Chiral-induced spin selectivity in photo-induced electron transfer: Investigating charge and spin dynamics in a master equation framework

The study of photo-induced electron transfer (PET) through a chiral bridge is recently receiving a growing amount of attention^1–8 as an ideal, simple test-bed for understanding Chirality-Induced Spin Selectivity (CISS). For instance, it has been shown that magnetic resonance techniques can be exploited to assess the spin wave function after PET of a radical pair generated in donor–chiral bridge–acceptor (D-χ-A) systems.^9,10 The simplicity of this kind of setup is due to the absence of metallic leads or substrates with large spin–orbit coupling, which otherwise could play an important role in spin selectivity detected via electron transport experiments.^11–13 This simplification will allow us to focus only on the role of the chiral bridge in the CISS effect both in the interpretation of experimental data and as a guide for a microscopic theoretical modeling of the phenomenon.

Moreover, a great interest toward spin-correlated radical pairs, potentially linked by a chiral bridge, is coming from the field of quantum technologies.^8,14–22 In particular, a setup consisting of a D-χ-A molecule linked to a spin qubit (D-χ-A-Q hereafter) has been recently proposed as a promising architecture for quantum computing, enabling high-temperature initialization and read-out of a single spin by exploiting spin-selective electron transfer.⁸ The high-temperature efficiency of the phenomenon is one of the important advantages it offers for quantum computing applications. For instance, the polarization of the transferred spin can be exploited for initialization of Q even at high temperature, where the qubit spin is completely unpolarized.

Another relevant issue that deserves investigation is the effect of CISS in charge recombination^6,7 through D-χ-A systems. Spin selectivity is expected to work in both directions, facilitating backward transfer for the opposite spin orientation than forward transfer. By linking spin to charge degrees of freedom, this property can be exploited to activate a spin-to-charge conversion mechanism that unfolds an even more promising quantum computing application, i.e., the readout of single spin.⁸ 

To gain a deeper understanding of these processes and implement realistic simulations, here, we consider a master equation describing spin-selective electron transfer reactions through suitable “Haberkorn” operators.²³ These reaction operators are defined as projectors on spin states with local spin polarization, expanding on what proposed in Ref. 7. This phenomenological model does not enter into the microscopic origin of the CISS effect and can, therefore, accommodate different forms of the reaction operators, according to developments in the fundamental understanding of CISS that are currently being carried out.^4,5,24

We simulate both forward and backward charge transfer processes in the presence of spin-selectivity. Remarkably, we find important effects arising from the interplay of the coherent spin dynamics and incoherent charge dynamics involved in the reaction. This allows us to highlight important consequences of introducing spin-polarizing reaction operators in Haberkorn equations and to identify limiting factors and a proper hierarchy of interactions to make these kinds of systems suitable for quantum computing applications.

Finally, we use the spin-selective master equation to simulate an entire quantum computing workflow, from initialization to gates and readout, following the proposal in Ref. 8. We show that the high-temperature initialization of a molecular spin qubit, such as a Cu²⁺ complex,²⁵ coupled to the acceptor spin is possible via a simple sequence of resonant microwave pulses, overcoming an important drawback of molecular qubits.²⁶ We also implemented one-qubit gates with high fidelity in times much shorter than decoherence times typical of molecular systems.^27–31 Finally, by inducing backward transfer via an external tool such as an electric field, we reproduce the steps needed to convert the spin information of the qubit to charge information on the donor, effectively enabling high-temperature readout of the quantum state of a single spin, a major technological challenge.

We stress that, for the proposed quantum computing applications, we are only interested in whether the CS state persists or charge recombination has occurred, independent of the spin state of CR, which acts only as a sink in our simulations. This corresponds to evaluating the charge on the donor, i.e., Tr ρ_CS. Hence, Eq. (1c) does not need to be explicitly solved and we omit it hereafter.

