Quantum Probe and Design for a Chemical Compass with Magnetic

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H = Hk = −γe Bk · Sk + Sk · λkj · Ikj (1) from FADH -O2− [28] for our calculations. k=XA,D Xk Xk,j (a) (b) 0.5 0.35 where γe = −geµB is the electron gyromagnetic ratio, 0.3 ˆ ~ ~ 0.45 λkj denote the hyperfine coupling tensors and Sk, Ikj 0.25 0.4 0.2 S

are the electron and nuclear spin operators respectively. V Φ 0.15 0.35 In our model, the magnetic field consists of two parts: 0.1 0.3 B~ k = B~ + L~ k, where the directional information about 0.05 0.25 0 B~ is what one wants to infer from the radical pair reac- 0 30 60 90 120 150 180 0 0.2 0.4 0.6 0.8 1 θ τ (µs) tion, and L~ k is the local gradient field applied to each radical and is independent of B~ . The spin relaxation and decoherence times resulting from the factors other FIG. 2: (Color online) Magnetic field sensitivity of a chemical than hyperfine interactions are assumed to be consid- compass enhanced by a gradient field. (a) Singlet yield ΦS erably longer than the radical pair lifetime [3, 11], to as a function of the angle θ of the weak magnetic field B~ maximize sensitivity to weak magnetic fields [21]. In (B = 0.46 G) with different gradient field strengths on the many photochemical processes, the radical pair is cre- acceptor, i.e. LA = 0 G (red, · · · o · · · ), 20 G (blue, ···⋄··· ), 40 G (green, ···△··· ), 80 G (purple, ···∗··· ), while LD = 0. ated in a spin-correlated electronic singlet state |Si = −1 The recombination rate k = 0.5µs . (b) Visibility V as a 1 (|↑↓i − |↓↑i) within the timescale of picoseconds. The √2 function of the radical pair lifetime τ = 1/k. The direction of nuclear spins start at thermal equilibrium, which under the gradient field L~ A is set as θA = 0. The same values of LA ambient conditions leads to an approximate density ma- are used as in (a). I trix as ρn(0) = j j /dj, where dj is the dimension of I the jth nuclear spinN and j is the identity matrix. The We define the molecular frame as the coordinate sys- Zeeman splitting from a magnetic field B~ as weak as the tem, and the weak magnetic field B~ can be represented geomagnetic field is much smaller than the thermal en- as B~ = B(sin θ cos φ, sin θ sin φ, cos θ). The gradient field ergy at ambient temperature. Nonetheless, the field can induces different local fields on two radicals. We as- influence the non-equilibrium electron spin dynamics and sume that the gradient field on the acceptor radical is thereby determine the ratio of the chemical product from L~ A = LA(sin θA, 0, cos θA) while L~ D = 0 for the donor the singlet or triplet recombination as long as the ther- radical. The strength of the weak magnetic field to be de- malization time is longer than the reaction time. tected is the same as the geomagnetic field, i.e. B =0.46 In experiments, one may measure different quantities G. To demonstrate the basic idea, we first consider φ = 0, that are dependent on the weak magnetic field B~ . Here and then generalize to arbitrary φ. we consider a simple first-order recombination reaction In Fig. 2 (a), we plot the singlet yield as a function of of the singlet radical pairs. We note that there is some the angle θ of the weak magnetic field B~ with different controversy over how to describe the radical pair reac- gradient field strengths LA =0G, 20 G, 40G, 80 G on the tions (see e.g. [4, 5, 22, 23]). Nevertheless, the con- acceptor. In the case of LA = 0, the directionality comes ventional phenomenological density matrix approach [20] only from hyperfine anisotropy. The gradient field clearly