Long-range non-equilibrium coherent tunneling induced by fractional vibronicresonances

R. Kevin Kessing,1, 2, 3 Pei-Yun Yang,4, 2 Salvatore R. Manmana,1, 5 and Jianshu Cao2

1Institut fur Theoretische Physik, Georg-August-Universitat Gottingen, 37077 Gottingen, Germany2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

3Institut fur Theoretische Physik, Universitat Ulm, 89069 Ulm, Germany4Beijing Computational Science Research Center, Beijing 100193, China

5Fachbereich Physik, Philipps-Universitat Marburg, 35032 Marburg, Germany(Dated: November 12, 2021)

We study the influence of a linear energy bias on a non-equilibrium particle on a chain withstrong coupling to local phonons using both a random-walk rate kernel theory and a nonperturba-tive, massively parallelized adaptive-basis algorithm. We uncover structured and discrete vibronicresonance behavior fundamentally different from both linear response theory and unbiased-chainpolaron dynamics. Remarkably, resonance between the phonon energy ~ω and the bias δε occursnot only at integer but also fractional ratios δε/(~ω) = m

n, which effect long-range n-bond m-phonon

tunneling. Potential applications range from molecular electronics to optical lattices and artificiallight harvesting via vibronic engineering of coherent quantum transport.

Introduction—The quantum dynamics of charge car-riers and excitations in molecular systems is of greatsignificance to a variety of research areas ranging fromphysics to material science, chemistry and biology. Theinfluence of nuclear or molecular vibrational degrees offreedom, which are quantized into phonons, can be ofgreat importance to such dynamics. The Holstein modelis a prototypical model for such electron–phonon (vi-bronic) coupling and has been widely used to describetransport in various systems such as polymers, molecu-lar aggregates, and semiconductors [1–8]. A further cru-cial ingredient to such dynamics is an energy gradientor “tilt” representing, for example, a voltage gradientin measurements of charge mobility or a natural energyfunnel [9, 10]. Such biased vibronic systems exhibit var-ious important coherent electron–phonon transport ef-fects which have recently attracted much attention [11],e.g., “phonon-assisted resonant tunneling” in inelastictunneling experiments [12–14] or the “Franck–Condon(FC) blockade” in quantum dot or molecular tunnel-ing experiments [15, 16], as well as vibrational enhance-ment of transport in antenna protein complexes [9, 17–19]. However, theoretical treatments of these phenomenahave used perturbative approaches to calculate steady-state currents through a single site [15], were restrictedto single-phonon transitions [20] or were limited to thelinear-response regime [21].

We report striking higher-order fractional and long-range resonances that occur within the “forbidden” FCblockaded regime when multiple sites are concatenatedand their full non-equilibrium dynamics calculated non-perturbatively. Phononless long-range resonant tun-neling has been studied theoretically and experimen-tally in cold-gas quantum simulators on tilted opticallattices [22–31], and such setups have been employedto investigate a plethora of phenomena such as quan-tum magnetism [22, 25, 30, 31], quantum dimer mod-

} δε

J

{hω

{ghνa)

site j − nsite j

site j + n

for δε = 1nhω

b)

j − 4 j − 2 site j j + 2 j + 4

c) local particle densityafter time t = 8 2π

ωδε = 0δε = 1

2 hω

FIG. 1. a) Sketch of the system: On the central site, aFranck–Condon (FC) excitation is induced, which tunnels toneighboring sites with amplitude J and interacts with localphonons ~ω via Holstein coupling g in the presence of a lin-ear potential bias δε. b) Resonant transitions are possible be-tween local eigenstates of the excited PES. The FC excitationis a superposition of many different local eigenstates, allow-ing for multiple resonant vibronic transitions to both sides.c) The numerically obtained density for an n = 2 vibronicresonance vs. the homogeneous case (for J = ~ω, g = 4~ω).

els [24], transport properties and dynamical phase tran-sition points [26, 27, 32] or the creation of anyons [33].We show that vibronic coupling naturally realizes andgeneralizes such resonant tunneling behavior.

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Model—Figure 1a illustrates our setup: a chain of Llinearly tilted molecular sites which interact via nearest-neighbor coupling J and couple linearly to local vibra-tions. These vibrational degrees of freedom are repre-sented by a single dominant mode, assumed to be har-monic with frequency ω, and each molecular site is cou-pled strongly to its own vibrational mode. The couplingto the vibrations is quantified by the parameter g, related

to the Huang–Rhys parameter as S =(g~ω)2

. A lineartilt shifts the energy of each site j relative to site j − 1by δε. The resulting Hamiltonian is:

H =

L−1∑

j=0

[− J

(|j〉〈j + 1|+ h.c.

