Single-molecule spectroscopy of radical pairs, a theoretical treatment and experimental considerations

ABSTRACT In this work, we propose to extend the scope of single-molecule spectroscopy to spin chemistry investigations of single radical pairs. A consistent theory of single-molecule spectroscopy on single radical pairs is proposed, which allows one to show that bunching phenomena are affected by singlet–triplet interconversion in radical pairs, which is, in turn, affected by local and external magnetic fields. A detailed study on the feasibility of such experiments with single flavin adenine dinucleotide, FAD, molecules is presented. We conclude that the observation of magnetic field effects on single FAD molecules is feasible but experimentally challenging for measurements on integrated fluorescence intensity and fluorescence event statistics. GRAPHICAL ABSTRACT


I. Introduction
Single-molecule spectroscopy (SMS) [1] was established in 1987 by Moerner and Carter who observed 'statistical fine structure' [2] in the absorption spectra of small molecular ensembles. Later on, SMS has evolved into a powerful tool [3] in the fields of physical chemistry, biophysics and quantum optics. SMS experiments are now feasible by probing optical absorption spectra as well as by fluorescence detection. In suitable molecular systems, also EPR (Electron Paramagnetic Resonance) and NMR (Nuclear Magnetic Resonance) experiments on single molecules are possible with indirect (optical) detection of magnetic resonance of molecular triplet states [4].
In this work, we propose to extend the SMS methodology to spin chemistry, specifically, we propose to detect CONTACT  single Radical Pairs (RPs) and to probe their properties by SMS techniques. Potentially, such an approach will combine the advantages of single-molecule EPR (performed on the triplet states of photo-excited molecules) and spin chemistry methods thus giving information on rate constants and RP structure. Determining rate constants of radiationless transitions in molecules is generally a complicated problem [5], which can be solved by combining SMS studies with measurements on molecular ensembles. Single-molecule experiments potentially allow one to access the spin dynamics of an RP in a specific nuclear spin state. Hence, SMS enables additional information to be obtained that can be lost after ensemble averaging. Furthermore, SMS experiments can pave the way to EPR detection of single RPs formed by intra-molecular electron transfer in photoexcited molecules. In this case, the molecule can absorb an additional photon and fluoresce only after the RP recombines back to the ground state. Since such a process is spin-dependent (can only occur from the singlet state), one can affect the singlet-triplet transitions, and thereby the statistics of fluorescent photons by applying external magnetic fields, either oscillating resonant fields or static fields. In this way, one can potentially access the EPR of single RPs by optical detection. Here we propose a theoretical description of this phenomenon and discuss possible experimental verification of our ideas. The paper thus consists of a theoretical part based on a full spin-dynamic model and an experimental feasibility study based on a kinetic RP model using measured experimental parameters. This allows discussion of the possible implementation of SMS experiments with single RPs.
Recently Liu et al. [6] discussed SMS on single RPs focusing on the Rabi oscillations that emerge due to light excitation. In this work, we generalise the theoretical treatment of SMS on single RPs. Specifically, we argue that the RP spin dynamics, which are sensitive to magnetic fields, reveal themselves in bunching phenomena. We discuss not only the second-order correlation function and its behaviour on different timescales but also Mandel's Q-parameter (see below) and present the results of Monte-Carlo modelling showing that magnetosensitive reaction stages are important for bunching phenomena.

A. Statistics of single-photon counts
In an SMS experiment the molecular system is continuously irradiated by light and events of single-molecule fluorescence are registered. There are several ways to assess the statistics of fluorescence events described in literature [7]. One of the possibilities is to analyze the second-order correlation function g (2) (t , t + t), which is defined as follows: Here P 1 (t )dt and P 1 (t + t)dt are the probabilities of single-photon counts within the time intervals is the joint probability of two photon counts within the same time intervals. Here we consider only stationary processes so that the statistic does not depend on t : In the case of single-molecule emission at t → 0 we always have g (2) (t) → 0 meaning that the probability to emit two photons at a small delay goes to zero (since the molecule needs to be excited again to emit another photon): this is the essence of the anti-bunching phenomenon. At long times, the emission of two photons in uncorrelated, hence, P 2 (t , t + t) ≈ P 1 (t )P 1 (t + t) and g (2) → 1. At intermediate times it is possible that g (2) ≥ 1, i.e. the light source is 'bunched'.
