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

Area of Science:

• Quantum Mechanics
• Geometric Phase Theory

Background:

• Geometric phase characterizes quantum system holonomy.
• Separating evolution operators is known for specific cases (Berry phase, adiabatic non-Abelian, nonadiabatic Abelian).
• The general nonadiabatic non-Abelian geometric phase separation remained an open problem.

Purpose of the Study:

• To solve the open problem of separating evolution operators for the most general case of nonadiabatic non-Abelian geometric phase.
• To unify the representations of all four types of geometric phase evolution.
• To provide a general approach for achieving purely holonomic evolution.

Main Methods:

• Demonstrated that the evolution operator can always be separated into holonomy and dynamic operators.
• Derived a matrix representation for this separation formula in cyclic evolutions.
• Established a necessary and sufficient condition for general evolutions to be purely holonomic.

Main Results:

• Successfully solved the long-standing problem of separating evolution operators in the general nonadiabatic non-Abelian geometric phase case.
• Developed a unified framework for representing geometric phase evolution.
• Identified conditions for purely holonomic evolution.

Conclusions:

• The study provides a general method to separate quantum system evolution operators into holonomic and dynamic parts.
• This work unifies existing theories on geometric phases and offers practical applications in quantum holonomy.
• The findings are crucial for advancing the understanding and application of quantum holonomy.