Geometric Phase of a Spin-1/2 Particle Coupled to a Quantum Vector Operator

TL;DR

This paper extends the concept of Berry's phase to a spin-1/2 system coupled to a quantum vector operator, revealing quantum effects and differences from classical phases through analytical and numerical analysis.

Contribution

It introduces a method to compute the geometric phase for a quantum-driven spin system and explores the quantum-classical relationship using Schmidt decomposition and concrete examples.

Findings

Quantum geometric phase depends on the commutator of field components.

The quantum phase differs from the classical phase, especially in the presence of noncommuting operators.

Results agree well with numerical simulations and highlight quantum effects in geometric phases.

Abstract

We calculate Berry's phase when the driving field, to which a spin-1/2 is coupled adiabatically, rather than the familiar classical magnetic field, is a quantum vector operator, of noncommuting, in general, components, e.g., the angular momentum of another particle, or another spin. The geometric phase of the entire system, spin plus "quantum driving field", is first computed, and is then subdivided into the two subsystems, using the Schmidt decomposition of the total wave function -the resulting expression shows a marked, purely quantum effect, involving the commutator of the field components. We also compute the corresponding mean "classical" phase, involving a precessing magnetic field in the presence of noise, up to terms quadratic in the noise amplitude -the results are shown to be in excellent agreement with numerical simulations in the literature. Subtleties in the relation between the quantum and classical case are pointed out, while three concrete examples illustrate the scope and internal consistency of our treatment.