)+ ~ω

(b†jbj +

1

2

)

+ gnexc.j

(bj + b†j

)+ δε jn

exc.j

], (1)

where |j〉 is the excitonic state localized on site j (where〈i|j〉 = δij); nexc.j = |j〉〈j| is the number of excitons

on this site (restricted to 0 or 1); bj (b†j) are bosonicannihilation (creation) operators for vibrational quanta

(phonons) on lattice site j; and ~ω(b†jbj + 1

2

)is the local

harmonic vibrational Hamiltonian on site j. We investi-gate the dynamics of a single Franck–Condon (vertical,diabatic) excitation from the vibrational and electronicground state, which is initially localized to the centralsite. To accommodate the strong vibronic coupling, wetruncate the vibrational Hilbert spaces at 128 phononsper mode, following [34].

The model allows for multiple interpretations in differ-ent contexts: In an excitonic model, |j〉 represents thefirst excited state at the j-th molecule with electronicground states on the remaining molecules. However, |j〉can also stand for a charged state arising from injectingan electron into the j-th molecular site, or for an atomplaced at a certain position in an optical lattice. Ourfindings can therefore be applied to various situationssuch as energy transfer in light harvesting, hopping ofa bosonic particle in a cold gas, charge mobility in or-ganic semiconductors, or non-equilibrium transport in amolecular junction with a potential bias.

Methods—To present a complete dynamical picture,we have developed both a numerical algorithm and an an-alytical approach. Our massively parallelized numericalmethod is based on dynamically adapted effective basissets, similar to a repeated dynamic use of “limited func-tional spaces” [35, 36], which can be applied in principleto arbitrary quantum dynamics problems. By exploitingHilbert-space localization of physically relevant states,the method computes dynamics only in the most relevantsubspace, which is adaptively reconfigured to follow theevolving wavefunction. The parallel nature of the linear-algebra problem is further leveraged by optimizing themethod to be run on graphics processing units (GPUs),

0.2

0.4

RM

SD

J = 110 hω

1.00

1.25

1.50

1.75J = hω

0.0 0.5 1.0Tilt parameter δε/(hω)

−0.02

−0.01

0.00

c.m

.pos

itio

nX−L 2

0.0 0.5 1.0 1.5Tilt parameter δε/(hω)

−0.4

−0.2

0.0

numericspert. theory

t = 1.5 2πω

t = 2.5 2πω

t = 3.5 2πω

t = 4.5 2πω

FIG. 2. The excitonic RMSD and movement of the aver-age exciton position in a tilted Holstein chain for a series oftilt parameter values δε after different times t. Left: pertur-bation theory (lines) and numerics (dots) for weak hoppingJ = 1

10~ω. Right: numerical data for strong hopping J = ~ω.

For all data: L = 9, g = 4~ω (strong vibronic coupling).

granting a substantial boost in runtime efficiency overexisting methods. See the supplemental material for fur-ther details [37]. To verify several important featuresof these numerical findings, we develop a random walkmodel with transition probabilities obtained from first-order-J perturbation theory and evaluated using pathintegrals, as described below.Resonance enhances transport—The Holstein dynam-

ics are calculated for a range of values of the tilt param-eter δε and two different values of J while keeping theremaining parameters fixed. In Fig. 2, the RMSD (thestandard deviation of the spatial distribution), defined as

RMSD =√σ2 :=

√√√√L−1∑

j=0

⟨nexc.j

⟩ [j −X

]2,

and the average position of the particle, X :=∑L−1j=0 j

⟨nexc.j

⟩, at different times are plotted as a func-

tion of the tilt parameter δε. Interestingly, the RMSDas a function of δε is highly non-linear, exhibiting strongspikes around integer and half-integer values of δε/(~ω).Additionally, smaller spikes appear around δε/(~ω) ∈{ 13 , 23 , 43}. Between these values, propagation is stronglysuppressed, suggesting transport-enhancing resonancesare realized at certain values of δε. Varying the remainingHamiltonian parameters reveals that the location of themaxima depends only on the ratio δε/(~ω) (SM [37], sec-tion 2). Note that the observed resonances for |δε| < ~ωlie within the “forbidden” FC blockade regime [15].Resonance produces spatially multimodal states—

Furthermore, Fig. 3 shows that the dynamics of the lo-

3

FIG. 3. The time evolution of the tilted Holstein chain excitons shown in Fig. 2 for the a) homogeneous (δε = 0) case, and theb) third-order resonant (δε = 1

3~ω) and c) second-order resonant (δε = 1

2~ω) cases. The insets show the exciton distribution at

the final time t = 8 2πω

. In the second- and third-order resonant tilted cases, the exciton does not simply propagate outwardsfrom a central peak into smooth tails, but instead excitonic density peaks form at the resonant sites, located n sites from theinitial site for δε = 1

n~ω.

cal density in the tilted chains behaves completely differ-ently than in an untilted chain. For δε = 0, the excitoninitially propagates into a symmetric, peaked state thatextends over a few lattice states, before self-trapping setsin and prevents further significant delocalization (Fig. 3a,cf. [34]). However, if δε is set to a resonant value, we findthat the final state is no longer a single-peak state, butnow exhibits multiple peaks and dips in its local excitondistribution. Further numerical results (SM [37], section3) explicitly show that the density n sites off-center ismaximized when δε is a multiple of 1

n~ω.

