Continuously parametrized quantum simulation of molecular electron transfer reactions

A comprehensive description of molecular electron transfer reactions is essential for our understanding of fundamental phenomena in bio-energetics and molecular electronics. Experimental studies of molecular systems in condensed-phase environments, however, face difficulties to independently control the parameters that govern the transfer mechanism with high precision. We show that trapped-ion experiments instead allow to reproduce and continuously connect vastly different regimes of molecular charge transfer through precise tuning of, e.g., phonon temperature, electron-phonon interactions, and electronic couplings. Such a setting allows not only to reproduce widely-used transport models, such as Marcus theory. It also provides access to transfer regimes that are unattainable for molecular experiments, while controlling and measuring the relevant observables on the level of individual quanta. Our numerical simulations predict an unconventional quantum transfer regime, featuring a transition from quantum adiabatic- to resonance-assisted transfer as a function of the donor-acceptor energy gap, that can be reached by increasing the electronic coupling at low temperatures. Trapped ion-based quantum simulations thus promise to enhance our microscopic understanding of molecular electron transfer processes, and may help to reveal efficient design principles for synthetic devices.


I. INTRODUCTION
Molecular electron transfer (ET) reactions are the fundamental steps in many chemical and biological processes [1][2][3][4][5][6]. A detailed understanding of the underlying physics is crucial in such diverse fields as catalysis [5], photosynthesis [7], biological signalling [8], molecular electronics [9] and energy conversion [10,11]. Marcus theory, the fundamental theory for thermally activated nonadiabatic ET rates, has been developed already in 1956 [1], and has since been generalized to include quantum or classical vibrations that modulate the energy gap or the coupling [1][2][3][4][5][6]12]. However, many ET reactions cannot be described by incoherent nonadiabatic rate theories. Currently, intensive research efforts are dedicated to non-standard transport regimes [11,13,14]. These include fast charge and energy transfer reactions on the timescales of vibrational relaxation and electronic dephasing, such as the primary reactions in photosynthesis [11], nonergodic and nonequilibrium vibrational dynamics [13,15], the transition from incoherent to coherent transport [16], the participation of quantum vibrations in promoting ultrafast ET, and the control of ET by external fields [15].
Understanding the particulars of a molecular ET reaction means deciphering how the molecular assembly, structure and dynamics determine the dominant transfer mechanism and the magnitude of the transfer rate. A common experimental approach is to chemically modify the molecular and/or solvent by laser-and radiofrequency fields. The versatility of trappedion arrays renders them ideal candidates to study elementary quantum transport processes under controlled conditions. Previous studies have focused on the transport of electronic (effective spins) and vibrational excitations (phonons) across ion chains [28][29][30][31], and on the influence of spin-phonon couplings on coherent exciton transport processes [32].
Here we show that analog trapped-ion quantum simulations with strong electron-phonon interactions allow to continuously map out the parameter space of a paradigmatic ET scenario. We introduce and discuss a minimal setting readily accessible for state-of-the-art experiments that is able to reproduce the predictions of Marcus theory, including the nonadiabatic inverted regime. We further establish how lowering the temperature allows to reach the nonadiabatic quantum transport regime, where vibronic (induced by the strong coupling of vibrational and electronic degrees of freedom) resonances determine the transfer rates. Finally, we demonstrate that the ion-trap platform makes unconventional ET regimes accessible that are not observed in molecular systems. We predict the crossover from conventional, nonadiabatic molecular ET to a quantum transfer regime where rates are limited by phonon lifetimes in the normal regime and heavily modulated by resonances in the inverted regime. These modulations can be understood as a consequence of trapped excited-state populations that do not participate in the adiabatic transport and only contribute to the transport on resonance.

