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

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Researchers developed a method to shape external electromagnetic fields that maximize quantum coherence in radical pairs by optimizing singlet yield in biochemical reactions. The team used a Schrödinger system with Zeeman and hyperfine coupling terms to model spin dynamics. The study applies the Pontryagin Maximum Principle to prove bang-bang optimal control structures in Hilbert space, establishing Fréchet differentiability. This mathematical framework confirms the optimality of discontinuous control fields for quantum coherence. A novel iterative Pontryagin Maximum Principle (IPMP) method was introduced to identify bang-bang controls, alongside gradient projection simulations. Numerical results demonstrated convergence, stability, and regularization effects in Hilbert spaces. Comparative analysis showed that filtering with continuous optimal fields versus non-filtering with bang-bang controls altered singlet yield maxima by less than 1%. This suggests functional equivalence in preserving quantum coherence. The findings pave the way for experimental studies on magnetoreception, offering potential insights into quantum biological phenomena like avian navigation and biochemical reactivity.

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

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AbstractThis 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 Schrödinger 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 Schrödinger system to a control field through filtering equations for the electromagnetic field. Fréchet 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 ($\textit{Abdulla et al, Quantum Sci. Technol., $\bf{9}$, 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.Popular summaryWe solve the quantum optimal control problem to find an external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state. In this context, the quantum coherence refers to the phase-coherent superposition of singlet and triplet states in a pair of free radicals, allowing them to exist simultaneously in both states in biochemical reactions. It is demonstrated that the Pontryagin Maximum Principle – a fundamental mathematical principle for the optimality of complex dynamical systems – turns out to be a fundamental principle for quantum coherence. We introduce a model of a filtered Schrödinger system, and demonstrate that trading off between the original non-filtered model with a bang-bang optimal magnetic field and the filtered model with a continuous in-time optimal magnetic field preserves quantum coherence. We develop a novel iterative regularization method for the identification of the bang-bang optimal control, and an associated simple continuous-in-time optimal magnetic field input. The method has some similarity with the gradient ascent method, with the difference being that instead of moving in the gradient direction in the full control space, it generates the movement in the manifold of bang-bang control vectors guided by the Pontryagin Maximum Principle.► BibTeX data@article{Abdulla2026quantumoptimal, doi = {10.22331/q-2026-05-06-2096}, url = {https://doi.org/10.22331/q-2026-05-06-2096}, title = {Quantum {O}ptimal {C}ontrol for {C}oherent {S}pin {D}ynamics of {R}adical {P}airs via {P}ontryagin {M}aximum {P}rinciple}, author = {Abdulla, Ugur G. and Rodrigues, Jose H. and Slotine, Jean-Jacques}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2096}, month = may, year = {2026} }► References [1] M. Bucci, C. Goodman, and T. L. Sheppard. ``A decade of chemical biology''. Nat. Chem. 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Hore. ``Spin relaxation of radicals in cryptochrome and its role in avian magnetoreception''. The Journal of Chemical Physics 145, 035104 (2016). https://doi.org/10.1063/1.4958624 [60] T. P. Fay, L. P. Lindoy, and D. E. Manolopoulos. ``Spin relaxation in radical pairs from the stochastic Schrödinger equation''. The Journal of Chemical Physics 154, 084121 (2021). https://doi.org/10.1063/5.0040519Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-06 06:32:05: Could not fetch cited-by data for 10.22331/q-2026-05-06-2096 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-06 06:32:05: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractThis 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 Schrödinger 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 Schrödinger system to a control field