Quantum Optimal Control for Coherent Spin Dynamics of Radical Pairs via Pontryagin Maximum Principle

TL;DR

This paper develops a quantum optimal control framework using Pontryagin Maximum Principle to optimize electromagnetic fields for spin dynamics in radical pairs, with applications in quantum biology.

Contribution

It introduces a new iterative Pontryagin Maximum Principle method for identifying bang-bang controls in quantum spin systems and demonstrates its effectiveness through numerical simulations.

Findings

The optimal control exhibits a bang-bang structure.

Numerical methods show convergence, stability, and regularization effects.

Filtering electromagnetic fields yields less than 1% change in singlet yield maxima.

Abstract

This paper aims to devise the shape of the external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state through maximization of the triplet-born singlet yield in biochemical reactions. The model is a Schr\"{o}dinger system with spin Hamiltonians given by the sum of Zeeman interaction and hyperfine coupling interaction terms. We introduce a one-parameter family of optimal control problems by coupling the Schr\"{o}dinger system to a control field through filtering equations for the electromagnetic field. Fr\'echet differentiability and the Pontryagin Maximum Principle in Hilbert space are proved, and the bang-bang structure of the optimal control is established. A new iterative Pontryagin Maximum Principle (IPMP) method for the identification of the bang-bang optimal control is developed. Numerical simulations based on IPMP and the gradient projection method (GPM) in Hilbert spaces are pursued, and the convergence, stability, and the regularization effect are demonstrated. Comparative analysis of filtering with regular optimal electromagnetic field versus non-filtering with bang-bang optimal field ({\it Abdulla et al, Quantum Sci. Technol., {\bf9}, 4, 2024}) demonstrates that the change of the maxima of the singlet yield is less than 1\%. The results open a venue for a potential experimental work on magnetoreception as a manifestation of quantum biological phenomena.