II. TRAPPED-ION SETUP
We begin with an outline of the elementary experimental platform for proof-of-principle quantum simulations of ET models. All relevant aspects of the dynamics can be discussed and experimentally demonstrated starting with two trapped ions, coupled by the Coulomb force while sharing the same trap potential. Extensions to larger numbers of ions or different trap geometries and dimensionalities [33] provide access to more complex situations which involve a larger number of electronic or phononic modes, giving rise, e.g., to intermediate bridge states [34].
The setup is sketched in Fig. 1 (a): The first, 'system', ion (blue dot) emulates the donor and acceptor states |D and |A , respectively, by means of two long-lived electronic levels [22][23][24]35], whereas the second, 'cooling' ion (red dot) is used exclusively to sympathetically cool the collective motion of the two-ion compound [35][36][37][38][39][40][41]. Choosing spectrally distinguishable ions or employing tightly focused laser beams prevents the cooling laser light from affecting the system ion's internal states.
The effective dynamics of the system ion's electronic, together with the compound's collective vibrational degree of freedom, is then described by the master equation with the effective Hamiltonian  Average phonon numbern ⇒ k B T = ω 0 / ln[(n + 1)/n]n 0.01 [35,38,[51][52][53] TABLE I. Correspondence between characteristic parameters of electron transfer (left) and their counterparts in trapped-ion simulations (middle), together with typical experimental values in ion-trap platforms (right). The notation in the left column is adopted from [6].
where {A, B} = AB + BA. The Hamiltonian (2) is formally identical to standard models of ET theory [2][3][4][5][6]: H decomposes into a purely vibrational part (first term), a (vibronic) coupling term between vibrational and electronic degree of freedom (second term), and two purely electronic parts (third and fourth term). The first term ω 0 a † a encodes the quadratic potential ∼ ω 0 z 2 in the dimensionless oscillator coordinate of the collective centerof-mass mode of the two ions. The second (coupling) term proportional to σ z (a + a † ) = 2σ z z, with σ z = |D D| − |A A| contributes a linear potential in the reaction coordinate z, with positive or negative sign depending on whether the system ion resides in the donor or acceptor state |D or |A . This displaces the harmonic oscillator vibrational potential minimum from the origin to z 1 = −g/2ω 0 (donor) and z 2 = g/2ω 0 (acceptor), respectively. The third term in Eq. (2) adds a statedependent energy shift ±∆E/2 of the vibrational harmonic oscillator potentials associated with the acceptor and donor levels, leading to a total energy offset ∆E (the donor-acceptor energy gap of the ET reaction) between donor and acceptor Born-Oppenheimer (BO) surfaces. Finally, the fourth term in Eq. (2) coherently couples the donor and acceptor states, through σ x = |A D| + |D A|, and will have the strongest impact on the vibrational energy landscape at the degeneracy of the donor's and the acceptor's BO surfaces [see Fig. 1 where it induces an avoided crossing of size 2|V|, associated with a coherent population transfer at Rabi frequency V/ . Adopting the definition ∆E = E A − E D , which follows the convention of ET theory, where a negative ∆E expresses that energy is released during the ET process [1][2][3][4][5][6], the vibrational energy landscape defined by (2) is depicted in Fig. 1 In the ion-trap setting, the collective mode frequency ω 0 is determined by the trap potential [35,43], an external magnetic field controls the system ion's energy splitting ∆E, and coherent laser driving [blue upward arrow in Fig. 1 (b)] induces the coupling Vσ x . The spin-phonon interaction term gσ z (a + a † )/2 between vibrational and electronic degree of freedom [blue downward arrow in Fig. 1 (b)] can be realized by bichromatic laser driving with suitably chosen detuning close to the trap frequency [32,[44][45][46][47]. The electron-phonon coupling constant g is then determined via the laser intensities and detunings [32,35,47]. An implementation with travelingwave light requires a rotating-frame description of the dynamics, which causes the trap frequency ω 0 to be replaced by the laser detuning. Throughout this article, we express all energy and time scales in units of ω 0 , which may represent an effective trap frequency.
In contrast to real molecular aggregates, all these relevant system parameters are continuously and precisely tunable over wide intervals. We summarize the correspondence between the most important ET parameters and their analogs in the proposed trapped-ion simulation in Tab. I.

