Professors Yu Xiaodong and Tong Dianmin, School of Physics of Shandong University, proved that “Evolution Operator Can Always be Separated into the Product of Holonomy and Dynamic Operators”, which has been published in Phys. Rev. Lett. 131, 200202,2023, recently.

Geometric phase is a fundamental quantity characterizing the holonomic feature of a quantum system undergoing a cyclic evolution. It is widely applied to various fields of physics. The evolution operator of a quantum system can be simply written as the product of Ablian ornon-Aeblian geometric phase and dynamical phase for anadiabatic cyclic evolution. However, for a nonadiabatic evolution, it is difficult to separate non-Abelian geometric phase from dynamic phase. How to separate a general evolution operator into the product of non-Abelian geometric and dynamical phases is a long-standing open problem.

In the present work, Yu and Tong solved this open problem. By introducing the notion of holonomy operator, they show that the evolution operator of a quantum system can always be separated into the product of holonomy and dynamic operators. Based on it, they further derive a matrix representation of this separation formula for cyclic evolution, and give a necessary and sufficient condition for a general evolution being purely holonomic. This finding is not only of theoretical interest itself, but also of vital importance for the application of quantum holonomy. It unifies the representations of all four types of evolution concerning the adiabatic/nonadiabatic Abelian/non-Abelian geometric phase, and provides a general approach to realizing purely holonomic evolution.

Link to the article:http://link.aps.org/doi/10.1103/PhysRevLett.131.200202