The radical-pair mechanism as a paradigm for the emerging science ofquantum biology

Iannis K. KominisDepartment of Physics, University of Crete, Heraklion 71103, Greece

The radical-pair mechanism was introduced in the 1960’s to explain anomalously large EPRand NMR signals in chemical reactions of organic molecules. It has evolved to the cornerstone ofspin chemistry, the study of the effect electron and nuclear spins have on chemical reactions, withthe avian magnetic compass mechanism and the photosynthetic reaction center dynamics beingprominent biophysical manifestations of such effects. In recent years the radical-pair mechanismwas shown to be an ideal biological system where the conceptual tools of quantum informationscience can be fruitfully applied. We will here review recent work making the case that the radical-pair mechanism is indeed a major driving force of the emerging field of quantum biology.

I. INTRODUCTION

Quantum biology is swiftly emerging as an interdis-ciplinary science synthesizing, in novel and unexpectedways, the richness and complexity of biological systemswith quantum physics and quantum information science.The intuitive disposition prevailing until recently, namelythat quantum coherence phenomena have no relevance tobiology, can be simply understood. Indeed, quantum co-herence has been clearly manifested in carefully preparedand well-isolated quantum systems, the experimental im-plementation of which was technically demanding. It wasthus plausible to assume that all timescales of interest incomplex biological matter are orders of magnitude largerthan the coherence time of any underlying quantum phe-nomenon. In other words, decoherence has been assumedto be detrimental merely due to the complexity of bio-logical systems and the vast environments they offer assinks of quantum information, environments that wereunderstood to be anything but ”carefully prepared”.

In recent years, however, a different physical picture isgradually emerging. Biological systems are indeed noth-ing like single atoms trapped in ultra-high vacuum, al-lowing the observation of coherences lasting for severalseconds [1] or even hours for solid-state nuclear spin sys-tems [2]. But they are also nothing like a classical systemwhere decoherence has obliterated any chance to detectoff-diagonal elements of the relevant density matrix [3].It appears that nature has optimized biological functionusing both worlds, i.e. taking advantage of quantum co-herence amidst an equally essential yet not detrimentaldecoherence.

In this review we focus on a major driving forceof quantum biology, namely the quantum dynamics ofthe radical-pair mechanism (RPM). The RPM involvesa multi-spin system of electrons and nuclei embeddedin a biomolecule and undergoing a number of physi-cal/chemical processes, like magnetic interactions and co-herent spin motion, electron transfer and spin relaxationto name just a few. For an overview of RPM’s long scien-tific history we refer the reader to a number of reviews ofthe earlier work on radical-ion pairs and spin chemistry[4–7].

Although the RPM has been known since the 1960s,the rich quantum physical underpinnings of the mecha-nism have been unraveled only recently. The first pa-per [8] discussing RPM in the context of modern quan-tum information theory introduced quantum measure-ments as a necessary concept to understand the quan-tum dynamics of RPM, leading among other things tothe fundamental spin decoherence process of radical-ion-pair (RP) reactions. In a number of papers that fol-lowed [9–19], the intricate quantum effects at play inRP reactions were further elucidated. Quantum mea-surement dynamics, phononics, in particular the phononvacuum, quantum coherence and decoherence, in partic-ular measurement-induced decoherence, coherence distil-lation, quantum coherence quantifiers, quantum corre-lations, quantum jumps and the quantum Zeno effect,concepts from quantum metrology and the quantum-communications concept of quantum retrodiction wereshown to be central for understanding RPM.

Were it not for these conceptual underpinnings, itwould be hard to justify why RPM constitutes aparadigm for quantum biology. It can of course be arguedthat radical-ion pairs involve spins, and spins are genuinequantum objects, but this argument points to the atom-istic, structural aspect of biology. That is, one could alsoargue that spins, e.g. singlet states in the helium atom,determine atomic structure and atomic structure is thebasis of biological structure, alas this is not what quan-tum biology is about. We hope, however, to convincethe reader that RPM’s position on the map of quantumbiology is indeed well-deserved.

This review will focus on our recent work concerningthe quantum foundations of RPM. This biochemical spinsystem is an open system and hence the relevant den-sity matrix ρ does not just obey Schrodinger’s equation,dρ/dt = −i[H, ρ], where H is the system’s Hamiltonian,but more terms are required to account for the ”open”character of RPs. These terms are compactly writtenas a ”reaction super-operator” L(ρ). The master equa-tion dρ/dt = −i[H, ρ] + L(ρ) is the starting point forall theoretical calculations involving RP reactions, hencethe foundational character of this discussion. Our spe-cific quest has been to find the exact form of the reaction

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super-operator L(ρ). Until we entered this field in 2008,the overwhelming majority of theoretical treatments inspin chemistry have used what we now understand is aphenomenological theory elaborated upon by Haberkorn[20] in 1976.

Returning to the chronological exposition, the first pa-per [8] illuminating the quantum-information substratumof RPM led to a resurgent interest of the spin chemistrycommunity on the foundational aspects of the field [21–30], and in parallel to a flurry of activity in the quantumscience community[31–57], even prompting some authorsto announce, more or less timely, ”the dawn of quantumbiology” [58]. The main motivation and excitement be-hind this work has largely been RPM’s relation to theavian magnetic compass. It is clear that concepts ofquantum measurements, quantum (de)coherence and en-tanglement take on a distinct scientific flavor when ap-plied to migrating birds. In this review we will only coverthe part of the literature relevant to our specific quest ofobtaining the master equation for dρ/dt. We feel that therest of the literature, especially papers discussing quan-tum coherence and entanglement in chemical magnetore-ception, address a very promising research venue, how-ever they are mostly based on the traditional descriptionof RP dynamics. Since the qualitative and quantitativeunderstanding of quantum coherence and entanglementis intimately tied to the master equation for dρ/dt, thesediscussions could be premature.

Focusing on the RPM, we will not cover other, equallyexciting venues of quantum biology, like quantum effectsin photosynthetic light harvesting and olfaction recentlyreviewed [59], coupling quantum light to retinal rod cells[60, 61], testing for electron spin effects in anesthesia [62],and studying ion transport in neuronal ion channels [63,64], to name a few.

The outline of this review is the following. In Section2 we briefly review the basic notions of spin chemistry,touching upon the traditional approach to the radical-pair mechanism. In Section 3 we present our work on thequantum foundations of RPM. In light of this discussion,we revisit the traditional approach in Section 4 and ex-plore its weaknesses and limits of validity. In Section 5 westudy single-molecule quantum trajectories, which allowus to obtain a deeper understanding of both approachesand test their internal consistency. In Section 6 we dis-cuss other competing theoretical approaches. We devoteSection 7 to discussing the nonlinear nature of our mas-ter equation, and its relation to foundational concepts ofquantum physics. We close with an outlook in Section 8.

II. SPIN CHEMISTRY

Spin chemistry [65–67] deals with the effect of electronand nuclear spins on chemical reactions. That such an ef-fect is possible in the first place is not too straightforwardto understand. Indeed, covalent and hydrogen bond en-ergies are on the order of 1000 kJ/mole and 10 kJ/mole,

respectively, while the electron’s Zeeman energy in a lab-oratory magnetic field of 1 kG is on the order of 1 J/mole,let alone earth’s field. Yet even at earth’s field, electronand nuclear spins can have quite a tangible effect in thisclass of chemical reactions. The explanation, to be revis-ited later, is an intricate combination of spin precessionand electron transfer timescales with spin angular mo-mentum conservation.

Although the first organic free radical, triphenyl-methyl, was discovered by Gomberg in 1900, the origin ofspin chemistry is set in the 1960s, when anomalously highEPR [68] and soon later NMR [69–71] signals were ob-served in chemical reactions of organic molecules, termedCIDEP (chemically induced dynamic electron polariza-tion) and CIDNP (chemically induced dynamic nuclearpolarization), respectively. The radical-pair mechanismwas introduced by Closs and Closs [72] and by Kapteinand Oosterhoff [73] as a reaction intermediate explainingthese observations (a recent editorial [74] briefly reviewsthe field’s history and early literature). Since then, spinchemistry has grown into a mature field of experimentaland theoretical physical chemistry [4]. In particular, theeffects related to CIDNP [75–91], which we will addressin Section 5.2, continue to attract a lot of interest sinceCIDNP has become a versatile tool to study photosyn-thetic reaction centers [92].

An intriguing result of early spin-chemistry work wasthe proposal by Schulten [93–95] that avian magnetore-ception is based on RP reactions. In this review we willnot touch upon applying spin chemistry to magnetore-ception, referring the reader to a representative part [96–114] of an extensive literature.

A. Introduction to the radical-pair mechanism

The quantum degrees of freedom of radical-ion pairsare formed by a multi-spin system embedded in abiomolecule, which can be either in the liquid or in thesolid phase. In particular, RPs are biomolecular ionscreated by a charge transfer from a photo-excited D∗Adonor-acceptor dyad DA, schematically described by thereaction DA → D∗A → D•+A•−, where the two dotsrepresent the two unpaired electron spins of the two rad-icals. The excited state D∗A is usually a spin zero state,hence the initial spin state of the two unpaired electronsof the radical-pair is also singlet, denoted by SD•+A•−.Now, both D and A contain a number of magnetic nucleiwhich hyperfine-couple to the respective electron. Nei-ther singlet-state nor triplet state RPs are eigenstatesof the resulting Hamiltonian, H, hence the initial for-mation of SD•+A•− is followed by singlet-triplet (S-T)mixing, i.e. a coherent oscillation of the spin state of

the electrons, designated by SD•+A•−H−− TD•+A•−.

Concomitantly, nuclear spins also precess, and hence thetotal electron/nuclear spin system undergoes a coherentspin motion driven by hyperfine couplings and the rest

3

of the magnetic interactions to be detailed later.This coherent spin motion has, however, a finite life-

time. Charge recombination, i.e. charge transfer fromA back to D, terminates the reaction and leads to theformation of the neutral reaction products. It is angu-lar momentum conservation at this step that empowersthe molecule’s spin degrees of freedom and their minus-cule energy to determine the reaction’s fate: there aretwo kinds of neutral products, singlet (the original DAmolecules) and triplet, TDA. As it turns out, their rela-tive proportion can be substantially affected by spin in-teractions entering the mixing Hamiltonian H. For com-pleteness we note that the reaction can close through theso-called intersystem crossing TDA → DA, mediated bye.g. spin-orbit coupling. A schematic diagram of theabove is shown in Fig. 1a.

Omitting the light excitation and initial charge trans-fer, which commence the reaction in a timescale usuallyfaster than the reaction dynamics, the reaction schemecan be simplified into Fig. 1b. As shown with the twoshaded boxes in Fig. 1b, the RP reaction consists oftwo physically very different processes working simulta-neously: (a) the unitary dynamics embodied by the mag-netic Hamiltonian H driving a coherent S-T oscillation ofthe RP spin state, and (b) the non-unitary reaction dy-namics reducing the RP population in a spin-dependentway. As it will turn out, the latter are the hardest tounderstand.

B. Definitions

The density matrix ρ describes the spin state of theRP’s two electrons and M magnetic nuclei located inD and A. Its dimension is d = 4ΠM

j=1(2Ij + 1), where4 is the spin multiplicity of the two unpaired electrons,ΠMj=1(2Ij + 1) is the spin multiplicity of the M nuclear

spins, and Ij is the nuclear spin of the j-th nucleus, withj = 1, 2, ...,M . The simplest possible RP contains justone spin-1/2 nucleus hyperfine coupled to e.g. the donor’sunpaired electron. In this case the density matrix has di-mension d = 8. Although unrealistic, this simple systemexhibits much of the essential physics without the addi-tional computational burden of more nuclear spins andmatrices of higher dimension, therefore it is frequentlyused as a model system.

