New approach to describe two coupled spins in a variable magnetic field

TL;DR

This paper introduces a novel method to analyze the evolution of two coupled spins in a time-dependent magnetic field, simplifying the solution process especially under adiabatic conditions.

Contribution

The paper presents a new approach that transforms the time-dependent Schrödinger equation using an instantaneous basis, enabling easier solutions for coupled spins in variable magnetic fields.

Findings

Exact solutions for hyperfine interaction with axial symmetry in constant fields.

Simplified solutions under adiabatic magnetic field variations.

Potential for perturbative treatment of other complex dynamical systems.

Abstract

We propose a method to describe the evolution of two spins coupled by hyperfine interaction in an external time-dependent magnetic field. We apply the approach to the case of hyperfine interaction with axial symmetry, which can be solved exactly in a constant, appropriately oriented magnetic field. In order to treat the nonstationary dynamical problem, we modify the time-dependent Schr\"odinger equation through a change of representation that, by exploiting an instantaneous (adiabatic) basis makes the time-dependent Hamiltonian diagonal at any time instant. The solution of the transformed time-dependent Schr\"odinger in the form of chronologically ordered exponents with transparent pre-exponential coefficients is reported. This solution is highly simplified when an adiabatically varying magnetic field perturbs the system. The approach here proposed may be used for the perturbative treatment of other dynamical problems with no exact solution.