In Secs. III and IV, we separately simulate charge separation and recombination by numerically solving this set of equations. To this aim, we will consider only a subset of Eqs. (1) at a time. This allows us to more clearly address the implications of charge and spin dynamics in the two separate reactions. This choice is made possible by the very different transfer rates usually characterizing charge separation and recombination in this kind of system. In fact, typical charge separation times are in the 100 ps-few ns range, while spontaneous recombination occurs on the μs and even longer time scales, thus being completely ineffective in the forward process.³⁶ 

In the simulations, we include reasonable values for all the parameters in the spin Hamiltonians, namely, a strong on-site isotropic exchange coupling on the donor J_PE = −1 eV, slightly different g-factors g_D = 2 − ɛ and g_A = 2 + ɛ (ɛ = 0.001) and the dipolar coupling J_CS computed for a donor–acceptor distance d = 20 Å within the point-dipole approximation. We assume fast photo-excited electron transfer k_F = 5 ns⁻¹, as occurs in spin-correlated radical pairs.³⁶ The magnetic field (parallel to the chiral bridge) is B = 1.05 T, large enough to ensure good spin state factorization in the CS stage, an important requirement for quantum computing applications (see Sec. V). The results of our simulations are shown in Fig. 2, where populations of ⟨S_z⟩ = ⟨S_z,1 + S_z,2⟩ = 0 spin states and local spin polarization ⟨S_z,i⟩ on both sites are reported as a function of time, up to 2 ns. The system was initialized in a photo-excited singlet spin state $|S〉=(|↑↓〉−|↓↑〉)/2$⁠, i.e., ρ_PE(0) = |S⟩_PE⟨S|_PE.

The most relevant result is that the final state is characterized by ⟨S_z,A⟩ = −⟨S_z,D⟩ = 0.5, i.e., maximum local spin polarization on both D and A, leading in turn to maximum overall spin polarization p = ⟨S_A,z⟩ − ⟨S_D,z⟩ = 1. Moreover, the charge is completely transferred from D to A. This could appear surprising since the evolution induced only by the Haberkorn reaction operators would lead to only a 50% charge transfer from D to A, as only half of the initial singlet population is transferred by the $P↑̂$ reaction operator. This complete charge transfer and maximum polarization are due to the fast coherent dynamics ruled by J_PE, which mixes |↓↑⟩_PE and |↑↓⟩_PE, thus ensuring that there is always sufficient population in the |↓↑⟩_PE state that eventually undergoes charge transfer. In order to highlight that this effect originates exactly from J_PE, we compare in Fig. 2 realistic simulations of Eq. (3) with J_PE = −1 eV with the hypothetical case of J_PE = 0. In the latter case the evolution is dominated by the reaction operator, which results, as anticipated, in incomplete charge transfer and polarization (dashed lines). Conversely, the coherent evolution in the CS stage (included in our simulations) is much slower and hence it does not affect the charge separation reaction. As discussed below, it plays a much more central role in recombination, given the longer time scale of this process.

Note that our choice for the reaction operator $P̂F=P̂↑=|↑↑〉〈↑↑|+|↓↑〉〈↓↑|$ is only an assumption due to the so far incomplete understanding of the microscopic origin of CISS in PET. Other choices are possible and can be included in Eq. (1). A reasonable alternative would be $P̂F=|↓↑〉〈↓↑|$⁠, as proposed in Ref. 7. The main difference introduced by this choice compared to the former would be excluding charge separation or recombination through ⟨S_z⟩ ≠ 0 spin states. Hence, no qualitative difference is expected for the simulations of Fig. 2, but a different dynamics could arise for initial spin states different from the singlet.

Figure 3 shows the results of numerical integration of Eq. (4), performed with the same J_CS as before, a recombination rate of k_R = 1 µs⁻¹ and starting from the fully charge-separated initial state $ρCS(0)=|σ0〉CS〈σ0|CS$⁠, in which $|σ0〉CS=(|↓↑〉CS+|↓↓〉CS)/2$⁠. This choice for the initial state allows us to investigate the effect of spin selectivity in charge recombination. To highlight spin selectivity effects, we omit spin relaxation and dephasing, which, in general, might be relevant on the studied time scale.

