A physicist-friendly primer on the Hamiltonian for quantum sensing in proteins: analytical expressions and insights for a toy model of the radical-pair mechanism

Abstract

Electron spin-dependent chemical reactions in proteins, often discussed under the ‘radical-pair mechanism’, have been studied for decades and remain the leading microscopic proposal for magnetic field sensing in biology. Yet the essential physics is often obscured by the complexity of realistic models. In this work, we present a physicist-friendly primer on the simplest radical-pair Hamiltonian that already captures many of the mechanism’s best-known qualitative features. The contribution of this work is fourfold. First, we place on record a complete analytical solution of this toy model, which has previously been studied extensively, mostly through numerical and partial analytical approaches. Working in the experimentally relevant singlet–triplet basis, we derive closed-form expressions for the instantaneous singlet population and for two related time-averaged singlet yields. Second, we introduce a new interpretation of these results that makes several familiar features of radical-pair physics transparent. In particular, we show that the dynamics admit a bright–dark decomposition (in the sense of spin mixing), similar to structures widely studied in atomic and optical physics, for example in electromagnetically-induced transparency. Third, through this bright–dark perspective, we clarify experimentally relevant features of the toy model. In particular, we show that the so-called ‘low-field effect’ arises from a coherence term between bright and dark sectors, and that the special role of zero field is best understood as a phase-locking phenomenon rather than merely as enhanced mixing. The same framework also makes it possible to explicitly identify the ‘pathway that opens’, as per the chemists’ language, once a nonzero field is applied. Fourth, we import methods from quantum magnetometr y, developed in the context of technological quantum sensing, to obtain further insight into the model. This allows us to clarify the role of initial state preparation and the trade-off between coherent phase accumulation and time-averaging penalties. The resulting toy model serves both as an analytically tractable benchmark and as a conceptual starting point for future work incorporating a true open quantum system treatment, unequal singlet and triplet decay rates, and fully directional magnetic field control.