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Quantum Probe and Design for a Chemical Compass with Magnetic Quantum probe and design for a chemical compass with magnetic nanostructures Jianming Cai Institut f¨ur Quantenoptik und Quanteninformation der Osterreichischen¨ Akademie der Wissenschaften, Innsbruck, Austria (Dated: September 12, 2018) Magnetic fields as weak as Earth’s may affect the outcome of certain photochemical reactions that go through a radical pair intermediate. When the reaction environment is anisotropic, this phenomenon can form the basis of a chemical compass and has been proposed as a mechanism for animal magnetoreception. Here, we demonstrate how to optimize the design of a chemical compass with a much better directional sensitivity simply by a gradient field, e.g. from a magnetic nanostructure. We propose an experimental test of these predictions, and suggest design principles for a hybrid metallic-organic chemical compass. In addition to the practical interest in designing a biomimetic weak magnetic field sensor, our result shows that gradient fields can server as powerful tools to probe spin correlations in radical pair reactions. Introduction.— Recently, there has been increasing in- in the vicinity of a hard ferromagnetic nanostructure [19], terest in quantum biology namely investigating quantum by applying a spatially uniform bias field that cancels the effects in chemical and biological systems, e.g., light har- field of the nanostructure in a small region of space. In vesting systems [1], avian compass [2–5] and olfactory essence, the strong gradient field at one spin can substi- sense [6]. The main motivation is to understand how tute for strong anisotropic hyperfine couplings required quantum coherence (entanglement) may be exploited for for a purely molecular compass. This geometry provides the accomplishment of biological functions. As a key step a more significant anisotropy and thereby shows much towards this goal, it is desirable to find tools that can larger directional sensitivity than does the conventional detect quantum effects under ambient conditions. The compass mechanism based only on anisotropic hyperfine ultimate goal of practical interest in studying quantum couplings. Without requiring extra nuclear spins, the biology is to learn from nature and design highly effi- present model can work merely with two electron spins cient devices that can mimic biological systems in order and thereby much simplifies quantum simulation of a to complete important tasks, e.g. collecting solar energy chemical compass with, e.g. quantum dots and Nitrogen- and detecting weak magnetic field. vacancy centers in diamond. With more freedom to tune As an example of quantum biology, the radical pair parameters in a better controllable environment, such mechanism is an intriguing hypothesis [7] to explain the kind of quantum simulations would be very helpful to ability of some species to respond to weak magnetic fields understand the recombination process of radical pairs, in [8–10], e.g. birds [11–13], fruit flies [14, 15], and plants particular, whether and how quantum measurement and [16]. A magnetochemical compass could find applications Zeno effect take place [4, 5]. in remote magnetometry, in magnetic mapping of mi- Chemical compass mechanism.— Many chemical pro- croscopic or topographically complex materials, and in cesses involve a radical pair intermediate, in which each imaging through scattering media [17]. It was demon- radical has an unpaired electron coupled to an external strated that a synthetic donor-bridge-acceptor compass composed of a linked carotenoid (C), porphyrin (P), and z fullerene (F) [18] can work at low temperature (193 K). It x is surprising that such a triad molecule is the only known z example that has been experimentally demonstrated to y be sensitive to the geomagnetic field (yet not at room AD x arXiv:1011.5495v3 [physics.chem-ph] 1 Mar 2011 temperature). It is currently not known how one might y construct a biomimetic or synthetic chemical compass that functions at ambient temperature. In this Letter, we approach to the goals of studying quantum biology in the context of chemical compass by demonstrating that a suitably designed gradient field can FIG. 1: (Color online) Left: A radical pair, coupled with the significantly improve the performance of a model chemi- surrounding nuclear spins (black arrow), in a weak magnetic cal compass (apart from increasing the intersystem cross- field B~ to be measured (yellow) and a strong magnetic gra- ing rate [19]), see Fig. 1. It also opens a possible route to dient L~ A (blue), due to e.g. a magnetic nanostructure. The probe spin correlations of radical pairs and thereby inves- outcome of a reaction depends on the direction of the weak tigate the role of quantum effects in spin chemistry. The field B~ . Right: The directions of B~ and the gradient field gradient field is strong at the location of one spin, and ap- at the location of the acceptor L~ A depicted in the molecular proximately zero at the other. Such a field can be created coordinate frame. 2 magnetic field and a few nuclei via the Hamiltonian [20] of European robins [21], see also [27]. Without loss of the essential physics, we take the hyperfine couplings (∼ G) ~ ~ ~ ˆ ~ . 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].
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