works well in most cases, in particular when the sin- enhances the amplitude of the direction-dependent com- glet and triplet recombination rates are the same (i.e. ponent of the magnetic field effect (MFE). To quantify kS = kT = k) [24]. We adopt this method and cal- the directional sensitivity, we use the magnetic visibility ∞ culate the singlet yield as ΦS = 0 f(t)PS(t)dt, where defined as [2] kt f(t)= ke− is the radical reencounterR probability distri- S S bution, and PS (t)= h |ρs(t)| i is the singlet fidelity for V = (max ΦS − min ΦS)/(max ΦS + min ΦS). (2) the electron spin state ρs(t) at time t. The integration of ΦS was performed following the method in [25, 26]. As the gradient field becomes larger, the sensitivity will Gradient enhancement of magnetic field sensitivity.— increase and approach to a saturate best value. Fig. 2 We starts from an optimally designed hyperfine compass (b) shows that for long radical pair lifetimes, the visibil- model, one radical has strong and anisotropic hyperfine ity with the gradient field LA =40 G is almost twice the interactions, and the other radical has no hyperfine cou- visibility without the gradient field. Usually, the radical plings [21]. We arbitrarily choose to call the first radical pair lifetime should be very long (microseconds) to max- the acceptor, A, and the second the donor, D, though imize the effect of weak magnetic field [21], and hence nothing that follows depends on this designation. Ritz performance, of the chemical compass [Fig. 2 (b)]. This . . and coworkers proposed that the radical pair FADH -O2− requirement places a severe constraint on the chemistry; meets this criterion, and they further speculated that this in typical radical pair reactions the lifetime is less than radical pair may be responsible for the magnetoreception 100 ns [20]. By increasing the overall magnitude of the 3 visibility, gradient-enhancement broadens the range of (a) (b) 0.35 0.5 L =0 L = 80 G candidate reactions for a chemical compass. 0.3 A A 0.45 Liquid crystal experiment.— In a uniaxially oriented 0.25 0.2 0.4 ΦS sample, the MFE is averaged over all values of the angle V 0.15 0.35 φ. Such a sample is prepared by, for instance, freezing 0.1 the molecules in a nematic liquid crystal in the presence 0.05 0.3 0 0 30 60 90 120 150 180 0.25 of a strong magnetic field [18]. The ensemble-averaged θ 0 30 60 90 120 150 180 A θ MFE depends on θ only and is characterized by 2π 1 FIG. 3: (Color online) Magnetic field sensitivity in a liquid hΦS(θ)i = ΦS (θ, φ)dφ (3) 2π Z0 crystal experiment. (a). Visibility of the average singlet yield hΦS (θ)i as a function of the angle θA of the gradient field It can be seen from Fig. 3 (a) that the enhancement of the LA = 80 G. The blue dashed curve represents the visibility sensitivity can still be observed with the average signal without the gradient field. (b) Ensemble average of the singlet 4 hΦS(θ)i by choosing appropriate values of θA. yield ΦS(θ) in Eq. (5) from a Monte Carlo simulation of 2×10 To induce the gradient field as above, one feasible way samples as a function of the angle θ of the weak magnetic is to use magnetic nanostructures [19]. We model the field B~ . The gradient field is LA = 40 G with θA =0 (purple, nanocrystal as a uniformly magnetized sphere, in which ···∗··· ), while LD = 0. The fluctuations of the local fields LD and LA are characterized by the 3-dimensional Gaussian case the external magnetic field is the same as that of a distributions with the variance σA = 2Gand σD = 0.1 G. For point dipole of magnetic moment m located at the center comparison, we