Environment-induced anisotropy and sensitivity of the radical pair mechanism in the avian compass

Several experiments over the years have shown that the earth's magnetic field is essential for orientation in birds' migration. The most promising explanation for this orientation is the photo-stimulated radical pair (RP) mechanism. In order to define a reference frame for the orientation task radicals must have an intrinsic anisotropy. We show that this kind of anisotropy and consequently the entanglement in the model are not necessary for the proper functioning of the compass. Classically correlated initial conditions for the RP, subjected to a fast decoherence process, are able to provide the anisotropy required. Even a dephasing environment can provide the necessary frame for the compass to work and also implies fast decay of any quantum correlation in the system without damaging the orientation ability. This fact significantly expands the range of applicability of the RP mechanism providing more elements for experimental search.

Figure 1

Angle sensitivity of the radical pair with an isotropic hyperfine tensor $A=a\mathbb{I}$ in the presence of the amplitude damping noise described by the Lindblad operator (4) for different increasing values of the coupling constant $\gamma$ from top to bottom. The first curve at the top is a reference curve for $\gamma =0,k=0.01\text{MHz}$, for the anysotropic hyperfine tensor and initial singlet state.

Figure 2

Concurrence (top oscillating plot) and quantum discord (lower oscillating plot) evolution for different values of the rates and magnetic field inclination angles. (a) Measurement rate $k=0.1\text{MHz}$, and dissipation rates $\gamma =2k$, with an angle $\theta =0$; in the inset an angle of $\theta =\pi /2$ was used. (b) Measurement rate $k=0.01\text{MHz}$, and dissipation rates $\gamma =10k$ with and angle $\theta =0$; in the inset an angle of $\theta =\pi /2$ was used. Is worth noting that the angle affects the amount of quantum correlations measured by the QD and the concurrence; for smaller angles the concurrence values are not only higher, but predict the same time of decoherence as the QD; however, for higher angles, the entanglement is less important than the classical correlations, making the QD to be higher and to predict longer correlation times. The decoherence was computed by means of the fidelity [30] between the evolving state and the initial singlet state.

Figure 3

Entanglement density for different values of the rates $\gamma$ and $k$, varying the angles $\theta$ of inclination of the earth's magnetic field $\left(47\mu T\right)$. The entanglement was measured by the use of concurrence. In the left panel, the rate values are $k=0.05\text{MHz},\gamma =6k$, and in the right panel the values are $k=0.1\text{MHz},\gamma =4k$. It is interesting to note that with smaller angles there are more entanglement sudden deaths and revivals, and that for angles around $\pi /4$ the decoherence time is shorter. The nuclear initial condition used in both cases was $|{\psi }_{\mathrm{Nuclear}}\rangle =\left(|0\rangle +|1\rangle \right)/\sqrt{2}$.

Figure 4

Perturbations in a singlet initial condition with an anisotropic hyperfine tensor without environmental noise. The red (lower continuous) curve represents the mean of 1000 perturbed singlet initial conditions. The mechanism appears quite robust, as the differences in the yield production remain unchanged when compared with the unperturbed singlet initial condition, given by the black (top continuous) curve in the figure). The dashed lines are four random yields taken from the sample.

Figure 5

Angle sensitivity for non singlet or triplet initial conditions, specifically ${\rho }_0=|\alpha \alpha \rangle \langle \alpha \alpha |$ with isotropic hyperfine tensors. Black (top curve): nuclear spin initially set to $|\uparrow \rangle$. Blue (lower curve): nuclear spin initially set to a mixed state $\left(|\uparrow \rangle +|\downarrow \rangle \right)/\sqrt{2}$. Red (middle curve): nuclear spin initially set to $|\downarrow \rangle$.

Figure 6

Angle sensitivity of isotropic Hamiltonians with $k=0.01\text{MHz}$. Black (middle curve): Evolution of the singlet yield for different angles with an entangled initial condition. Blue (lower curve): Evolution of the singlet yield for different angles with an entangled initial condition and $\gamma =2k$. Red (top curve): Evolution of the singlet yield for different angles with the unentangled initial condition for both electronic and nuclear spins ${\rho }_0=\frac{1}{4}\left(|\alpha \beta \rangle \langle \alpha \beta |+|\beta \alpha \rangle \langle \beta \alpha |\right)\otimes \left(|\uparrow \rangle \langle \uparrow |+|\downarrow \rangle \langle \downarrow |\right)$ and with $\gamma =0$.