It is angular momentum conservation at the recom-bination process that forces the decomposition of theRP’s spin space into an electron singlet and an elec-tron triplet subspace, defined by the respective projec-tors QS and QT. These are d × d matrices given byQS = 1

41d − sD · sA and QT = 341d + sD · sA, where

sD and sA are the spin operators of the donor andacceptor electrons written as d-dimensional operators,e.g. the j-th component of sD would be written assjD = sj⊗12⊗12I1+1⊗12I2+1...⊗12IM+1, where the firstoperator in the Kronecker product refers to the donor’selectron spin, the second to the acceptor’s electron spin

DA

hν

D*A

TDA

SD + A TD + A

ISC

kSkT

(a)

DA TDA

SD + A

TD + A

kTkS

unitary dynamics

(b)

non-unitary dynamics

H

H

FIG. 1. (a) Radical-pair reaction dynamics. A donor-acceptordyad DA is photoexcited to D∗A and a subsequent chargetransfer produces a singlet state radical-pair SD•+A•−, whichis coherently converted to the triplet radical-pair, TD•+A•−,due to intramolecule magnetic interactions. Simultaneously,spin-selective charge recombination leads to singlet (DA) andtriplet neutral products (TDA). The latter can intersystemcross into DA and close the reaction cycle. (b) Simplified ver-sion of (a) neglecting the photoexcitation and charge transfersteps. Both diagrams could be misleading if taken too liter-ally, since they might suggest that e.g. only singlet radical-pairs recombine to singlet neutral products. This is not thecase, since a radical-pair in a coherent S-T superposition canrecombine into e.g. a singlet neutral product (see Section 3.7).

and the rest to the nuclear spins. By s we denote theregular (2 dimensional) spin-1/2 operators and by 1m

the m-dimensional unit matrix. The projectors QS andQT are complete and orthogonal, i.e. QS + QT = 1d andQSQT = QTQS = 0.

The RP’s singlet subspace has dimension ΠMj=1(2Ij+1)

while the triplet subspace has dimension 3ΠMj=1(2Ij + 1).

The electron-spin multiplicity 1 in the former corre-sponds to the singlet state

|S〉 =1√2

(|↑↓〉 − |↓↑〉

), (1)

while the electron-spin multiplicity of 3 in the latter

4

stems from the triplet states

|T+〉 = |↑↑〉 (2)

|T0〉 =1√2

(|↑↓〉+ |↓↑〉

)(3)

|T−〉 = |↓↓〉 (4)

The coupled basis |S〉 , |T+〉 , |T0〉 , |T−〉 describes thetwo unpaired electron spins. For example, for an RPcontaining n spin-1/2 nuclei, it would be d = 2n+2 andthe projectors would be written as QS = |S〉 〈S| (⊗12)n

and QT = (|T+〉 〈T+|+ |T0〉 〈T0|+ |T−〉 〈T−|)(⊗12)n.There are also two rates to consider, the singlet and

triplet recombination rates, kS and kT, respectively.These are defined as follows: at t = 0 we prepare anRP ensemble having no magnetic interactions (H = 0) inthe singlet (triplet) electron state (and any nuclear spinstate). Then its population would decay exponentially atthe rate kS (kT). In general, during a time interval dt,the measured singlet and triplet neutral products will be

drS = kSdtTrρQS (5)

drT = kTdtTrρQT (6)

Indeed, if all RPs were in the singlet or triplet state,the fraction of them recombining in the singlet or tripletchannel during dt would be kSdt and kTdt, respectively.If they are in the general state described by ρ, then kSdtand kTdt have to be multiplied by the respective fractionof singlet and triplet RPs.

C. Initial state

Radical-ion pairs are usually produced from an elec-tron spin zero neutral precursor, so their initial state isthe electron singlet state. The thermal proton polariza-tion at room temperature and at magnetic fields as largeas 10 kG is about 10−5, which for all practical purposescan be approximated by zero. The RP initial state havingzero nuclear polarization and being in the singlet electronspin state is

ρ0 =QS

TrQS(7)

Indeed, since Q2S = QS, it is Trρ0QS = 1. If Ii,j is

the j-th component (j = x, y, z) of the i-th nuclear spin,then Trρ0Ii,j = 0 since Ii,j are traceless operators.

D. Hamiltonian evolution

The magnetic interactions included in H drive the uni-tary RP dynamics. The simplest way to understand S-Tmixing is by using the classical vector model and consid-ering first an imaginary RP consisting of just two elec-tron spins having Larmor frequencies different by δω. As

S

t=0 t=π/δω

= 12

T0 = +

ω

ω+δω12

Py (C16H10)

DMA (C8H11N)

(b)

(a)

FIG. 2. (a) Classical vector model describing S-T conver-sion of a two-electron spin state driven by a difference inthe electrons’ Larmor frequencies. (b) Example of a radical-pair, Py•+DMA•−. The different nuclear spin environmentof Py•+ and DMA•− is the basis of singlet-triplet mixing.

shown in Fig. 2a, if the initial state is the singlet state|S〉, after a time τ = π/δω the two spins will have devel-oped a π phase difference, their state becoming the triplet|T0〉. In reality this model is encountered in the so-called∆g mechanism operating at high magnetic fields [116],where an actual difference in the g-factor of the two elec-trons is responsible for their different Larmor frequency.In most cases, however, S-T mixing is caused by the dif-ferent nuclear spin environment coupled to each electronthrough hyperfine interactions. To illustrate hyperfine-induced S-T mixing, we consider the example of Py-DMA(Pyrene-Dimethylaniline), shown in Fig. 2b. Here Py isthe electron donor and DMA the acceptor, i.e. the RPis Py•+DMA•−. In this example, the donor radical has10 proton spins hyperfine coupled to its unpaired elec-tron, while the acceptor radical has 12 spins, 11 protonsand one nitrogen, coupled to its unpaired electron. Thisdifference in nuclear spin environments ”seen” by the un-paired electrons of Py•+ and DMA•− drives S-T mixing.

To come back to the earlier discussion of the very possi-bility of spin-dependent chemical reactions, note that the

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S

0 20 40 60 80 100

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time in units of 1/Ω time in units of 1/Α

time in units of 1/Α2

0 20 40 60 80 100

time in units of 1/Α4

0 20 40 60 80 100

(b)

(c) (d)

FIG. 3. Examples of S-T mixing driven by various magneticHamiltonians. (a) RP with just two electrons with differentLarmor frequencies. (b) RP with one nuclear spin isotropi-cally coupled to the donor’s unpaired electron with hyperfinecoupling A. (c) RP with one nuclear spin in each radical,isotropically coupled with hyperfine couplings A1 and A2. Forthe plot A2 = 10A1. (d) RP with 4 nuclear spins in one radicaland none in the other. The hyperfine couplings are A1 = 1,A2 = 2, A3 = 3 and A4 = 10.

diffusion constant of Py/DMA in typical solvents used in[93] is D = 10−5 cm2/s, while the inter-radical distancewhere the RP recombination reaction is appreciable is[115] on the order of 10 nm, hence the reaction dynamicstake place at a timescale of about 100 ns. The electronspin precession time at a hyperfine magnetic field of 10 Gis about 30 ns, short enough for spin motion to influencethe reaction yields.

We will now visualize a few among many possibili-ties of S-T mixing, by plotting in Fig. 3 the Hamil-tonian evolution of the RP spin state for four differ-ent Hamiltonians. We plot the expectation value of QS

given by 〈QS〉 = TrρQS, where ρ evolves just uni-tarily, dρ/dt = −i[H, ρ]. In all cases we start with ρ0

given in (7). In Fig. 3a we consider an RP with justtwo unpaired electron spins having different Larmor fre-quencies and residing in an external magnetic field alongthe z-axis, so that H = ωDszD + ωAszA. In this ex-ample ρ is 4-dimensional. Singlet-triplet mixing occursat the mixing frequency Ω = |ωD − ωA|. In Fig. 3bwe consider an RP with just one spin-1/2 nucleus cou-pled to e.g. the donor electron, so the Hamiltonian readsH = AsD · I, where A is the isotropic hyperfine couplingand I the nuclear spin operator. In this case d = 8.Moving to higher-dimensional examples, in Fig. 3c weconsider two spin-1/2 nuclei, one coupled to the donorand one to the acceptor electron. Now the HamiltonianisH = A1sD ·I1+A2sA·I2, and d = 16. Finally, in Fig. 3dwe consider a 64-dimensional case where 4 nuclear spinscouple to the donor electron, hence H =

∑4j=1AjsD · Ij .

Unlike the previous cases, the evolution is seen to beaperiodic [42], due to a superposition of multiple S-T os-

cillations[? ]. Although not of concern for this review, wenote that other sorts of interactions can be included inH,like spin-exchange and long-range dipolar [7]. Moreover,hyperfine couplings can also be anisotropic [40].

E. Traditional master equation for radical-pairreactions

Besides the unitary evolution of the RP spin densitymatrix ρ, the reaction super-operator describing the spin-dependent population loss into neutral products mustalso be taken into account. The majority of theoreticalcalculations were so far performed with Haberkorn’s mas-ter equation, also termed ”traditional” in recent yearswhen other approaches came along:

dρ

dt= −i[H, ρ]− kS

2

(QSρ+ ρQS

)− kT

2

(QTρ+ ρQT

)Haberkorn′s reaction super−operator

(8)Obviously, the master equation for ρ is fundamental forspin chemistry, being the basis for predicting all exper-imental observables. Surprisingly, a microscopic first-principles derivation of (8) was given only recently [23] asa response to our work challenging it (see Section 6.2). Asinput, this or any contending master equation requires allmagnetic interactions entering the Hamiltonian H, andthe two rates kS and kT. As output, it can theoreticallypredict any experimental observable. A few examples of-ten encountered are: (A) The singlet and triplet reactionyields follow by integrating (5) and (6), Yx =

∫∞0drx,

with x = S,T. Given that the initial density matrix isproperly normalized, Trρ0 = 1, it will be YS + YT = 1.(B) The reaction rate, given by the decay rate of the RPpopulation, dTrρ/dt = −kSTrρQS− kTTrρQT. Inthe special case kS = kT ≡ k, the RP population decaysexponentially at the rate k, i.e. dTrρ/dt = −kTrρ,which follows from the previous equation since[? ]QS +QT = 1. (C) The magnetic field effect [117] is aboutthe magnetic field dependence of the time evolution ofTrρ, and is defined as MFE = TrρB − TrρB=0.

To summarize, the particular form of the reactionsuper-operator of (8) or any contending master equationis imprinted in all physically accessible observables. Im-portantly, the reaction super-operator does not only re-duce the RP population, Trρ, but also induces a changein the spin-state character embodied by ρ when kS 6= kT.To understand this, consider a balanced mixture of sin-glet and triplet RPs which are magnetically frozen, i.e.there are no magnetic interactions, H = 0 (such idealizedscenarios will be used frequently in order to gain physicalintuition on the dynamics). If kS = kT, the balance isnot distorted as the RPs disappear at the same rate irre-spective of their total electron spin. On the other hand,if kS 6= kT, the ensemble of surviving RPs will becomemore singlet (triplet) if kT > kS (kT < kS). As an ex-ample, we plot in Fig. 4 the time dependence of (a) the

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0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

trρ

tr

ρQ

SkS=Ω/5kΤ=Ω/5

kS=Ω/10kΤ=Ω/2

kS=Ω/2kΤ=Ω/10

(b)

(a)

time in units of 1/Ω

FIG. 4. For an imaginary RP with just two electrons andfor the same Hamiltonian as in Fig.3a, we use the traditionalmaster equation (8) and plot the time evolution of (a) 〈QS〉and (b) the RP population Trρ, for three different cases:(i) kS = kT, (ii) kS < kT and (iii) kS > kT.

singlet character of ρ, TrρQS, and (b) the RP popula-tion, Trρ, for the Hamiltonian used in Fig. 3a, but nowcalculated from the full master equation (8). It is clearlyseen that the time evolution of both observables dependson the relative magnitude of kS and kT, and of course onthe particular form of the reaction super-operator in (8).