This brings us to our main insight: When the parame-ters are resonant as δε/(~ω) = m

n with n and m small [38]integers, then an excitation that is initially localized atsite j will tunnel to sites j ± n (and from there to sitesj±2n, etc.). In other words, the nearest-neighbor tunnel-ing behavior of an unbiased chain is replaced by phonon-mediated (vibronic) tunneling over n sites, leading tospatially multimodal and inhomogeneous states—or ef-fectively suppressing the tunneling if n is too large orδε/(~ω) /∈ Q. This n-site tunneling is the origin of theRMSD spikes seen in Fig. 2.

Remarkably, certain resonant shifts δε increase theoverall diffusivity, e.g., δε = ~ω

2 in Fig. 1c and Fig. 2.Propagation is then enhanced in both the “uphill” and“downhill” direction, even for iterated resonant tunnel-ing processes (j → j ± 2 → j ± 4). We consider this tohave potential applications in nano- and mesoscale sys-tems, which one may, by exploiting this phenomenon,vibronically engineer to optimize energy transport.

The resonance mechanism—The observed resonanttunneling is due to matching energies between vibronicstates at different sites. The resonant vibronic statesmust be considered relative to the electronically excited-state potential energy surface (PES), whose vibrationaleigenstates we denote as |ν′〉 while those of the the ground

state PES are denoted |ν〉. As shown in Fig. 1, whenexpressed in the excited PES eigenbasis, the Franck–Condon |ν = 0〉 state is a g-dependent superposition ofmany different |ν′〉 eigenstates, given by the coherentstate formula [39] with α = g

~ω . Then, if δε = ~ω, everyconstituent |ν′〉 state of a Franck–Condon state is isoen-ergetic with the state |ν′ + 1〉 at the downhill neighboringsite and |ν′ − 1〉 at the uphill neighbor, opening up mul-tiple resonant transition pathways. More generally, forδε = m

n ~ω, the resonant transitions are mediated by m-phonon and n-fold hopping matrix elements. Further,once the particle has tunneled from a state |ν′〉 at site jto |ν′ ±m〉 at site j ± n, the process can be successivelyrepeated, tunneling to sites |ν′ ± 2m〉 at j ± 2n, etc.

We can compare this behavior to the dynamics of atilted Bose–Hubbard chain in the Mott insulating phaseJ � U with equal filling n0 at each site [22, 25, 29].When the tilt constant δε is a simple fraction of the Hub-bard interaction, δε = U/n, a single boson can tunnelfrom any site in the downhill direction by n sites to forma state that is isoenergetic with the initial state [29, 31].The resonant states can be mapped to dipoles and canbe used to construct effective spin or quantum dimermodels [22, 24, 31]. The resonant tunneling we observein the tilted Holstein model is similar to that in tiltedBose-Hubbard systems, but differs in some key aspects:(i) The multiple excited |ν′〉 states that constitute theFranck–Condon excitation allow for tunneling in both thedownhill and uphill direction, whereas the Mott insulat-ing state only allows for downhill transitions. (ii) Our lo-calized initial state causes resonant tunneling to initiallystart only from a single site. (iii) Most importantly, theequidistant spacing of the vibrational QHO levels meansthat a repeated tunneling process is possible, since a pri-mary tunneling event is energetically equivalent to an it-erated secondary tunneling, giving rise to the secondary

4

resonance peaks in Fig. 3c. In contrast, in the Hubbardmodel, a doublon is bound to the first resonant site [22].

Random-walk rate kernel model—To further supportour findings, we verify the numerical results using an ana-lytic approach to determine effective rate kernels for hop-ping events between adjacent sites. In contrast to previ-ous studies where the Markov approximation is adoptedto yield rate constants, the current analysis goes beyondthe Markov approximation and focuses on transient dy-namics [40, 41]. The resulting kinetics is equivalent toa continuous-time random walk, i.e., a generalization ofPoisson kinetics on networks [42, 43].