plot the singlet yield with no gradient field of the sphere [29]. We denote the position relative to the (red, · · · o · · · ), and the one with the gradient field LA = 40 G center of the sphere by the vector r, and assume that (θA =0) without fluctuations (blue, ···△··· ). In both panels both r and m lie along the z-axis. The magnetic field at the radical pair lifetime is 2µs and B = 0.46 G. r is µ m B(r)= 0 ˆr, (4) L~ + ∆ , L~ + ∆ respectively. We have used Monte 2πr3 A A D D Carlo simulations to calculate the above ensemble aver- 7 2 where µ0 = 4π × 10− N·A− is the permeability of free age in Eq.(5). In Fig. 3 (b), we see that the enhancement space, the magnetic moment m = M̺Ω with M the spe- from the gradient field can still be observed. 4 3 cific magnetization, ρ is the material density, Ω = 3 πR Probe spin correlations in a chemical compass.— Be- is the volume of the particle and R is its radius. The sides the significant enhancement of the directional sen- parameters for the typical magnetic material Fe3O4 are sitivity offered by gradient fields, we now examine how 2 1 3 M = 43 A·m · kg− , ̺ = 5210 kg·m− [19]. For they can provide new insights into the quantum dynamics molecules with a separation rAD between two radicals of radical pair reactions. For the present model chemical a few nanometers [30], it is sufficient for a nanoparti- compass, if the gradient field on the acceptor L~ A domi- cle to induce a large local field imbalance (∼ 10 G) on nates over the hyperfine couplings and the weak magnetic the donor and acceptor. For example, using a Fe3O4 field B~ , the singlet yield can be written as [26] nanoparticle with the radius R = 15 nm, it is possible to 1 1 induce the local field difference as large as ∼ 40 G be- Φ (L~ , B~ )= − hAˆ ⊗ Vˆ i (6) S A 4 4 tween the position rA = 35 nm and rD = rA +rAD = 38.5 nm (assuming rAD = 3.5 nm). By generating an addi- where the expectation value is calculated over the ini- tional homogenous field to compensate the field at the tial state, and Aˆ = |u0ihu0| − |u1ihu1| (with {|u0i, |u1i} position rD, we can effectively obtain the gradient field ~ ~ ˆ ˆ the eigen states of LA · SA), V = hUD† AUDi with on the donor and acceptor molecule as LA ≃ 40 G and UD = exp(iγetB~ · S~D) and the average taken over time LD = 0 G respectively. weighted by f(t). By choosing L~ A in the direction ofx ˆ, To see whether the effect of the gradient field shown yˆ, andz ˆ, the corresponding operator Aˆ will be Xˆ, Yˆ , Zˆ above can manifest with experimental imperfections, we ~ ~ respectively (which are the Pauli operators). Moreover, take into account the fluctuations of LA and LD by mod- for each Aˆ, one can choose Bˆ also in the direction ofx ˆ, eling the fluctuation as the three-dimensional Gaussian 2 yˆ, andz ˆ such that the operators of Vˆ (as a linear com- 1 ∆i distribution f(∆i) = 2 3/2 exp(− | 2| ) (i = A, D) (2πσi ) 2σi bination of Pauli operators) are linear independent, see with σA = 2 G and σD =0.1 G. Therefore, the ensemble [26]. The singlet yields corresponding to these choices average of hΦs(θ)i in Eq. (3) is of LÂ and Bˆ lead to nine independent equations, from which we can infer the spin correlations hMˆ ⊗ Nî for Φ (θ)= hΦ (θ)i |∆ ∆ f(∆ )f(∆ )d∆ d∆ (5) the radical pair state, where M,ˆ Nˆ = Xˆ, Yˆ or Zˆ. With S Z S A, D A D A D these correlations, one may check whether the radical where hΦS(θ)i |∆A,∆D is the average singlet yield when pair state violates Bell inequalities [31]; or obtain lower the local fields on the acceptor and donor molecules are entanglement bounds of the radical pair state, see Ref. 4

