Engineering the uncontrollable: Steering noisy spin-correlated radical-pairs with coherent and incoherent control

Abstract

The quantum control of spin-correlated radical pairs (SCRPs) holds promise for the targeted manipulation of magnetic field effects, with potential applications ranging from the design of noise-resilient quantum information processors to genetically encodable quantum sensors. However, achieving precise handles over the intricate interplay between coherent electron spin dynamics and incoherent relaxation processes in photoexcited radical-pair reactions requires tractable approaches for numerically obtaining controls for large, complex open quantum systems. Employing techniques relying on full Liouville-space propagators becomes computationally infeasible for large spin systems of realistic complexity. Here, we demonstrate how a control engineering approach based on the Pontryagin Maximum Principle (PMP) can offer a viable alternative by reporting on the successful application of PMP-optimal control to steer the coherent and incoherent spin dynamics of noisy radical pairs. This enables controls for prototypical radical-pair models that exhibit robustness in the face of relevant noise sources and paves the way to incoherent control of radical-pair spin dynamics.

Figures (4)

• Figure 1: Schematic of our approach, utilising an algorithm based on the Pontryagin Maximum Principle to generate control modulations $u(t)$. These are generated for a system of electron spins $\mathbf{\hat{S}}_1$ and $\mathbf{\hat{S}}_2$ of a radical-pair, formed from radicals $\mathrm{A}^{\bullet}$ and $\mathrm{B}^{\bullet}$, which are subject to interradical interactions, hyperfine coupling with nuclear spins $\mathbf{\hat{I}}$, and Zeeman interaction with a static magnetic field $\vec{B}_{0}$. In addition, the spins can experience relaxation processes due to random field noise $\hat{\hat{\mathcal{R}}}_{\mathrm{RF}}$, arising from magnetic field fluctuations, and singlet-triplet dephasing noise $\hat{\hat{\mathcal{R}}}_{\mathrm{STD}}$, arising, for example, from recombination and fluctuations in the interradical coupling. We consider both coherent control, via a time-dependent field $\vec{B}_{1}(t)$ perpendicular to $\vec{B}_{0}$, and incoherent control over $\hat{\hat{\mathcal{R}}}_{\mathrm{RF}}$ and $\hat{\hat{\mathcal{R}}}_{\mathrm{STD}}$. These controls influence the interconversion between singlet $^\mathrm{S}[\mathrm{A}^{\bullet}\cdot\cdot\cdot \mathrm{B}^{\bullet}]$ and triplet $^\mathrm{T}[\mathrm{A}^{\bullet}\cdot\cdot\cdot \mathrm{B}^{\bullet}]$ spin configurations and subsequent chemical outcomes, with rates $k_{\mathrm{b}}$ and $k_{\mathrm{f}}$, guided by a reaction yield cost function. Shown here is an example $[\mathrm{FADH}^{\bullet} / \mathrm{Z}^{\bullet}]$ radical-pair, with FADH$^{\bullet}$ (blue) and its most significant anisotropic N$5$ hyperfine coupling highlighted (silver), and a Z$^{\bullet}$-radical devoid of hyperfine couplings represented by a red sphere.
• Figure 2: Minimising the singlet recombination yield via coherent control of the prototypical radical-pair system from mae (described in the text) in the presence of a) uncorrelated random-field noise (URF), b) correlated random-field noise (CRF), and c) singlet–triplet dephasing noise (STD), respectively. Solid curves represent the best-performing control field from 9 replications, while shaded bands indicate the region between the maximum and the 80th percentile of the recombination yield for the various control attempts. Noise rates used were $\gamma^{(\mathrm{URF})}=\gamma^{(\mathrm{CRF})}=0.5~\mu\mathrm{s}^{-1}$, $1~\mu\mathrm{s}^{-1}$, and $2~\mu\mathrm{s}^{-1}$ for URF and CRF; and $\gamma^{(\mathrm{STD})}=1~\mu\mathrm{s}^{-1}$, $5~\mu\mathrm{s}^{-1}$, $10~\mu\mathrm{s}^{-1}$, and $20~\mu\mathrm{s}^{-1}$ for STD, with the arrows pointing in the direction of increasing noise rate.
• Figure 3: Anisotropic magnetic field effects optimised by application of controlled directional noise for various noise scenarios as a function of the exchange interaction $j_{\mathrm{ex}}$. For a simple one-nitrogen radical-pair, the spread of the recombination yield as a result of reorienting the geomagnetic field direction has been optimised by application of timed directional noise in the absence (left) and presence of uncorrelated random field noise of rate 1$~\mu\mathrm{s}^{-1}$(middle) and 4$~\mu\mathrm{s}^{-1}$ (right). We optimized the added noise assuming a) correlated pairwise control (CPC) noise of both electrons, b) uncorrelated pairwise control (UPC) noise, and c) uncorrelated independent control (UIC) noise, as described in Table \ref{['tab:noise_models']}. The control noise amplitude was capped at $6 \mu$s$^{-1}$ and we controlled the initial $2 \mu$s using $2000$ piecewise constant control steps ($t_1 = 2 \mu$s, $t_0 = 0 \mu$s).
• Figure 4: Modulations without (left) and with the presence of URF noise of rate 1$~\mu\mathrm{s}^{-1}$ (middle) and 4$~\mu\mathrm{s}^{-1}$ (right) with $j_{\mathrm{ex}}/(2\pi) = 2 \text{MHz}$ in $[\mathrm{FADH}^{\bullet}/ \mathrm{Z}^{\bullet}]$ for incoherent control using uncorrelated pairwise control (UPC) noise where $u_1$ modulates the equatorial component of $\hat{\hat{\mathcal{R}}}_{\mathrm{UPC}}$ and $u_2$ modulates its axial component with control amplitudes $u_1 , u_{2} < 1$, the maximal amplitudes corresponding to a noise rate of $6 \mu$s$^{-1}$. Noticeably, the first $0.5\,\mu\mathrm{s}$ appears as a crucial point to exert control, likely reflecting the period of highest coherent interconversion and significant radical-pair population.