Quantum simulation of charge and exciton transfer in multi-mode models using engineered reservoirs

Domain: physics
Published: 2025-12-05
Editorial outcome: minor_revision

Abstract
Quantum simulation enables studies of open-system dynamics in non-perturbative regimes by programming electronic, vibrational, and environmental interactions on comparable energy scales. Trapped ions offer this capability, combining spins, phonons, and tunable dissipation on one platform. We demonstrate an open-system quantum simulation of charge and exciton transfer in a multi-mode linear vibronic coupling model. Using tailored spin-phonon interactions with reservoir engineering, we emulate a system with two dissipative vibrational modes coupled to donor and acceptor sites and track its non-equilibrium dynamics. We continuously tune the system from the charge transfer regime to the vibrationally assisted exciton transfer regime and find that degenerate modes enhance transfer rates at large energy gaps, while non-degenerate modes activate pathways that reduce the energy-gap dependence. Thus, the presence of one additional vibration introduces interfering pathways and reshapes non-perturbative excitation transfer. Our results establish a scalable, hardware-efficient route to simulate vibronic processes with engineered environments.

Introduction
Molecular vibrations drive a wide range of phenomena in charge and energy transfer in complex chemical and biological systems. Understanding these processes requires modeling the interactions among the electronic, spin, and vibrational degrees of freedom, which cannot be treated independently, especially when the Born-Oppenheimer (BO) approximation breaks down. Although the BO approximation is the cornerstone of structural chemistry, it fails in cases where nuclear and electronic motions become strongly coupled, as occurs in nitrogen fixation and photosynthesis. The fully simultaneous quantum treatment of electrons and vibrations is still a daunting task for existing numerical methods1–5.

Quantum rate phenomena generally require a model that accounts for both fast, intramolecular vibrational modes and slower, environment-mediated modes. Linear vibronic coupling models (LVCMs) offer the simplest framework for describing such systems by assuming that electronic states couple to multiple vibrational modes only in a linear fashion6–8. LVCMs have been widely used to model many complex processes, including singlet fission, triplet–triplet annihilation, and charge transfer in organic photovoltaics9–11.

The high degree of control and tunability of programmable quantum platforms, such as trapped ions8, superconducting qubits12, and photonic devices13, potentially provides an alternative approach to large-scale classical computations for studying condensed-phase chemical dynamics through direct quantum simulation. In particular, trapped-ion analog and analog-digital simulators have already realized a variety of chemical dynamics with fully programmable system parameters and accurate time-resolved features14–18. With engineered reservoirs on their native spin and bosonic degrees of freedom, trapped-ion systems have also been recently used to study non-equilibrium quantum reaction dynamics19–23.

In this work, we present the first trapped-ion simulation in which unitary spin-phonon couplings and mode-selective dissipation via cooling are independently programmed, enabling real-time open-system emulation of excitation transfer dynamics in a two-mode LVCM. Emulating spin-phonon systems with multiple bosonic modes, combined with environment engineering, represents an important step toward realizing models of molecular systems in which multiple intramolecular (fast) modes coexist with long-wavelength, low-frequency environment-mediated modes; both of which play crucial roles in excitation-energy transfer24,25. Here, by using both ground-state and optical qubits, we achieve simultaneous control over mode frequencies, vibronic couplings, and dissipation rates. We characterize the low-temperature transfer rate across a wide parameter range, highlighting the roles of mode degeneracy and vibronic coupling strength. We focus on two transfer regimes (see Fig. 1a): charge transfer (CT) and vibrationally assisted exciton transfer (VAET), which correspond to strong and weak vibronic coupling, respectively. The distinction is made based on the differing phenomenological effects of the vibronic coupling on typical excitation and charge transfer processes6,26. When the vibrational modes are strongly coupled to the electronic states of the system, they reshape the potential energy landscape and actively influence the charge transfer reaction27. By contrast, weak couplings between the vibrational and electronic degrees of freedom primarily allow phonons to assist coherence between the electronic eigenstates28. We note that charge transfer can also occur in the weak vibronic coupling regime29. However, in this study, we use this term to refer specifically to the strong vibronic coupling regime, with which it is more commonly associated.

Results
The minimal multi-mode LVCM considers two vibrational modes and is described by the following Hamiltonian (ℏ=1)30:

H=Vσ + ΔE σ + σ + iσ a +ay +ωaya , (1)
x 2 z 2 z i i i i i
i=1

where σ are the Pauli operators acting on the donor and acceptor electronic sites, |D⟩ ≡ |↑⟩ and |A⟩ ≡ |↓⟩, with an energy difference ΔE and an electronic coupling strength V. For ΔE > 0, the Hamiltonian describes an exothermic reaction when the excitation is transferred from the donor site to the acceptor site. Each harmonic oscillator i with a vibrational energy ω is associated with creation (ay) and annihilation (a) operators, and it is linearly coupled to the electronic sites at a rate g.
i i

In this model, the CT regime is realized when the vibronic coupling is comparable to or larger than the harmonic frequency (g ≳ ω, also known as strong coupling in quantum optics), which is characteristic of many situations involving charge motion in chemical and biological reactions, ranging from redox catalysis to solvent-induced electronic delocalization27,31–33. The donor and acceptor electronic sites in this regime are described by uncoupled two-dimensional potential energy surfaces with respect to the (y1, y2) spatial coordinates, where y = y (a +ay) = 2 with y = 1=2mω and m being the particle mass. The electronic coupling V mixes the two potential energy surfaces by opening an avoided crossing with a gap of 2V at their intersection. A strong vibronic coupling (g ≳ ω) distorts the energy landscape by inducing a displacement of g/ω between the donor and acceptor potential energy surfaces along y, providing the electronic coupling with a dependence on the overlaps of the displaced oscillator states20,27,34. During CT, the donor population undergoes crossing of the energy barrier along an effective reaction coordinate, roughly defined by the axis y1 + y2 in the degenerate case. The total reorganization energy of the system is defined as λ = λ = g2=ω, which is the energy required to displace the wave packet on an uncoupled potential energy surface by the distances of |g/ω| and |g /ω | along the y and y axes, respectively. When the electronic coupling is small (V≈0), the reorganization energy λ and energy difference between the two surfaces ΔE determine the classical activation energy, defined by U=(ΔE+λ)2/4λ20,27.