180 180 0.5 0.5 perfect anti-correlation of the spins for any direction of 150 150 0.45 0.45 L~ , while for the classically correlated state this is true 120 120 A 0.4 z 0.4 ~

θ 90 θ 90 only in a certain direction of LA (i.e. thez ˆ direction). 0.35 0.35 60 x 60 Summary.— We have demonstrated that a gradient 0.3 y 0.3 30 30 0.25 0.25 field can lead to a significant enhancement of the perfor- 0 0 0 60 120 180 240 300 360 0 60 120 180 240 300 360 mance of a chemical compass. The gradient field also pro- φ φ vides us a powerful tool to investigate quantum dynamics 180 180 0.5 0.5 of radical pair reactions in spin chemistry. In particular, 150 150 0.45 0.45 120 120 it can distinguish whether the initial radical pair state is 0.4 z 0.4

θ 90 θ 90 in the entangled singlet state or in the classically corre- 0.35 0.35 60 x 60 0.3 y 0.3 lated state, even in the scenarios where such a goal could 30 30 0.25 0.25 not be achieved before. These phenomena persist upon 0 0 0 60 120 180 240 300 360 0 60 120 180 240 300 360 φ φ addition of partial orientational averaging and addition of realistic magnetic noise. The eﬀects predicted here 180 180 0.5 0.5 150 150 may be detectable in a hybrid system compass composed 0.45 0.45 120 120 of magnetic nanoparticles and radical pairs in an oriented 0.4 z 0.4

θ 90 θ 90 0.35 liquid crystalline host. Our work offers a simple method 0.35 x 60 60 0.3 y 0.3 to design/simulate a biologically inspired weak magnetic 30 30 0.25 0.25 field sensor based on the radical pair mechanism with a 0 0 0 60 120 180 240 300 360 0 60 120 180 240 300 360 φ φ high sensitivity that may work at room temperature. Acknowledgements.— We are in debt to Nan Yang and Adam Cohen for valuable suggestions and beneficial com- FIG. 4: (Color online) The gradient field as a tool to test the munications. We thank Kiminori Maeda, Gian Giacomo initial radical pair state. The singlet yield ΦS as a function of the angles (θ, φ) of the weak magnetic field B~ with B = 0.46 Guerreschi and Otfried Gühne for helpful discussions, G. The gradient field on the acceptor is LA = 80 G, on the and Hans Briegel for continuous support in this work. donor it is LD = 0. The angle of the gradient field θA with The work is supported by FWF (SFB FoQuS). π π respect to the z-axis is 0 (upper), 4 (middle) and 2 (lower). The patterns of the singlet yield over (θ, φ) from the initial singlet (left) and classically correlated state (right) are quite π π similar for θA = 0, but are very different for θA = 4 and 2 . The radical pair lifetime is chosen as 2µs. [1] G. S. Engel, et al. Nature 446, 782 (2007); H. Lee, et al. Science 316, 1462 (2007); E. Collini and G. D. Scholes, Science 323, 369 (2009); M. Mohseni, et al. J. Chem. 129 [32]. The above idea can be extended to monitor the Phys. , 174106 (2008); M. B. Plenio, S. F. Huelga, New J. Phys. 10, 113019 (2008); F. Caruso, A. W. Chin, dynamics of spin correlations suppose one can switch on A. Datta, S. F. Huelga, and M. B. Plenio, J. Chem. Phys. gradient fields during the reaction. 131, 105106 (2009); M. Sarovar, A. Ishizaki, G. R. Flem- As an example, we show that gradient fields can dis- ing, K. B. Whaley, Nature Physics 6, 462 (2010). tinguish the singlet and the classically correlated initial [2] J.-M. Cai, G. G. Guerreschi, and H. J. Briegel, Phys. Rev. 104 state ρ = (|↑↓i h↑↓|+|↓↑i h↓↑|)/2. For systems where the Lett. , 220502 (2010) arXiv: 0906.2383 (2009). c [3] E. Gauger, E. Rieper, J. J. L. Morton, S. C. Benjamin, V. radical pair lifetime is much longer than the decoherence Vedral, Phys. Rev. Lett. 106, 040503 (2011). time, the conventional hyperfine-mediated MFE does not [4] I. K. Kominis, Phys. Rev. E 80, 056115 (2009). strongly depend on the initial states and thus can not al- [5] J. A. Jones, P. J. Hore, Chem. Phys. Lett. 488, 90 (2010). low one to achieve this goal, see e.g. [2]. If the gradient [6] L. Turin, J. Theor. Biol. 216(3), 367 (2002); J. C. Brookes, field is along the z-axis, the singlet yields are quite simi- F. Hartoutsiou, A. P. Horsfield, A. M. Stoneham, Phys. lar for the singlet and the classically correlated state [see Rev. Lett. 98, 038101 (2007). Fig. 4 (upper)]. However, if we vary the direction of the [7] K. Schulten, C. E. Swenberg and A. Weller, Z. Phys. Chem NF111 ~ , 1 (1978). gradient field LA, then the visibility for the singlet state [8] R. Wiltschko and W. Wiltschko, Bioessays 28, 157 (2006). will be much larger than for the classically correlated ini- [9] S. Johnsen and K. J. Lohmann, Nature Rev. Neurosci 6, tial state [Fig. 4 (middle, lower)]. In particular, for the 703 (2005). classically correlated state, the singlet yield is insensitive [10] C. T. Rodgers and P. J. Hore, Proc. Natl. Acad. Sci 106, π 353 (2009). to the angles (θ, φ) for θA = 2 while for the singlet state the angular sensitivity persists [Fig. 4 (lower)]. The dif- [11] T. Ritz, S. Adem and K. Schulten, Biophys. J 78, 707 (2000). ference originates from the essential boundary between [12] W. Wiltschko and R. Wiltschko, J. Exp. Biol 204, 3295 classical and quantum correlation (entanglement). The (2001). large gradient field can be viewed as a measurement of the [13] T. Ritz et. al, Nature 429, 177 (2004). acceptor spin along L~ A: the singlet state demonstrates [14] R. J. Gegear, A. Casselman, S. Waddell and S. M. Rep- 5

