Chiral-induced spin selectivity augments quantum coherence in avian compass

Coherence is a resource for quantum technologies, and its existence and utility for the biological processes occurring at ambient conditions have been extremely intriguing. Avian magnetoreception is one such biological process where the radical pair spin dynamics, along with the role and utility of quantum coherence, has been investigated from several aspects [1, 2, 3, 4, 5, 6, 7, 8]. The radical pair mechanism is based on a spin-sensitive chemical reaction that is mediated by a protein molecule  [9, 10, 11]. Owing to the chirality of protein molecules, the chiral-induced spin selectivity (CISS) effect could play an essential role in the electron transport part of the reaction. The origin of CISS is attributed to the spin-orbit interaction and the electrostatic potential provided by the chiral molecules [12, 13, 14, 15, 16, 17, 18, 19]. It was shown by Fay et al.  [20] that chirality in conjunction with spin-orbit reaction in electron transfer reaction generates coherence locally. It was done for the electron spin echo experiment. It was also shown in  [21] that the prerequisite for forming a radical pair for avian magneto-reception is the transfer of electrons. This transfer or transport of electrons between donor and acceptor occurs in a chiral molecule contributing toward CISS and affecting the operation of the avian compass.

Coherence quantifiers are based on the non-diagonal/coherence element of the density matrix of a quantum state [22, 23, 24]. It has been suggested in references [1, 25] that global coherence rather than local or electronic coherence might enhance the compass sensitivity. Therefore, we correlate the yield of the spin-selective chemical reaction with coherence in a chiral medium for the avian-magnetoreception.

In this work, we make use of relative entropy of coherence [22] and total coherence [3] as coherence quantifiers to answer the following questions: i) how does the CISS affect the local and global coherence in avian compass, ii) how do dipolar and exchange interaction in conjunction with CISS affect the total local and global coherence measures, iii) how does the environmental decoherence affect the multi-nuclei radical pair mechanism, and more importantly iv) does the quantum coherence play any utilitarian role for the avian compass? If yes, in what form? We have considered the case of three nuclei each on flavin adenine dinucleotide FAD and tryptophan TrpH radicals. The FAD act as a donor entity, whereas TrpH act as an acceptor entity. The hyperfine interaction values of these nuclei have been taken from ref. [9].

The manuscript has been organized as follows: Section II discusses the methodology followed for analysis. Section III discusses the results, wherein subsection III.1 discusses the effect of CISS on quantum coherence, and subsection III.2 explores the impact of electron-electron interactions on system sensitivity along with CISS. Section  III.3 illustrate an increase of coherence due to CISS at various rate combinations. Section IV demonstrates the effect of environmental decoherence on the system. Section V examines the utilitarian aspect of the quantum coherence in the avian compass.

In the radical pair model of the avian compass, an electron is photo-excited in the acceptor molecule, creating a vacancy in the ground state. Another electron from a neighboring donor molecule travels in the chiral medium to fill this vacancy. It results in the formation of a radical pair where the spin state of the electron on the donor molecule is $\hat{S}_{D}$ and on the acceptor molecule is $\hat{S}_{A}$ .[21, 26].

The spin state of the above formed radical is governed by Hamiltonian given by [27, 28, 21]

where $\omega=g\bar{\mu_{B}}\bar{B}$, $\bar{B}=B_{0}{((cos\theta cos\phi)\bar{x}+(cos\theta sin\phi)\bar{y}+(cos\theta)\bar{z}})$. $B_{0}$ corresponds to the earth’s magnetic field. $\theta$ and $\phi$ correspond to the orientation of the magnetic field with respect to hyperfine tensor [4]. $J$ and $D$ are the exchange and dipolar interactions. $A$ is the hyperfine tensor depicting interactions between electrons and neighboring nuclear spins.

The spin state of the radical pair evolves under Zeeman and hyperfine interactions. Along with this evolution of the spin state, the radical pair also recombines back as shown in Fig. 1. The recombination either happens back to the ground state or to the signaling state (via protonation with $H^{+}$ of the acceptor radical, cf. Fig. 1). The CISS plays role in the formation and recombination of the radical pair as ther involve electron transport through the chiral medium. Therefore, the effect of CISS is captured by the initial state $P_{I}$ and recombination state $P_{R}$, also shown with red arrows in Fig. 1. The signaling state does not involve the transfer of electrons (only $H^{+}$ involved), therefore, CISS is not involved in its formation (shown with the blue arrow in Fig. 1). We define [21]:

Them the initial density matrix is given as: $P_{I}=\ket{\psi_{I}}\bra{\psi_{I}}\otimes\frac{I}{Z}$, where $\frac{I}{Z}$ corresponds to the mixed state of nuclei, and $Z$ is size of the Hilbert space of the nuclei. The recombination operator $P_{R}={\ket{\psi_{R}}\bra{\psi_{R}}}$ accounts for recombination to the ground state where $\ket{\psi_{R}}$ is:

The CISS parameter $\chi\in[0,\frac{\pi}{2}]$ depends on the spin selectivity of the reactants medium; $\chi=0$ corresponding to no CISS and $\chi=\pi/2$ correspoding to the maximum CISS. The master equation governing the state evolution of the system is given as:

Where $k_{F}$ is the protonation rate to the signalling state, and $k_{R}$ is the recombination rate (back to the ground state) [21, 26].