The basis of the approach is to first consider a dimer,L = 2, and then separate the total HamiltonianH (Eq. 1)

into the hopping operator T = −J(|0〉〈1| + |1〉〈0|

)and

H0 := H − T . Then the initial state is taken to be lo-calized to |0〉 and the time-dependent transition proba-bility is calculated as q(t, δε) := 〈1|U (1)(t)|0〉 using thefirst-order Dyson operator U (1)(t). After tracing out thephononic degrees of freedom and evaluating the propa-gators using path integrals, we obtain

q(t, δε) =

∣∣∣∣J

~

∣∣∣∣2 ∫ t

0

dτ1

∫ t

0

dτ2 exp

{2S(e−iω(τ1−τ2) − 1

)

+ 2iS [sin(ωτ1)− sin(ωτ2)]− i δε~

(τ1 − τ2)

}, (2)

where S =(g~ω)2

is the Huang–Rhys factor. These tran-sition probabilities are extended to a chain of lengthL > 2 by applying the dimer transition probability oneach bond. In the tilted case, we obtain the transitionprobability in the opposite direction by taking δε → −δε.

To explain the vibronic resonance, we first examinethe properties of q(t, δε) in Eq. 2. Define F (τ1, τ2) as theintegrand of q(t, δε = 0), i.e.,

q(t, 0) =:

∫ t

0

∫ t

0

F (τ1, τ2) dτ1 dτ2 .

Then F (τ1, τ2) is periodic in both arguments with a pe-riod T = 2π

ω . Now, for general δε ∈ R, we can writethe transition probability in Eq. 2 as a modified Fouriertransform of F (τ1, τ2):

q(t, δε) =

∫∫

R2

e−i~ δετ1e

i~ δετ2χ[0,t]2F (τ1, τ2) dτ1 dτ2 ,

where χ[0,t]2 is the two-dimensional boxcar function onτ1, τ2. In the absence of χ[0,t]2 , the transfer probability

q(t, δε) would be exactly the Fourier transform F of F ,which would then vanish unless the Fourier frequencymatches the periodicity of F :

δε~

= m2π

T= mω for m ∈ N0.

This is the first-order vibronic resonance condition anddemonstrates how the resonance peaks arise. We cangain further insight into the structure of q(t, δε) by tak-ing into account the boxcar function χ[0,t]2 , which actsas a 2D convolution in Fourier space. Using the Diraccomb structure of F , and considering only the Fourierfrequency mω that is closest to the given value of δε/~,we then obtain

q(t, δε) ≈ 2cm1− cos

(t(δε~ −mω

))(δε~ −mω

)2 , (3)

where cm is the Fourier coefficient of F at frequenciesω1 = −ω2 = mω. Equation 3 determines the structureof a resonance peak at δε = m~ω and shows the oscil-latory structure of the transient side-peaks around themain resonance peaks. Furthermore, a Taylor expansionof cosine shows that

q(t, δε)

∣∣∣∣δε≈m~ω

≈ cm(t2 − t4

12

(δε~−mω

)2),

which demonstrates the quadratic growth of the increas-ingly sharp main peaks as time t increases. Both theoscillatory side-peaks and the quadratic growth of themain peaks are confirmed in Fig. 2a.

Physically, the first-order perturbation describes thetransition between adjacent sites, which differ in energyby δε. In accord with energy conservation, the transi-tion is allowed if the vibrational energy difference m~ωmatches the tilt energy, m~ω = δε. More generally, tocapture long-range tunneling over n bonds, an n-th orderperturbative expansion would be required, leading to thegeneralized fractional resonance condition m~ω = nδε.

In Fig. 2, we compare the perturbative dynamics tothe numerical results for J = 1

10~ω, which shows ex-cellent agreement between the results. Both methodsshow first-order tunneling spikes around integer multi-ples of ~ω. The weak J suppresses higher-order tunnel-ing events and the associated spikes at rational fractionsmn ~ω (though transient side-peaks appear, predicted byEq. 3). A small but persistent second-order tunneling ef-fect is observable only in the full numerical calculations(SM [37], section 4). For δε / 0.1~ω, we observe linearresponse behavior for short times (SM [37], section 5).Conclusion & Outlook—We have uncovered resonance-

dependent transport behavior in tilted vibronic chains.Tunneling over n bonds is allowed for δε/(~ω) = m

n , cor-responding to m-phonon and n-th order tunneling tran-sitions. To study this problem, we have developed bothan analytical and a numerical method.

This generalizes the resonant tunneling found in Mottinsulators on tilted Bose–Hubbard chains [22, 25, 29], aslong-range, repeated hopping in both directions is nat-urally obtained. Vibronic coherence has also emergedas an active mechanism in light-harvesting systems,

5

molecule semiconductors, and molecule electronics. Ourdiscovery of long-range tunneling resonances may havean important bearing on the “phonon antenna” mecha-nism [44, 45], a proposed type of environment-assistedquantum transport [46]. Prospective technological appli-cations are to exploit the bias-dependent resonance peaksfor optimization or selective switching of quantum trans-port, or to enable nanoscale sensing of structural param-eters, e.g., as an extension of inelastic electron tunnel-ing spectroscopy (cf. [47]). Future efforts will aim at ex-tensions from the chain configuration to thin films, nan-otubes, and quantum networks, and determine the influ-ence of coupling to bath modes (cf. [48]): Calculationsof noisy driven energy transfer in a dimer suggest thatsuch behavior is indeed robust [19]. Furthermore, we ex-pect this behavior can be realized in quantum simulatorsusing tilted optical lattices [49–51] or superconductingqubits [52–54].