Another possibility of characterising single-molecule emission is to measure the number of photon counts N(t) in the time interval (0, t). Knowing N(t) and N 2 (t) one can also introduce Mandel's Q-parameter [8,9]: For a stationary process, one can find a relation between g (2) and Q [7]. The Q value can be negative (sub-Poissonian statistics of photon counts), zero (Poissonian statistics) or positive (super-Poissonian statistics). Suband super-Poissonian statistics indicate that the light source is either bunched or anti-bunched. Using Q one can also detect anti-bunching behaviour of the photon emission statistics of a single molecule [7]. The typical time-dependence of the functions introduced above is presented in Figure 1 for a two-level system, in which transitions between the ground state and excited state are driven by light and fluorescence from the excited state is measured. One can see that the g (2) function starts from zero at short times (anti-bunching takes place), then increases having an oscillatory component, which corresponds to the Rabi beats driven by light, and eventually goes to 1 at long times. The Rabi beats can be seen only when the light intensity (giving rise to the transition rate L ) is high enough so that L k f . When the light intensity is low, the Rabi beats are strongly suppressed due to decoherence (as the fluorescence process also results in fast decay of the coherence). Hereafter, we always assume moderate or low light intensity, where the Rabi oscillations do not manifest themselves. The N function (which is essentially the integration of the Rabi oscillations) exhibits an increase with time, which is almost perfectly linear at long times but is non-linear at short times. The Q-parameter exhibits sub-Poissonian behaviour at short times followed by super-Poissonian behaviour at long times.

B. Reaction scheme
In this work, we do not limit ourselves to a simple twolevel system and treat the scheme depicted in Figure 2. A single molecule in the ground state |a is excited to state |b by a continuous light source; the induced transitions |a ↔ |b are dynamic and occur at a frequency L . However, unless one performs the experiment under special conditions (at cryogenic temperatures and special matrices or in the solid state), light excitation can be considered as an incoherent process [10]. Hence, except for the example shown in Figure 1, we chose the parameters ( L and the decoherence time) such that the Rabi oscillations are completely suppressed. We also assume that there is a spontaneous transition |b → |a , which gives rise to photon emission (fluorescence). The fluorescence rate is denoted as k f . In addition, we assume that from the excited state an RP can be formed in the singlet state, |S , at a rate k S . The singlet-state RP can recombine to the ground state, |a , at a rate k R . In addition, we take into account interconversion, i.e. singlet-triplet spin state mixing in the RP, which is of a coherent nature. For the sake of simplicity, we introduce only the mixing between the |S state and the central triplet state, |T 0 , although Here the molecule is excited by light from the ground state |a to the excited state |b ; the rate of the induced transitions, |a ↔ |b , is denoted as L . We also take into account spontaneous transitions |b → |a occurring at a rate k f , which give rise to observable fluorescence. From the |b state an RP can also be formed in the singlet state at a rate k S ; the RP can recombine to the ground state at a rate k R also from the singlet state. In the RP, we take into account coherent singlet-triplet mixing; in the simplest case this is S − T 0 mixing occurring at a frequency δω.
S − T ± can also be taken into account using the same theoretical formalism. This limitation is not of principal importance: where necessary, our treatment can be generalised to take into account transitions between any states of the RP. The approximation of S − T 0 mixing corresponds to high magnetic fields, where interconversion is due to the difference, δω, in the Larmor precession frequencies of the two radical centres. In turn, δω = 0 is due to hyperfine couplings with magnetic nuclei or due to the difference, g, in the g-factors of the radical centres [11]. Here we assume that RP is a 'dark state' of the system, (i) which gives no fluorescence and (ii) in which the system is trapped for a relatively long time. The latter assumption corresponds to the situation L , k f k S , k R .
