non-commuting $\Lambda$-type Hamiltonians.