We thank Fabian Heidrich-Meisner and Martin Pleniofor insightful discussions and feedback.

RKK acknowledges generous scholarships and travelfunds provided by the Studienstiftung des deutschenVolkes. PYY acknowledges support from National Natu-ral Science Foundation of China (Grant No. U1930402).RKK and SRM acknowledge funding by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) – 217133147/SFB 1073, project B03. JC ac-knowledges support from the NSF (Grants No. CHE1800301 and No. CHE 1836913).

RKK acknowledges computational resources providedby the state of Baden-Wurttemberg through bwHPC andthe DFG through grant No. INST 40/575-1 FUGG (JUS-TUS 2 cluster), and gratefully acknowledges GPU re-sources provided by the Institute of Theoretical Physicsin Gottingen and financed by the DFG and the Bun-desministerium fur Bildung und Foschung (BMBF).

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Supplemental Material for:Long-range non-equilibrium coherent tunneling induced by fractional vibronic

resonances

R. Kevin Kessing,1, 2, 3 Pei-Yun Yang,4, 2 Salvatore R. Manmana,1, 5 and Jianshu Cao2

1Institut für Theoretische Physik, Georg-August-Universität Göttingen, 37077 Göttingen, Germany2Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

3Institut für Theoretische Physik, Universität Ulm, 89069 Ulm, Germany4Beijing Computational Science Research Center, Beijing 100193, China

5Fachbereich Physik, Philipps-Universität Marburg, 35032 Marburg, Germany(Dated: November 10, 2021)

CONTENTS

1. Numerical method: ParMuDA S11.a. Motivation: Most basis states contribute

negligibly S11.b. Overview of the method S11.c. Relation to existing methods S3

2. Resonance is between δϵ and ℏω S3

3. Gaps of length n occur for δϵ = mn ℏω S3

4. Numerics show second-order tunneling for smallJ S5

5. Linear response for small δϵ S5

References S5

1. NUMERICAL METHOD: PARMUDA

1.a. Motivation: Most basis states contributenegligibly

The ParMuDA method is an extension of a simplematrix-vector time evolution algorithm in which the “rel-evant” parts of the Hilbert space are adaptively addedand removed. The motivation for this is that most of aHilbert space is irrelevant to a calculation [1]; however,the parts of the Hilbert space that are most relevant atone point in time may not be the most relevant at anotherpoint in time.

A simple illustrative example is a Gaussian state in aharmonic trap (directly equivalent to the phonons underconsideration): In the energy eigenbasis, the Gaussianstate at (q = 0, p = 0) is exactly described by a singleeigenstate. However, upon displacing the state to a dif-ferent (q, p), this initial eigenstate may no longer be ap-preciably occupied, and instead the occupation will shiftmostly to another set of eigenstates. This principle offollowing the basis states occupied by the state vector ateach given time is the basic idea underlying ParMuDA.

0 1 2basis state |n〉 ×107

10−19

10−15

10−11

10−7

10−3

wei

ght|〈n|ψ〉|2

99 %99.9 %99.99 %

101 103 105 107

basis state |n〉

t = 5/ωt = 10/ωt = 25/ωt = 50/ω

Distribution of expansion coefficients (sorted by weight)

FIG. S1.1. The distribution of the expansion coefficients for aParMuDA calculation of the Holstein model at several times,in a lin-log plot (left) and a log-log plot (right). The majorityof basis states contribute negligibly to the total wavefunction.The distribution is between an exponential and a power lawand becomes more fat-tailed over time. The shaded/hatchedareas show how many coefficients are necessary to capture agiven weight of the t = 50/ω state. Parameters: L = 25,g = 4ℏω = 4J , δϵ = 0.

In Fig. S1.1, we show the idea of the relevant subspacein practice by considering the distribution of expansioncoefficients of the Holstein state vector, which falls offvery rapidly. We see that less than half of the coefficientsaccount for 99.99% of the total weight of the fat-tailed fi-nal state’s distribution. Thus, assuming this falloff to beexponential, we could therefore keep only n basis statesand neglect the remaining basis states (potentially in-finitely many: dim(H) − n) and incur an error only oforder O

(e−λn

), which is precisely the underlying idea of

the adaptive Hilbert space.