Within the CT regime, there exist two electronic coupling regimes: the strictly nonadiabatic (or perturbative) regime with |V|<λ/4 and the strongly adiabatic (or non-perturbative) regime with |V|~λ/420,27,35. In the former, Fermi’s golden rule, where we treat the electronic coupling as a perturbation to the uncoupled donor-acceptor system, predicts that transfer dynamics occur when the quantized vibronic energy levels of the donor and acceptor surfaces match, resulting in transfer rate resonances at:

ΔE = ℓω + ℓ ω , (2)
1 1 2 2

with ℓ and ℓ being integers (see Section IIIA of the Supplementary Information). On the other hand, at large electronic coupling (|V|~λ/4), the avoided crossing becomes appreciable, leading to the hybridization of the two-dimensional donor and acceptor potential energy surfaces into upper and lower adiabatic energy surfaces (see Fig. 1c). This results in delocalized eigenstates that are superpositions of donor and acceptor vibronic states20,27. In this regime, Fermi’s golden rule no longer applies, which provides strong motivation to study these transfer conditions experimentally.

Conversely, weak vibronic coupling (g ≪ ω) is more characteristic of VAET, as occurs between pigments in light-harvesting compounds and their reaction centers14. In this regime, excitation transfer between donor and acceptor sites with an energy separation ΔE is enabled by the electronic coupling term Vσ, resulting in two eigenenergies of the total electronic system that differ by √ΔE2+(2V)2. Unlike in the CT regime, where strong vibronic couplings displace the donor and acceptor energy surfaces and thereby define the vibronic states that participate in excitation transfer, the couplings between the electronic system and the vibrational modes in the VAET regime are weak. From Fermi’s golden rule, these weak couplings perturbatively facilitate quantized energy exchange between the oscillators and the electronically coupled excitations itesthemselves, leading to transfer resonances at:

ΔE = √(ℓω + ℓ ω )2+(2V)2, (3)
1 1 2 2

with ℓ and ℓ being integers. At resonance, the combined vibrational energy ℓω + ℓ ω provided by the two modes exactly bridges the electronic energy gap, thereby assisting the transfer14,21,28 (see Section IIIB of the Supplementary Information).

In most realistic chemical situations, the interaction between the donor-acceptor vibronic system and the external environment causes the vibrational modes to undergo incoherent dissipation. Thus, the full Hamiltonian, Htotal = H + Hb + Hsb, also includes the bath degrees of freedom Hb, described by a large collection of continuous harmonic oscillators, as well as a linear coupling between the bath and the system’s vibrational modes, given by Hsb. The correlations of the environment and their influence on the system can be characterized by a continuous spectral density function J(ω). It is worth noting that the system contains both the electronic sites and the vibrational modes in Eq. (1). On the other hand, J(ω) does not represent the vibrational modes themselves, but rather the environment that causes them to dissipate. Under the assumptions of an Ohmic environment27,36 and vibrational dissipation rates that are weaker than the vibrational and thermal energies (γ ≪ ω, γ ≪ k T)37, the dissipative dynamics of the system can be effectively described by a Lindblad master equation:

∂ρ = -i[H,ρ] + γ (n +1)L [ρ]+γ n L [ρ] , (4)
∂t i i i a i i ay
i=1 i i

where L [ρ]=cρcy-1/2{cyc,ρ}, and n is the average thermal phonon number that describes the temperature of the environment for each vibrational mode ω with k T =ω=log(1+1=n). The vibrational excitations continuously evolve to equilibrate with the bath at a rate γ, leading to an irreversible population transfer from the donor site to the acceptor site20,27.

To experimentally realize the multi-mode LVCM, we use a dual-species chain of one 171Yb+ ion and one 172Yb+ ion trapped in a linear Paul trap20. We encode the electronic degrees of freedom in the two hyperfine clock states of the 171Yb+ ground-state qubit, |2S 1/2, F=1, mF=0⟩ ≡ |↑⟩ and |2S 1/2, F=0, mF=0⟩ ≡ |↓⟩ with the frequency splitting of ωhf = 2π×12.642 GHz (see Fig. 2a). In this work, we use the two radial out-of-phase modes (also referred to as tilt modes), each along an orthogonal radial principal trap axis, to represent ω and ω (see “Methods”).

Our experiment is performed in a driven rotating frame, where we use two π/2 pulses to map the z spin basis in Eq. (1) onto the y basis. This method allows us to use the ion-light interactions between the 171Yb+ qubit and the 355 nm Raman laser tones to independently engineer individual terms of the Hamiltonian in Eq. (1) with precise control. Two laser tones resonant with the qubit frequency generate the single-qubit operations (Vσ and (ΔE/2)σ terms), while two other laser tone pairs at frequencies ±μ =±(ω +δ) from the qubit resonance realize the vibronic coupling and harmonic terms in Eq. (1). Here, δ is the detuning from its respective radial tilt mode at frequency ω , which determines the vibrational energy ω through δ ≡ -ω20,38.

