Quantum-limited biochemical magnetometers designed using the Fisher information and quantum reaction control

Radical-ion pairs and their reactions have triggered the study of quantum effects in biological systems. This is because they exhibit a number of effects best understood within quantum information science, and at the same time are central in understanding the avian magnetic compass and the spin transport dynamics in photosynthetic reaction centers. Here we address radical-pair reactions from the perspective of quantum metrology. Since the coherent spin motion of radical pairs is effected by an external magnetic field, these spin-dependent reactions essentially realize a biochemical magnetometer. Using the quantum Fisher information, we find the fundamental quantum limits to the magnetic sensitivity of radical-pair magnetometers, arriving at a sensitivity $\delta B=2\mathrm{pT}/\tau \left[1\mu \text{s}\right]\sqrt{{\nu }_0\left[{10}^{12}\right]}$, given in terms of radical-pair lifetime $\tau$ and number of radical pairs ${\nu }_0$. We then explore how well the usual measurement scheme considered in radical-pair reactions, the measurement of reaction yields, approaches the fundamental limits. In doing so, we find the optimal hyperfine interaction Hamiltonian that leads to the best magnetic sensitivity as obtained from reaction yields. This is still an order of magnitude smaller than the absolute quantum limit. Finally, we demonstrate that with a realistic quantum reaction control reminding one of Ramsey interferometry, here presented as a quantum circuit involving the spin-exchange interaction and a recently proposed molecular switch, we can approach the fundamental quantum limit within a factor of 2. This work opens the application of well-advanced quantum metrology methods to biological systems.

Figure 1

(a) Radical-pair reaction dynamics. A charge transfer following the photoexcitation (not shown here) of a donor-acceptor dyad DA produces a singlet-state radical pair ${}^{\mathrm{S}}{D}^{•+}{A}^{•-}$, which is coherently converted to the triplet radical pair, ${}^{\mathrm{T}}{D}^{•+}{A}^{•-}$, due to intramolecule magnetic interactions embodied in the spin Hamiltonian ${\mathcal{H}}_B$. Simultaneously, spin-selective charge recombination leads to singlet (DA) and triplet neutral products $\left({}^{\mathrm{T}}DA\right)$. (b) Quantum metrology aspect of the radical-pair reaction, where an initial spin state is transformed into a final spin state, which depends on the magnetic field $B$ through the spin Hamiltonian. The measurement of the final state conveys information about $B$.

Figure 2

Magnetic sensitivity $1/\delta B$ in units of $1/k$ $(1/\delta B$ is better visualized than $\delta B)$, determined from the singlet reaction yield for a radical pair with one nuclear spin-1/2, having equal recombination rates $k_{\mathrm{S}}=k_{\mathrm{T}}=k$. Isotropic hyperfine coupling $\left(A_x=A_y=A_z=A\right)$ and initial state (a) $|\mathrm{S}\rangle \otimes |\Uparrow \rangle$, (b) $|\mathrm{S}\rangle \otimes |\Downarrow \rangle$, (c) an equal mixture of the previous two. (d) Maximally anisotropic hyperfine coupling $(A_x=A$ and $A_y=A_z=0$, or $A_y=A$ and $A_x=A_z=0)$. In this case $\delta B$ is the same for all three initial states. (e), (f) When a time-resolved measurement of the reaction yield is possible, $\delta B$ is calculated by Eq. (10), resulting in (e) and (f) for the isotropic and maximally anisotropic case, respectively, where the mixed singlet initial state [as in (c)] was used for both.

Figure 3

According to the proposal of [45], radical-pair reactions can be controlled by binding the two radicals to the two ends of a molecular switch, the conformation of which can be laser controlled. We here consider that apart from the Zeeman and hyperfine coupling term in the magnetic Hamiltonian, in the cis conformation there is also a finite-exchange coupling, which the authors in [45] take to be infinite.

Figure 4

Time dependence of $g_t(A,B)={\partial }_B\mathrm{Tr}\left\{{\rho }_t{\mathrm{Q}}_{\mathrm{S}}\right\}$, where ${\rho }_t=e^{-i{\mathcal{H}}_{\mathrm{trans}}t}{\rho }_0{e}^{i{\mathcal{H}}_{\mathrm{trans}}t}$, for a mixed singlet initial state ${\rho }_0={\mathrm{Q}}_{\mathrm{S}}/\mathrm{Tr}\left\{{\mathrm{Q}}_{\mathrm{S}}\right\}$, and (a) an isotropic and (c) a maximally anisotropic hyperfine interaction. In (b) and (d) we depict the corresponding probability per unit time, $d{p}_r/dt=k{e}^{-kt}$, for the molecular switches to close by the reaction control laser pulses, which are tuned to coincide with the positive swings of $g_t$. In the middle of trace (b) there are a number of pulses missing, since there the corresponding $g_t$ is on average zero and hence will not contribute to the singlet-yield magnetic sensitivity. For all plots it was $A=352k$ and $B=17.6k=A/20$.

Figure 5

Quantum circuit for a quantum-limited biochemical magnetometer, approaching the absolute quantum limit on $\delta B$ by a factor of 2. The two unpaired electron spins of the radical pair start out in the singlet state. Circuitwise, this follows from $|\downarrow \downarrow \rangle$ by application of a Hadamard and a controlled-not gate. The nuclear spin is unpolarized, compactly denoted by the $2\times 2$ unit matrix divided by 2, $\mathbb{1}/2$ (equivalently, there are two such circuits for each of the two nuclear spin states). The Hamiltonian ${\mathcal{H}}_{\mathrm{cis}}$, including a finite-exchange interaction, acts upon the initial radical-pair state for a time ${\tau }_1$. A reaction control laser pulse opens all molecular switches, and the magnetometric Hamiltonian ${\mathcal{H}}_{\mathrm{trans}}$ acts for a time ${\tau }_2$. Another reaction control pulse closes some switches, and upon closure the radical-pair spin state evolves again by ${\mathcal{H}}_{\mathrm{cis}}$ until it recombines. The singlet recombination yield, i.e., the measurement of the singlet projector ${\mathrm{Q}}_{\mathrm{S}}$, carries the magnetic field information.

Figure 6

(a) Magnetic sensitivity of the singlet reaction yield, ${\mathrm{\Lambda }}_B=|d{Y}_{\mathrm{S}}/dB|$, and (b) uncertainty $\delta B$ in the estimation of the magnetic field, normalized by the absolute quantum limit $\delta B_F$. The blue dashed line reproduces the result of the reaction control scheme of [45], while the red solid line is the result of this work. Our reaction control scheme approaches $\delta B_F$ within a factor of 2. (c) The minimum of the red solid line in (b) is plotted as a function of the exchange coupling $J$. For $J=0.65A=229k$, we obtain $\delta B_{\min }=2\delta B_F$. But nearby values of $J$ are induced by molecular vibrations; hence averaging trace (c) leads to the realistic uncertainty 2.2 times away from $\delta B_F$.