1.b. Overview of the method

The basic idea of the presently used method is inspiredby the idea underlying tDMRG: To efficiently truncatethe high-dimensional Hilbert space into a smaller Hilbertspace in a time-dependent manner [2, 3]. However, whileDMRG methods are based on singular value decom-positions of the local matrices to determine the effec-

S2

Heff(t) |ψ(t)⟩ not possible

repeat

evolve

U(δt)

adapt

FIG. S1.2. The basic idea of the presently employed method: Construct an effective Hilbert space, evolve exactly within thiseffective Hilbert space, truncate and repeat (in this schematic, |ψ⟩ is not truncated). Each circle represents a basis state.Dark blue: the effective Hilbert subspace; light blue: non-zero components of the state vector; yellow highlighting: basis statesaffected by a change.

tive Hilbert space, the present method uses a distinctweighting and adaptation algorithm to construct effec-tive Hilbert space and state vectors.

Given a Hilbert space with an arbitrary countable or-thonormal basis set {|n⟩}, the basic algorithm of a singletime evolution step is (see Fig. S1.2):

Input: Start with a given state vector (= wavefunction)|ψ(t0)⟩ which has non-zero basis coefficients ψn =⟨n|ψ⟩ for a limited number of basis states |n⟩, then:

1. construct a Hilbert subspace, or “effective Hilbertspace”, Heff of the entire Hilbert space; this Hilbertsubspace should capture both

(a) the current state vector,(b) and “neighboring” basis states into which the

state vector may evolve;

2. construct the Hamiltonian matrix acting on Heffand evolve the state vector |ψ(t)⟩ using the sub-space Hamiltonian;

3. if necessary, truncate the resulting state vector toprevent a memory overflow in the following itera-tion.

Output: the new time-evolved wavefunction |ψ(t0 + δt)⟩.Repeat from step 1 until the desired time has beenreached.

According to its defining features, we call this methodthe parallelized multi-step-dynamically adapted basis setmethod, or ParMuDA. Below, we elaborate on some keydetails of these features. As the name suggests, eachof these steps is tailored towards easy numerical paral-lelization, as most of the computational steps consist of

linear algebra in the form of sparse matrix-vector mul-tiplication, which results in a highly time-efficient algo-rithm that was implemented on graphics processing units(GPU) [4].

The effective Hilbert space. The dynamically adaptedeffective Hilbert space Heff(t) is constructed as the spanof a select set of basis states. These basis states consistof all of the basis states that comprise the current statevector as well as “neighboring” basis states. The neigh-boring basis states of the state |ψ(t)⟩ are defined via theimage of the Hamiltonian matrix acting on |ψ(t)⟩: Givena Hamiltonian H, a total Hilbert space H ⊇ Heff spannedby a basis {|n⟩} and a state |ψ(t)⟩, we define a minimalset of basis states M(t) supporting the state |ψ(t)⟩, i.e.,

M(t) := supp{|ψ(t)⟩} = {|n⟩ ∈ H | ⟨n|ψ(t)⟩ = 0} .

Then the set of k-th degree neighboring states of M isdefined as

N (k)M :=

{|n⟩ ∈ H

∣∣ ⟨n|Hk|m⟩ = 0 ∀ |m⟩ ∈ M}.

We use k ≤ 2 for the effective Hilbert space in our calcu-lations, i.e., we use

Heff(t+ δt) = span{M(t) ∪N (1)

M(t) ∪N (2)M(t)

}.

Restriction of the Hamiltonian. The Hamiltonian ma-trix is explicitly constructed via restriction to Heff. Thatis, the effective Hamiltonian is given by matrix elements

⟨m|H|n⟩ ∀m,n ∈ Heff(t).

This means that all matrix elements that map to or froma basis state not contained within Heff(t) are discarded(set to zero). Due to the nature of the Hamiltonian inquestion, we are able to employ a highly efficient sparsematrix representation of these effective Hamiltonians.

S3

State vector truncation. The truncation of the statevector after each time evolution step retains only theN highest-weighted expansion coefficients ψn, where theweighting wn of the expansion coefficient ψn at time t isskewed towards the future time t+τ by using a weightingfunction

wn :=

∣∣∣∣ ⟨n|(1− iτ

ℏH

)|ψ(t)⟩

∣∣∣∣2

.

1.c. Relation to existing methods

The present method belongs broadly to the family ofadaptive basis set methods, which deal with the expo-nential growth of the Hilbert space by choosing a small,truncated Hilbert space and systematically adapting thebasis set to the evolution of the state vector.

A variety of adaptive basis set methods have been de-veloped over the last decades, mostly in quantum chem-istry. Many of these employ moving Gaussian basis sets,e.g., combined with Ehrenfest dynamics or a variationalansatz (cf. [5–11]). However, unlike these approaches,our ansatz employs stationary basis states that are adap-tively added or removed. Two other methods that useadaptive stationary basis sets are the adapted trajectory-guided (aTG) scheme [12] and the earlier “wavepacketpropagation in a distributed Gaussian basis” (proDG)method [13, 14]. The most obvious difference betweenthese methods and ours is that we employ an orthog-onal basis set that is not over-complete [15], namelythe phonon eigenbasis and the local excitonic states (or-thonormal by construction).