III. QUANTUM FOUNDATIONS OF THERADICAL-PAIR MECHANISM

We will now lay out the main premise of this review,demonstrating that the RPM is a biochemical system sur-prisingly rich in quantum-information-science conceptsand effects. The need for such concepts appears whenone recognizes the physical model behind the RPM andattempts to translate this model into a master equation.

A. Physical model of the radical-pair mechanism

The diagrams depicting the reaction dynamics (Fig.1a and Fig. 1b) convey a gross picture but miss im-portant details. In Fig. 5 we present a more detailedenergy level structure that is critical in unraveling thequantum foundations of RPM. We brought into the pic-ture the vibrational excitations of the neutral productmolecules. At high excitation energy, these states forma quasi-continuous reservoir quasi-resonant with the RP

DA

magneticinteractions

Singlet Reservoir

TripletReservoir

Radical-Ion-Pair

SD + A TD + A

TDA

Population loss (rate kS)

Decoherence (rate kS/2)

Population loss (rate kT)

Decoherence (rate kT/2)

Open and Leaky Quantum SystemEnergy

HεS

FIG. 5. Radical-pair energy levels including the vibrationalexcitations of the neutral product states. Due to these reser-voir states, RPs (i) lose population by recombination into theneutral ground states through two real and consecutive tran-sitions, from the RP to the quasi-resonant reservoir statesfollowed by the decay of the latter to the ground state, and(ii) simultaneously suffer S-T decoherence due to virtual tran-sitions from the RP to the reservoir states and back.

DBA

D BA+ -

(DBA)*

hν

FIG. 6. Reproduction of Fig. 1 of [122], where hν is thephoton energy exciting the donor-bridge-acceptor DBA to(DBA)∗, and the horizontal red arrows denote the intramolec-ular charge transfers, the creation of the charge separatedstate D+BA− (1), and the recombined state DBA (2).

energy levels. The physics resulting from this reservoirare at the core of our work. Before proceeding, we shouldnote that the level structure presented in Fig. 5 is typ-ical for electron-transfer studies [118–121] and was wellknown before our work. For example, in Fig. 6 we re-produce a figure appearing in a study on supermolecularcharge separation [122]. The authors state (Section IIof [122]) that charge recombination D+BA− → DBA isrealized with a vibronic level of the charge-transfer statedecaying into a vibrationally excited ground-state mani-fold. What we have done in Fig. 5 is to adapt exactly thismodel and also include spin conservation by consideringtwo different manifolds for the general case of singlet andtriplet recombination products.

While the model of Fig. 5 was understood beforeour work, the physical picture following from taking this

7

model at face value was not. We have shown that radical-pairs are both leaky and open quantum systems, hencetheir non-unitary dynamics consist of two processes un-folding simultaneously and independently: (a) singlet-triplet decoherence and (b) spin-dependent populationdecay. Both processes are intimately linked to the vibra-tional reservoirs.

B. Quantum phononics: the role of phononvacuum in radical-pair reactions

The phonon degrees of freedom in RP reactions arereminiscent of another emerging field, quantum phonon-ics. Coherent manipulation of phonons [123] is witness-ing a rapid growth in recent years both due to the fun-damental interest of reaching the quantum limit in me-chanical degrees of freedom [124] and due to the poten-tial of developing novel phonon-based technologies [125].Regarding biological systems, it would at first sight ap-pear that vibrations just represent a classical noise sourceadding to the complexity of the molecular scaffold sup-porting biological dynamics. Again, the counterintuitivepicture is currently emerging that nature is also able toharness quantum degrees of freedom of vibrations in in-tricate ways. In photosynthesis, phonons are understoodto crucially regulate energy transport in light-harvestingcomplexes [126–130]. What we will demonstrate next isthat the phonon vacuum is the fundamental source ofS-T decoherence in RP reactions, much like the photonvacuum is for atomic coherences.

C. Singlet-triplet decoherence in the radical-pairmechanism: null quantum measurement of

recombination products

Null quantum measurements are well-known in quan-tum optics. A vivid example [131] is the single atomsurrounded by an array of unit-quantum-efficiency pho-todetectors. If the atom is initially in a coherent su-perposition of the ground and excited states, |ψ0〉 =

(|g〉+ |e〉)/√

2, and if no photon is detected until time t,the atomic state will be (apart from a normalization con-stant) |ψt〉 ∝ e−Γt/2|e〉+ |g〉, where Γ is the spontaneousdecay rate. The interpretation is that even the absenceof a photon detection can condition our knowledge of theatom’s state, e.g. if no photon is ever detected then theatom’s state cannot contain an excited state component.

In analogy, consider a single radical-ion pair in the spinstate |ψ0〉 at t = 0. Suppose that within the followingtime interval dt there is no detection of a neutral recom-bination product. What is |ψdt〉 ? In our first publication[8] we provided a first-principles derivation of the masterequation for non-recombining (or surviving) RPs, that is,the case of no detection of recombination products.

We have shown that until an RP charge-recombines,it does not just undergo Hamiltonian evolution, as was

intuitively expected, but it suffers random projectionsto the singlet or triplet electron spin state. In the en-semble picture, this represents a fundamental S-T deco-herence process in the sense that it cannot in any waybe mitigated. In contrast, technical decoherence can inprinciple be suppressed in an experiment designed care-fully enough. To give an exaggerated example, inadver-tently radiating RPs with radio-frequency noise (as it ispurposefully done in studying bird disorientation [114]),would cause spin decoherence. This can obviously besuppressed by removing rf noise sources from the exper-iment’s proximity, hence rf noise is an example of tech-nical decoherence. The physical process we will presentnext is not such a technical source of decoherence, butan unavoidable one, intimately linked to the radical-pairrecombination mechanism.

D. Derivation of the master equation describingS-T decoherence

When dealing with open quantum systems it is crucialto correctly identify the open system and its environ-ment, so that one can apply the standard treatment ofevolving the combined system and tracing out the en-vironment. Based on Fig. 5, we have identified as the”system” the spin degrees of freedom of the radical-ionpair, described by the RP density matrix ρ. The ”en-vironment” is represented by the two reservoirs, the ex-cited vibrational manifolds of the neutral ground statesDA and TDA. We stress that these vibrational states areempty, or unoccupied, as long as the RP has not recom-bined. They should not be confused with the thermallyexcited vibrational states of the radical-pair itself. Whenthe back electron-transfer from the acceptor A to thedonor D neutralizes the radical-pair, the neutral moleculefinds itself in an excited vibrational state. These are thereservoir states, and their quantum state during the RP’slifetime is the vacuum.

The state evolution of surviving RPs was based on a2nd-order perturbation treatment of the spin-selective RPcoupling with the unoccupied vibrational reservoirs, pre-sented in [8] and from a slightly different perspective in[17], the main points of which we recapitulate here. Con-sider for the moment just the singlet reservoir. Since theamplitude for the transition to the i-th singlet reservoirstate, D•+A•− → SDA

∗i , should be proportional to the

singlet character of the RP state, the coupling Hamil-

tonian reads V =∑i

(hi + h†i

), where hi = uia

†i cQS.

The operator QS projects the RP state onto the electron-

singlet subspace, while the raising operator a†i producesa single occupation of the i-th reservoir level. The tran-sition amplitude is ui, and the operator c describes theoccupation of the acceptor’s electron site, i.e. c†c = 1(c†c = 0) means the electron is localized at the acceptor

(donor). The hermitian conjugate h†i describes the re-

verse process SDA∗i → SD•+A•−. As explained in [8, 11],

virtual transitions D•+A•− → SDA∗i → SD•+A•− driven

8

by hi followed by h†i , produce the random singlet projec-tions. Tracing out the reservoir degrees of freedom wearrive at (tilde denotes interaction-picture):

dρ(t)

dt=∑ij

∫ ∞0

dτ eije−iεSτ

[QS(t)ρ(t)Q†S(t− τ)− ρ(t)Q†S(t− τ)QS(t)

]+ h.c., (9)

where we also traced out the c degrees of freedom aswe are interested in the time evolution of the RP state,for which 〈c†c〉 = 1. The reservoir operators are Ei =

uieiεita†i , and the expectation value eij = 〈Ei(0)E†j (τ)〉 is

calculated in the phonon vacuum state. Finally, the tran-sition amplitudes uj are composed of an electronic matrixelement and a nuclear overlap matrix element, uj = vjχj[132]. We consider these to be independent of j in thevicinity of εS (see Fig. 5), and introduce the Franck-Condon averaged density of states g(ε) = |χ(ε)|2d(ε),which takes into account both χ(ε), the nuclear wavefunction overlap integral, and d(ε), the density of vibra-tional states at the energy ε. Identifying the recombina-tion rate kS with the expression 2π|v|2g(εS) we obtaindρ/dt = −kS2 (QSρ + ρQS − 2QSρQS). Adding into thepicture the triplet reservoir and the unperturbed unitaryevolution, we arrive at the Lindblad-type master equa-tion

dρ

dt

∣∣∣decoh

=− i[H, ρ]

− kS

2(QSρ+ ρQS − 2QSρQS)

− kT

2(QTρ+ ρQT − 2QTρQT) (10)

Due to the relation QS + QT = 1, the above can take themore compact form

dρ

dt

∣∣∣decoh

= −i[H, ρ] +DJρK, (11)

where the S-T decoherence super-operator is

DJρK = −kS + kT

2(QSρ+ ρQS − 2QSρQS) (12)

E. Null quantum measurement of recombinationproducts in terms of single-molecule quantum

trajectories

Typically, a Lindblad-type equation describes the ef-fect of a continuous quantum measurement [133] on thesystem’s density matrix. What is measured in this caseand who performs the measurement? The measured ob-servable is the singlet character of the RP’s spin state,i.e. the observable QS. The measurement is performedby the molecule’s own vibrational reservoir, i.e. by the

existence of the phonon vacuum and the possibility theRP has to occupy these phonon reservoir states. Thisis an intra-molecule continuous quantum measurement.The density matrix evolving according to (11) describesan ensemble of radical-pairs. What about the equivalentpicture in terms of quantum trajectories? Consider anRP in the pure state |ψ〉 at time t. Assume that no re-combination product was observed within dt. What is|ψ〉 at time t+ dt? There are three different possibilitiesfor single-molecule state evolution:(i) projection to the singlet RP state

|ψS〉 =QS|ψ〉√〈ψ|QS|ψ〉

,

taking place with probability dpS = (kS+kT)dt2 〈ψ|QS|ψ〉.

(ii) projection to the triplet RP state

|ψT〉 =QT|ψ〉√〈ψ|QT|ψ〉

,

taking place with probability dpT = (kS+kT)dt2 〈ψ|QT|ψ〉.

(iii) unitary evolution driven by H, taking place withprobability 1− dpS − dpT.