In general, spin state mixing in the RP is expected to be sensitive to external magnetic fields, static or oscillating [11,12]. The reason is that at high static fields B 0 only S − T 0 mixing is possible, while at low fields S − T ± transitions are operative as well due to non-secular hyperfine couplings. Furthermore, the S − T 0 mixing frequency also depends on the field, namely, δω = g · γ e · B 0 (here γ e is the electronic gyromagnetic ratio). When oscillating fields are applied, they can drive EPR transitions in the radical pair and also affect singlet-triplet mixing. In this way, the EPR spectra of RPs can be detected: this idea of indirect EPR detection has given rise to the family of RYDMR (Reaction Yield Detected Magnetic Resonance) methods [13].

C. Calculation method
To evaluate the quantities of interest, i.e. g (2) , N and Q, we proceed as follows. First, we write down the equation for the density matrix of the molecular system: HereĤ is the Hamiltonian of the molecule, while the super-matrixR takes into account transitions between the states |a , |b and |S . The density matrixρ is introduced in the basis of six quantum states: Hereafter, we split the basis into two groups of states (two states |a , |b and four states of the RP). In this basis of states the Hamiltonian is block-diagonal: HereĤ 1 is the 2 × 2 Hamiltonian of the states |a and |b , which takes into account the induced transitions, whilê H 2 is the 4 × 4 Hamiltonian of the RP. The blocks of the Hamiltonian are as follows: δω 2Ŝ 2z (7) HereŜ 1,2 are the electron spin operators of the two radical centres. In theĤ 2 Hamiltonian only the following two elements are non-zero: We assume fast decoherence between the states |a , |b and the RP states; hence the density matrix is block-diagonal: Finally, transitions |b →|a (fluorescence), RP formation and recombination are taken into account byR, which has the following elements: In these expressions we assume that fluorescence, RP formation and RP decay give rise to decoherence. The decoherence rate is taken to be equal to half of the rate at  (2) (t) (solid red line) and Q(t) (magenta dashed-dotted line). The reaction scheme is given by Figure 2; here k f = 0.04 ns −1 ; which the state population (diagonal element of the density matrix) decays. This assumption is consistent with previous considerations of decoherence associated with chemical processes [14]. A similar model was used Liu et al. [6] to simulate the signal from a single NV − centre sensor in a diamond crystal. Using a Hamiltonian similar to that given by Equation (4), the authors discussed spin evolution on the microsecond timescale corresponding to the timescale where Rabi oscillations show up. Here we are interested also in the anti-bunching and bunching effects occurring on different timescales ( Figure 3) and in the Magnetic Field Effect (MFE) on the correlation functions g (2) (t) and Q(t) and their behaviour on different timescales.
To solve Equation (4) forρ, we write down the density matrix in Liouville space (where the density matrix becomes a column-vector and super-operators become square matrices) and solve it for the pertinent initial condition, ρ 0 = ρ(t = 0), in the usual way To evaluate the quantities of interest, we introduce ρ 0 assuming that the system goes back to the ground state |a after the previous emission event. Hence, ρ 0 has only one non-zero matrix element: Using this initial condition, we can define g (2) (t) as the probability to find the system in state |b at time t divided by ρ bb at t → ∞, hence [7]: Division by ρ bb (t) provides correct normalisation so that g (2) (t) → 1 at t → ∞. To calculate the Q parameter, we proceed as follows [7,9]: Here double integration appears comes from calculation of N 2 .