One method originating from the physics communityand which is related to the presently employed one isthat of limited functional spaces (LFS) [16], first intro-duced by Bonča et al. [17]. The ansatz in LFS is alsoto determine an effective Hilbert space via the n-fold ap-plication of the Hamiltonian to a given quantum state.However, our present method is, to our knowledge, thefirst method to apply this principle dynamically, therebycontinuously recalculating the effective Hilbert space.

2. RESONANCE IS BETWEEN δϵ AND ℏω

Since we have four different Hamiltonian parameters(ω, J , g and δϵ), we demonstrate here that the resonanceis only between δϵ and ℏω, whereas changing the value oft and g does not shift the position of the resonance peaks.Recall that the main numerical results in the main textwere obtained for g = 4J and J = ℏω (δϵ was varied toprobe the resonance behavior). For purposes of brevity,set ℏω in this configuration to the unit energy, i.e., thereference parameters are J = ℏω = 1 and g = 4. Wethen vary each of g, J and ω independently and keep the

remaining parameters fixed to the aforementioned values:In Fig. S2.1, we set J = 0.85; in Fig. S2.2, we set g = 3; inFig. S2.3, we set ℏω = 1.25. These should be comparedto Fig. 2 (right side) in the main text. All of the resultsshown here use only the numerical ParMuDA methodand are not based on perturbation theory.

We see in all three plots that the position of the reso-nance peaks as a function of δϵ/(ℏω) is unchanged. Theintensity of the peaks, however, is changed, which is ex-pected due to the effect of changing the remaining pa-rameters: Decreasing J decreases the overall particle mo-bility and especially the probability of long-range tun-neling events; decreasing the exciton–phonon coupling gdecreases polaronic trapping effects and increases over-all mobility; and increasing ω decreases the displacementbetween the excited and ground state PES minima, alsodecreasing polaronic effects and slightly increasing mo-bility (equivalently, increasing ω increases the energy ofmany–phonon states, making them less unfavorable andthus decreasing polaronic behavior).

3. GAPS OF LENGTH n OCCUR FOR δϵ =mnℏω

In the main text, we described the appearance of localpopulation spikes at sites C±n for δϵ = m

n ℏω, where siteC is the initial and central site—in other words, we pro-posed that the higher-order (fractional) resonance peaksare due to tunneling events over more than one site,where the denominator of the energy ratio δϵ

ℏω dictatesthe tunneling length. We shall present additional infor-mation supporting this claim by systematically investi-gating the local populations at sites C−2 and C−3, whereour hypothesis proposes we should find density spikes forδϵℏω = m

2 and δϵℏω = m

3 , respectively, with m ∈ N. Thisanalysis reveals more information than the RMSD alone:For example, an increase in RMSD cannot differentiatebetween strong build-up of density at sites C±2 or weakerbuild-up at sites C ± 3. The change in local densities isshown in Fig. S3.1.

We see that the data supports our hypothesis and thatthe local density at sites C −m is maximized when δϵ isan integer multiple of ℏω

m . In particular, we see that thethird-order resonance peaks at δϵ

ℏω = m3 present as dips

in the local density at C − 2, and that their increasedRMSD is due to build-up at C − 3. On the other hand,the apparent increase in density at C − 3 for δϵ

ℏω = m2 is

attributable to the tail of the strong local density spikesat C − 2 (notice the difference in absolute magnitudes,cf. Fig. 3 of the main text). The same also applies toδϵℏω ∈ Z, which enhances tunneling across every bond.

S4

0.8

1.0

1.2

1.4

1.6R

MSD

atsp

ecifi

cti

mes

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75Tilt parameter δε/(hω)

−0.4

−0.3

−0.2

−0.1

0.0

c.m

.dis

plac

emen

tX−L 2

t = 1.50 2πω

t = 2.50 2πω

t = 3.50 2πω

t = 4.50 2πω

FIG. S2.1. The RMSD and center-of-mass deviation, but withJ = 0.85 instead of J = 1. We see that the position of thepeaks is unaffected by this change. The remaining parametersare the same: L = 9, ℏω = 1, g = 4.

1.2

1.4

1.6

1.8

2.0

2.2

RM

SDat

spec

ific

tim

es

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75Tilt parameter δε/(hω)

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

c.m

.dis

plac

emen

tX−L 2

t = 1.50 2πω

t = 2.50 2πω

t = 3.50 2πω

t = 4.50 2πω

FIG. S2.2. The RMSD and average position with g = 3 in-stead of g = 4. Due to the decreased polaronic trapping,L = 15 was chosen here. The remaining parameters are thesame: J = ℏω = 1.