In an ensemble of RPs, these single-molecule events areunobservable, so we have to average over (i)-(iii), trans-forming an initially pure into a mixed state. This averageexactly reproduces the master equation (11). In otherwords, writing

ρt+dt = dpSρS

+ dpTρT

+ (1− dpS − dpT)(ρt − idt[H, ρt]), (13)

leads to (11) for dρ/dt = ρt+dt−ρtdt , ρt = |ψ〉〈ψ|, ρS =

|ψS〉〈ψS| and ρT = |ψT〉〈ψT|.The physical significance of the sum kS + kT appear-

ing in the probabilities dpS and dpT and also appearingin (11) is the fact that both singlet and triplet reservoirscontinuously ”measure” the same observable, namely QS.The result of this measurement is either 1 or 0, corre-sponding to the singlet and triplet projections, respec-tively. In particular, the singlet reservoir measures theobservable QS at the rate kS/2. The ”yes” result ofthis measurement corresponds to QS 7→ 1 and the sin-glet projection, while the no/null result corresponds toQS 7→ 0 and the triplet projection. Similarly, the tripletreservoir measures the observable QT = 1 − QS at therate kT/2. The ”yes” result of this measurement cor-responds to QS 7→ 0 and a triplet projection, while theno/null result corresponds to QS 7→ 1 and the singletprojection. Equivalently, QS is measured at the totalrate (kS + kT)/2. These measurement results are unob-servable, each radical-pair has its own random historyof projections, hence the ensemble exhibits a decay ofS-T coherence. In Fig. 7b we plot an example (repro-duced from [18]) of the evolution described by (11). Thedecaying amplitude of the 〈QS〉-oscillations is what this

9

time (units of 1/A)5 100 15 20 25

Monte Carlodρdt decoh

singlet projectionstriplet projections

0.0

QS

0.8

0.4

0.6

0.2

1.0

(b)

(a)

0.0

QS

0.8

0.4

0.6

0.2

1.0

FIG. 7. Time evolution of 〈QS〉 for a model RP with onenuclear spin, taking into account only S-T decoherence andS-T mixing driven by H = ω(sDz + sAz) + AsD · I. We tookω = A/10 and kS = kT = A/4. (a) single-RP quantumtrajectory, depicting singlet and triplet projections at randominstants in time. The initial RP state for this trajectory is|S〉 ⊗ |⇑〉. (b) average of 20,000 such trajectories (red solidline), half of which have initial state |S〉 ⊗ |⇑〉 while the otherhalf have initial state |S〉 ⊗ |⇓〉. The prediction of the masterequation (11) is shown by the black dashed line. The initialstate of the density matrix was ρ0 given by (7).

decoherence describes. A single-RP quantum trajectoryis shown in Fig.7a. This trajectory consists of the Hamil-tonian evolution interrupted at random instants by theaforementioned singlet or triplet projections. Averaginga large number of such trajectories exactly reproducesthe evolution of Fig. 7b.

F. The two roles of the coupling Hamiltonian

In contrast to S-T dephasing produced by virtual tran-sitions to the reservoir states and back, charge recombi-nation proceeds by real transitions to the reservoir states,followed by their decay to the ground state. The latterhappens at ps timescales, so the rate limiting process isthe former. Within 1st-order perturbation theory and byusing Fermi’s golden rule we find that the recombina-tion rate will be kS = 2π|〈f |V|i〉|2d(εS) = 2π|v|2g(εS),where as initial state |i〉 we chose a pure singlet stateof the radical-pair, and as a final state |f〉 one among

the near-resonant and quasi-continuum reservoir statesdescribed by the density of states d(E). So the recom-bination rates calculated from Fermi’s golden rule havethe exact same form identified in the previous derivationof dρ

dt

∣∣decoh

. This is not a coincidence, since the Hamil-tonian V coupling the radical-pair with the vibrationalreservoir is responsible for both processes, which are oforder |V|2. Indeed, the coupling V appears two times in(9), and the rate kS depends on the square of V throughFermi’s golden rule. Hence both processes lead to first-order reaction kinetics, dρ ∝ dt.

G. Reaction terms

The master equation (11) is trace-preserving, i.e. un-der the evolution DJρK the normalization Trρ remainsconstant. We will now describe the second process, thespin-dependent recombination, leading to neutral reac-tion products and the concomitant decay of Trρ. How-ever, charge recombination does not only lead to a decayof Trρ. Radical-ion pairs are in some quantum stateat the moment of their recombination, hence their dis-appearance from the RP ensemble leads to yet anotherstate change of the ensemble of surviving RPs (see Sec-tion 2.5).

Given kS and kT, we can calculate the number ofRPs that recombine during dt in the singlet and tripletchannel. These are drS = kSdtTrρQS and drT =kTdtTrρQT, already introduced in (5) and (6). Equiv-alently, drS (drT) is the probability that a single RPwill recombine during dt in the singlet (triplet) channel.However, just knowing these probabilities does not tellus what the state of the RP was before it recombined.Put differently, the physical information we have at handis the detection of neutral recombination products (theground states DA or TDA of Fig. 5), the number ofwhich reflect these probabilities. Based on this informa-tion, we have to find the pre-recombination state of theRPs recombined during dt.

With this discussion we motivate the formal calcula-tion [18] of the reaction terms. This is accomplished us-ing a number of quantum-information concepts. First, wedefine a quantum coherence quantifier, a measure map-ping the RP density matrix ρ into a number quantifyingS-T coherence. Second, we need the concept of quantumretrodiction, developed in the theory of quantum com-munications.

1. Radical-pair recombination, S-T coherence, and quantumretrodiction

Quantum retrodiction [135–139] is the reverse of pre-diction, which we regularly deal with. That is, given adensity matrix describing a physical system, it is straight-forward to predict the probabilities of possible measure-ment outcomes. What is less straightforward is to retro-

10

dict the system’s preparation state given a specific mea-surement outcome. This is relevant to quantum commu-nications [134], since Bob, the receiving end of a com-municating party, attempts to reconstruct the quantumstate delivered to him by Alice, the sender, based onspecific outcomes of quantum measurements he decidesto perform.

To demonstrate the need of quantum retrodiction inRP reactions, consider an RP ensemble and a time in-terval dt so small that only one RP recombines duringdt. If we knew the state |ψ〉 of this RP just before itrecombines, we would subtract |ψ〉〈ψ| from the ensembledensity matrix to account for this recombination event.But at any given moment, the density matrix of an RPensemble consists of a mixture of pure states that havebeen produced by the magnetic Hamiltonian H, but alsoundergone a number of singlet or triplet projections atrandom and unknown times. Given the neutral recombi-nation product, which is either the singlet or the tripletground state, one cannot unambiguously retrodict thepre-recombination state |ψ〉. The theory of quantumretrodiction allows us to probabilistically do so, in a waythat depends on S-T coherence of the RP ensemble at thetime we detect the particular recombination products.

To give a few simple examples of why this is so, con-sider a maximally incoherent mixture of n singlet and mtriplet RPs, described by ρt = n|S〉〈S| + m|T0〉〈T0|, thenormalization being Trρt = n + m. With this infor-mation at hand, and given the detection of e.g. a singletneutral product within dt, it is certain that the neutralproduct originated from an RP in the state |S〉〈S|. Toaccount for this recombination event, one would sim-ply use the reaction term dρ = −|S〉〈S|, and henceρt+dt = ρt+dρ = (n−1)|S〉〈S|+m|T0〉〈T0|. This mixturehad zero S-T coherence. Another example is a mixture ofn radical-pairs in the state |ψ1〉 = (|S〉+ |T0〉)/

√2 and m

radical-pairs in the singlet state |ψ2〉 = |S〉. This mixturehas partial S-T coherence. Assume that within dt we de-tect a singlet neutral product. Did this originate from anRP in the state |ψ1〉 or |ψ2〉? Intuitively one could arguethat it is the former with probability n/(n + 2m) andthe latter with probability 2m/(n+ 2m). But how do we

find the answer when we are just given a density matrixρ? Given an arbitrary density matrix ρ, we can left andright multiply ρ with 1 = QS + QT and thus write

ρ = ρSS + ρTT + ρST + ρTS, (14)

where ρxy = QxρQy, with x = S,T. The last two terms,ρST + ρTS, embody S-T coherence. In fact, the decoher-ence super-operator DJρK exactly damps ρST + ρTS, i.e.QSρ+ ρQS − 2QSρQS = ρST + ρTS. In other words, vir-tual transitions RP → vibrational reservoir → RP pushρ towards an incoherent mixture ρSS + ρTT. The extentto which this is accomplished also depends on how fastS-T coherence is generated by the Hamiltonian H.

It is the closure relation 1 = QS + QT that automati-cally led to (14), forced upon the RP dynamics by angularmomentum conservation, which breaks up the RP’s statespace into eigenstates of either QS or QT. On the otherhand, the magnetic Hamiltonian H mixes the eigenstatesof QS and QT, producing S-T coherence. Which are thedensity matrix elements describing S-T coherence in theterm ρST +ρTS? To address this question, we note that ρalso describes the spin state of a (possibly large) numberof nuclear spins. Hence ρST + ρTS is in general a block-off-diagonal matrix. We will give a few simple examplesto elucidate this point. Considering first an imaginaryRP with just two electrons, the density matrix ρ (of di-mension 4) and it’s coherent part, ρST + ρTS, written inthe basis |S〉 , |T+〉 , |T0〉 , |T−〉, would be

ρ =

ρSS ρST+ρST0

ρST−

ρT+S ρT+T+ρT+T0

ρT+T−

ρT0S ρT0T+ρT0T0

ρT0T−

ρT−S ρT−T+ρT−T0

ρT−T−

ρST + ρTS =

0 ρST+ρST0

ρST−

ρT+S 0 0 0ρT0S 0 0 0ρT−S 0 0 0

(15)

For an 8-dimensional radical-pair with one spin-1/2 nu-cleus the basis is |S〉 ⊗ |⇑〉,|S〉 ⊗ |⇓〉,|T+〉 ⊗ |⇑〉,|T+〉 ⊗|⇓〉,|T0〉 ⊗ |⇑〉,|T0〉 ⊗ |⇓〉,|T−〉 ⊗ |⇑〉,|T−〉 ⊗ |⇓〉, and

ρST + ρTS =

0 0 ρST+⇑⇑ ρST+⇑⇓ ρST0⇑⇑ ρST0⇑⇓ ρST−⇑⇑ ρST−⇑⇓0 0 ρST+⇓⇑ ρST+⇓⇓ ρST0⇓⇑ ρST0⇓⇓ ρST−⇓⇑ ρST−⇓⇓

ρT+S⇑⇑ ρT+S⇑⇓ 0 0 0 0 0 0ρT+S⇓⇑ ρT+S⇓⇓ 0 0 0 0 0 0ρT0S⇑⇑ ρT0S⇑⇓ 0 0 0 0 0 0ρT0S⇓⇑ ρT0S⇓⇓ 0 0 0 0 0 0ρT−S⇑⇑ ρT−S⇑⇓ 0 0 0 0 0 0ρT−S⇓⇑ ρT−S⇓⇓ 0 0 0 0 0 0

(16)

How do we quantify how coherent is an ensemble of RPsdescribed by some density matrix ρ? In other words,how do we measure the ”strength” of ρST + ρTS ? We

call this measure pcoh and we treat it as a probabilitymeasure taking values between 0 (zero S-T coherence)and 1 (maximum S-T coherence).