In addition to the outlined quantum-mechanical model, we used a Monte-Carlo approach to model the statistics of photons emitted by the single-molecule. In this approach, we assume the initial condition given by Equation (11) and evaluate g (2) (t) using Equation (12). To evaluate the time dependence of ρ bb (t) we proceed as follows. First, we set a small time step, t (much smaller than all other characteristic times related to the problem under consideration). When at the time instant t the molecule is in the state |i , at time (t + dt) we evaluate the probabilities that the molecule remains in the state |i or goes to another state |j . If the transition |i →|j is due to the transitions given by the rates k f , k S , k R (these transitions are accompanied by a complete loss of quantum correlation) the corresponding transition probability is equal to k i→j · t · ρ ii (t). If we consider dynamic transitions, e.g. |a ↔ |b or singlet-triplet transitions, the corresponding block is multiplied by the corresponding evolution operator to evaluate ρ jj (t + t). This sequence of events is repeated many times with small t; each time when the radiative transition |b → |a occurs an emitted photon is counted. From the statistics of the instants of time when photons are emitted we can calculate g (2) as well as Q in the standard way. One should note that this method is not completely equivalent to the dynamic model described above because Monte-Carlo methods lead to additional decoherence, i.e. to faster decay of some of the off-diagonal elements ofρ, since in Monte-Carlo modelling essentially projective 'quantum measurements' [15] are performed.
As we demonstrate below, concepts presented in this subsection can be used to describe SMS experiments on realistic systems and to predict the size of MFEs probed in SMS studies of RPs.

A. Statistics of single-molecule fluorescence
Calculation results for the functions of interest, g (2) and Q, are shown in Figure 3. One can see that short-time behaviour corresponds to anti-bunching as both g (2) and Q are equal to zero. After an increase on the time-scale of 1/ L both functions exhibit an oscillatory behaviour coming from S − T 0 mixing in the off-state (or dark state), i.e. in the RP state. One can see that due to the spin dynamics in the off-state at long times g (2) can be greater than 1, which corresponds to bunching of the light source.
In order to elucidate the effect of spin state mixing on the photon statistics, we calculated the g (2) and Q functions at different δω values. The calculation result is shown in Figure 4.
One can clearly see that short-time behaviour of g (2) is not affected by singlet-triplet spin mixing, whereas long-time behaviour strongly depends on δω Specifically, bunching phenomena, which correspond to the fact that g (2) > 1 at long times, are affected by the singlet-triplet mixing being sensitive to the δω value.
In addition to calculations of the g (2) and Q functions, which characterise the statistics of photon counts, we performed Monte-Carlo modelling of photon counts and switching of the system between the on-states and offstates. The results are shown in Figure 5 for two different values of δω. In this plot we show the time dependence of the total populations of the |a and |b states, i.e. onstates, in which the system can emit photons, and indicate the instants of time when photons are emitted by asterisks. When the system goes to the RP states, i.e. off-states, it cannot emit the photons until it goes back to the onstates. The population of the on-states is equal to the sum {ρ aa + ρ bb } calculated at each instant of time; this population switches between the two values, 0 and 1. One can see that both the distribution of the off-times and the statistics of photon counts strongly depend on δω.
In particular, when δω is very small (singlet-triplet state mixing is inefficient) on average the system spends less time in the off-states. This means that photon bunches occur more frequently and the delay between the bunches is shorter. When δω is large, the system can be trapped in the RP state for a longer period of time because singlet-triplet mixing brings the molecule to non-reactive triplet states. These considerations coming from Monte-Carlo modelling are consistent with the results of the dynamic model used here.
Hence from the statistics of photon counts one can see that 'blinking' of the single-molecule, i.e. the length of the off-periods indeed depends on the singlet-triplet mixing efficiency. So, the Monte-Carlo analysis supports the calculation results for the g (2) function. An additional option to study single RPs, which is not considered in detail here, is given by resonant EPR excitation of the transitions between the RP state as it is done in RYDMR. Such transitions affect singlet-triplet mixing as well; hence they can alter the statistics of photon counts paving the way to EPR on single RPs.