The most striking difference between Figs. 2 and 3 is the loss of population from the |↓↑⟩_CS state [Fig. 3(a)], which should be prevented by the specific form of $P↓̂$⁠. However, the spin–spin interaction in $ĤCS$ couples and mixes |↓↑⟩_CS and |↑↓⟩_CS, similarly to J_PE in the charge separation reaction. Since one of the two coupled states is affected by the reaction operator $P↓̂$⁠, this continuous mixing effectively leads to a loss of population from |↓↑⟩_CS and, therefore, to Tr ρ_CR > 1/2 [Fig. 3(b)] and a corresponding loss of spin polarization [Fig. 3(c)]. This effect disappears if we remove J_CS from the simulations (dashed lines in Fig. 3).

We note that with the set of parameters above, this harmful non-spin-selective recombination is significantly slower than k_R, and hence, it only yields a limited loss of spin polarization. Nevertheless, several parameters can improve or hinder the effectiveness of this channel. In the specific case of Fig. 3, the small Δg = g_A − g_D and slow k_R are both favorable conditions for the onset of a non-spin polarizing recombination. In order to preserve spin polarization in charge recombination, we can adopt one of the following approaches:

1. Reducing the intensity of the spin–spin interaction J_CS by increasing the distance d between the donor and acceptor. This directly limits the coherent spin dynamics of the CS state but also suppresses k_F and k_R rates.³⁷ 

2. Increasing the energy gap between the spin states |↑↓⟩_CS and |↓↑⟩_CS coupled by J_CS, thus effectively suppressing their evolution. This can be achieved by increasing the difference between the two g-factors, as well as by introducing additional interactions in the spin Hamiltonian (see below).

3. Accelerating the recombination process. Indeed, unwanted spin recombination will be less effective if charge recombination takes place in a shorter time scale than coherent spin dynamics. Moreover, for k_R ≫ J_CS/ℏ, spin selective recombination would be restored by the quantum Zeno effect, which results in a significant slow-down of coherent evolution of a quantum state in the presence of much faster incoherent processes (see below).

Although the loss of spin polarization may appear as a severely limiting factor for potential applications in quantum technologies (e.g., reducing spin selectivity for state readout), in Sec. V, we show that including a molecular spin qubit coupled to the acceptor spin, can entirely suppress the detrimental effect of J_CS.

Before moving to quantum computing applications, we examine an interesting aspect of these simulations, namely, the possibility of exploring the onset of the quantum Zeno effect (QZE) by varying the recombination rate k_R. Due to QZE, continuous measurement of a quantum system halts its coherent evolution. Analogously, QZE appears also in the case of continuous strong coupling to a noisy environment or in electron transfer described through Haberkorn reaction operators, as already predicted in Ref. 33.

This mechanism was not relevant in the charge separation step described in Sec. III, since the large J_PE/ℏ ≫ k_F led to extremely fast coherent spin dynamics. Conversely, the typically slow coherent evolution induced by J_CS in the CS stage opens up the possibility of observing QZE. The impact of QZE on charge recombination is explored in Fig. 4, where we present spin state populations and spin polarizations on D and A as a function of k_Rt for both k_R = 1 µs⁻¹ and the much faster rate k_R = 1 ns⁻¹. The other parameters and the initial state are the same as in Fig. 3. We note the onset of QZE for large values of k_R, combined with the slower coherent dynamics typical of the CS stage. Indeed, the coherent evolution induced by J_CS is practically suppressed when k_R ≫ J_CS/ℏ, thus avoiding the population of the |↑↓⟩_CS state and, hence, preserving spin polarization during charge recombination. This is clearly shown in Fig. 4 (right). Besides their fundamental interest to explore the QZE, these considerations are also important for applications in quantum technologies, where spin-selective charge transfer is important.