pert, Nature 454, 1014 (2008). see also [26] for a comparison with the present model. [15] R. J. Gegear, L. E. Foley, A. Casselman and S. M. Rep- [25] B. Brocklehurst, J. Chem. Sot. Faraday Trans. 72, 1869 pert, Nature 463, 804 (2010). (1976). [16] M. Ahmad, P. Galland, T. Ritz, R. Wiltschko and W. [26] J.-M. Cai, The calculation details are presented in the Wiltschko, Planta, 225, 615 (2007). following Appendix. [17] N. Yang, Y. Tang, and A. E. Cohen, Nano Today, 4, 269 [27] I. A. Solov’yov, K. Schulten, Biophys. J 96, 4804 (2009). (2009). [28] F. Cintolesi, T. Ritz, C. W. M. Kay, C. R. Timmel, P. J. [18] K. Maeda et. al, Nature 453, 387 (2008). Hore, Chem. Phys. 294, 385 (2003). [19] A. E. Cohen, J. Phys. Chem. A 113, 11084 (2009). [29] D. J. Griﬃths, Introduction to Electrodynamics, 3rd Edi- [20] U. E. Steiner, T. Ulrich, Chem. Rev 89, 51 (1989). tion, Prentice Hall Inc. (New Jersey 1999). [21] T. Ritz et al., Biophys. J 96, 3451 (2009). [30] M. Di Valentin, A. Bisol, G. Agostini and D. Carbonera, [22] K. L. Ivanov, M. V. Petrova, N. N. Lukzen and K. Maeda, J. Chem. Inf. Model. 45, 1580-1588 (2005). J. Phys. Chem. A, 114, 9447(2010). [31] R. Horodecki, Phys. Lett. A. 210, 223 (1996). [23] A. I. Shushin, J. Chem. Phys. 133, 044505 (2010). [32] K. M. R. Audenaert, M. B. Plenio, New J. Phys. 8, 266 [24] In this case, the singlet yields calculated from diﬀerent (2006). approaches are expected to be very similar [4, 5, 22, 23],

Appendix

Calculation of singlet yield.— We adopt the method as in [1] to calculate the singlet yield. For the self- completeness, here we present a simple outline of this method. The Hamiltonian for the system (two electron spins, one of which is coupled with several surrounding nuclear spins ) is as follows

~ ~ ~ ˆ ~ H = Hk = −γe Bk · Sk + Sk · λkj · Ikj (7) kX=1,2 Xk Xk,j

In our calculations of the main text, we have neglected the Zeeman interactions between the nuclear spins and the external magnetic field. We have included these interactions, and verified that the induced difference is very small (as the gyromagnetic ratio for a nucleus H and N is much smaller than γe). The numbers of hyperfine couplings are take from Ref.[2]. We calculate the singlet yield as [3]