B. Experimental considerations
While magnetic fields on photochemical reactions have been widely observed, the additional restrictions required to observed effects on single molecules largely limit the choice of systems available. Here we discuss some of the key requirements and identify a possible candidate RP. We then consider more quantitatively the feasibility of observing magnetic field effects on the candidate RP and the key characteristics for designing an improved candidate.
One of the primary restrictions for observing single molecule magnetic field effects (SM-MFEs) is that the reaction system undergoes completely cyclic photochemistry. It is necessary to continuously observe the reaction cycling of a single RP generating photochemical reaction. The vast majority of photochemical reactions that exhibit magnetic field sensitivity are non-cyclic, yielding photochemical products that differ from the starting materials, or include minor side reactions that equivalently lead to photobleaching. Some solid-state materials can perform well over many reaction cycles, but experimental observation of single molecules in these materials can be difficult as the charge carriers are mobile and cannot be constrained to a single location. A logical choice is a molecular system that efficiently generates a diradical, typically through an intramolecular electron transfer reaction and to anchor the molecule so that it cannot move. Indeed, some excellent candidate molecules have been synthesised, including the C-P-F triad (carotenoid-porphyrin-fullerene) used to first observe anisotropic magnetic field effects [16]. This and similar molecules, might prove good candidates for attempting to study SM-MFEs. Another possibility is to look to biology, although to date, reliable observations of the magnetic field dependence of photobiological processes are extremely few [17][18][19]. Of particular interest are the cryptochromes, which have been proposed as the potential magnetosensor underpinning animal magnetoreception. For a recent review of animal magnetoreception and the role of radical pairs, see [20]. Given that avian magnetoreception is believed to take place in low light conditions and many questions remain as to how RPs can reliably detect both the magnitude and direction of the extremely weak (approximately 30-50 μT) geomagnetic field, one can hypothesise that individual RPs in the retina may show much more pronounced anisotropic low field responses in specific hyperfine configurations. If a bird is able to discern single molecules distributed across the retina, it can ignore the RPs that show little magnetic field response and accumulate the signal from only those retinal spots that are modulated most, as it moves its head in the field. This idea means that SM-MFEs may be important to the mechanism of animal magnetoreception which provides additional motivation for their experimental realisation. To date, magnetic field responses have been published in purified cryptochrome proteins in vitro from a plant [18] (Arabadopsis Thaliana) and an insect [17] (Drosophila Melanogaster). However, in both cases, the rate of return of the dark state (in this case, the RP state) to the fluorescing ground state is extremely slow, which would rule out the observation of single photon blinking on a reasonable time scale. As a compromise, we consider in more detail the molecule flavin adenine dinucleotide (FAD). FAD is the blue light absorbing cofactor and electron acceptor in cryptochromes and is known to show substantial MFEs in isolation in acidic solution [5] (ideally around pH 2.3). Woodward's laboratory has extensively studied FAD photochemistry using microspectroscopy and had demonstrated that a single microscopic sample can be continuously photoexcited for a day with no observable photodegradation [21]. Furthermore, FAD is an excellent candidate for fluorescence microscopy due to its absorption and emission in the visible region of the spectrum, preferred for microscope optics [22]. Excitation is simple with CW 450 nm blue laser diodes, which are extremely stable, and fluorescence is most intense in the green region of the spectrum which is an excellent match for many high performance EMCCD and sCMOS cameras and also single photon detectors. The main drawbacks associated with using FAD are (i) that its absorption coefficient is not as large as many of the molecules used in existing single molecule studies, (ii) it's fluorescence quantum yield is low at only 0.13 and its quantum yield of non-radiative return to the ground state is high. Here we consider the experimental conditions required to observe MFEs on single FAD RPs and highlight the essential characteristics of an improved candidate.
First, we calculate the Signal to Noise Ratio (SNR) associated with observing the fluorescence from single FAD molecules, in order to investigate whether single molecule level detection of FAD is feasible. Second, to investigate whether it may be possible to observe MFEs on single FAD molecules, we calculate the extent, to which the blinking of FAD changes in an applied magnetic field, using the reaction scheme for FAD at low pH proposed by Murakami et al. [5]. The results of these calculations indicate that it is theoretically possible to detect FAD single molecules undergoing RP generation and MFEs thereon, using commercial optics and detectors. However, in order to realise these measurements, highly precise experimental techniques are required.

