Dark state population determines magnetic sensitivity in radical pair magnetoreception model

What is the real role of the quantum coherence and entanglement in the radical pair (RP) compass, and what determines the singlet yield have not been fully understood. In this paper, we find that the dark states of the two-electron Zeeman energy operator (TEZE) play an important role in the RP compass. We respectively calculate the singlet yields for two initial states in this dark state basis: the coherent state and the same state just removing the dark state coherence. For the later there is neither dark state coherence nor entanglement in the whole dynamical process. Surprisingly we find that in both cases the singlet yields are the same, and based on this result, we believe that the dark state population determines the singlet yield completely, and the dark state coherence and entanglement have little contribution to it. Finally, we also find that the dark state population as well as the singlet yield anisotropy is fragile to the vertical magnetic noise. However, the orientation is robust and is even enhanced by the parallel magnetic noise because the dark states expand a decoherence-free subspace. The dark state population as well as the orientation is more robust to the hyperfine coupling noise.