Operators for the Aharonov-Anandan and Samuel-Bhandari Phases

TL;DR

This paper develops operators for the Aharonov-Anandan and Samuel-Bhandari phases, linking geometric phases to quantum evolution and constructing a quantum clock operator with canonical properties.

Contribution

It introduces operators for geometric phases in quantum systems, including non-cyclic evolutions, and connects these to quantum clocks with canonical commutation relations.

Findings

Operator for Aharonov-Anandan phase for time-independent Hamiltonians.

Derived operator for Samuel-Bhandari phase for non-cyclic evolutions.

Constructed a quantum clock operator with a canonical commutator with the Hamiltonian.

Abstract

We construct an operator for the Aharonov-Anandan phase for time independent Hamiltonians. This operator is shown to generate the motion of cyclic quantum systems through an equation of evolution involving only geometric quantities, i.e. the distance between quantum states, the geometric phase and the total length of evolution. From this equation, we derive an operator for the Samuel and Bhandari phase (SB-phase) for non cyclic evolutions. Finally we show how the SB-phase can be used to construct an operator corresponding to a quantum clock which commutator with the Hamiltonian has a canonical expectation value.