Interradical motion can push magnetosensing precision towards quantum limits

Abstract

Magnetosensitive spin-correlated radical-pairs (SCRPs) offer a promising platform for noise-robust quantum metrology. However, unavoidable interradical interactions, such as electron-electron dipolar and exchange couplings, alongside deleterious perturbations resulting from intrinsic radical motion, typically degrade their potential as magnetometers. In contrast to this, we show how structured molecular motion modulating interradical interactions in a live chemical sensor in cryptochrome can, in fact, increase sensitivity and, more so, push precision in estimating magnetic field directions closer to the quantum Cramér-Rao bound, suggesting near-optimal metrological performance. Remarkably, this approach to optimality is amplified under environmental noise and persists with increasing complexity of the spin system, suggesting that perturbations inherent to such natural systems have enabled them to operate closer to the quantum limit to more effectively extract information from the weak geomagnetic field. This insight opens the possibility of channeling the underlying physical principles of motion-induced modulation of electron spin-spin interactions towards devising efficient handles over emerging molecular quantum information technologies.

Figures (8)

• Figure 1: Magnetosensing modulated by interradical motion. (a) Schematic of a magnetosensitive array (in the eye-frame $(X, Y, Z)$), composed of magnetoreceptor units (gray spheroids; molecular frame $(x,y,z)$) distributed with fixed orientations along the retina. Incident blue light photoexcites radical-pairs within the magnetoreceptors. (b) Blue-light-activated magnetoreceptor protein cryptochrome. (c) Within cryptochrome, an FAD$^{\bullet -}$/W$_{\mathrm{C}}^{\bullet+}$ radical-pair is created in the singlet state ($^{1}[\cdot]$) and interconverts with triplet states ($^{3}[\cdot]$) via the interradical-separation-modulated Hamiltonian $\hat{H}(r(t))$, which includes Zeeman, hyperfine, exchange and electron-electron dipolar (EED) interactions. Spin-selective recombination occurs with rate $k_{\mathrm{b}}(t)$, while signaling products form at a rate $k_{\mathrm{f}}$. The Zeeman interaction imparts magnetosensitivity depending on the field orientation ($\theta$, $\phi$) in the molecular frame. (d) Angular dependence of the singlet yield $\Phi_{\mathrm{S}}$ and relative anisotropy $\Gamma = [\max(\Phi_{\mathrm{S}})-\min(\Phi_{\mathrm{S}})]/\overline{\Phi}_{\mathrm{S}}$ sampled over $37\times37$ magnetic field orientations ($\theta\ \& \ \phi \in [0, \pi]$) for a static FAD$^{\bullet -}$/W$_{\mathrm{C}}^{\bullet+}$ pair (fixed $\mathbf{r}(t)$) including EED interactions with exchange coupling $J_{0}/(2\pi)=0\,$MHz, and N$5$ and N$1$ hyperfine couplings on FAD$^{\bullet -}$ and W$_{\mathrm{C}}^{\bullet+}$, respectively. (e) Harmonic driving of the interradical separation at frequency $\nu_{\mathrm{d}}$ enhances the relative anisotropy, as shown for $48\times100$ combinations of $\nu_{\mathrm{d}}$ ($1\leq \nu_{\mathrm{d}}\leq 10$ MHz) and $J_{0}$ ($-50\leq J_{0}/(2\pi) \leq 50$ MHz) with EED interactions included.
• Figure 2: Approach to the quantum Cramér-Rao bound (QCRB) under driven dynamics and increasing interaction complexity in a simple radical-pair model comprising the N$5$ nucleus, representing the dominant hyperfine coupling of FAD$^{\bullet -}$/W$_{\mathrm{C}}^{\bullet+}$. The approach is assessed via the ratio of classical to quantum Fisher information $F_{\theta}/\mathcal{F}_{\theta}$, plotted as a function of $209\times200$ combinations of driving frequencies $\nu_{\mathrm{d}}$ and exchange couplings $J_{0}$ spanning $0.01\leq \nu_{\mathrm{d}}\leq 100$ MHz and $-100\leq J_{0}/(2\pi) \leq 100$ MHz, respectively. Each point corresponds to the maximum ratio achieved over 180 magnetic field orientations ($\theta \in [0, \pi]$ at fixed $\phi=0$). Panels show scenarios of increasing system complexity: (a) exchange only; (b) inclusion of electron-electron dipolar coupling (EED) interactions; (c) inclusion of both EED, and random-field noise (RFR). Circles indicate closest approach to the