III. ELECTRON TRANSFER THEORY
To understand the ET process from donor to acceptor (|D → |A ) governed by the open-system dynamics (1), it is most intuitive to consider the potential landscape depicted in Fig. 1 (b), with the BO surfaces E D (z) and E A (z). From this picture, it becomes evident that a key parameter for the transport process is the reorganization energy λ. It is the energy needed to displace the reaction coordinate on the donor BO surface from its minimum position to that of the acceptor BO surface [1][2][3][4][5][6]. In the geometry determined by Eq. (2), this amounts to λ = ω 0 (z 1 − z 2 ) 2 = g 2 /ω 0 . A large reorganization energy compared to the phonon frequency signifies that such displacement will be associated with strong vibrational excitations. The phonon mode is thus strongly coupled to the electronic degree of freedom, and actively contributes to the ET reaction (i.e., it is ET-active). This is in contrast to models of vibrationally-assisted transport of excitations, where the coupling to phonons is typically less pronounced, and the transport is predominantly coherent and therefore reversible [8].
Irreversible population transfer from |D to |A is mediated by classical or quantum mechanical mechanisms, depending on the dominant energy scale. These are the thermal phonon energy k B T , the phonon energy ω 0 , the coupling strength V, the energy gap ∆E, and the reorganization energy λ. On the one hand, the ratio of temperature and phonon energy determines whether the system is operated in a quantum or classical parameter regime. On the other hand, the relative (as compared to reorganization and phonon energy, respectively) strength of the avoided crossing of size 2|V| between the two BO surfaces controls the transition from nonadiabatic (between two distinct, diabatic BO surfaces) to adiabatic (between the minima of the same, lower BO surface) transfer.
In the following, we explore different transfer regimes by controlling individually the thermal phonon energy k B T , the (effective) trap frequency ω 0 , and the coupling strength V in numerical simulations of the time evolution governed by Eq. (1). We always consider cases where the initial and final electronic states are well localized in different regions of the molecule, known as localized polaron states with |V| < λ/4 [1][2][3][4][5][6].

IV. CLASSICAL NONADIABATIC TRANSPORT
The classical regimes involve ET-active vibrational frequencies with ω 0 λ and temperatures with ω 0 k B T . Many molecular ET reactions in condensed phases (i.e., in solution, cellular or molecular-junction environments rather than in vacuum) belong to this category because low frequency environmental vibrations couple to ET. In this regime the Landau-Zener parameter γ LZ = π 3/2 |V| 2 /( ω 0 √ λk B T ) distinguishes the nonadiabatic (weak electronic coupling, γ LZ 1) and adiabatic (strong electronic coupling, γ LZ 1) limits [54].
The classical nonadiabatic case (CN) is described by Marcus theory. Classical, thermally activated vibrations can tune the electronic states |D and |A to resonance, such that ET can take place at the crossing point of the two diabatic BO surfaces, while preserving the total energy of the system. The ET rate is then given by the expression [1][2][3][4][5][6] where U = (∆E + λ) 2 /(4λ) is the activation energy to the resonance conformation on the donor BO surface [see Fig. 1 This transfer rate is proportional to |V| 2 and reaches a maximum in the activationless limit (U = 0), when |∆E| = λ, such that the acceptor BO surface intersects the minimum of the donor surface [see Fig. 2 (a), panel C]. The dependence of ln k CN on the detuning ∆E is an inverted parabola-the seminal Marcus parabola, cf. the orange line in Fig. 2 (c). This parameter regime can be reached in the trapped-ion system for a thermal phonon population with a relatively low ω 0 . For our numerical simulations of the trapped-ion dynamics, we initialize the system at time t = 0 in |D D| ⊗ ν p ν |ν ν|, where |ν is a Fock state of ν phonons, weighted by a thermal distribution p ν at temperature T . We extract the ET rate k by fitting an exponential decay a exp(−kt) to the time evolution of the donor state population p D (t) after a transient evolution of ω 0 t/2π = 5 that is dominated by coherent dynamics (see also Appendix A). Typical evolutions are displayed in Fig. 2 10.5 ω 0 and γ LZ 10 −3 . The corresponding average phonon number ofn = 10 can be reached without requiring efficient cooling. Figure 2 (c) shows the extracted transfer rates from the trapped-ion simulation (dark blue circles). These follow nearly perfectly the Marcus prediction (orange line), Eq. (4), which we plot without adjustable parameters. Both the normal and the inverted transport regimes can be simulated. Small deviations near the activationless case may be due to a broadening of the phonon resonances caused by the laser cooling process that is not captured by Marcus theory (4).