C. Signal-to-noise ratio of single FAD fluorescence
There is a published work that claims to demonstrate imaging of single molecules of FAD [23]. While this is very promising for our experimental approach, the work considers measurements at high pH in which any MFEs will be very small, and in addition, some question marks remain over the measurement, as the authors claim that the observed FAD molecules are not bound to the microscope slide surface and are undergoing free diffusion. For free diffusion of FAD in aqueous solution at room temperature, we estimate that the molecules will travel an average distance of a few microns during the 50 ms sampling period used in this work and thus it is hard to imagine how single FAD molecules can be imaged under such conditions. We calculate the SNR based on the method described in references [24]. First, the number of photons N detected from single molecule is as follows: where η system is the collection efficiency of the experimental system, A is the rate of absorption, QY is the intrinsic quantum yield of the material, and τ int is the integration time. Also, the rate of absorption A is written as A = σ I/ ω (σ : absorption cross-section, I: excitation intensity, ω: angular frequency of excitation light). Next, a general expression for the SNR of a CCD or CMOS camera is SNR = N N + n back + n dark + n 2 read (15) where n back is background noise, n dark is dark current noise, and n read is readout noise. According to this expression, the shot noise √ N exists in all cases. Here, we consider the upper limit SNR MAX to simplify discussion, and apply it to the case of FAD: The normal method to calculate the collection efficiency of a fluorescence microscope yields (see Supporting Materials for explanation): η system ≈ 0.27 (17) for the custom built Total Internal Reflection Fluorescence (TIRF) microscope in the Woodward laboratory.
In addition, the absorption cross-section and intrinsic quantum yield of FAD at the excitation wavelength 450 nm are 4.321 × 10 −17 cm 2 and 0.13 (at pH 2.3) respectively [25]. Using these values and realistic imaging condition values (I = 100 W/cm 2 , τ int = 100 ms), we can obtain the upper limit of signal to noise ratio of FAD following as: In addition, calculating with the noise values of our commercial sCMOS camera (ORCA Flash 4.0 V3, Hamamatsu), we can obtain a more realistic upper value of: A SNR of approximately 2 or more is sufficient for the detection of single molecules. So, if the detection of single FAD molecules is performed in a matrix with low scattering, such detection is theoretically possible. However, it is important to emphasise the need to immobilise the FAD so that its fluorescence can be integrated for 100 ms or preferably longer. This is perhaps the most critical challenge to overcome experimentally. In support of this insight, there is one further existing publication, in which the fluorescence of single molecules of FAD has already been detected as a reporter for immobilised enzymes [26], but for long signal integration times.

D. Estimation of RP blinking rate in FAD
Having established that single-molecules of FAD can potentially be imaged in a TIRF microscope, we consider whether MFEs on single-molecule blinking are likely to be measurable for realistic FAD photochemistry. Using the FAD photochemical reaction scheme at low pH ( Figure 6) proposed by Murakami and et al. [5], we determine the change of the average blinking rate of FAD in the presence and absence of an applied magnetic field.
In the Supporting Materials we present a calculation of the steady-state populations of the S 0 , S 1 , triplet and four RP states using the rates introduced in Figure 6. In this calculation, for the sake of simplicity, we limit ourselves to a rate description of the processes, neglecting all coherent effects.