However, fast recombination rates are not typical for this kind of system, in which recombination usually takes place over the 1–10 µs time scale.^20,38 Nevertheless, since charge transfer rates are determined by the energy level structure of the system, they can be modulated by external stimuli, such as laser pulses³⁹ or electric fields. The dependence of charge transfer rates on an applied electric field has been theoretically estimated in Ref. 40 through Marcus’ theory for adiabatic electron transfer, obtaining an exponential growth of charge transfer rates with an increasing electric field. Hence, it is reasonable to assume a swift growth of charge transfer rates over multiple orders of magnitude, leading to effects comparable to those simulated in Fig. 4. The capability of controlling recombination rates is also crucial for applications in the field of quantum computing, as expanded upon in Sec. V.

The applications of CISS-mediated electron transfer in quantum technologies proposed in Ref. 8 involve the addition of a molecular spin qubit Q (S_Q = 1/2) to the D-χ-A system. While individual molecular spins usually need extremely low temperatures (⁠$∼10$ mK) for initialization and are extremely difficult to read, this D-χ-A-Q hardware exploits CISS and its effectiveness at high temperatures to circumvent these problems. In fact, by using resonant microwave pulses, it is possible to transfer spin polarization from the acceptor to the qubit, exploiting spin transitions of the system in the CS stage. In addition, qubit state readout can be implemented by first transferring spin polarization back from qubit to the acceptor and then exploiting the spin-selective charge recombination through a chiral bridge to transform the spin information initially stored in the qubit to charge information on the donor site, much easier to assess.

Results of the full simulation are reported in Fig. 5, where ⟨S_z,A(t)⟩ and ⟨S_z,Q(t)⟩ are shown in the top panel and occupancy of the three charge transfer stages is plotted in the bottom panel as a function of time. We now describe in more detail the five steps of the simulation in Fig. 5, emphasizing the strengths and weaknesses of the proposed model and architecture:

• Charge separation is simulated exactly as shown in Fig. 2, leading to CISS-induced full polarization of the acceptor within 3 ns.

• Qubit initialization is accomplished with two consecutive resonant pulses addressing the spin transitions |↓↑⟩_CS ⊗ |↑⟩_Q ↔ |↓↓⟩_CS ⊗ |↑⟩_Q and |↓↑⟩_CS ⊗ |↓⟩_Q ↔ |↓↑⟩_CS ⊗ |↑⟩_Q, effectively transferring spin polarization from the acceptor to the qubit as explained in Refs. 8 and 9. Note that during this step, the |↓⟩_A spin state of the acceptor starts being populated, thus activating slow spin selective charge recombination, as can be seen from the slight population variation in Fig. 5 (bottom) in steps II–IV. However, this charge recombination is very slow compared to coherent spin manipulation and does not lead to a significant error at the end of the simulation.

• One qubit gate can now be implemented on the initialized qubit via resonant pulses. In Fig. 5, we show a R_y(π) gate inverting the polarization of the qubit, obtained with two resonant pulses between the spin states |↓↑⟩_CS ⊗ |↓⟩_Q ↔ |↓↑⟩_CS ⊗ |↑⟩_Q and |↓↓⟩_CS ⊗ |↓⟩_Q ↔ |↓↓⟩_CS ⊗ |↑⟩_Q, implemented in parallel for optimal performance.

• Reverse initialization. After qubit manipulations, we swap the qubit state with the acceptor. This can be achieved by performing the two pulses of step II in reverse order. Since in step (III) we performed a π gate on the qubit, here we obtain inverse polarization of the acceptor spin with respect to the initial CISS polarization.

• Charge recombination. Once the state of the qubit has been swapped to the acceptor, we induce spin-selective charge recombination to convert spin into charge and, hence, access the former state of the qubit by measuring the final charge on the donor. In particular, the ideal CISS effect will only allow the |↓⟩ component of the wave-function to recombine (through the reaction operator $P̂↓$⁠). As a consequence, the two states of the qubit we want to readout (|↑⟩_Q or |↓⟩_Q at the end of point III) will result in different charge states of D and A—either neutral if recombination had occurred or both still charged if recombination is blocked by CISS. Hence, a measurement of the donor’s charge state (with a charge detector close to D) allows us to access the former spin state of the qubit.

