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The geometric phase is a fundamental quantity characterizing the holonomicfeature of quantum systems. It is well known that the evolution operator of aquantum system undergoing a cyclic evolution can be simply written as theproduct of holonomic and dynamical components for the three special casesconcerning the Berry phase, adiabatic non-Abelian geometric phase, andnonadiabatic Abelian geometric phase. However, for the most general caseconcerning the nonadiabatic non-Abelian geometric phase, how to separate theevolution operator into holonomic and dynamical components is a long-standingopen problem. In this work, we solve this open problem. We show that theevolution operator of a quantum system can always be separated into the productof holonomy and dynamic operators. Based on it, we further derive a matrixrepresentation of this separation formula for cyclic evolution, and give anecessary and sufficient condition for a general evolution being purelyholonomic. Our finding is not only of theoretical interest itself, but also ofvital importance for the application of quantum holonomy. It unifies therepresentations of all four types of evolution concerning theadiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides ageneral approach to realizing purely holonomic evolution.

Evolution Operator Can Always be Separated into the Product of Holonomy and Dynamic Operators