Knowing the state populations, we can simplify the description by introducing the populations of the fluorescent state and off-state. Thus, in single molecule experiments, by observing fluorescence and non-fluorescence, we can consider a two-state system. To analyze FAD photochemistry from this system we define a fluorescencing state (on-state) and a non-fluorescencing state (off-state) as follows: Here P ON and P OFF represent the populations of the on-state and off-state respectively. Reduction of the real multi-level system to this two-state system is sketched in Figure 7, as well as the statistics of photon counts.  The rate equation is then written as follows: Here, by using kinetic equations presented in Supporting Materials, we can re-write the equations as follows: We can then express the on-time τ ON and off-time τ OFF constants with the reaction rates of the proposed FAD photochemistry scheme by comparing the coefficients, which are calculated in the Supporting Materials. Here, we discuss only the steady-state in the proposed FAD photochemistry scheme. Using the treatment given in the Supporting Materials, we can express the on-time τ ON and off-time τ OFF under steady-state conditions (steady-state values are denoted by an overbar), by using the reaction rates of the proposed FAD scheme. The τ OFF time can be introduced as follows: The explicit expression for τ OFF contains a large number of terms and is reproduced in the supporting information. Calculated using the reaction rate of FAD determined by Murakami et al. [5], τ ON is independent of the magnetic field and τ OFF is magnetic field dependent.
Numerical values of τ OFF at zero field and in the presence of a 0.2 T field are determined as: Here, we adopted the numerical values of reaction rates from [5]. Also, the numerical values of τ ON are determined from Supporting Materials as follows: Next, to investigate whether an MFE is detectable by observing the blinking of FAD, we calculate the number of photons detected during the average on-time, n. Using the above results, we can express the number of photons detected during the average on-time, n, of FAD as follows: where φ ISC is the quantum yield of intersystem crossing. Since k ex is much smaller than k ISC , we ignore the first term of the above equation. Using the values of φ F and φ ISC from the Supporting Materials, we obtain n = 0.08. This indicates that the on-state of FAD cannot be distinguished from the off-state by fluorescence ( Figure 8). We now consider two different possible approaches to resolving MFEs in this system experimentally despite the low photon counts. The first is based on measuring the integrated fluorescence intensity and the second is based on the fluorescent photon statistics as realised through the Mandel Q-parameter discussed in the first part of this paper.

E. MFEs manifest on total integrated fluorescence for single FAD molecules
First, the number of detected photons emitted by FAD under steady-state conditions is expressed as follows: where τ ON τ int /(τ ON + τ OFF ) is an expression taking into account the dead time of the excitation light. Then, we define the MFE of the number of detected photons emitted by FAD under steady conditions as follows: Here, to demonstrate the feasibility of the experiment, we estimate the observable of the number of detected photons N obs int as follows: where δN int is the error of the detected photons, and N sample is the number of samples. And, we estimate the observable of the MFE defined in terms of the number of detected photons as follows: (31) The number of detected photons emitted during the integration time by steady-state FAD and the estimated observable of MFE thereon are shown in Figures 9 and  10. Although the result in the Figures shows that a higher MFE can be obtained by using a higher excitation intensity, there is an upper limit to the number of photons emitted, because the fluorescence emission rate saturates the number of photons emitted. As the saturation intensity giving the upper limit, I sat , can be estimated by I sat ≈ ω/σ τ f , the saturation intensity of FAD is estimated as I sat ≈ 36 kW/cm 2 [24]. Hence, the MFE from a single FAD RP is about 12% at this intensity ( Figure 10). Also, as shown in Figures 9 and 10, to measure such observables requires 10-100 samples (1-10 s in case of an integration time of 100 ms) to achieve appropriate levels of error. This result shows it theoretically possible to observe MFEs on single FAD molecules by integrating blinking FAD fluorescence, although some background issues might arise due to additional scattering light as a result of the high laser excitation intensity. In addition, some integrating time issues might arise due to photobleaching as a result of the high laser excitation intensity.

