Spin in a General Time Varying Magnetic Field: Generalization of the Adiabatic Factorization of Time Evolution

TL;DR

This paper extends the adiabatic factorization of the quantum spin evolution operator to general time-varying magnetic fields, introducing a new operator to account for non-adiabatic effects and generalizing magnetic resonance conditions.

Contribution

It provides a generalized factorization of the spin evolution operator for arbitrary magnetic field variations, including a new operator for non-adiabatic transitions, independent of spin representation.

Findings

Derived an explicit expression for the angular velocity of the non-adiabatic operator N(t)

Identified conditions under which N(t) can be explicitly determined

Generalized the magnetic resonance condition for non-adiabatic scenarios

Abstract

An extension of the adiabatic factorization of the time evolution operator is studied for spin in a general time varying magnetic field $B(t)$. When $B(t)$ changes adiabatically, such a factorization reduces to the product of the geometric operator which embodies the Berry phase phenomenon and a usual dynamical operator. For a general time variation of $B(t)$, there should be another operator $N(t)$ in the factorization that is related to non-adiabatic transitions. A simple and explicit expression for the instantaneous angular velocity of this operator is derived. This is done in a way that is independent of any specific representation of spin. Two classes of simple conditions are given under which the operator $N(t)$ can be made explicit. As a special case, a generalization of the traditional magnetic resonance condition is pointed out.