V. QUANTUM REGIMES
Let us now turn to the discussion of quantum regimes for ET. When k B T ≈ ω 0 the quantum nature of the vibrational mode has to be accounted for when modelling the transfer process. We first consider the parameter regime |V| λ/4, where V and γ are of the same order of magnitude, which is described by quantum nonadiabatic (QN) ET theory [1,2,4,6]. In Fig. 1 (b), this situation is sketched by indicating the respective energies E D,ν and E A,ν of the uncoupled donor and acceptor vibronic eigenstates |D |ν and |A |ν as horizontal lines. Population transfer is then facilitated by resonances be- tween pairs of such states, as expressed by Fermi's golden rule: where the p ν describe the initial thermally distributed phonon populations and FC ν,ν = | ν|e g(a † −a)/ω 0 |ν | 2 is the Franck-Condon factor between two displaced harmonic oscillators of identical frequency. This ET transfer regime can be simulated on the ion-trap platform with standard parameters, i.e., by cooling to smaller phonon numbers as compared to those discussed in the classical regime above. Figure 3 (a) shows the vibronic energy levels in the quantum nonadiabatic regime for different val-ues of ∆E. The time evolution of the donor state population is fitted to an exponential decay [after a transient evolution of ω 0 t/2π = 100], see Fig. 3 (b), forn = 0.01 (k B T 0.217 ω 0 ). Figure 3 (c) depicts the variation of the obtained transfer rates with ∆E. The classical Marcus parabola is now heavily modulated by vibronic resonances which allow for fast ET at sharply defined values of the donor-acceptor energy splitting. The transfer rates are extremely well described by the predictions of nonadiabatic theory, Eq. (5), which is plotted for comparison [55]. We note that this vibronic modulation of the resonances, which is readily accessible in an ion-trap quantum simulation, is washed out for molecular ET experiments in condensed-phase environments by uncontrolled disorder and additional coupling to low-frequency vibrations. Figures 2  and 3 demonstrate that current ion-trap technology can simulate well-known ET rate regimes. As such, these results define a benchmark for the experimental ability to simulate the ET Hamiltonian.
With increasing electronic coupling V (and all other parameters fixed) -which is controlled by the injected laser power -the above picture of resonant transitions between diabatic states is washed out by the opening of an appreciable avoided crossing between the donor and acceptor BO surfaces [see Fig. 4 (a)], ultimately defining two new, excited and ground state adiabatic potential surfaces which do not intersect any more.
This trend towards a dominant role of V manifests in a marked change of the dependence of the transfer rate on the energy gap: As one enters the regime |V| γ, strong direct coupling V quickly transfers initial population from the donor to the acceptor -where it can only remain if the phonons created in the processes are promptly removed by dissipation (implemented by the cooling laser with rate γ). In the above limit, however, population is transferred back and forth between donor and acceptor many times before dissipation renders the process irreversible. Consequently, γ becomes the limiting factor which controls the transfer rate, and a simple model restricted to only the initial state |D |0 (corresponding to the the zero-temperature limit) and a single resonant vibronic state |A |ν yields a transfer rate of (see Appendix B) which is nothing but the decay rate of the νth vibrational level, according to the dissipator (3). This regime of transfer rates limited by γ, where the direct coupling V no longer affects the transfer rate as in Eq. (5), is not readily accessible in a condensed phase environment where vibrational relaxation is fast. Equation (6) neglects coupling to off-resonant states, which becomes important when the direct coupling V approaches the phonon energy ω 0 (leading to avoided crossings with widths on the order of phonon energies), as well as the effect of V on the phonon lifetime due to hybridisation of vibrational levels near resonance. Nevertheless, it allows us to understand the basic transport mechanism as mediated by the vibrational excitations of the emerging adiabatic BO ground state -which can therefore be considered as a quantum adiabatic (QA) transfer regime.
We show examples of such quantum adiabatic time evo- lution in Fig. 4 (b), and the variation of the transfer rate as a function of the donor-acceptor energy gap ∆E in Fig. 4 (c), with the same parameters as in Fig. 3, except for the much larger electronic coupling strengths [V/( ω 0 ) = 0.25, 0.50, 0.75]. Whereas in Fig. 3 the transfer rate is limited by the Franck-Condon factors of the transition from ν = 0 to ν phonons, see Eq. (5), and thus peaks at |∆E| = 5 ω 0 (the classically activationless case), here the adiabatic rate is merely limited by the cooling rate, and on resonance keeps increasing with ∆E. In the normal transfer regime (|∆E| < 5 ω 0 ) the resonances broaden with increasing direct coupling V, and the transfer rates eventually converge to the adiabatic limit (6), see yellow and pink lines in Fig. 5 (a). The driving-induced broadening of the resonances induces the rate to shrink at resonance as V is further increased, while the off-resonant rates grow, which explains the slight decrease at very large values of V. Furthermore Fig. 5 (b) demonstrates that k/γ scales approximately linearly with ν, as predicted by Eq. (6).
Yet another structural feature comes into play in the inverted regime (|∆E| > 5 ω 0 ): As apparent in particular in the limit of large V [≥ 0.50ω 0 in Fig. 4 (c)], the transfer resonances broaden less strongly, and, incidentally, sharpen asymmetrically. Furthermore, the transfer rates do not saturate with increasing V, as to be expected from Eq. (6), but instead decay after reaching a maximum for an optimal value of the coupling, see Fig. 5 (a). Closer inspection of the spectral structure of the Hamiltonian (2) indeed reveals that the dominant eigenstate in the time evolution of the here selected initial state (approximately |D |0 ) is trapped and strongly localized in the excited adiabatic BO surface. In contrast to the population on the lower BO surface, the excited population remains trapped unless transfer is facilitated by a resonance, as we can clearly observe in Fig. 4 (c) for V = 0.75ω 0 (see Appendix B 3). These observations imply that experimental control over the initial state, which can be achieved with high accuracy in trapped-ion experiments, defines yet another handle to control the ET efficiency in this largely unexplored parameter regime.