Note that spontaneous recombination would require multiple μs. Hence, in this step, we consider a fast-changing k_R that exponentially varies over $≈1$ ns from k_R = 1 µs⁻¹ to k_R = 1 ns⁻¹, compatible with a quick variation of an applied electric field.⁴⁰ This allows us to significantly speed-up charge recombination, thus completing it in times much shorter than the acceptor spin relaxation and of non-spin-selective recombination (also thanks to the quantum Zeno effect). This is needed for the readout scheme to work properly. Below, we discuss the implications of longer or non-spin-selective recombination times on the effectiveness of the scheme. Final readout of the charge state of the donor can then be realized by a sensitive electrometer,⁴¹ whose specific realization is beyond the scope of these simulations.

In summary, the calculation based on realistic parameters documents an efficient readout with complete recombination and an error less than 1%. The inclusion of dephasing effects with reasonable T₂ (a few hundred of ns on D and A, a few μs on Q) only marginally increases this error. This approach to qubit readout can be extended to a full tomography of the quantum state by adding rotations on the qubit before steps IV and V. Indeed, the high fidelity of coherent manipulation with microwave pulses ensures a full control of the qubit in step III.

By simulating charge separation as in Sec. III and implementing the reaction operator defined in Eq. (6) with p_CISS = 0.9, we obtain a maximum polarization on the acceptor ⟨S_z,A⟩ = 0.4. Hence, the value of p_CISS has a dominant impact on qubit initialization and, consequently, on qubit manipulation and readout. Nonetheless, experimental evidences of CISS are very promising, and even better performances for these systems can be predicted when taking into account further developments in determining the most suitable molecules. In addition, the proposed scheme theoretically compares well to molecular qubit readout alternatives, for which efficient readout is still a major challenge to overcome.^46–48 

By performing numerical simulations of both charge and spin dynamics of D-χ-A and D-χ-A-Q systems, we show CISS to be promising as an enabling technology for quantum applications. We find a good performance of the simulated system for both qubit state initialization and readout when using realistic parameters.

To achieve this, we have used a simple extension of Haberkorn equations to treat forward and backward charge transfer in the presence of spin selectivity. Exploiting a large external magnetic field parallel to the chiral axis we obtain the spin state factorization needed to preserve spin polarization over time in the charge separated state, which would otherwise quickly oscillate under the influence of the transverse component of spin-spin coupling in presence of a smaller field. Moreover, the coupling of the donor-acceptor spin system to a molecular qubit induces a further quenching of the coherent evolution responsible for non-spin-selective charge recombination. In addition, the theoretical framework of Haberkorn’s equations, given its phenomenological nature, is very flexible in accommodating alternative forms of spin-selective reaction operators. This will allow to adapt their form once microscopic models of CISS in electron transfer will be developed.⁵ 

Our findings on the interplay of incoherent and coherent dynamics and the subsequent emergence of the quantum Zeno effect are useful in quantum computing applications. In addition, they could be relevant to rationalize experiments for the detection of CISS in electron transfer.² As an example, time-resolved EPR has already been proposed^4,7,9 as a key experimental tool in this field, with some experiments already being performed on D-χ-A systems.¹⁰ Given the technique sensitivity only to the charge-separated state and its capability to probe the spin wave function of the system as a function of time, trEPR spectra are influenced by both charge and spin dynamics, making our considerations in Sec. IV useful in data interpretation.

This work has received funding from the Horizon Europe Program through the ERC-Synergy project CASTLE (Project No. 101071533), from the European Union–NextGenerationEU, PNRR MUR Project No. PE0000023-NQSTI, from the Italian Ministry of Education and Research (MUR) through PRIN Project 2017 Grant No. CR5WCH Q-chiSS “Quantum detection of chiral-induced spin selectivity at the molecular level,” from the European Union's Horizon 2020 research and innovation program (FET-OPEN project FATMOLS) under Grant Agreement No. 862893 and from Fondazione Cariparma.

The authors have no conflicts to disclose.

E. Macaluso: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). A. Chiesa: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). P. Santini: Methodology (equal); Supervision (equal); Writing – review & editing (equal). R. Bittl: Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). S. Carretta: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.