The engineered dissipation of the vibrational modes is realized by driving the narrow transitions from the ground-state manifold |g⟩ ≡ |2S 1/2⟩ to the optical metastable-state manifold |o⟩ ≡ |2D 3/2⟩ of the 172Yb+ ion, detuned by -ω from resonance, using a total of four laser tones to address all the involved Zeeman sub-levels. Together with a 935 nm repumper beam to optically reset the electronic excitation to the ground states, this results in sympathetic cooling on both radial tilt modes of the chain with independently tunable dissipation rates. This is equivalent to generating a structured bath of continuous harmonic oscillators with two Lorentzian spectral densities centered at ω and ω , with full widths at half maximum γ and γ , respectively21,37 (see Section IV of the Supplementary Information). In this setup, all the system parameters, including the bath properties, are determined by the frequency and power of the laser tones used to generate the corresponding ion-light interactions. Therefore, they can be precisely tuned and independently calibrated20, with the magnitudes of the system parameters used in our study being much larger than the decoherence rates caused by experimental imperfections (see Section II of the Supplementary Information).

In this work, we focus on the non-perturbative quantum regime27, where the electronic coupling strength is strong (|V|~λ/4), and the bath temperatures are low (n ≲ 0.1–0.2, see “Methods”). We compare the excitation transfer behaviors of the two-mode systems with those of their single-mode counterparts, which have been experimentally realized on trapped-ion simulators in refs. 14, 20, 21 to various extents. In the following, we characterize the transfer dynamics by measuring the inverse lifetime of the donor population, PD = (〈σz〉+1)/2, which is described by refs. 20, 27, 34, 35:

k = R P (t)dt : (5)
T tP (t)dt
D

This choice accounts for both the rate of dynamical equilibration and the steady-state population, which is analogous to the transfer efficiency, crucial for studying energy conversion in chemical processes.

We first study non-perturbative CT with g ≳ ω in two different cases: degenerate (ω = ω ≡ ω) and non-degenerate (ω > ω ), where the Fermi’s golden rule analysis breaks down. In the degenerate case, we observe in Fig. 3a that, in the two-mode model, the exothermic region characterized by monotonically increasing transfer rates with respect to the energy offset ΔE, caused by both the broadening effect of strong electronic coupling to off-resonant states and by the dissipation rates that limit the transfer rates (|V|>γ)20,27, extends to a higher energy gap, ΔE ≈ 4ω, compared to its single-mode counterpart (ΔE ≈ 3ω). This extension can be attributed to the larger number of state configurations available when two vibrational modes (two dimensions) are present, rather than just one (one dimension). We note that it is not possible to adjust the Hamiltonian parameters of a single-mode model to reproduce this observation.

Moreover, we remark that resonant peaks at ΔE ≈ ℓω, with ℓ being an integer, appear when the energy difference is sufficiently high (ΔE ≳ 5ω). Under this energy gap condition, the initially localized donor state has significant overlaps with the eigenstates of the upper hybridized surface, leading to population trapping, as explained in refs. 20, 27. At these resonant peaks, the trapped population is released from the upper hybridized surface to the lower hybridized surface during the evolution. Owing to the large energy offset and the increased number of state configurations in the two-mode degenerate case (relative to the single-mode case), the steady states of these resonant transfers exhibit greater overlaps with the acceptor states, as shown in Fig. 3b, c, where the final donor populations, PD(tsim), are closer to zero than the steady-state populations of the single-mode counterpart (see Supplementary Fig. 7b). This results in enhanced transfer rates at large energy gaps ΔE ≳ 5ω, compared to the single-mode case.

Conversely, when ω > ω , the donor-acceptor energy landscape supports highly delocalized states along y , as the system lies deeply in the adiabatic regime (|V|>λ/4)20,27. This introduces multiple states delocalized along the y direction, in addition to the existing delocalization along y provided by the fast vibrational mode levels. Together, these enable additional transfer channels among the highly delocalized two-dimensional vibronic states across the donor-acceptor energy gap. Thus, the transfer process in the non-degenerate case is less sensitive to the donor-acceptor energy offset, making it more robust to variations in this parameter. Due to our experimental resolution (see Section II of the Supplementary Information), we cannot resolve the resonances associated with these transfer channels and instead observe a smooth transfer rate curve in Fig. 3d. In addition, unlike the degenerate case, the transfer rates at the resonances with large ΔE are not evidently enhanced relative to those of the single-mode system. We emphasize that the non-degenerate two-mode spectrum observed in Fig. 3d cannot be obtained by tuning the parameters of a single-mode model.

To verify that the additional transfer pathways, provided by the slow vibrational mode, are responsible for the smooth transfer profile shown in Fig. 3d for the non-degenerate CT case, we decrease the electronic coupling strength and the vibronic coupling strength to the slow vibrational mode in Fig. 4a-c (see Section VII of the Supplementary Information for more details on VAET-CT crossover). Choosing a weaker electronic coupling (|V| < λ/4) while maintaining strong vibronic coupling to the fast mode (g > ω) place the system closer to the nonadiabatic CT regime along the fast mode direction. In this case, the donor and acceptor populations remain localized on their respective potential energy surfaces along y, and resonant excitation transfers occur between well-defined donor and acceptor vibronic states at ΔE ≈ ℓω, with ℓ being an integer20,27 (the one-dimensional case of Eq. (2)). In the single-mode scenario, where only y is considered, this results in the manifestation of the vibrational mode structure in the transfer rates spectrum (solid blue curve in Fig. 4a). Therefore, choosing a weaker electronic coupling allows us to distinguish the influences of the two vibrational degrees of freedom on the transfer dynamics when the second vibrational mode is involved. At the same time, lesser vibronic coupling to the slow mode reduces the distortion of the potential energy surfaces along y , making the transfer processes involving the slow mode to enter the VAET regime and introducing additional transfer rate resonances energetically enabled by the slow mode.

