Retrodictive derivation of the radical-ion-pair master equation and Monte Carlo simulation with single-molecule quantum trajectories

Radical-ion-pair reactions, central in photosynthesis and the avian magnetic compass mechanism, have been recently shown to be a paradigm system for applying quantum information science in a biochemical setting. The fundamental quantum master equation describing radical-ion-pair reactions is still under debate. Here we use quantum retrodiction to formally refine the theory put forward in the paper by Kominis [I. K. Kominis, Phys. Rev. E 83, 056118 (2011)]. We also provide a rigorous analysis of the measure of singlet-triplet coherence required for deriving the radical-pair master equation. A Monte Carlo simulation with single-molecule quantum trajectories supports the self-consistency of our approach.

Figure 1

Simplified energy level diagram depicting radical-ion-pair reaction dynamics. A donor-acceptor dyad is photo excited and a subsequent charge transfer produces a singlet radical-ion pair. Magnetic interactions within the radical pair induce coherent singlet-triplet mixing, while spin-dependent charge recombination leads to singlet and triplet neutral products at the respective reaction rates $k_{\mathrm{S}}$ and $k_{\mathrm{T}}$. The reaction can, in principle, close through intersystem crossing from the triplet to the singlet ground state.

Figure 2

Detailed energy level structure of radical-ion pairs. The vibrational excitations of the singlet (DA) and the triplet $({}^{\mathrm{T}}\text{DA}$) ground state form a reservoir that probes the electron spin state of the RP, leading to an intramolecule measurement of ${\mathrm{Q}}_{\mathrm{S}}$. Virtual transitions (rates $k_{\mathrm{S}}/2$ and $k_{\mathrm{T}}/2)$ to the reservoir levels and back to the RP lead to S-T decoherence, while real transitions (rates $k_{\mathrm{S}}$ and $k_{\mathrm{T}})$ to the reservoir states followed by their decay to the ground state lead to recombination.

Figure 3

The time evolution of $\langle {\mathrm{Q}}_{\mathrm{S}}\rangle$ for a model RP with one nuclear spin, taking into account only S-T decoherence and S-T mixing driven by the Hamiltonian $\mathcal{H}=\omega (s_{1z}+s_{2z})+A{\mathbf{s}}_1·\mathbf{I}$, where the Larmor frequency is taken $\omega =A/10$ and the recombination rates are $k_{\mathrm{S}}=k_{\mathrm{T}}=A/4$. These parameters represent a typical RP at earth's field with a hyperfine coupling on the order of 1 mT and recombination times on the order of 20 ns. (a) Single-RP quantum trajectory, depicting singlet and triplet projections at random instants in time. The initial RP state for this trajectory is $|S\rangle \otimes |\uparrow \rangle$. (b) Average of 20 000 such trajectories (red solid line), half of which have initial state $|S\rangle \otimes |\uparrow \rangle$ while the other half have initial state $|S\rangle \otimes |\downarrow \rangle$. The time axis was split into 10 000 steps $dt$, in every one of which one out of the three possibilities outlined in Sec. 3 was realized. The prediction of the trace-preserving master equation (1) is shown by the black dashed line. The initial state for the density matrix was the usually considered singlet state with unpolarized nuclear spin, $\rho ={\mathrm{Q}}_{\mathrm{S}}/\mathrm{Tr}\left\{{\mathrm{Q}}_{\mathrm{S}}\right\}$.

Figure 4

Time evolution of $\mathrm{Tr}\{\tilde{\rho }{\mathrm{Q}}_{\mathrm{S}}\}$ (red solid line) and S-T coherence $\mathcal{C}\left(\tilde{\rho }\right)$ (black dashed line) for the same RP considered in Fig. 3, taking into account only S-T mixing driven by the Hamiltonian $\mathcal{H}$, i.e., $d\tilde{\rho }/dt=-i[\mathcal{H},\tilde{\rho }]$. The singlet state obviously corresponds to zero S-T coherence, while the state in-between the extrema of $\mathrm{Tr}\{\tilde{\rho }{\mathrm{Q}}_{\mathrm{S}}\}$ corresponds to an S-T superposition and hence maximum S-T coherence.

Figure 5

Time evolution of $\langle {\mathrm{Q}}_{\mathrm{S}}\rangle$ including S-T mixing, S-T decoherence, and recombination for the same RP Hamiltonian used in Figs. 3 and 4, with $k_{\mathrm{S}}=k_{\mathrm{T}}=A/4$. (a) Example of a single-RP quantum trajectory with initial state $|S\rangle \otimes |\uparrow \rangle$. (b) Monte Carlo simulation (red solid line) using 10 000 trajectories (two initial states $|S\rangle \otimes |\uparrow \rangle$ and $|S\rangle \otimes |\downarrow \rangle$, with 5000 trajectories for each), prediction of the master equation of this work (dashed line), and the earlier theory (solid line) introduced in [5]. The corresponding measure of S-T coherence $p_{\mathrm{coh}}$ is shown with the blue dotted line. The Monte Carlo and the theoretical prediction of this work coincide.

Figure 6

Similar plots as Fig. 5, but with asymmetric recombination rates. (a) $k_{\mathrm{S}}=0,k_{\mathrm{T}}=A/4$. (b) $k_{\mathrm{S}}=0,k_{\mathrm{T}}=A/2$.

Figure 7

Comparison of the three theories for the case presented in Fig. 6, i.e., $k_{\mathrm{S}}=0$ and $k_{\mathrm{T}}=A/2$. (a) The S-T coherence, embodied by the amplitude of the oscillation of $\langle {\mathrm{Q}}_{\mathrm{S}}\rangle$, decays faster in the Jones-Hore theory, slower in our theory, and even slower in the traditional theory. (b) The corresponding decay rate ${\gamma }_c/k$, where $k_{\mathrm{T}}=2k$.