11

However, there are a few complications. First, ρ car-ries information about all sorts of coherences, e.g. nu-clear spin coherences. But we are interested in quantify-ing the coherence between the electron singlet and tripletsubspace, irrespective of the nuclear spin state. For ex-ample, consider the two-nuclear-spin radical-pair in thestate |S〉 ⊗ 1√

2(|⇑⇑〉+ |⇓⇓〉). This state contains nuclear

spin coherence and, in fact, also nuclear spin entangle-ment, however it is an eigenstate of QS, and hence haszero S-T coherence. We need a measure quantifying threetypes of singlet-triplet coherence that can in general ex-ist, S-T+, S-T0 and S-T−. In [18] we defined

C(ρ) =∑j=0,±

√TrρST|Tj〉〈Tj|ρTS, (17)

where |Tj〉〈Tj | is the projector[? ] to one of thethree triplet subspaces. This definition is motivated byconsidering the most general pure state of a radical-pair,written as |ψ〉 = α|S〉 ⊗ |χS〉 +

∑j=0,± βj |Tj〉 ⊗ |χj〉,

where α, β+, β0 and β− are complex amplitudes,and |χS〉, |χ+〉, |χ0〉 and |χ−〉 are arbitrary andnormalized spin states of multiple nuclei. It isC(|ψ〉 〈ψ|) = |αβ1| + |αβ0| + |αβ−1|, hence indeed C(ρ)describes the combined ”strength” of all three coher-ences. For the example states given in (15) and (16) itwould be C(ρ) = |ρST+

| + |ρST0| + |ρST− | and C(ρ) =∑

j=0,±√|ρSTj⇑⇑|2 + |ρSTj⇑⇓|2 + |ρSTj⇓⇑|2 + |ρSTj⇓⇓|2,

respectively.A second complication has to do with the normaliza-

tion of ρ. Since C(ρ) is linear in the matrix elements of ρ,scaling as Trρ, which is a decaying function of time dueto charge recombination, C(ρ) is also a decaying functionof time. Therefore C(ρ) has to be normalized by Trρin order to define the following probability measure.

To extract such a measure from C(ρ), we must nor-malize C(ρ) by its maximum. In [18] we calculatedthe maximum of C(ρ) when only the Hamiltonian evo-lution is taken into account, i.e. maxC(ρ), wheredρ/dt = −i[H, ρ]. So in [18] we defined

pcoh =1

TrρC(ρ)

maxC(ρ)(18)

In retrospect, this definition of the normalization is notsatisfactory, because if one is given a density matrix ρdescribing an RP ensemble without any reference to H,the measure pcoh cannot be calculated. Surely, in mostcalculations where one propagates the density matrix,the evolution is driven by a known Hamiltonian. But inprinciple it would be desirable to have a measure pcoh

defined by just providing the density matrix ρ. An-other way to normalize C(ρ) based solely on ρ is de-rived from the most general pure state |ψ〉 given before.We define the maximally S-T coherent pure state as thestate having equal amplitudes in all four terms of |ψ〉,|α| = |β+| = |β0| = |β−|. Due to normalization it fol-lows that |α| = |β+| = |β0| = |β−| = 1/2 and hence

C(|ψ〉〈ψ|) = 3/4. We can thus define

pcoh =4

3

C(ρ)

Trρ(19)

A few comments are in order: (i) the definition of C(ρ)has the proper (linear) scaling with the density matrix el-ements to qualify [144] for a proper measure of coherence.(ii) we defined the pure state of maximum coherence asin [144], where it is stated that in a space of dimensionN spanned by the vectors |j〉, where j = 1, 2, ..., N , the

state of maximum coherence is 1√N

∑Nj=1 |j〉. (iii) in ide-

alized theoretical scenarios, e.g. when one considers a2-dimensional radical-pair spanned by just two states,|S〉 and |T0〉, the normalization constant 3/4 changes ac-cordingly, i.e. it would become 1/2. (iv) when we firstintroduced the need for an S-T coherence measure [11] wedefined pcoh in a way that it scales quadratically with thematrix elements of ρST + ρTS, and this is not an accept-able measure according to [144]. The definition (17) al-leviates this problem. (v) the different normalizations in(18) and (19) produce acceptable numerical differences,but a more thorough study is required to fully understandthe normalization of C(ρ).

2. Coherence Distillation

There is one more step before proceeding with thetheory of quantum retrodiction. As mentioned previ-ously, the general RP density matrix can be written asρ = ρSS +ρTT +ρST +ρTS. Having defined the coherencemeasure pcoh(ρ), we can introduce two new matrices, themaximally incoherent and maximally coherent version ofρ, denoted by ρincoh and ρcoh, respectively, and given by

ρincoh = ρSS + ρTT (20)

ρcoh = ρSS + ρTT +1

pcoh(ρST + ρTS) (21)

While the interpretation ρincoh is obvious, the interpre-tation of ρcoh is more subtle. Reminiscent of the con-cept of entanglement distillation [140, 141], ρcoh alludesto the coherence-distilled version of ρ, since pcoh(ρcoh) =1. To give a simple example, consider like in Section3.7.1 a mixture of n radical-pairs in the state |ψ1〉 =

(|S〉 + |T0〉)/√

2 and m radical-pairs in the singlet state|ψ2〉 = |S〉. It is pcoh = n/(n + m), so pcoh should bethe probability that picking an RP out of the ensembleit will have maximum S-T coherence. In general, us-ing (20) and (21) we can write any density matrix ρ asρ = (1− pcoh)ρincoh + pcohρcoh.

3. Quantum retrodiction

In Fig. 8 we show that RP reactions can be viewedas a biochemical realization of quantum communication.

12

DA

H SD + A TD + A

TDABob

prediction

DA TDABob

retrodiction

(a) (b)

SD + A TD + AH

FIG. 8. Quantum communication model of the radical-pairmechanism. Alice holds the quantum state of the radical-pair.Bob only has access the the recombination products, whichare either singlet or triplet. In the predictive direction (a) weknow the radical-pair state and calculate the probabilities ofBob’s measurement outcomes. In the retrodictive direction(b) we have access to a specific recombination product andtry to estimate the pre-recombination radical-pair state.

Indeed, the RP spin state might be considered as thequantum state held by Alice, the sender. Bob just de-tects singlet or triplet neutral products. From these hemight infer the information stored in Alice’s states, forexample the value of the applied magnetic field. Quan-tum retrodiction probabilistically answers the questionof what was the pre-recombination state of a radical-pair, the neutral product of which is detected by Bob.The relevant formalism [137, 138] is used in [18] to cal-culate the conditional probabilities that upon detectinga singlet (triplet) product, the pre-recombination stateof the radical-pair was maximally coherent or maximallyincoherent, P (incoh|S) = P (incoh|T) = 1 − pcoh, andP (coh|S) = P (coh|T) = pcoh. To arrive at the re-action terms, we need to subtract the estimated pre-recombination state of the recombined radical-pair fromthe density matrix of the surviving RPs. This cruciallydepends on S-T coherence [18]. For a maximally coherentdensity matrix it is

δρ1Scoh = δρ1T

coh = − ρ

Trρ,

whereas for a maximally incoherent density matrix it is

δρ1Sincoh = − QSρQS

TrρQS, δρ1T

incoh = − QTρQT

TrρQT

Stated in words, these rules mean that in the extremeof maximum coherence we have to subtract the full pre-recombination state from ρ, whereas in the opposite ex-treme we subtract either a singlet or a triplet RP. Theprevious update rules hold for a single molecule. Duringdt we will detect drS singlet and drT triplet products,hence the reaction terms read

dρ

dt

∣∣∣recomb

= drSδρ1S + drTδρ

1T, (22)

where

δρ1S = P (incoh|S)δρ1Sincoh + P (coh|S)δρ1S

coh

δρ1T = P (incoh|T)δρ1Tincoh + P (coh|T)δρ1T

coh

H. Radical-pair master equation

As already mentioned, singlet-triplet dephasing andspin-dependent recombination are two physical processesrunning simultaneously and independently, while stem-ming from the same interaction Hamiltonian betweenradical-pair and vibrational reservoir. Using (11) and(22), we thus arrive at the sought after master equation

dρ

dt=dρ

dt

∣∣∣decoh

+dρ

dt

∣∣∣recomb

= −i[H, ρ] +DJρK +RKJρK (23)

where RKJρK is the reaction super-operator

RKJρK =− (1− pcoh)(kSQSρQS + kTQTρQT

)− pcoh

drS + drT

dt

ρcoh

Trρ(24)

The term −i[H, ρ] generates S-T coherence, which is dis-sipated by DJρK, while RKJρK accounts for the state-dependent recombination. As expected, the normaliza-tion of ρ decreases by the neutral products appearing inthe time interval dt, i.e. from (23) we find dTrρ =−drS − drT. Given the particular Hamiltonian H andrecombination rates kS and kT, and using the definitionof pcoh, it is straightforward to propagate ρ starting froma specific initial condition.

I. Examination of special cases

1. Recombination without S-T mixing

Consider an imaginary RP having a non-reactivetriplet state and no S-T mixing, i.e. take kT = 0 andH = 0.

(i) singlet initial state, ρ0 = QS/TrQS. We expectthe RP population to decay exponentially at a rate kS,the surviving RP state remaining the same as at t = 0.Indeed, it is DJρ0K = 0 and pcoh = 0. Plugging theseand H = 0, kT = 0 into (23) leads to dρ/dt = −kSρ, soρt = e−kStQS/TrQS, as expected.

(ii) maximally incoherent S-T mixture[? ], i.e. ifρ0 = αQS/TrQS + βQT/TrQT, with α, β real andα + β = 1. We find that dρ/dt = −kSQSρQS, so thatρt = αe−kStQS/TrQS+ βQT/TrQT, as expected.

(iii) maximally coherent superposition |ψ0〉 = (|S〉 +

|T0〉)/√

2. Now the master equation cannot be solvedanalytically, since at t = 0 we have a maximally co-herent state with pcoh = 1, but right in the follow-ing interval dt the S-T dephasing super-operator DJρK

13

will produce a partially coherent state having pcoh < 1,hence all terms of RKJρK will start contributing. How-ever, as a preamble to quantum trajectories to be dis-cussed in Section 5, we note four possibilities for stateevolution during the interval dt: (1) singlet projection,with probability dpS = kSdt/4, (2) triplet projectionwith probability dpT = kSdt/4, (3) singlet recombina-tion with probability drS = kSdt/2, and (4) no evo-lution, i.e. the state remains |ψ0〉, with probability1 − dpS − dpT − drS = 1 − kSdt. Those RPs pro-jected at some point to the non-reactive triplet statewill remain there. Their total number as t → ∞ iskSdt/4+(1−kSdt)kSdt/4+(1−kSdt)

2kSdt/4+ ... = 1/4.At first sight this is a weird result challenging our un-

derstanding of quantum measurements. Starting with|ψ0〉, in which the probability for the RP to be in thetriplet state is 1/2, we find that only 25% of the RPsare locked in the non-reactive triplet state at the end ofthis reaction. The apparent contradiction with our intu-ition is due a cavalier phrasing in the previous argument.Indeed, if we prepare an RP ensemble in the state |ψ0〉and immediately perform a global projective measurementon the ensemble, we will find 50% of the RPs in |S〉 and50% in |T0〉. But the radical-pair reaction is not a globalprojective measurement performed at once on all RPs.Instead, it proceeds as a continuous intra-molecule mea-surement leading to the already discussed random pro-jections. These enhance the singlet yield, since those RPsprojected to the singlet state will definitely recombine atsome point.

2. Equal recombination rates

The case kS = kT ≡ k is especially simple, sincethe reaction is spin-independent, so RKJρK should nothave any effect on the RP state. Indeed, now it isRKJρK = −kρ, hence dρ/dt = −i[H, ρ]− k(ρQS + QSρ−2QSρQS) − kρ. Defining ρ = e−ktR, it follows thatdR/dt = −i[H, R] − k(RQS + QSR − 2QSRQS), i.e. thestate of the RPs evolves just as if they never recombine,of course with their total population decaying exponen-tially. If we now consider the case H = 0 and writeR(t) = RSS(t) +RTT(t) +RST(t) +RTS(t) we easily findR(t) = RSS(0)+RTT(0)+e−kt

(RST(0)+RTS(0)

), mean-

ing that the initially present coherence ρST(0) + ρTS(0)decays at the rate 2k, while the population decays a therate k. That is, coherence is damped due to the decayingpopulation, but also due to the intrinsic S-T dephasingprocess.