F. MFEs manifest in Mandel's Q-parameter for single FAD molecules
Finally, we show the feasibility of observing Mandel's Q-parameter for single RPs by using realistic FAD photochemistry. To estimate Mandel's Q-parameter for single FAD molecules, we calculated the mean of the waiting time τ and the mean of the squares of the waiting time τ 2 . The waiting time, τ , is the time to the first emission event time from the instant when the molecule absorbs a photon. The next emission event can be regarded as independent of the previous emission event. Therefore the waiting time is the time between adjacent photon emission events [7,27]. For long integration times relative to the waiting time, the number of photon counts N(τ int ) , which we defined in the first part of this paper, can be approximated as follows [28];  where O is the Landau symbol. The square of the number of photon counts N 2 (τ int ) can be approximated as follows [28]; Also, Mandel's Q parameter can be approximated as [7,28]; Also, the standard deviation of waiting times σ (τ ) is as follows: The excitation intensity dependence of the mean of the waiting time τ and the standard deviation of waiting times, σ (τ ) are shown in Figure 11. Both show excitation intensity dependences that are clearly magnetic field dependent. To provide a preliminary demonstration of the feasibility of the experiment, we estimate the value of Mandel's Q parameter for observation of 100% of emitted photons. The observable of the number of photon counts N(τ int ) can be estimated as follows: The observable of the variance of the number of photon counts V(N(τ int )) following a Normal or Poisson distribution can be estimated as follows when N sample is large: We estimate the value of Mandel's Q parameter as follows:   Also, we define the magnetic field effect on Mandel's Q parameter (MFE(Q)) as follows: Then, we estimate the value of MFE(Q) as follows: To calculate the above values, we select the shortest integration time sufficient such that the approximation remains appropriate. We then consider the number of samples necessary to achieve sufficient error levels to make realistic measurements. The accuracy of our calculations of Mandel's Q parameter and MFE(Q) are provided in the Supporting Materials. Based on this optimisation procedure, we set the integration time to 1 ms. Mandel's Q parameter and MQ-MFE for single FAD molecules are plotted against excitation intensity in Figures 12 and 13. In Figure 12, the value of Mandel's Q-parameter for FAD increases with both excitation intensity and magnetic field.
In Figure 13, it is demonstrated that using MFE(Q), we can theoretically extract the MFE from single FAD blinking, while it is impossible to obtain it directly from the change of blinking rate or the average waiting time. However, the observation of MFE(Q) require 10 6 samples (10 3 s in case of integration time of 1 ms) to achieve reasonable levels of statistical error ( Figure 14). In short, we have demonstrated that it is, in principle, possible to observe FAD blinking statistics with commercially available equipment and that the MFE can be extracted from these statistics through the use of Mandel's Q-parameter. FAD is an excellent molecule in which to observe ordinary MFEs, but it is less ideal for observing MFEs on single RPs. This is primarily due to its photophysics. In the quest to find a good system in which to observe single RP MFEs, FAD is a good template molecule, but replacement of the isoalloxazine ring with a chromophore which has a much higher absorption cross section along with reducing internal conversion as much as possible would allow integrated fluorescence measurements at more reasonable laser intensities. Our determination of the magnetic field sensitivity of Mandel's Q parameter promises the possibility of experimentally observing single RP MFEs in FAD at lower laser powers, but requires long averaging times to build sufficient statistics to resolve the effects.

IV. Conclusions
In this work we have demonstrated, using spin-dynamic simulations, that SM-MFEs can be manifest in observable fluorescence blinking statistics. In addition, we identified FAD as a target molecule undergoing cyclic, magnetic field dependent photochemistry that is a starting point for exploring experimental realisation of SM-MFEs. Using an established kinetic scheme for FAD photochemistry, calculations based on existing experimental data demonstrate that SM-MFEs are potentially measurable in FAD using integrated fluorescence measurements at high irradiation intensities. By experimentally measuring Mandel's Q-parameter, MFEs can in principle be experimentally observed at much lower laser intensities, but at the expense of extremely long measurement times. We argue then that SM-MFEs should be realisable experimentally and continued theoretical and experimental work is underway to achieve this goal. If a system can be established in which routine measurement of SM-MFEs is possible, more complex SMS experiments with single RPs can be undertaken, for example RYDMR experiments.