through filtering equations for the electromagnetic field. Fréchet 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 ($\textit{Abdulla et al, Quantum Sci. Technol., $\bf{9}$, 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.Popular summaryWe solve the quantum optimal control problem to find an external electromagnetic field that drives the spin dynamics of radical pairs to a quantum coherent state. In this context, the quantum coherence refers to the phase-coherent superposition of singlet and triplet states in a pair of free radicals, allowing them to exist simultaneously in both states in biochemical reactions. It is demonstrated that the Pontryagin Maximum Principle – a fundamental mathematical principle for the optimality of complex dynamical systems – turns out to be a fundamental principle for quantum coherence. We introduce a model of a filtered Schrödinger system, and demonstrate that trading off between the original non-filtered model with a bang-bang optimal magnetic field and the filtered model with a continuous in-time optimal magnetic field preserves quantum coherence. We develop a novel iterative regularization method for the identification of the bang-bang optimal control, and an associated simple continuous-in-time optimal magnetic field input. The method has some similarity with the gradient ascent method, with the difference being that instead of moving in the gradient direction in the full control space, it generates the movement in the manifold of bang-bang control vectors guided by the Pontryagin Maximum Principle.► BibTeX data@article{Abdulla2026quantumoptimal, doi = {10.22331/q-2026-05-06-2096}, url = {https://doi.org/10.22331/q-2026-05-06-2096}, title = {Quantum {O}ptimal {C}ontrol for {C}oherent {S}pin {D}ynamics of {R}adical {P}airs via {P}ontryagin {M}aximum {P}rinciple}, author = {Abdulla, Ugur G. and Rodrigues, Jose H. and Slotine, Jean-Jacques}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2096}, month = may, year = {2026} }► References [1] M. Bucci, C. Goodman, and T. L. Sheppard. ``A decade of chemical biology''. Nat. Chem. Biol. 6, 847–854 (2010). https://doi.org/10.1038/nchembio.489 [2] P. Ball. ``The dawn of quantum biology''. Nature 9, 10–18 (2011). https://doi.org/10.1038/474272a [3] N. Lambert and et al. ``Quantum biology''. Nat. Phys. 9, 10–18 (2013). https://doi.org/10.1038/nphys2474 [4] T. Ritz and et al. ``Magnetic compass of birds is based on a molecule with optimal directional sensitivity''. Biophysical journal 96, 3451–7 (2009). https://doi.org/10.1016/j.bpj.2008.11.072 [5] T. Ritz, P. Thalau, J. B. Phillips, R. Wiltschko, and W. Wiltschko. ``Resonance effects indicate a radical-pair mechanism for avian magnetic compass''. Nature 429, 177–80 (2004). https://doi.org/10.1038/nature02534 [6] C. Niessner and et al. ``Magnetoreception: activated cryptochrome 1a concurs with magnetic orientation in birds''. Journal of the Royal Society, Interface 10, 20130638 (2013). https://doi.org/10.1098/rsif.2013.0638 [7] R. J. Usselman, C. Chavarriaga, P. R. Castello, M. Procopio, T. Ritz, E. A. Dratz, D. .J. Singel, and C. F. Martino. ``The quantum biology of reactive oxygen species partitioning impacts cellular bioenergetics''. Sci. Rep. 6, 38543 (2016). https://doi.org/10.1038/srep38543 [8] J. Cai, F. Caruso, and M. B. Plenio. ``Quantum limits for the magnetic sensitivity of a chemical compass''. Phys. Rev. A 85, 040304 (2012). https://doi.org/10.1103/PhysRevA.85.040304 [9] J. Cai and M. B. Plenio. ``Chemical compass model for avian magnetoreception as a quantum coherent device''. Phys. Rev. Lett. 111, 230503 (2013). https://doi.org/10.1103/PhysRevLett.111.230503 [10] F. Cintolesi, T. Ritz, C. W. M. Kay, C. R. Timmel, and P. J. Hore. ``Anisotropic recombination of an immobilized photoinduced radical pair in a 50-${\mu}$T magnetic field: a model avian photomagnetoreceptor''. Chemical Physics 294, 385–399 (2003). https://doi.org/10.1016/S0301-0104(03)00320-3 [11] L. Turin. ``A spectroscopic mechanism for primary olfactory reception''. Chemical Senses 21, 773–791 (1996). https://doi.org/10.1093/chemse/21.6.773 [12] L. Turin, E. M. C. Skoulakis, and A. P. Horsfield. ``Electron spin changes during general anesthesia in Drosophila''. Proceedings of the National Academy of Sciences 111, E3524–E3533 (2014). https://doi.org/10.1073/pnas.1404387111 [13] M. Mohseni and et. al. ``Quantum effects in biology''.

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

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