As shown by the resolved resonances in Fig. 4a, the additional peaks around the nonadiabatic transfer resonances of the fast mode, which coincide with the single-mode spectrum (solid blue curve), are induced by the second mode and located at ΔE = √(ℓω + ℓ ω )2+(2V)2 = ℓω + ℓ ω , where the ℓω contribution comes from the VAET process bridging the energy gap between vibronic states defined by the fast mode (see the discussion of the VAET regime below). The processes associated with ℓ = 1 dominate and are more clearly observable because of the perturbative nature of VAET. With either a larger electronic coupling strength V or a stronger vibronic coupling strength to the slow vibrational mode g , the additional resonances caused by the presence of the slow mode broaden and merge with the fast-mode transfer resonances into a smooth spectrum, as shown, for example, in Fig. 3d. This highlights the crucial role of the simultaneous presence of fast and slow vibrational modes in reducing the CT dependence on the donor-acceptor energy gap through the additional transfer pathways.

Unlike in the CT regime, where strong vibronic coupling distorts the donor-acceptor potential energy landscape that defines the vibronic eigenstates of the system, in the VAET regime (g ≪ ω), the vibrational modes are weakly coupled to the electronic degree of freedom and therefore only act as facilitators of the exothermic transfer, where units of vibrational energy are exchanged with the electronic sites during the excitation transfer at specific ΔE resonances, given by Eq. (3) (see Fig. 5)28. Here, we also consider systems in the non-perturbative regime, characterized by strong electronic coupling (|V|~λ/4). For the vibrationally degenerate case (ω = ω ≡ ω), shown in Fig. 5a, the transfer rate of the first resonance in the two-mode model is not enhanced compared to the single-mode case because only a single phonon from either degenerate mode can contribute to the transfer process at a time. In fact, the transfer rate becomes slightly slower due to the additional broadening from the dissipation of the second mode. On the contrary, at ΔE = √(2ω)2+(2V)2, VAET processes in which the total vibrational energy of 2ω is supplied by a linear combination of the energy quanta from the two degenerate modes (2ω = 2ω = ω + ω ) interfere constructively. This leads to enhanced transfer rates compared to the single-mode case, where only the process involving a vibrational energy input of 2ω can assist the transfer.

In the non-degenerate case (ω > ω ), the second-order processes occur at different donor-acceptor energy offsets, and their corresponding transfer resonances can be resolved experimentally (see Fig. 5c). As such, the presence of the slow mode does not affect the transfer rates at the resonances involving ω and 2ω energy inputs. This is because, when we choose ω/ω to be non-integer, there exists no non-trivial resonance involving an energy input of ℓ ω from the slow mode, where ℓω + ℓ ω = ℓω. However, additional resonances emerge instead from processes involving energy inputs given by linear combinations of the two non-degenerate vibrational energies. In Fig. 5c, we observe three additional resonances beyond the two existing resonances provided by the fast vibrational mode for the two-mode case. These peaks correspond to the processes involving vibrational energy inputs of ω , 2ω , and ω + ω .

It is worth noting that, in both cases of mode degeneracy, the second-order processes associated with a vibrational energy contribution of ω + ω consist of two sets of coherent pathways (one phonon from the fast mode, then another from the slow mode, and vice versa) that interfere constructively (see Fig. 1b). This coherent addition of the two rate amplitudes leads to an approximate two-fold enhancement of the peak at ΔE ≈ 1.63ω in Fig. 5c with respect to the neighboring peaks at ΔE ≈ 1.29ω and ΔE ≈ 1.97ω that are associated with the second-order processes involving 2ω and 2ω energy inputs, respectively (see Section V of the Supplementary Information). This observation suggests that approximately half of the enhancement at ΔE ≈ 1.95ω in the two-mode degenerate case with respect to the single-mode case is attributed to the ω + ω pathways (see Section V of the Supplementary Information). While constructive interference of the two vibrational modes requires full coherent control over both degrees of freedom, our results also demonstrate that this coherent enhancement remains resilient in the presence of dissipation.

Methods
Experimental setup
The experimental system used for this study is thoroughly described in ref. 20, where the dynamics of the single-mode LVCM in the CT regime are realized with four Raman 355 nm and two 435.5 nm laser tones. In this work, we include two additional tones to each beam to generate the terms associated with the second vibrational mode, resulting in a total of ten laser tones on the 355 nm and 435.5 nm lasers, over which we have full control of both amplitudes and frequencies. We also use the collective modes of the two-ion chain along both radial directions of the trap, specifically the y and z tilt modes (ω ≡ ω = 2π×3.151 MHz and ω ≡ ω = 2π×3.740 MHz), to encode the vibrational degrees of freedom in the two-mode LVCM Hamiltonian. The unused radial collective modes are the center-of-mass modes with ω ≡ ω = 2π×3.318 MHz and ω ≡ ω = 2π×3.882 MHz. Since the frequency separations among the available radial collective modes are set to be sufficiently large (Δω ≳ 2π×140 kHz), undesired off-resonant spin-phonon interactions, which are proportional to g/(μ - ω ), can be neglected23,52. Here, μ is the 355 nm laser frequency detuning from the qubit resonance, used to generate the spin-dependent optical force on the target radial tilt mode i, and ω is the relevant collective radial frequency considered for the off-resonant spin-phonon drive (j≠i).