∞ ΦS = f(t)PS (t)dt (8) Z0

kt where f(t)= ke− is the radical re-encounter probability distribution, and PS (t) = hS|ρ(t)|Si is the singlet fidelity for the electron spin state ρ(t) at time t. Eq. (8) can be obtained from the conventional Haberkorn approach [4] in the case that the singlet and triplet recombination rates are the same, i.e. kS = kT = k. The singlet yield is calculated following the method in [1]. We first write the singlet fidelity as I iHt j iHt P (t) = Tr[e− (ρ )e (|SihS| I )] (9) S 0 d j Oj j Oj I j i(ω ω )t = hm|(ρ )|ni · hn|(|SihS| I )|mi · e− m− n (10) 0 d j Xm Xn Oj j Oj where ρ0 is the initial state of the radical pair, and the initial state of the nuclear spins at room temperature can be I approximated as ρb(0) = j j /dj , where dj is the dimension of the jth nuclear spin, and {|mi} and {|ni} denote the eigen states of the HamiltonianN H in Eq.(1). After some calculations, we have the singlet yield as

∞ kt k 1 Φ = ke− P (t)= ρ A (11) S Z S d mn nm k + i(ω − ω ) 0 Xm Xn m n I I where d = di , ρm,n = hm|(ρ0 j j )|ni and An,m = hn|(|SihS| j j )|mi. Q N N

Comparison between Haberkorn approach and quantum measurement master equation.— In the main text, we consider the radical pair reaction with the same singlet and triplet recombination rate, i.e. kS = kT = k. The 6 method we use to calculate the singlet yield is based on the Haberkorn approach [4] that describes the recombination of radical pairs dρ k k = −i[H,ρ] − S (Q ρ + ρQ ) − T (Q ρ + ρQ ) (12) dt 2 S S 2 T T where QS and QT are the projection operators for the singlet and triplet electronic states of the radical pair. There are alternative master equations based on quantum measurement that have been proposed to describe the recombination of radical pairs [5, 6]. Under the condition kS = kT = k, these master equations [5, 6] can be written in the following form dρ = −i[H,ρ] − (k + k )ρ + k Q ρQ + k Q ρQ (13) dt S T S T T T S S From Eq.(12) or Eq.(13), one can obtain the density matrix of the radical pair state ρ(t) at time t and thus calculate the singlet yield as

∞ ΦS = k Tr [QSρ(t)] dt (14) Z0 We compare the results of the singlet yield from these two approaches. For the simplicity of calculation, we take the three most significant hyperfine interactions in FADH., i.e. those for the nitrogens N5 and N10 and the proton H5 [2]. It can be seem from Fig. 5 (for the long lifetime τ = 1/k = 2µs) and Fig. 6 (for the short lifetime τ = 1/k = 50ns) that the difference between the results from two approaches is very small (around 1 ∼ 2%).

(a) (b) 0.39 0.45 L = 0 A HAB QM L = 20 G 0.37 A HAB QM 0.35 0.4 0.33 S S 0.35 Φ 0.31 Φ 0.29 0.3 0.27 0.25 0.25 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ θ

0.4 0.4 S S Φ Φ 0.35 0.35 0.3 0.3 HAB QM 0.25 0.25 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ θ

FIG. 5: (Color online) A comparison between the singlet yield from Haberkorn appraoch (HAB, red) and quantum measurement master equation (QM, blue). The radical pair life time is τ = 1/k = 2µs. The other parameters are the same as Fig.2 (a) in the main text.