This section is divided into three subsections. In the first subsection, we analyze the effect of CISS on the local and global coherence in the radical pair system. Interestingly, we observe that the CISS enhances both local and global coherences in the radical pair system. In the second subsection, we examine the effect of the dipolar interactions on global and local coherence along with CISS in the RP model. In the last subsection, we study the coherence in RP system as a function of recombination and protonation rates ($k_{R}$ and $k_{F}$).

In this section, we take into consideration the decoherence effect of the surrounding system. We modify Eq. 4 to add spin decoherence operators in the Lindblad formalism.

In Eq. 13, $Decohrence$ corresponds to the spin decoherence occurring due to surrounding environment. Mathematically we take six decoherence operators: $C_{1}=\sigma_{x}\otimes I_{E2}\otimes I_{N}$, $C_{2}=\sigma_{y}\otimes I_{E2}\otimes I_{N}$, $C_{3}=\sigma_{z}\otimes I_{E2}\otimes I_{N}$, $C_{4}=I_{E1}\otimes\sigma_{x}\otimes I_{N}$,$C_{5}=I_{E1}\otimes\sigma_{y}\otimes I_{N}$ and $C_{6}=I_{E1}\otimes\sigma_{z}\otimes I_{N}$. $I_{E1}$ correspond to the mixed state of electron on FAD^⋅ radical while $I_{E2}$ correspond to the mixed state of electron on TrpH^⋅ radical. $I_{N}$ is the combined mixed state of the nuclei and $k$ is the decoherence rate.

In Fig. 7, we plot total global ($M_{G}$) and local cohrence ($M_{L}$). This calculation has been done at $(k_{F},k_{R})=(10^{6},10^{8})~{}s^{-1}$ and $J=0$ and $D=0$. We observe that at full CISS, coherence is maximum even under decoherence. We have also listed $\triangle M_{G}$ and $\triangle M_{L}$ with various decoherence rates $k$ in Tab. 5. We observe a reduction in value of $\triangle M_{G}$ and $\triangle M_{L}$ as decoherence rate $k$ increases. However, interestingly, we observe increment in coherence due to CISS even at high decoherence rates.

In this section, we correlate coherence to the forward product (signalling state) yield of the reaction described in Fig. 1, demonstrating the utilitarian role of coherence. We use the correlation coefficient to show a statistical correlation between signaling state yield and the radical pair spin coherence (both local and global). We use numerous orientations of radical pair with respect to the external magnetic field to show this correlation in a four nuclei system at $(k_{F},k_{R})=(10^{6},10^{8})~{}s^{-1}$. The forward (signaling state) yield is defined as:

Where $\hat{\rho(t)}$ is the solution of the master equation Eq. 4, $Tr$ is the trace over the state density matrix $\rho$. $k_{F}$ is the rate associated with the signaling state.

In Fig. 8.(a), we have plotted the total global coherence ($M_{G}$) and the signaling state yield ($\phi_{F}$) for 2500 combinations of $\theta$ and $\phi$. We have taken the values where $\theta\in\{0^{o},180^{o}\}$ and $\phi\in\{0^{o},360^{o}\}$. The calculation is performed for no CISS case i.e. $\chi=0$. Similarly, in Fig. 8.(b), we have plotted total local coherence ($M_{L}$) and signaling state yield ($\phi_{F}$). The $R$ value (red font) corresponds to the correlation coefficient between the coherence measure and signaling state yield. The red line is the linear fit line corresponding to the scattered points. Similar plots have been plotted for intermediate CISS case (i.e. $\chi=\frac{\pi}{4}$) in Fig. 9 and full CISS case (i.e. $\chi=\frac{\pi}{2}$) in Fig. 10.

The total local coherence ($M_{L}$) has no clear correlation with the signaling state yield ($\phi_{F}$) for three values of $\chi$. Total global coherence ($M_{G}$) shows a high correlation with the yield of the forward signaling state i.e. as the degree of CISS increases, the correlation parameter $R$ between total global coherence and signaling state yield increases. In the full CISS case, the value is near unity showing a high correlation of global coherence with the forward signaling state. This forward signaling state is responsible for sending signals to neurons or brains. Hence, global coherence exhibit strong utilitarian role for the compass action than the local coherence i.e. a higher spin selectivity due to chirality leads to a higher correlation of global coherence with forward signaling state yield.

In conclusion, chiral-induced spin selectivity (CISS) effect causes sustained coherence in the radical pair mechanism of avian magnetoreception. It hints towards the possibility that spin coherence might be sustained in a realistic system despite many nuclei for significant time. Moreover, we also observe that the global coherence in the CISS-assisted avian compass is strongly correlated with the signaling state yield. This indicates that unlike local coherence. global coherence has strong utilitarian role in the compass action. We also observe that dipolar and exchange interactions are generally detrimental to the coherence of the avian compass, but their effect can be countered by CISS. All these conclusions confirm the significance of CISS in the avian compass spin dynamics.