IV. HABERKORN’S MASTER EQUATION

If the master equation (23) we arrived at is a deepertheory capturing the quantum dynamics of RP reactions,then Haberkorn’s master equation must be some sortof limiting case of our theory. It is readily seen that

Haberkorn’s master equation (8) can be recast in theform dρ/dt = −i[H, ρ] + DJρK + RHJρK, where RHJρKis Haberkorn’s reaction operator:

RHJρK = −kSQSρQS − kTQTρQT (25)

If by hand we force pcoh = 0 in our reaction opera-tor RKJρK, we exactly retrieve RHJρK. In other words,the S-T dephasing process DJρK is latently built intoHaberkorn’s master equation, however the reaction terms(25) skew the state evolution of RPs by retrodicting thepre-recombination state always assuming zero S-T coher-ence.

In his 1976 paper Haberkorn had qualitatively exam-ined various forms of the master equation, one being thedecoherence master equation (11). However, he correctlydismissed it since it is trace-preserving and hence ”cor-responds to spin relaxation without reaction”. We nowunderstand that S-T dephasing is just one aspect of thedynamics running simultaneously with the other, the RPrecombination. In any case, Haberkorn’s 1976 paper is atribute to the power of simple physical arguments leadingto important insights.

A. When is the traditional master equation anadequate theory?

The traditional master equation (8) has been used inmost theoretical considerations in the field of spin chem-istry since the late 1960s, adequately accounting for manyexperiments. So a natural question to ask is: why do weneed a new theory when the one at hand, even if not fun-damentally sound, appears to be good enough? Towardsthe answer we first remind the reader of the early era ofatomic physics, when atom-light interactions were suc-cessfully described by Einstein’s rate equations, whichjust considered the transfer of atomic populations be-tween atomic states, the transfer being triggered by (atthe time incoherent) light sources such as lamps. Afterthe invention of the laser, the field of coherent atomic in-teractions exploded, since atomic coherences could thenbe excited and probed. Einstein’s rate equations werenot sufficient to describe experiments, which requiredthe introduction of the atomic density matrix, i.e. thecoupled time evolution of atomic populations and coher-ences. The parameter deciding which approach is themost suitable is the ratio of the Rabi frequency of theexciting light to the relaxation rate, which in this case isthe spontaneous decay rate.

In spin chemistry, many experiments so far have beenapparently dominated by spin relaxation, which amongother things, also damps S-T coherence quantified bypcoh. If these relaxation phenomena push pcoh to-wards zero much faster than the fundamental decoher-ence mechanism we have considered, then our masterequation quickly converges to the traditional theory. Thiswill be further quantified in the following subsection.

14

We can thus arrive at the conclusion that, notwith-standing fortuitous agreement with experiments broughtabout by relaxing environments, understanding spinchemistry experiments at the fundamental level is notpossible within the traditional theory. Moreover, with-out a fundamental theory it is neither possible to predictthe full range of physical effects that can be in principleobserved, nor design new experiments and explore newfronts of the field. For example, understanding the aviancompass mechanism at the quantum level, e.g. address-ing the fundamental heading error of the compass andits magnetic sensitivity [13] is a promising venue of bio-chemical quantum metrology, which however cannot beconclusively explored without the fundamental theory ofRPM. This is because even if radical-pairs participatingin the natural avian compass are plagued by relaxation,future biomimetic sensors need not be.

Finally, there are cases where any agreement with ex-periments is deluding, i.e. Haberkorn’s incorrect reac-tion super-operator forces other system parameters (likeHamiltonian couplings) to take on incorrect values in or-der for the theory to match the data. As will be elab-orated in Section 5.2, such an example are the spin dy-namics in CIDNP, which cannot be understood withinHaberkorn’s approach, since the lifetimes of the involvedRPs are too short for relaxation to set in.

1. Additional relaxation processes mask the underlyingquantum dynamics of radical-pairs

A general relaxation process can be described by a setof Kraus operators Kn, satisfying

∑nK

†nKn = 1 and

transforming the density matrix according to ρ → ρ′ =∑nKnρK

†n, i.e. the master equation (23) will be aug-

mented with the term KJρK =(∑

nKnρK†n − ρ

)/dt.

Note that DJρK is a special case of KJρK resulting from thefollowing three Kraus operators: K1 =

√1− λQS, K2 =√

1− λQT and K3 =√λ1, with λ = 1− (kS + kT)dt/2.

The simplest way to make our point is to assume thatbesides the fundamental S-T dephasing process, we havean independent relaxation mechanism of rate β, beingdescribed by the same Kraus operators K1, K2 and K3.Defining[? ] the super-operator CJρK = QSρQT+QTρQS,we use (23) to find that CJρK obeys the equation

dCJρKdt

= −iCJ[H, ρ]K− ΓcCJρK, (26)

where Γc = β + kS

(12 + 〈QS〉

)+ kT

(12 + 〈QT〉

), and we

defined 〈Qx〉 = TrρQx/Trρ with x = S,T. Takingthe trace of (23), we find that dTrρ/dt = −κTrρ,where κ = kS〈QS〉+ kT〈QT〉. We finally define the ”gen-uine” S-T decoherence rate as γc = Γc−κ. This describesthe decay of S-T coherence due to all effects other thanthe changing normalization of ρ. For the considered ex-ample it is γc = (kS + kT)/2 + β. It is obvious that if

β Ω, (kS + kT)/2, where Ω is the rate at which S-T coherence is generated by the Hamiltonian, CJρK willquickly (compared to the reaction time) decay to zero,and thus pcoh(ρ) ≈ 0. Hence the traditional theory willprovide a consistent description of the dynamics.

B. Haberkorn’s reaction operator contradicts thephysics of quantum state reconstruction

The form of Haberkorn’s reaction operator (25) canbe simply phrased as follows: singlet products originatefrom singlet radical-pairs, and triplet products originatefrom triplet radical-pairs. This reasoning obliterates thevery concepts of quantum superposition and quantummeasurement and is unphysical on many grounds, onerelated to the well understood quantum state reconstruc-tion. Suppose we are provided with a number of qubitsall prepared in the state |ψ〉 = (|0〉 + |1〉)/

√2, which

however is not disclosed to us. Suppose then that weperform a measurement of each qubit either in the basis

B1=|0〉, |1〉 or in the basis B2= |0〉+|1〉√2, |0〉−|1〉√

2. Mea-

suring in B1, the measurement outcomes will be 0 or 1and we should conclude that the preparation state was|0〉 or |1〉. Measuring in B2, the outcome will alwaysbe 0, and we should conclude that the prepared statewas |ψ〉. Repeating this many times we should con-clude that we are provided with qubits in all three states|ψ〉, |0〉 and |1〉. In other words, based on the dictumof identifying the pre-measurement state with the post-measurement state, we would never be able to use thestatistical-interpretation of quantum measurements andreconstruct the original state preparation.

C. Haberkorn’s master equation coherentlygenerates entanglement starting out with none

We will again consider an RP ensemble initially pre-pared in the state |ψ0〉 = (|S〉 + |T0〉)/

√2, not under-

going any S-T mixing (H = 0) and having kT = 0.Haberkorn’s theory predicts that the RP state remainspure at all times, with the ensemble consisting at time tof (1 + e−kSt)/2 radical-pairs in the state

|ψt〉 =1√

1 + e−kSt

(e−

kSt

2 |S〉+ |T0〉)

(27)

At t = 0 it is |ψ0〉 = |↑↓〉, which is not entangled. How-ever, it is seen from (27) that without any coherent op-eration on the radical-pairs, they spontaneously acquirea non-zero entanglement while coherently evolving to themaximally entangled and non-reactive triplet state |T0〉.This is unphysical. The caveat, as will be explained in de-tail in Section 6.2, is that an irreversible reaction causingpopulation leakage is wrongly understood to coherentlyoperate on the amplitudes of the RP quantum state. Incontrast, in our approach an RP in the state |ψ0〉 is (i)

15

either projected to |S〉, or (ii) projected to |T0〉, or (iii)recombines to a singlet neutral product, or (iv) stays putat |ψ0〉. The probabilities of (i) and (ii) are the same (seeSection 3.8.1), so together they represent a balanced mix-ture of maximally entangled states, the mixture carryingzero entanglement. Event (iii) produces net entangle-ment, but it does so by an incoherent operation, relatedto measurement-induced entanglement, to be addressedin more detail elsewhere.

V. HABERKORN’S MASTER EQUATION ISNOT CONSISTENT WITH THE

QUANTUM-TRAJECTORY PICTURE

In Section 3.5 we discussed the interpretation of (11) interms of single-molecule quantum trajectories involvingthree possible events, (i) singlet projection with probabil-ity dpS, (ii) triplet projection with probability dpT, and(iii) Hamiltonian evolution with probability 1−dpS−dpT.

However, the full quantum dynamics include both S-Tdephasing and recombination. To account for the lat-ter, we have to augment the quantum trajectories, whichare now formed by 5 possible events taking place withindt: (E1) singlet projection with probability dpS, (E2)triplet projection with probability dpT, (E3) singlet re-combination with probability drS, (E4) triplet recombi-nation with probability drT, and (E5) Hamiltonian evo-lution with probability 1− dpS − dpT − drS − drT. Doesthe average of many quantum trajectories reproduce ourmaster equation? Moreover, which are the quantum tra-jectories in Haberkorn’s approach, and do they reproduceHaberkorn’s master equation?

A. Quantum trajectories in Haberkorn’s approach

The concept of quantum trajectories, well known inquantum optics [145, 146], was only recently introduced[18] in spin chemistry. In any case, in [19] we presenteda significant constraint to be met when one attempts aquantum-trajectory analysis of Haberkorn’s master equa-tion.

The intuitive understanding in spin chemistry was thatRPs evolve unitarily under the action of H until the in-stant they recombine into neutral reaction products. Wehave shown in [19] that this physical picture must be ad-vocated if Haberkorn’s theory is to be consistent in thespecial case kS = kT. First of all, what is so special aboutthis case? The rates kS and kT are RP-specific parame-ters entering the master equation, which obviously mustbe valid for all radical-pairs, those for which kS = kT andthose for which kS 6= kT. In fact the latter are abundantin photosynthetic reaction centers, as will be presentedin the following section. Moreover, quantum dynamicsare non-trivial in this case of asymmetric recombinationrates. In contrast, RP quantum dynamics simplify a lotwhen kS = kT ≡ k, where RP population decays expo-

Singlet projection

Triplet projection

Singlet recombination

Triplet recombination

Event Probability of eventin Kominis’ approach

kS+kT2dpS=dt QSψ ψ

kS+kT2dpT=dt QTψ ψ

drS=dt kS QSψ ψ

drT=dt kT QTψ ψ

Hamiltonian evolution 1-dpS-dpT-drS-drT

Probability of eventin Haberkorn’s approach

0

0

drS=dt kS QSψ ψ

drT=dt kT QTψ ψ

1-drS-drT

FIG. 9. Possible events forming quantum trajectories in theapproach of Kominis and Haberkorn.

nentially at a rate k, without the decay affecting the stateof the surviving RPs. Haberkorn’s master equation (8)now becomes:

dρ/dt = −i[H, ρ]− kρ (28)

Defining a new density matrix R by ρ = e−ktR, it isdR/dt = −i[H, R]. Thus, apart from an exponentialdecay of the normalization Trρ, the density matrix ρevolves unitarily. Switching to quantum trajectories, wecan exactly retrieve (28) if we assume that a single RPfaces just three possibilities along its state evolution dur-ing dt: (i) singlet recombination with probability drS,(ii) triplet recombination with probability drT, and (iii)Hamiltonian evolution with probability 1 − drS − drT.Indeed, the single-molecule state will be ρ/Trρ, henceaveraging (i)-(iii) we get:

ρ+ dρ = drS

(ρ− ρ

Trρ

)+ drT

(ρ− ρ

Trρ

)+ (1− drS − drT)

(ρ− i[H, ρ]dt

)(29)

In the above equation we update ρ by removing asingle-molecule density matrix, weighing the result bythe respective probability for singlet (first term) andtriplet (second term) recombination, then weighing in theHamiltonian evolution occuring if none of these two pos-sibilities materialize. Taking into account that drS =kdtTrρQS and drT = kdtTrρQT, and further thatdrS + drT = kdtTrρ due to the completeness relationQS + QT = 1, equation (29) readily leads to (28).