0.8

1.0

1.2

1.4

1.6

1.8

2.0

RM

SDat

spec

ific

tim

es

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75Tilt parameter δε/(hω)

−0.6

−0.4

−0.2

0.0

c.m

.dis

plac

emen

tX−L 2

t = 1.50 2πω

t = 2.50 2πω

t = 3.50 2πω

t = 4.50 2πω

FIG. S2.3. The RMSD and average position with ℏω = 1.25instead of ℏω = 1. The position of the peaks as shown hereis unchanged because the x axis shows δϵ/(ℏω); in absoluteterms, the position of the resonant values of δϵ is shifted ac-cording to the change in ℏω. As in Fig. S2.2, L = 15 toaccommodate smaller polaronic effects. J = 1, g = 4.

0.05

0.10

0.15

0.20

dens

ity

2off

-cen

ter:〈n

2(t

)〉

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75Tilt parameter δε/(hω)

0.02

0.04

0.06

0.08

dens

ity

3off

-cen

ter:〈n

1(t

)〉 t = 1.50 2πω

t = 2.50 2πω

t = 3.50 2πω

t = 6.50 2πω

FIG. S3.1. The same calculations as in Fig. 2 (right side) ofthe main text, but instead of the excitonic RMSD, the localexciton density 2 sites off-center (site 2, top) and 3 sites off-center (site 1, bottom) is shown. L = 9, J = ℏω = 1, g = 4.Data shown is numerics with interpolation (no perturbativeresults).

S5

0.2 0.3 0.4 0.5 0.6 0.7 0.8Tilt parameter δε/(hω)

0.05

0.06

0.07

0.08

0.09

0.10

0.11

0.12R

MSD

t = 1.5 2πω

t = 2.5 2πω

t = 3.5 2πω

t = 4.5 2πω

numericspert. theory

FIG. S4.1. Zoom-in on the top left of Fig. 2 from the maintext: A comparison between first-order perturbation theoryand numerical results for L = 9, J = 1

10ℏω, g = 4ℏω, showing

the build-up of a second-order tunneling event only in thenumerics.

4. NUMERICS SHOW SECOND-ORDERTUNNELING FOR SMALL J

In the main text, we introduced a first-order (in thetunneling J) perturbation theory (PT) approach thatprovides excellent agreement with the numerical reultswhich take into account all orders of of the hopping op-erator. This agreement is valid for small values of J ,which mostly precludes the occurrence of higher-ordertunneling events which are not captured by a first-orderperturbative approach.

Our hypothesis for the resonant tunneling mechanismpostulates that the resonances at fractional values of theenergy ratio δϵ

ℏω = mn , the resonant tunneling transi-

tions are due to m-phonon and n-th order tunneling pro-cesses. Therefore, to further validate our hypothesis, weinvestigate whether we observe an increase in RMSD forδϵℏω = 1

2 , corresponding to second-order tunneling events,in Fig. S4.1. This Figure shows, as expected, that theoverall excellent agreement for the results is slightly lessgood around specifically the point δϵ

ℏω = 12 , where the

numerics provide slightly higher RMSD values than thefirst-order PT. This is in line with the hypothesis thatδϵℏω = 1

2 enables resonant second-order tunneling pro-cesses, which are consequently captured by the numerics,but not by a first-order PT.

5. LINEAR RESPONSE FOR SMALL δϵ

In the bottom left of Fig. 2 of the main text, we dis-played the mobility, i.e., the displacement of the center of

0.00 0.05 0.10 0.15 0.20Tilt parameter δε/(hω)

−0.00150

−0.00125

−0.00100

−0.00075

−0.00050

−0.00025

0.00000

0.00025

c.m

.dis

plac

emen

tX−L 2

t = 1.5 2πω

t = 2.5 2πω

t = 3.5 2πω

t = 4.5 2πω

numericspert. theory

FIG. S5.1. Zoom-in on the bottom left of Fig. 2 from themain text. The dashed gray lines are extrapolations of theslope at the origin and correspond to a linear response, whichis approximately valid for small times t and small gradientshifts δϵ. The parameters are the same as in Fig. S4.1: L = 9,J = 1

10ℏω, g = 4ℏω.

mass X from the initial position. For small values of δϵ,the response of the system to a weak perturbative exter-nal gradient can be modeled as linear response, X = cδϵ.We show in Fig. S5.1 that our numerical and analyti-cal findings also exhibit this linear behavior around theorigin, with better agreement with linear response forsmaller times t and a smaller range of agreement forlonger times. Furthermore, the slopes of the linear re-sponses (c in the above equation) are proportional to t2,in agreement with Eq. 4 of the main text.

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S6

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