Experimental sequence
The experimental sequence (see Fig. 2b) begins with Doppler cooling and Raman-resolved sideband cooling on all four collective radial modes of the chain, which results in an initial phonon population of both radial tilt modes, n ≲ 0.1–0.2, which is set to match the independently measured n. We then apply a π/2 pulse to map the z qubit basis to the y basis and two consecutive displacement operations via the spin-dependent optical force to prepare the system in the donor state |D⟩ ≡ |↑⟩, where ρ = |↑⟩⟨↑| ⊗ |n1⟩⟨n1| ⊗ |n2⟩⟨n2|, and |n i⟩ = D(±g/2ω) |n i⟩ are displaced Fock states associated with vibrational mode i. For simulating the open-system LVCM dynamics described by Eq. (4), we simultaneously apply the six 355 nm, four 435.5 nm, and two 935 nm laser tones. After evolving the system for time tsim, we apply another π/2 pulse to map the quantum state in the y basis back to the z qubit basis and measure the probability of the system being in the donor state PD = (〈σz〉+1)/2 via state-dependent fluorescence.

Transfer rate data analysis
We experimentally simulate the LVCM dynamics from t = 0 ms to t = tsim, where the finite simulation time tsim ranges from 2 to 8 ms, corresponding to 24–65 vibrational cycles of the fast mode (defined by ωt/2π), within which the system undergoing the first transfer resonance reaches equilibrium. Since Eq. (5) defines the transfer rate in the limit of t → ∞, an offset correction is required when applying it to finite-time dynamics. Particularly, in the case of a strictly localized donor population, PD(t)=constant, Eq. (5) still yields a non-zero transfer rate of k = 2/tsim, which approaches zero only as t → ∞. Therefore, we need to remove the undesired background from the finite-time evaluation of the non-zero steady-state donor population in the transfer rate calculations, as follows20,35:

k = R tsim PD(t)dt / (R tsim t PD(t)dt - tsim/2) : (6)
T 0 0

We note that this background correction does not alter the characteristic features of the transfer rate spectra. By interpolating the donor population probability PD(t) for both the experimental and numerical data using the same time steps and applying the modified formula above, we obtain the transfer rates of the dynamics reported in the main text. As in ref. 20, we also use a resampling (bootstrapping) procedure at each time step of the PD(t) measurements to estimate the uncertainties of the extracted transfer rates. For each time step, we treat the experimental error as the standard deviation of a normal distribution centered at the measured mean value, from which the resampled datasets are drawn. We then use the standard deviation of the transfer rates calculated from the resampled datasets as the uncertainty associated with the reported rates.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The manuscript by So et al. directly builds on the authors’ previous work published in science advances (DOI: 10.1126/sciadv.ads8011). From an experimental standpoint, there is almost no change or additional expansion beyond using a second motional mode - a very minor, incremental change. This can nicely be seen by comparing Figure 1 of the present manuscript with Figures 1 and 2 of their previous work.\nAs the experimental part appears (almost) identical (as do the figures), this leaves the application side to be evaluated. Here I find a shift in framing of the scientific question from electron transfer (previously) -> charge transfer (now) and the addition of another application (VAET) corresponding to another operating regime (slight tuning of laser / trap parameters).\nThe present manuscript effectively frames the use of their simulator in the context of chemistry, which also differs from their previous work.\nOverall, however, I do not see sufficient novelty in any way to warrant publication in nature communications and am rather disappointed in the significant overlap in figures, equations and presentation. I therefore recommend the submission to be rejected and would recommend to the authors to reconsider how much novelty this presents and to be more open about the significant overlap (incremental aspect) of their present work.", "reviewer_2": "The manuscript \"Quantum Simulation of Charge and Exciton Transfer in Multi-mode Models with Engineered Reservoirs\" reports a quantum simulation of excitation transfer in a chemically motivated model that couples spins to bosonic modes, capturing the electronic and vibrational degrees of freedom of molecular systems. In contrast to previous realizations, this work successfully implements a two-mode model with a high degree of tunability. This represents a significant experimental challenge, as it requires precise control over multiple degrees of freedom in several trapped ions, as well as a multi-species setup capable of handling different isotopes.\nThe simulation demonstrates impressive control over key parameters, including independent tuning of dissipation rates for both vibrational modes and of the couplings between electronic and vibrational degrees of freedom. This level of control enables the authors to explore real-time dynamics in physically relevant regimes, including cases with degenerate vibrational modes. As a result, the setup allows for a detailed investigation of charge transfer mechanisms and vibrationally assisted exciton transport, phenomena that are central to many biological processes and have been widely studied in theory.\nThe manuscript is well written, and the data are analyzed and presented clearly. Overall, this work marks a significant milestone in the relatively underexplored area of quantum simulations of molecular electron and exciton transfer. It opens up several compelling theoretical questions—such as the potential role of interacting multimode environments in enhancing energy transport, and is likely to inspire further experimental studies in this direction.\nFor these reasons, I believe the manuscript fits well within the scope of Nature Communications, and I support its publication.", "reviewer_3": "This manuscript “Quantum Simulation of Charge and Exciton Transfer in Multi-mode Models with Engineered Reservoirs” used trapped-ion quantum simulation to model charge and exciton transfer in a multi-mode (2 modes) vibronic system, tuning between charge-transfer and vibrationally assisted exciton-transfer regimes via controlled vibronic couplings. The results show how degenerate and non-degenerate vibrational modes differently influence transfer efficiency, offering insights for designing advanced molecular electronics and photovoltaics. In this draft, the results align well with the theoretical model developed in this study and support the paper’s stated hypotheses, making it a thorough and commendable work. However, the explanations lack clarity for non-specialist readers, the figures have low resolution, and the novelty is limited since the experimental method and theory closely follow Ref. 20 and Ref. 28, respectively. For publication in Nature Communications, the authors should better highlight the novelty, present a clearer explanation, and address the following questions:\n1. On page 3, the authors state, “we present the first … in a two-mode LVCM.” Is this claim of being the “first” sufficiently justified? The experimental method appears quite similar to that in Ref. 20, and although Ref. 20 is from the same group, the method itself has already been published.\n2. In page 3, “We focus on two transfer regimes … which correspond to strong and weak vibronic coupling, respectively”. Readers need more physical context and explanation on why CT and VAET correspond to strong and weak vibronic coupling, respectively. There is only a footnote here; appropriate citation is necessary.\n3. The resolution of all the figures is quite low.\n4. Fig. 1A: Wouldn’t it be clearer to draw the same picture (2 parabolas representing energy surfaces) but with different parameter regimes for VAET and CT? See Fig. 3a and 4a of [F. Schlawin et al., PRX Quantum 2, 010314 (2021)], for example. As the only difference between VAET and CT seems to be the vibronic coupling strength, it is hard to understand why reaction coordinate (y) is not shown for VAET. Also, avoided crossing is hardly visible in the CT panel.\n5. The labels in Fig1 C are too small to read\n6. In pages 4 and 5 “In this model, the charge transfer (CT) regime is…”: While the concepts such as avoided crossing and reorganization energy are well explained, this paragraph lacks explanation on what phenomena happen, such as how the donor population transfer depends on the energy gap. This contrasts with the next paragraph that explains the VAET regime, which clearly mentions that energy exchange is facilitated at certain resonance frequencies. It is thus hard to understand what to expect for the CT regime as opposed to the VAET regime.\n7. In pages 4 and 5, the authors present one page of theoretical discussion, but the connection between this theory and the experimental results is not clearly established. I suggest condensing this section and integrating the theoretical explanations alongside their corresponding experimental results or calling back the theory when discussing the experimental data.\n8. In page 6, Right above Eq. 2: “Under the assumptions of linear coupling between the vibrational modes and a Markovian environment with Ohmic spectral density” is confusing. For anyone who did not read Ref. 31, it is hard to understand that the Ohmic spectral density here corresponds to the external bath that couples to the reaction coordinate, as typically the reaction coordinate (represented as the vibrational mode) also constitutes the “environment”.\n9. In page 7, Right above Eq. 3: This paragraph starts with “In this work, we focus on the non-perturbative regime, where the electronic coupling strength is strong (|V|~λ/4). However, later in the Transitioning to VAET section, suddenly weak electronic coupling regime (|V|<λ/4) is considered.\n10. Overall, it is hard to understand why Transitioning to VAET section should consider the weak electronic coupling regime, if the difference between CT and VAET is the vibronic coupling strength. This is especially confusing as the difference of energy transfer dynamics between the CT and VAET regimes is poorly explained in the introductory paragraphs.