Probe spin correlations with gradient fields.— We assume that the gradient field on the acceptor L~ A is much larger than the hyperfine couplings and the weak magnetic field B~ . To calculate the singlet yield, for simplicity, the Hamiltonian can be approximated as [7]

H ≃−γe(L~ A · S~A + B~ · S~D) (15) 7

(a) (b) 0.414 L = 0 A HAB QM L = 20 G HAB QM 0.46 A 0.41

0.406 0.45 S S Φ Φ 0.402 0.44 0.398

0.394 0.43 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ θ

0.48 S

Φ 0.48 Φ 0.47 0.47 HAB 0.46 QM 0.46 0 30 60 90 120 150 180 0 30 60 90 120 150 180 θ θ

FIG. 6: (Color online) A comparison between the singlet yield from Haberkorn appraoch (HAB, red) and quantum measurement master equation (QM, blue). The radical pair life time is τ = 1/k = 50ns. The other parameters are the same as Fig.2 (a) in the main text.

We denote the density matrix of the initial radical pair state as ρ, and use the eigen states of L~ A · S~A ({|u0i, |u1i}), namely L~ A · S~A|u0i = LA|u0i and L~ A · S~A|u1i = −LA|u1i, as the spin basis of the acceptor. The initial state can then be written as

mn m,n ρ = |umiAhun|⊗ ρD where ρD = hum|ρ|uni (16) m,nX=0,1 The singlet ﬁdelity at time t is

~ ~ ∞ 00 11 ΦS (LA, B) = f(t) · {hS| |u0ihu0|⊗ (UDρD UD† ) |Si + hS| |u1ihu1|⊗ (UDρD UD† |Si}dt (18) Z0 h i h i 1 ∞ 00 11 = f(t) · Tr(ρD UD† |u1ihu1|UD + ρD UD† |u0ihu0|UD)dt (19) 2 Z0

∞ I Aˆ ⊗ U † AUˆ = f(t) · Tr[( − D D )ρ]dt (20) Z0 4 4 1 1 = − hAˆ ⊗ Vˆ i (21) 4 4

ˆ ˆ ∞ ˆ ~ where A = |u0ihu0| − |u1ihu1| and V = 0 dtf(t)UD† AUD. By choosing LA in the direction ofx ˆ,y ˆ, andz ˆ, the corresponding operator Aˆ will be Xˆ, Yˆ , ZˆRrespectively (which are the Pauli operators). As an example, we assume that Aˆ = Zˆ, it can be seen that if we choose B~ in the direction ofx ˆ,y ˆ, andz ˆ respectively, the corresponding operators ˆ ˆ ˆ ˆ ˆ ˆ ∞ 2 2 2 ∞ of V are cZ − sY , cZ + sX, Z, with c = 0 dtf(t)cos(γetB) = k /[k + (γeB) ] and s = 0 dtf(t) sin (γetB) = 2 2 kγeB/[k + (γeB) ]. These operators are linearR independent. From the singlet yields correspondingR to these choices 8 of LÂ and Bˆ, we can have three independent equations, following which we can infer the spin correlations hZˆ ⊗ Xî, hZˆ ⊗ Yˆ i, hZˆ ⊗ Zî. In a similar way, we can choose Aˆ = X,ˆ Yˆ and obtain the spin correlations hXˆ ⊗ Xî, hXˆ ⊗ Yˆ i, hXˆ ⊗ Zî, and hYˆ ⊗ Xî, hYˆ ⊗ Yˆ i, hYˆ ⊗ Zî.

[1] B. Brocklehurst, J. Chem. Sot. Faraday Trans. 72, 1869 (1976). [2] F. Cintolesi, T. Ritz, C. W. M. Kay, C. R. Timmel, P. J. Hore, Chem. Phys. 294, 385 (2003). [3] U. E. Steiner, T. Ulrich, Chem. Rev 89, 51 (1989). [4] R. Haberkorn, Mol. Phys. 32, 1491 (1976). [5] I. K. Kominis, Phys. Rev. E 80, 056115 (2009). [6] J. A. Jones, P. J. Hore, Chem. Phys. Lett. 488, 90 (2010). [7] The approximation mainly results in a correction to the singlet ﬁdelity in the second line of Eq.(17), which nonetheless will not aﬀect the singlet yield due to its fast oscillation.