VI. CONCLUSIONS
We have shown that trapped-ion analog quantum simulators provide a versatile testbed for studying phonon-mediated electron transfer under well-controlled, adjustable conditions. Numerical simulations demonstrate that a rather simple system composed of two ions coupled to a single phononic mode is able to reproduce essential features of donor-acceptor molecular ET. Continuously tuning the parameters allows us to connect and study a great variety of largely unexplored transport regimes which are not readily accessed by molecular ET experiments in a continuous way. When operated in a hightemperature regime, the predictions of Marcus theory can be verified for classical nonadiabatic ET with a high level of control over individual parameters. In the low-temperature regime, it is possible to study the emergence of quantum nona- diabatic vibronic transfer resonances. By further increasing the electronic coupling, quantum adiabatic transport is observed only in the normal regime, whereas the inverted regime gives rise to a transfer mechanism mediated by resonances due to trapped excited-state populations. It is impossible to realize such a controlled crossover between adiabatic and nonadiabatic regimes in molecular systems. Our work points out that incoherent features in the longtime dynamics of quantum simulators can give rise to interesting transport phenomena that are of high relevance, e.g., in molecular chemistry. These features become visible in the decay rates, after the coherent phenomena have largely been damped out by the engineered dissipation of phonons. Our ap-proach is therefore complementary to predominantly coherent quantum simulations of the short-time dynamics that provide access to strongly entangled many-body systems by trying to mitigate decoherence [23,24]. At longer time scales, the natural decoherence of the trapped-ion system becomes increasingly relevant. However, the time evolutions considered here are on the order of few ms. This is still shorter than realistic quantum error correction cycles [27], which in turn are subject to much stronger restrictions in terms of fidelity and coherence than our simulations.
Possible extensions of our proposal include coherent laser couplings of the ions to multiple motional modes [46] to enable vibronic transfer on a more complex vibrational backbone. The envisioned setup can further include larger numbers of ions in one-or two-dimensional trap geometries [33,56], to study the influence of intermediate bridge states [34] on transfer processes between different ions with tunable interaction range [57,58]. Other studies may target the influence of different initial states, including nonequilibrium or even nonclassical vibrations [15,51,[59][60][61].
In many molecular ET reactions in condensed-phase environments a large number N vib of vibrational modes µ are ETactive with λ µ / ω µ > 1 and k B T/ ω µ ≥ 1. Quantum effects play a relevant role in the dynamics of such modes, but their exact treatment is usually intractable [62]. Instead, numerical ET simulations use approximations which treat a large fraction of ET-active modes as classical vibrations. Trapped-ion quantum simulations provide an alternative to large-scale numerical simulations and semiclassical approximations, by reproducing the fully quantum ET dynamics in a well-controlled setup. Experimental quantum simulations of ET may thus complement theoretical, computational and experimental efforts on molecular ET systems, and may be used to benchmark approximate models. This opens up new pathways towards the microscopic understanding of ET processes, the identification of novel transfer regimes, and the design of highly efficient ET systems. F. S. and M. G. contributed equally to this work.