\n11. In page 8, under the Charge transfer session, “In the degenerate case … This extension can be explained by the larger phase-space volume of the adiabatic energy surfaces when there are two modes available rather than just one.” What does this broadening effect mean and what does it affect? The transfer rates are monotonically increasing w.r.t. which variable? Why do we care about whether the two-mode case extends this effect from 3ω to 4ω?\n12. Can the upper boundary of the exothermic region (ΔE≈4ω for two modes, 3ω for one mode) quantitatively be explained? Why 4ω and 3ω?\n13. In page 8, “Moreover, we remark that resonant peaks at ∆E ≈ lω, with l being an integer, appear when the energy difference is sufficiently high (∆E >5ω)” I think the author means even at sufficiently high energy gap (E>5w) the transition speed is still large compared to other cases (non-degenerate ). Right? But this current explanation is not expressing in this way, it sounds like the resonance only appears at high energy gap.\n14. Same paragraph as question 13: It is hard to understand what the comparisons are against. This issue is the most severe in this paragraph but is persistent throughout the entire manuscript. (i) “greater overlaps with the acceptor states” – I assume the authors mean that the overlaps are greater for the two-mode case than for the one-mode case, but this is not shown in Figs. 2B and 2C because one-mode case is simply not shown. (ii) “This results in enhanced transfer rates at large energy gaps” – enhanced compared to what? Single-mode case? Smaller energy gaps?\n15. In page 8, “upper hybridized surface”, “lower hybridized surface”, “phase-space volumn” are not common phrases for general audience. It’s better to explain them or using figures to demonstrate them for better understanding.\n16. In page 8, “Conversely, … delocalized two-dimensional vibronic states.” These explanations are not easy to understand. The author refers to Fig 1A but Fig 1 A doesn’t have enough labels to help to understand, for example, there is no y2 or \"multiple states delocalized along y2\" explicitly shown in the figure.\n17. In page 9, “Due to our experimental resolution (see Supplementary Information II)”: I checked SI but have no idea where the experimental resolution is explained.\n18. Second paragraph of Transitioning to VAET section: Why are the peaks at ΔE≈lω_1+ω_2? (Why coefficient of ω_2 is 1?)\n19. Fig. 4: It would be nice to label the ΔE values used in Fig. 4B at Fig. 4A with the corresponding colors. Similar for Fig. 4D and 4C.\n20. Eq. 4 – Despite the explanation given, I couldn’t understand why 2/t_sim should be subtracted. Why simulating t = 2-8 ms instead of 0-8 ms? This is different from Eq. 3.\n21. Supplementary Information II: The choice of jump operation for the motional dephasing seems weird; c_im=a_i^† a_i is more natural? See Ref. 21 for example.\n22. Supplementary Information VII: To my understanding, “nonadiabatic” and “adiabatic” refer to the weak and strong electronic coupling regime, respectively. So, it was hard to understand why “the system progressively evolves into the nonadiabatic regime of CT” as the vibronic coupling strength g/ω is increased beyond 0.5.\n23. Supplementary Information VII: From looking at Fig. S6B it is hard to judge why g/ω=0.5 has to be the crossover point between VAET and CT. Could there be a metric for defining this crossover? At least, a qualitative justification of the crossover point is needed.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We would like to resubmit our revised manuscript, entitled “Quantum Simulation of Charge and Exciton Transfer in Multi-mode Models using Engineered Reservoirs,” for consideration in Nature Communications. We have carefully considered the Reviewers’ comments provided in the peer review and have made substantial revisions to address those concerns. Enclosed with this submission is a detailed point-by-point response to the reviews, outlining the changes made to strengthen the manuscript. We also attach a version of the manuscript with major changes tracked for clarity. We believe that this revision better highlights the impact and novelty of our work, making it suitable for publication in Nature Communications.\n\nReviewer 1\n[...]\nWe thank the Reviewer for taking the time to evaluate our work. Her/his critiques helped us realize that the impact and novelty of our study were not sufficiently emphasized in the previous version of the manuscript. Therefore, we would like to clarify the motivation behind our presentation and highlight the distinct advances of this manuscript.\nFirstly, we deliberately chose to maintain a similar presentation of our setup and results to highlight the continuity between this work and our previous study (Ref. [20]). Despite some similarities, we respectfully disagree with the Reviewer’s assessment that adding one vibrational mode with controlled coherent drive and engineered dissipation represents only “a very minor, incremental change.” The present work is substantially more experimentally challenging than the setup in Ref. [20], as it requires precise control of twelve laser frequency tones and extensive calibration of eight fully tunable parameters governing the quantum evolution (see Eqs. (1) and (4)). While the addition of another mode, including reservoir engineering, might appear straightforward, it is far from trivial in practice. As rightly noted by Reviewer 2, this advance constitutes a significant milestone in quantum simulation experiments of non-equilibrium chemical dynamics, as it demonstrates a clear pathway toward scalability for studying systems with a larger number of engineered bosonic modes.\nSecondly, the new experimental capabilities demonstrated in our work have enabled us to pose new scientific questions and observe phenomena unique to two-mode systems, which cannot be readily inferred from the single-mode model investigated in Ref. [20]. Specifically:\n• We carefully mapped out how transfer dynamics in the charge/electron transfer regime are influenced by the energy degeneracy of vibrational modes, with transfer rate spectra that cannot be reproduced by simply renormalizing a single-mode model.\n• As the Reviewer also noted, we further studied the system in another operating regime, vibrationally assisted exciton transfer, thereby providing deeper insight into the role of vibrational coherence in the dynamics.\n• We experimentally observed transfer peaks stemming from vibrational coherence between the two phonon modes. Crucially, this result is both new and somewhat unexpected, given that both bosonic modes are subject to damping.\n• We showed in the Supplementary Information that our results can also be extended to the same class of multi-mode systems with more than two vibrational modes, which are inaccessible to study using an experimental setup featuring only a single mode.\nFinally, we emphasize that low-frequency modes associated with protein scaffold dynamics in light-harvesting complexes play a crucial role in energy transfer. Unlike intramolecular modes localized on pigments, low-frequency modes can influence multiple sites, sustaining correlated energy fluctuations across them. In our work, we simulate two tunable intramolecular modes that are coupled to an Ohmic bath, which typically models the protein environment in light-harvesting complexes. Therefore, our study, comprising Hamiltonian and reservoir engineering with precisely controlled coupling strengths and dissipation rates, represents an important step toward the simulation of realistic molecular systems.\nIn light of these points, we have revised the manuscript to emphasize more clearly the additional experimental challenges, the new scientific insights, and the distinctions from our previous work, so that these contributions are now clearer to the readers. Specifically:\n• We have added a sentence to the first paragraph of page 3 to explain the relevance of our experimental realization in the broader context of excitation-energy transfer. The sentence reads as follows: “Engineeringspin-phononsystemswithmultiplebosonicmodes, combined with environment engineering, represents an important step toward realizing models of molecular systems in which multiple intramolecular (fast) modes coexist with long-wavelength, low-frequency environment-mediated modes—both of which play crucial roles in excitation-energy transfer.”