As promised, we have arrived at a significant con-straint limiting our freedom to define Haberkorn’s quan-tum trajectories, namely that in the special case kS = kT,Haberkorn’s master equation is consistent with quantumtrajectories in which a radical-pair just evolves unitarilyuntil it recombines. At this point we have to forfeit ourgoal to conclusively explore what sort of quantum tra-jectories Haberkorn’s proponents might stipulate in thegeneral case kS 6= kT. Starting from the physical modelof the radical-pair, depicted in Fig. 5, and pinning down

16

the spin-selective interaction Hamiltonian V coupling theradical-pair with the vibrational reservoirs, one is unam-biguously led to the singlet and triplet projections of ourapproach. Being unable to imagine what would be afirst-principles derivation of some different sort of quan-tum trajectories, which would not disturb RP state evo-lution in case kS = kT, we make the working assumptionthat for any values of kS and kT the quantum trajecto-ries Haberkorn’s proponents would pick are the same asfor the special case kS = kT. In Fig. 9 we summarizeboth pictures of quantum trajectories to be used in thefollowing consistency check.

B. Chemically induced dynamic nuclearpolarization

As mentioned in the introduction of Section 2, CIDNPis a versatile tool to study photosynthetic reaction cen-ters with significantly enhanced NMR signals resultingfrom the dynamics of spin transport taking place alongwith charge transport. It is beyond the scope of thisreview to address the rich field of CIDNP. An excellentrecent review is [92]. In general, nuclear spin dynamicsin CIDNP are regulated by the radical-pair mechanism,and similarly with EPR [147], from the measured NMRspectra one can obtain rich structural information aboutthe reaction centers, e.g. electronic structure of the pho-tochemically active cofactors. We have recently shown[19] that CIDNP is exquisitely sensitive to the quantumdynamics of radical-pair reaction kinetics, since it inter-estingly turns out that for a large class of bacterial andhigher-plant reaction centers, the recombination ratesof the participating donor-acceptor systems are highlyasymmetric, kT kS [87]. In this regime non-trivialquantum effects in the state evolution driven by the re-action super-operator are most dominant.

Instead of comparing absolute quantitative predictionsof Haberkorn’s theory against our theory, in [19] we usedCIDNP to test the internal consistency of both theories.It is known [145, 146] that the predictions of the masterequation must exactly coincide with the predictions re-sulting from the average of many single-molecule quan-tum trajectories. The results of this test performed in[19] demonstrate that while our approach is largely albeitnot perfectly consistent, Haberkorn’s approach is highlyinconsistent, and hence is ruled out. We used a Hamil-tonian of the same form considered in CINDP works like[89],

H =∆ω

2sAz−

∆ω

2sDz +ωIIz +AsAzIz +BsAzIx, (30)

where the relevant parameters are outlined in [19]. Im-portantly, we use asymmetric rates kT kS perti-nent to photosynthetic reaction centers [92]. We cal-culate the nuclear spin deposited to the reaction prod-ucts, dIz, as a function of time [90]. When propagat-ing the density matrix, the properly normalized density

dark

light

FIG. 10. Schematic of a solid-state photo-CIDNP experiment (the NMR signal was takenfrom http://ssnmr.lic.leidenuniv.nl/files/ssnmr/10-03.Matysik%20CIDNP.pdf). A solid sample of reactioncenters is placed in a high-field NMR magnet. Upon illu-mination, an NMR signal is observed, enhanced by severalorders of magnitude above thermal equilibrium.

matrix of the singlet and triplet neutral products is[?] ρx = QxρQx/TrρQx, while their number producedduring dt is drx = kxdtTrρQx, with x = S,T. Hencethe total (singlet + triplet) ground-state nuclear spin ac-cumulated during dt is

dIz = drSTrIzρS+ drTTrIzρT

= dt(kSTrIzQSρQS+ kTTrIzQTρQT

)(31)

When propagating quantum trajectories, the properlynormalized state of the singlet and triplet reaction prod-uct is |ψx〉 = Qx|ψ〉/

√〈ψ|Qx|ψ〉, and the nuclear spin

deposited to the reaction product is 〈ψx|Iz|ψx〉 for atrajectory terminating with a x-recombination, wherex = S,T. We consider two different initial conditions,|ψ0〉 = |S〉⊗ |⇓〉 and |ψ0〉 = |S〉⊗ |⇑〉, the results of whichwe average in order to simulate the realistic scenario ofstarting with unpolarized nuclear spins. Finally, we in-tegrate the averaged traces in order to find

∫dIz, which

we normalize by Ithermal ≈ 10−5, the Boltzmann equi-librium nuclear spin. The ratio

∫dIz/Ithermal is the so-

called CIDNP enhancement, directly measurable in ex-periments.

The results of the consistency check are shown in Fig.11a and Fig. 11b (reproduced from [19]). The level ofinconsistency of Haberkorn’s approach is evident by vi-sually inspecting Fig. 11b. Even more impressive is theresult for the enhancement

∫dIz/Ithermal, shown in the

insets of Fig. 11(a,b). The traditional theory is incon-sistent both in the sign and the magnitude of the effectat the level of 740%, whereas in our approach (Fig. 11a)the two results are of the same sign and different in mag-nitude by 50%.

As expected, some observables are not as sensitive toreaction kinetics as others. For example, in Fig. 11(c,d)we plot Trρ as a function of time. Again, by qualita-tively looking at Fig. 11d, Haberkorn’s theory is imme-diately ruled out. However, at early times the inconsis-tency is rather innocuous in this case. For example, thereaction time (1/e of the initial population) read off theplot is about 6/kT and 8/kT for the Monte Carlo and the

17

dIz/

I ther

mal

dIz Ithermal

Monte Carlo

Master Equation

(a)+(b)2

0

5

-5

-10

-15

-2410

-3603

dIz Ithermal

Monte Carlo

Master Equation

(a)+(b)2

505

-4250

10

0

-10

-20

(a)

(b)

time (in units of 1/kT)0 4 8 12 16 20

(d)

0.0

0.20.4

0.6

0.8

1.00.0

0.2

0.4

0.60.8

1.0 (c)

trρ

trρ

dIz/

I ther

mal

FIG. 11. Monte Carlo (grey) and master equation (black) cal-culation of (upper plots) the nuclear spin dIz deposited to theneutral reaction products normalized by Ithermal, and (lowerplots) the population of the RP ensemble, Trρ, as a functionof time. (a,c) Kominis’ and (b,d) Haberkorn’s approach. Inthe insets of (a) and (b) we show the integral of the respec-tive traces, reflecting the total nuclear spin of the reactionproducts at the end of the reaction. The Monte Carlo is theaverage of 2× 105 and 2× 103 quantum trajectories for (a,b)and (c,d), respectively, generated as explained in [19]. Muchfewer trajectories are required for simulating Trρ, since ineach trajectory, Trρ = 1 until the time of recombinationand zero afterwards, while the nuclear spin deposited to thereaction products is zero everywhere except at the time ofrecombination.

master equation, respectively, which are not alarminglydifferent.

To summarize, using Haberkorn’s master equation toextract physical parameters of photosynthetic reactioncenters from CIDNP data will lead to significantly skewednumbers, since both the structure of the Hamiltonian

and the relevant couplings will be forced by the incor-rect reaction terms of (8) to attain unphysical values inorder to match the data. The level of the discrepancycan be gauged by comparing our Monte Carlo result for∫dIz/Ithermal (-2410) with Haberkorn’s master equation

result (-4250). If the scientific community studying pho-tosynthetic reaction centers is content with somewhatbetter than order-of-magnitude estimates of the relevantphysical parameters, Haberkorn’s approach is adequate.If precision is of interest, the conventional approach isinadequate.

Finally, our master equation succeeds in matching ourMonte Carlo, which precisely captures the full quantumdynamics at the single molecule picture, at the level of50% for

∫dIz. This constitutes significant progress in

establishing a fundamentally sound understanding of RPquantum dynamics, but this level of consistency is stillnot good enough. The weakness of our master equationis traced to the definitions of pcoh and ρcoh, which will berefined in forthcoming works.

VI. COMPETING THEORETICALAPPROACHES

Soon after our first paper [8] challenging the tradi-tional theory, yet another master equation based onquantum measurement considerations [31], and a micro-scopic derivation of the traditional master equation [23]appeared in the literature. To these was recently addeda study [50] using the formalism of quantum maps.

A. The derivation of Jones & Hore

In [31] the authors use the operator sum approach toevolve the RP density matrix, encapsulating the physicsof the atom being ”watched” by an array of photo-multipliers (Section 3.3), namely the assumption that aradical-pair that could recombine in e.g. the singlet chan-nel, and was not observed to do so, is projected to thetriplet state. This assumption derives from consideringthe RP spin degrees of freedom to be directly coupledto a physically observable environment, like the atom iscoupled to the observable radiation field.

Put differently, considering Heisenberg’s measurementchain depicted in Fig. 12, where a quantum system in-teracts with an apparatus, which interacts with anotherapparatus and so on and forth until registering an actualmeasurement, the approach of Jones & Hore is tanta-mount to cutting the chain right after the first appara-tus. According to our understanding this is physicallyimpossible, because the first apparatus is realized by theunobservable vibrational excitations. These are coupledto the observable phonon bath (second apparatus), so wecan physically register the recombination event only withthe detection of a phonon resulting from the decay of thevibrational excitations to the ground state. The absence

18

QuantumSystem

MeasurementApparatus 1

MeasurementApparatus 2

....

Heisenberg’s cut

Jones-Hore Approach

KominisApproach

FIG. 12. The Jones-Hore approach assumes that RPs are di-rectly ”observed” by their recombination products. In ourapproach this is physically impossible because of the interme-diate vibrational excitations of Fig. 5, which stand betweenthe RP spin degrees of freedom and the neutral ground states.Thus the two approaches cut Heisenberg’s measurement chainat different places.

of such a detection cannot unambiguously condition theradical-pair’s state as in the atom watched by photode-tectors, because of the unobservable possibility of virtualtransitions to the vibrational reservoir and back to theRP.

Thus, in our approach, the presence of the interme-diate ”apparatus” between the radical-pair and the fi-nal product detection ameliorates the influence of the”null recombination-product measurement” on the RP’sstate. In the Jones-Hore approach, this influence is inten-sified due to the absence of the intermediate apparatus.This is why the Jones-Hore theory predicts a higher S-T dephasing rate than our theory. A recent experiment[25] performed pulsed-EPR spectroscopy on a carotenoid-porphyrin-fullerene triad to find the S-T dephasing rateto be lower than the Jones-Hore prediction, ruling outthe theory, since the theoretically expected fundamen-tal relaxation rate always sets a lower bound to mea-sured rates. The S-T dephasing rate measured in [25]is consistent with Haberkorn’s prediction, apart from apostulated extra relaxation mechanism attributed to g-anisotropy. The magnitude of this extra relaxation rateis roughly estimated in [25] to be factor of 2 or 3 (depend-ing on temperature) higher than Haberkorn’s dephasingrate. Even if this attribution turns out to be correct,it is already understood that in the presence of stronghyperfine relaxation, Haberkorn’s theory is expected tobe a good approximation. Hyperfine relaxation is for S-T mixing in radical-pairs what Doppler broadening is foratoms. In order to study natural broadening of an atomictransition, Doppler broadening must be eliminated. Inradical-pairs, in order to study the fundamental S-T de-phasing we have introduced, one needs to experimentwith molecules not suffering hyperfine relaxation withintheir lifetime. The molecule used in [25] is not one ofthose.