Appendix A: Determination of the decay rate
The information about the decay rates is contained in the long-time dynamics of the ion-trap system. In our simulations, these were obtained by propagating for sufficiently long times. Alternatively, if it is hard to extract the decay constant by an exponential fit (e.g., due to shorter propagation times), the decay rate can be efficiently approximated using [64] where the integral excludes a transient initial time interval that is dominated by coherent effects. In the case of a pure exponential decay exp(−kt), the decay rate exactly reduces to the decay constant k.

Appendix B: Quantum adiabatic dynamics
To understand the transition from the Fermi golden rule result (5) to stronger interaction strengths V in the quantum transport regime, we consider more closely the resonance between the initial state |D |0 and an acceptor state |A |ν . Such a resonance is achieved for ν = |∆E|/( ω 0 ) [see Fig. 1 (b)]. The decay of the excited state can be described by the effective non-Hermitian Hamiltonian whereγ is the decay rate of the acceptor state. Here, we assumed that the electronic coupling V is further modulated by the Franck-Condon factor FC 0,ν and neglected the coupling to other states. The donor population decays as with d = γ 2 − 4|V| 2 FC 0,ν . This can be mapped to the Jaynes-Cummings model on resonance, with a cavity mode that is coupled to a reservoir, leading to a Lorentzian spectral density [42]. Inserting Eq. (B2) into (A1) yields the decay rate k 0ν QA =γ where the parameter η =γ/(|V| FC 0,ν ) determines the ratio of the phonon decay rate and the coherent coupling strength.

Weak coupling
The weak coupling limit, |V| FC 0,ν γ, is characterized by η 1. We obtain Withγ = γ, this reproduces the Marcus result pertaining to the one transfer channel on resonance, and the delta-function broadened to a Lorentzian with width γ. It coincides with the nonadiabatic result (5) on resonance.

Eigenstate width in the inverted regime
To understand the survival of the sharp resonances in the inverted regime, we consider the "eigenstate width" [65] of the initial state where p j ≡ j|ρ| j is the overlap of the initial density matrix ρ = |D D| ⊗ ν p ν |ν ν| with the eigenstate | j of the full Hamiltonian (2) and we use the convention p ln p = 0 for p = 0. Intuitively, a larger value of W indicates an increasingly complex dynamical evolution that, in a superposition of many different energy eigenstates, explores a larger fraction of the state space. This increases the likelihood to populate states that respond strongly to the laser cooling process and thus lead to an irreversible transfer of population from |D → |A .
The width (B6) is shown in Fig. 6 for three different interaction strengths V, corresponding to the three panels in Fig. 4. We observe a clear resonance structure that is less and less pronounced as the interaction strength is increased. However, the broadening arises asymmetrically: while the resonances are strongly washed out in the range ∆E/ ω 0 = 3, . . . , 5, the eigenstate width remains more clearly peaked at integer values of ∆E/ ω 0 . In particular, the resonances at ∆E/ ω 0 = 6 and 8 remain very sharp, reflecting the behavior of the transfer rate in the simulation in Fig. 4 (c). Resonances at ∆E/ ω 0 = 7 and 9 are broadened more strongly, again, in agreement with the transfer rate simulations. In summary, the analysis of the eigenstate width hence reveals (a) that in the inverted regime efficient transport |D → |A relies on the resonance of diabatic states, and (b) that the shape of the resonances is related to the distribution of energy eigenstates in the initial density matrix.