\n• We have separated Fig. 1 into two figures: one emphasizing the new aspects of the two-mode models with additional details on the two vibronic coupling regimes, and the other focusing solely on the experimental aspects.\n• We have emphasized that our results cannot be reproduced by tuning parameters in single-mode models in multiple places.\n• We have pointed out that the enhancement in the transfer rates, attributed to coherence in the transfer pathways, is resilient to dissipation and represents a new observation.\n• We have listed the additional calibrations required for the study of the two-mode models in Supplementary Information I to emphasize that this work is a significantly more complex undertaking than our previous study.\nWith these considerations in mind, we would greatly appreciate it if the Reviewer could reevaluate the impact and novelty of our work, and we hope she/he will conclude that our manuscript is suitable for publication in Nature Communications.\n\nReviewer 2\n[...]\nWe thank the Reviewer for recognizing the quality and importance of our work and for supporting its publication in Nature Communications.\n\nReviewer 3\n[...]\nWe thank the Reviewer for raising the concern regarding our claim of “we present the first ... in a two-mode LVCM.” Although we employed the same technique for open-system quantum simulation of a single-mode LVCM in Ref. [20], the current work represents the first experimental application of the technique to multi-mode systems, which is why we used that wording.\nThere are two reasons we want to emphasize that this work is the first experimental demonstration of a multi-mode LCVM with independently programmable unitary evolution and mode-selective dissipation:\n1. Applying the technique used in Ref. [20] to two bosonic modes while retaining independent control of both the Hamiltonian and dissipation parameters is a non-trivial experimental achievement, as it requires much more extensive calibrations for both the system’s evolution and the state preparation.\n2. These new capabilities enabled us to pose new scientific questions and observe phenomena unique to two-mode systems, ranging from the role of mode degeneracy to the observation of vibrationally coherent addition of transfer pathways, which cannot be accessed by a single-mode system.\nTaking these considerations into account, we believe it is reasonable to maintain our claim.\n[...]\n(Responses to points 2-23 omitted for brevity in this example, but would be included verbatim here)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "I appreciate the work the authors put into revising their manuscript in light of all the referees’ comments, which certainly improved the manuscript overall.\nHowever, I still remain unconvinced that the work passes the bar for publication in Nature Communications as it does not mark a significant (experimental) advance or insight over the authors’ previously published work.\nAdding coherent control over a second motional mode may have been a significant advance for the authors, advancing their experimental capability. However, this was already demonstrated in the following manuscripts, in one case even using the same ion species:\nValahu, C.H., Olaya-Agudelo, V.C., MacDonell, R.J. et al. Direct observation of geometric-phase interference in dynamics around a conical intersection. Nat. Chem. 15, 1503–1508 (2023). https://doi.org/10.1038/s41557-023-01300-3\nWhitlow, J., Jia, Z., Wang, Y. et al. Quantum simulation of conical intersections using trapped ions. Nat. Chem. 15, 1509–1514 (2023). https://doi.org/10.1038/s41557-023-01303-0\nAlbeit, slightly different, coherent vibrational transfer between phonon modes has also previously been demonstrated, including transfer enhancement under noise - just not presented with a quantum chemistry framing:\nEngineering Vibrationally Assisted Energy Transfer in a Trapped-Ion Quantum Simulator\nDylan J Gorman1, Boerge Hemmerling1,*, Eli Megidish1, Soenke A. Moeller1, Philipp Schindler2, Mohan Sarovar3, and Hartmut Haeffner1\nPhys. Rev. X 8, 011038 – Published 7 March, 2018\nDOI: https://doi.org/10.1103/PhysRevX.8.011038\nEnvironment-Assisted Quantum Transport in a 10-qubit Network\nChristine Maier1,2, Tiff Brydges1,2, Petar Jurcevic1,2, Nils Trautmann3,†, Cornelius Hempel1,2,4, Ben P. Lanyon1,2, Philipp Hauke5,6, Rainer Blatt1,2, and Christian F. Roos1,2,*\nPhys. Rev. Lett. 122, 050501 – Published 8 February, 2019\nDOI: https://doi.org/10.1103/PhysRevLett.122.050501\nIn summary, what’s new in the present manuscript is the use of previously demonstrated (by other and the same authors) techniques in the context of Charge and Exciton transfer. As to whether this alone constitutes a novel-enough advance to warrant publication in Nature Communications, I defer to the editors judgement.", "reviewer_2": "I continue supporting publication of the manuscript in Nature Communications.", "reviewer_3": "Revision is excellent. I especially appreciate that the flow from CT to “Transitioning to VAET” to VAET is now clearly explained and well-motivated. Besides the remarkable experimental implementation, this manuscript’s thorough explanation of the physics of vibronic model in various regimes is valuable in its own right. I support the revised version for publication.\nJust a few remaining comments:\n1. Fig.1A, the horizontal axis is y1/y_10, while the axis in Fig.1C is y1/y_01. Please keep them consistent if these are not intended. Also, the blue and red color assignment of Fig.1A and 1C are inconsistent.\n2. Line 12 of pg 3 – I also suggest citing [N. Lorenzoni et al., Sci. Adv. 11(40), eady6751 (2025)].\n3. Above Eq. 4 – While this part is somewhat clarified, I suggest clarifying even more, explicitly stating that here the “system” is vibronic (containing both the electronic states and the vibrational modes), and that J(\\omega) does not represent the vibrational modes but rather the environment that causes the vibrational modes to dissipate. Also, I don’t understand “Markovian environment with Ohmic spectral densities” – Markovian environment by definition has a flat (constant) spectral density.\n4. Fig.2A, the direction of the purple arrow does not indicate Raman transition between up and down states. They should be one goes up and another one goes down. Or both double-head arrows.\n5. 7th-to-last line of pg.10 – “Fermi golden-rule analysis” would be more specific than “approximate analysis”.\n6. Fig. 4 caption - \\gamma_{im}=0.0100\\omega_1 has too many significant digits. Hard to believe that motional dephasing rate is characterized this precisely. Also, last sentence should go to caption for (A) panel.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "We would like to resubmit our revised manuscript, entitled “Quantum Simulation of Charge and Exciton Transfer in Multi-mode Models using Engineered Reservoirs,” for consideration in Nature Communications. We have carefully considered the Reviewers’ comments provided in the peer review and have made revisions to address those concerns. Enclosed with this submission is a detailed point-by-point response to the reviews.\n\nReviewer 1\n[...]\nWe thank the Reviewer for taking the time to evaluate our work. However, we respectfully disagree with the Reviewer’s assessment that our study lacks the experimental advances or insights required for publication in Nature Communications and stand by the arguments presented in our previous response. We are also aware of the following works and have already cited them in the appropriate context; they appear in our main text as Refs. [15, 16, 14, 19], respectively. In those studies, the setups use only one ion species and do not include vibrational dissipation, an element unique to our system. As Reviewer 3 noted, our work is also distinctive in demonstrating charge transfer, vibrationally assisted exciton transfer, and their crossover with engineered vibrational dissipation on two independent modes within a single framework. Moreover, these capabilities enable us to probe the role of mode degeneracy, which was impossible in our previous work.\n[...]\n\nReviewer 2\n[...]\nWe thank the Reviewer for their continued support for the publication of our work in Nature Communications.\n\nReviewer 3\n[...]\nWe thank the Reviewer for their kind words on the revision and for endorsing its publication in Nature Communications. We have addressed the Reviewer’s remaining comments as follows:\n[...]\n(Responses to points 1-6 omitted for brevity)"}]

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