In contrast, another experiment [26, 27] reported evi-

dence of a non-trivial state evolution of non-recombiningradical-pairs, the rate of which appears to be consistentwith the Jones-Hore prediction. A global and more de-tailed analysis of these two conflicting experiments willbe undertaken elsewhere.

B. The derivation of Ivanov and co-workers

The microscopic derivation of Ivanov et al. in [23]seamlessly arrives at Habekorn’s master equation. Theproblem is that this derivation describes a system otherthan radical-pairs. This is because the authors considerthe neutral reaction products to be coherently coupledto the radical pairs. However, from the electron-transferpicture of Fig. 5 and Fig. 6, it is clear that the presenceof the continuous manifold of vibrational excitations ofthe neutral products prohibits any coherence between re-actants and products, as long as by ”product” we iden-tify the physically observable neutral ground state. So isthe master equation a matter of semantics, i.e. definingwhat is the reaction product? Not really. Again, it isthe physics of the virtual transitions to the intermedi-ate states and back to the RP that do not allow one toproclaim the reaction is over just counting on a non-zeroamplitude of the intermediate states.

These considerations have been analyzed in detail in[12] with the visualizing help of Feynman diagrams shownin Fig. 13. The coherent reactant-product coupling pos-tulated [23] to describe electron-transfer reactions (Fig.13a) is shown in Fig. 13b. This represents an ”effective”theory neglecting the existence of the intermediate states,much like the early theory of nuclear β-decay neglectedthe existence of electroweak bosons. In reality, the inter-mediate states prevent the build-up of reactant-productquantum coherence, because the amplitudes for such acoupling summed over all intermediate states interfereaway to zero (Fig. 13c). This is because of the energydenominator appearing in perturbation theory, which isan odd function of the energy difference between reac-tant and intermediate states [12]. The first non-zero am-plitude for coherently coupling reactants with productscomes from 4th-order perturbation theory (Fig. 13d) andis irrelevant for reaction kinetics. In Fig. 13e we showa diagram responsible for radical-pair S-T dephasing,which involves a virtual transition to the intermediatestates (1) and back (1’). Finally, the product is formedby two consecutive real transitions, (1) from the reactantto the intermediate state, and (2) from the intermediatestate to the product (Fig. 13f). To summarize, omittingintermediate states in irreversible reactions can lead to

conceptual rigmaroles like states |ψ〉 =(| 〉+ | 〉

)/√

2,

i.e. coherences between an exploded and un-explodedbomb.

We can now make a point regarding our approach. Di-agrams (e) and (f) of Fig. 13 have different final states,

hence do not interfere, and this is why to calculate dρdt

19

reaction coordinate

Ureactant D+A-

neutral product DA

R P*

P

1

2

phonon

R

P

i

R

P*

P

Σi

i =0i

R

P*

P

P* P

phonon

phonon

ii

R

P*

P phonon

i

R

P*i

i

R

1’

1

1’

1 2

(b) (c)

(d)

(a)

(e) (f)

...

FIG. 13. (a) Energy level diagram of an electron-transferreaction. (b) Effective reactant-product coupling neglectingintermediate states |P∗

i 〉. (c) Within 2nd-order perturbationtheory the coherent reactant-product coupling interferes tozero. (d) Leading (4th) order coherent reactant-product cou-pling. (e) Reactant dephasing, resulting from a virtual tran-sition to the intermediate states and back. (f) Product for-mation, resulting from a real transition from the reactant tothe intermediate state and the decay of the latter to the prod-uct state. Essentially, (f) consists of two separate diagramsmaterializing consecutively, i.e. |P∗

i 〉 is here a real and not avirtual state like in (c-e).

in our master equation (23) we add dρdt

∣∣decoh

, represented

by Fig. 13e, with dρdt

∣∣recomb

, represented by Fig. 13f.

C. The derivations of Briegel and co-workers

Briegel and co-workers considered the description ofRP quantum dynamics from the perspective of quantummaps in [43] and more comprehensively in [50], where theauthors attempt a generalized formulation of the prob-lem, at the cost though, of not offering a clear-cut an-swer to the question an experimentalist, for example onedetecting solid-state NMR signals from photosyntheticreaction centers, might rightfully ask: ”is Haberkorn’sapproach adequate to account for the data and if not,which is the master equation one should use?”. The an-swers given in [50] are ”maybe” and ”it depends”. Wereply ”no” and Eq. (23).

In [50] the authors invent an abstract environment,without specifying where this environment comes fromand what are the microscopic physics coupling the RP’sspin degrees of freedom to this environment. The authorsthen go on to provide several master equations given interms of decay and dephasing amplitudes, which how-ever, the ”user” is supposed to specify. For example,given the typical scenario of a solid-state CIDNP exper-iment, i.e. an immobilized RP characterized by the tworates kS and kT, and in the absence of any technical deco-herence, it is not clear in [50] what is the master equation

one ought to use.We stress the physical scenario of immobilized radical-

pairs, because the authors in [50] elevate molecular re-encounters to a fundamental physical effect, somethingthat we think is more perplexing than illuminating. Thisis because the recombination rates kS and kT are inprinciple known functions of the distance between thetwo radicals. So if one has solved the immobilized RPcase, one can readily generalize to a liquid state scenario,where the two radicals diffuse and re-encounter, andwhere kS and kT would become functions of time throughtheir dependence on inter-radical separation. To summa-rize, the considerations of [50] should become more pre-cise in order that (i) it is discernible if and where theydiffer from other approaches, (ii) they are amenable tocriticism, and (iii) they are useful to experimentalists.

VII. THE PHYSICS OF THE NONLINEARMASTER EQUATION

The nonlinear nature of our master equation (23) isa subtle point requiring a thorougher examination thanwe can afford here. Although we sympathize with theunease of many quantum physicists with nonlinear mas-ter equations, we understand the nonlinearity of (23) tobe a necessary evil forced upon us by the physics of theproblem. We will elucidate this point by a few thoughtexperiments. In all of them we consider no S-T mixing(H = 0) and an unreactive triplet state (kT = 0).

Suppose that we prepare two RP ensembles, one inthe maximally coherent state |ψ1〉 = (|S〉+ |T0〉)/

√2, or

equivalently ρ1 = |ψ1〉〈ψ1|, and the other in the max-imally incoherent mixture of singlet and triplet states,ρ2 = 1

2 |S〉 〈S|+12 |T0〉 〈T0|. In Section 3.9.1 we presented

the counterintuitive result that in the first ensemble weend up with 25% of the original RPs locked in the non-reactive triplet state, while in the second we will end upwith 50% of the original RPs locked in the non-reactivetriplet state. Both ensembles have the same ρSS + ρTT,but different ρST + ρTS. Hence the reaction depends onρST + ρTS. That this dependence is nonlinear is seen byconsidering two different ensembles that will occupy therest of the discussion.

Consider the ensemble E1 prepared in the state |ψ1〉 =

(|S〉 + |T0〉)/√

2 and E2 in the state |ψ2〉 = (|S〉 −|T0〉)/

√2. The two ensembles just differ in the sign of

ρST + ρTS. The end result is the same for both, that is,at t 1/kS we are left with 25% of the original RPslocked in the non-reactive triplet state. It is thus clearthat the reaction depends on the ”absolute value” of S-Tcoherence, which is a non-linear operation.

Now consider mixing the initial contents of E1 and E2

and offering the box E1+2 to a good experimenter whocan quickly, before any recombination takes place, es-tablish that the density matrix of the system is ρ1+2 =(|S〉〈S| + |T0〉〈T0|

)/2. Indeed, in spin chemistry exper-

iments individual RPs are not accessible. At the high-

20

est laboratory magnetic fields, the typical (e.g. for anEPR transition) frequency is on the order of 1011 Hz,the corresponding wavelength and resolvable volume be-ing 1 mm and 1 mm3, respectively. This volume con-tains a macroscopic number of RPs. Hence any spin-state measurement on the box E1+2 (assumingly fast andnon-destructive) will be a global measurement on all RPsinside the box, and it will be consistent with the maxi-mally incoherent density matrix ρ1+2. What will be thefinal population of non-reactive triplet RPs that will re-main in the box? If we know the specific preparation, weconclude it will be 25%, like it was for E1 and E2 indi-vidually. If we don’t, like the experimenter who is offeredthe box E1+2, we conclude it will be 50%, as expectedfrom ρ1+2.

Contradiction? Not quite. The issue at hand has todo with proper versus improper mixtures [148–150]. Thecase in which the experimenter is unaware of the spe-cific state preparation is referred to as a proper mixture.There is maximum S-T coherence at the single moleculelevel both in E1 and in E2, but due to the mixing thewhole box E1+2 appears maximally incoherent. In con-trast, if there is a genuine decoherence process, like S-Tdephasing, leading to the same mixed state, one talks ofan improper mixture. In this case there is no coherenceat the single molecule level, and the box E1+2 indeedcontains a mixture of singlet and triplet RPs.

There is an illuminating analogy with Young’s doubleslit experiment [11]. Suppose that a light source emitsperfectly coherent light and we register photons on theobservation screen for a time τ . An interference pat-tern will be formed. Suppose then that we keep register-ing photons for a consecutive time interval τ , this timeputting a π phase shifter in front one of the two slits. Ifwe were to observe the two runs individually, we wouldsee two perfect interference patterns shifted with respectto each other. On the other hand, if we wait to regis-ter all photons from the two consecutive runs, and thenlook at the observation screen, there will be no interfer-ence pattern. If one is presented with the final picture,one will conclude that the light source was incoherent.This is the analogy of the proper mixture discussed be-fore. In contrast, if one performs this experiment, againwith perfectly coherent light, and installs a measurementapparatus after the slits acquiring complete which-pathinformation, the interference pattern will genuinely dis-appear. This is the analog of the improper mixture dis-

cussed above.Going back to the bewildered experimenter who is pro-

vided the mixture ρ1+2, while all his global measurementswill lead him to expect 50% unreacted triplet RPs, he willbe surprised to finally measure 25%. To our understand-ing, so far proper and improper mixtures were more of anepistemological curiosity rather than of practical utility.That is, we are unaware of any quantum science exper-iments able to differentiate proper from improper mix-tures. We will close this review with a conjecture, theproof of which we leave as a problem to be addressed inthe more abstract context of the foundations of quantumphysics. Leaky quantum systems, in which the leakagedepends non-linearly on the quantum-state, can differen-tiate between proper and improper mixtures.

VIII. OUTLOOK

We hope to have convinced the reader that radical-pairreactions indeed constitute an ideal paradigm for quan-tum biology, since they require for their understandingthe whole conceptual toolset of quantum information sci-ence, even touching upon the foundations of quantumphysics. The complex quantum dynamics of radical-pairreactions emerge from the interplay of coherent spin dy-namics and incoherent electron transfer reactions, ren-dering this system both open and leaky. To our knowl-edge, this is rather unusual in quantum science, and con-ceptually quite challenging, explaining in part the cur-rent lack of consensus in the scientific community on thequantum foundations of RPM.

It is true that the last several years have witnessed sev-eral exciting discoveries making the case of quantum biol-ogy. We feel that the synthesis of the two seemingly dis-joint fields that dominated 20th century science, ”quan-tum” and ”bio”, will lead to breakthroughs beyond ourcurrent ability to forecast. Hence we will be more thancontent if this review inspires many other researchers tojoin this tremendously exciting scientific endeavor.

IX. ACKNOWLEDGMENTS

We acknowledge support from the European Union’sSeventh Framework Program FP7-REGPOT-2012-2013-1 under grant agreement 316165.

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