QCRB; crosses indicate highest chemical contrast $\Gamma$. For (a) the maximum ratio $78.3\%$ occurs at $J_{0}/(2\pi) = 41\,$MHz, $\nu_{\mathrm{d}} = 47\,$MHz, with $NF_{\theta} = 0.216$ (angular precision $\Delta \theta =2.76^{\circ}\times10^{-2}$--$8.72^{\circ}\times10^{-2}$). Incorporating EED and RFR broadens the region of high ratios, reaching $96.4\%$, while maximum $\Gamma$ values ($0.456$, $0.342$, and $0.197$) also occur at higher QCRB saturation ($12\%$, $56.1\%$, and $78\%$). Across all these cases, absolute Fisher-information magnitudes remain significant ($NF_{\theta}\geq 0.011$), corresponding to angular precisions better than $\Delta \theta =1.23^{\circ}$. Closer approaches to the QCRB with EED and RFR thus occur within practical levels of precision and reflect more efficient extraction of the available information.
• Figure 3: Approach to the quantum Cramér-Rao bound (QCRB) under driven dynamics in a representative radical-pair model of FAD$^{\bullet -}$/W$_{\mathrm{C}}^{\bullet+}$ comprising N$5$ and N$1$ hyperfine couplings, electron-electron dipolar and exchange interactions. (a) The QCRB is assessed via the ratio of classical to quantum Fisher information $F_{\theta}/\mathcal{F}_{\theta}$, plotted as a function of $48\times100$ combinations of driving frequencies $\nu_{\mathrm{d}}$ and exchange couplings $J_{0}$ spanning $1\leq \nu_{\mathrm{d}}\leq 10$ MHz and $-50\leq J_{0}/(2\pi) \leq 50$ MHz, respectively. Each point corresponds to the maximum value achieved over $37\times37$ magnetic field orientations ($\theta\ \& \ \phi \in [0, \pi]$). The closest approach to the QCRB is indicated by a circle while the cross indicates highest chemical contrast $\Gamma=0.161$ with $86.8\%$ QCRB saturation and $NF_{\theta}=0.161$ (angular precision $\Delta\theta=0.101^{\circ}$--$0.319^{\circ}$).(b) The singlet yield $\Phi_{\mathrm{S}}$ profile for the closest approach to the QCRB of $F_{\theta}/\mathcal{F}_{\theta} = 89.5\%$, which occurs at $J_{0}/(2\pi) = 39\,$MHz, $\nu_{\mathrm{d}} = 5.98\,$MHz, with $NF_{\theta}=0.015$ ($\Delta\theta=0.327^{\circ}$--$1.04^{\circ}$) is shown and compared to the static system, with relative anisotropy $\Gamma$ indicated. (c) Singlet probability $p_{\mathrm{S}}$ over time that gives rise to the maximum and minimum singlet yields. Overall, driving provides close approaches to the QCRB, providing a larger contrast in yield and better defined scanning axis, originating from a restoration of coherent oscillations in the singlet probability at the driving frequency.
• Figure 4: Comparison of quantum-controlled interradical fluctuations that maximize yield contrast, harmonically driven fluctuations, and the static system. (a) Relative anisotropy ($\Gamma$), with inset showing the modulated interradical distance for both control and driven cases at the optimal exchange coupling value $J_{0}/(2\pi)=39\,$MHz. (b) the maximal Classical Fisher information ($NF_{\theta}$), and (c) the maximal ratios ($F_{\theta}/\mathcal{F_{\theta}}$), calculated over $37\times 37$ combinations of $\theta \ \& \ \phi \in [0, \pi]$ across the exchange coupling range $-50\leq J_{0}/(2\pi) \leq 50\,\mathrm{MHz}$. Quantum-controlled fluctuations generate the strongest enhancements, followed by the driven system, both of which significantly outperform the static system.
• Figure 5: Robustness of approach to the quantum Cramér-Rao bound (QCRB) under driven dynamics of FAD$^{\bullet -}$/W$_{\mathrm{C}}^{\bullet+}$ with four hyperfine couplings (N$5$ and N$10$ in FAD; N$1$ and H$1$ in Trp) electron-electron dipolar and exchange interactions. Approach is assessed by the maximal $F_{\theta}/\mathcal{F}_{\theta}$ for the case of $J_{0}/(2\pi)=0\,$MHz with $200$ driving frequencies in the range $1\leq \nu_{\mathrm{d}} \leq 100\,$MHz, while insets extend the exchange coupling to $200$ values in the range $-100\leq J_{0}/(2\pi) \leq 100\,$MHz for select driving frequencies $\nu_{\mathrm{d}}=1.5,\ 8, \ \& \ 35\,$MHz. The maximal $F_{\theta}/\mathcal{F}_{\theta}$ is evaluated for orientations maximizing and minimizing yield difference in the static case. This shows that approaches to the QCRB are maintained on increase of hyperfine complexity.