State transitions and decoherence in the avian compass

The radical pair model has been successful in explaining behavioral characteristics of the geomagnetic compass believed to underlie the navigation capability of certain avian species. In this study, the spin dynamics of the radical pair model and decoherence therein are interpreted from a microscopic state transition point of view. This helps to elucidate the interplay between the hyperfine and Zeeman interactions that enables the avian compass and clarify the distinctive effects of nuclear and environmental decoherence on it. Three regimes have been identified for the strength of the hyperfine interaction with respect to that of the geomagnetic Zeeman. It is found that the compass is likely to function in the large hyperfine interaction regime. Using a quantum information theoretic quantifier of coherence, we find that nuclear decoherence induces new structure in the spin dynamics for intermediate hyperfine interaction strength. On the other hand, environmental decoherence—modeled by two different noise models—seems to disrupt the compass action.

Figure 1

Singlet $\left(S\right)$ and triplet $(T_0,T_{+},T_{-})$ yields vs geomagnetic field orientation for various hyperfine coupling strengths. The subfigures illustrate how various spin transition pathways among $|s\rangle ,|t_0\rangle ,|t_{+}\rangle ,|t_{-}\rangle$ make the final spin yields field dependent. Simultaneous Zeeman and anisotropic hyperfine interactions play collaborative roles in inducing these transitions, which ultimately lead to magnetosensitive spin yields. (The key transitions induced by anisotropic hyperfine and Zeeman interactions are shown in the state transition diagram, Fig. 2.) Additionally, we observe a conspicuous peak appearing in $S$ yield for a range of hyperfine interaction strengths. This peak corresponds to the dip in $T_{+}$ and $T_{-}$ yields. We recognize that this peak appears because of decoherence due to the nucleus.

Figure 2

Key spin-state transitions induced by various terms of the Hamiltonian given by Eq. (1).

Figure 3

The normalized singlet yield (black) and the normalized relative entropy of coherence (red [gray]) as a function of magnetic field orientation $\left(\theta \right)$ at $a/\gamma B_0=1$. The relative entropy of coherence serves as a measure of coherence [35]. The dip in coherence corresponding to the peak in singlet yield indicates that the peaks in singlet yield in Fig. 1 are due to nuclear decoherence. This fact is further affirmed by the singlet yield curve obtained when the nuclear and the radical pair spin states are disentangled, cf. Appendix pp2.

Figure 4

Singlet yield as a function of geomagnetic field orientation in presence of Gauger environmental noise for the noise rates of $\mathrm{\Gamma }=$ 0, ${10}^4,5\times {10}^5$, and ${10}^7$ ${\text{s}}^{-1}$. The figure shows that if noise rate is equal to or greater than $k$, the orientation dependence of the singlet yield is destroyed.

Figure 5

Singlet yield as a function of geomagnetic field orientation in presence of Walters environmental noise for the noise rates of $\mathrm{\Gamma }=$ 0, ${10}^4,5\times {10}^5$, and ${10}^7$ ${\text{s}}^{-1}$. For this noise model, the noise rate greater than $k$ disrupts the coherent evolution of the radical pair state. Therefore, the singlet yield becomes flat and does not show the peak arising due to nuclear interaction.

Figure 6

Singlet $\left(S\right)$ and triplet $(T_0,T_{+},T_{-})$ yields vs geomagnetic field orientation for various hyperfine coupling strengths. Now the nuclear and radical pair spin states are disentangled. Unlike Fig. 1, the peaks in singlet yield do not appear here. This confirms that the peaks in singlet yield (Fig. 1) are direct manifestation of nuclear decoherence.

Figure 7

Singlet $\left(S\right)$ and triplet $(T_0,T_{+},T_{-})$ yields vs geomagnetic field orientation in presence of Gauger environmental noise with noise rate $\left(\mathrm{\Gamma }\right)$ = ${10}^4$ and ${10}^5{\text{s}}^{-1}$. The figure shows that spin yields tend to become uniform as the environmental noise rate is increased. Thus environmental noise opens the spin transition pathways between all four spin states of the radical pair.