Quantum Transport of Rydberg Excitons with Synthetic Spin-Exchange Interactions
aa r X i v : . [ qu a n t - ph ] J u l Quantum Transport of Rydberg Excitons with Synthetic Spin-Exchange Interactions
Fan Yang, Shuo Yang, ∗ and Li You
1, 2, † State Key Laboratory of Low Dimensional Quantum Physics,Department of Physics, Tsinghua University, Beijing 100084, China Beijing Academy of Quantum Information Sciences, Beijing 100193, China
We present a scheme for engineering quantum transport dynamics of spin excitations in a chainof laser-dressed Rydberg atoms, mediated by synthetic spin-exchange arising from diagonal van derWaals interaction. The dynamic tunability and long-range interaction feature of our scheme allowsfor the exploration of transport physics unattainable in conventional spin systems. As two concreteexamples, we first demonstrate a topological exciton pumping protocol that facilitates quantizedentanglement transfer, and secondly we discuss a highly nonlocal correlated transport phenomenonwhich persists even in the presence of dephasing. Unlike previous schemes, our proposal requiresneither resonant dipole-dipole interaction nor off-diagonal van der Waals interaction. It can bereadily implemented in existing experimental systems.
Developing controlled large-scale quantum systemsconstitutes a central goal of quantum simulation andquantum computation [1, 2]. Among the variety of physi-cal realizations, neutral atoms present several unique ad-vantages [3], their inherent qubit identity, long coherencetime, flexible state maneuverability, as well as tunablequbit-qubit interactions, for instance mediated by Ry-dberg states [4]. The continued progresses in Rydberg-atom studies offer great potential for probing many-bodydynamics [5–9]. With improved operation fidelity [10]and increased system size [11], quantum simulation onthe Rydberg atom based platform [8, 12] is becoming in-creasingly attractive.The transport of particle or spin via quantum state-changing interactions is essential for understanding en-ergy or information flow. Emulating such problems ona quantum simulator constitutes a focused thrust withinthe broad quantum physics community [13–17]. Earlierefforts based on Rydberg-atom systems have providedfirst insights [18–24], where transport of spin excitationis facilitated typically by resonant dipole-dipole interac-tion (DDI) or by off-diagonal van der Waals (vdW) flip-flop interaction between Rydberg states. They includedirect spin-exchange between different Rydberg states[18–20], second-order exchange between ground state andRydberg state [21–23], and a fourth-order process insideground internal state manifolds [24–26].In this Letter, we propose a simpler yet as effectivemethod for engineering exciton transport dynamics ina Rydberg-atom system. The use of resonant DDI orflip-flop vdW interaction is avoided. Instead, our mainidea relies on a perturbative spin-exchange process byoff-resonantly dressing the ground state to a Rydbergstate. Capitalizing on the diagonal vdW interaction-induced Rydberg level shift, perturbations from differ-ent pathways collectively contribute to a net exchangeinteraction between the ground and the Rydberg states.When exciton-exciton interaction as well as dephasingare included, our model system is shown to be capa-ble of simulating various transport dynamics unattain- able in conventional spin systems. In the first example,we establish an interesting topological pumping proto-col, whereby the exciton experiences a quantized center-of-mass motion. In the second example, we show thatthe long-range interaction between excitons permits theformation of high-order magnon bound state, which ex-hibits nonlocal correlations even when ballistic transportturns into classical diffusion due to dephasing.
Model .—The system we study is an array of individu-ally trapped cold atoms, dressed by laser fields that cou-ple the ground state | g i to a Rydberg state | r i [7, 10]. Itis modeled by the Hamiltonianˆ H = X i Ω i σ ix + X i ∆ i ˆ σ irr + X i
EffectiveRigorous - - Site index i (a) (b)(c) (d) ( M H z ) ( M H z ) J ( r ) J ( r ) FIG. 1. (a) Level structure for the proposed atomic system.We consider Rb atom with | g i = | S / , F = 2 , m F = − i , | r i = | S, J = 1 / , m J = − / i [10]. (b) Illustration ofthe mechanism for generating synthetic spin-exchange inter-action. (c) Synthetic exchange strength J ( r ) versus distance r for Ω / π = 5 MHz and ∆ / π = ±
50 MHz. The dashed linesdenote the facilitation condition ∆ + V ( r ) = 0. (d) The leftpanel shows the concurrence for the first two nodes [ C (ˆ ρ )]and the two end nodes [ C (ˆ ρ )]. The right panel shows thefidelity to the target state ( | r g · · · g i − i | g g · · · r i ) / √ H and ˆ H eff , respectively) and values of dressing param-eters (lower figure) with V ( d ) = 3∆ (∆ >
0) and d = 4 . µ m.The numerical data represent averages over 500 calculations,assuming a Gaussian distribution of atomic position along thechain direction with 0 . µ m standard deviation. The errorbars mark one standard deviation intervals. to the original model [Eq. (1)] and dropping the constant − P j Ω j / j , we arrive at an effective Hamiltonian [28]ˆ H eff = X i (cid:18) ∆ i + Ω i i (cid:19) ˆ σ irr + X i = j I ij ˆ σ irr ˆ σ jgg + J ij ˆ σ i + ˆ σ j − , (2)where ˆ σ i + = | r i ih g i | and ˆ σ i − = | g i ih r i | are spin raisingand lowering operators for the i -th atom. The Ising-type interaction I ij and the spin-exchange interaction J ij respectively take the following forms I ij = Ω j V ( r ij )4∆ j [∆ j + V ( r ij )] , J ij = X β = i,j Ω i Ω j V ( r ij )8∆ β [∆ β + V ( r ij )] . In contrast to earlier dressing schemes [25, 26], the spin-exchange interaction we find constitutes a pure syntheticinteraction as the initial Hamiltonian Eq. (1) containsonly diagonal vdW interactions. It exhibits different r -dependence compared with previous schemes [Fig. 1(c)]and is highly tunable in terms of Ω i and ∆ i . This effectivemodel is not restricted to any particular type of lattice,and this work considers the simplest one-dimensional pe-riodic chain with a spacing d .The exciton transport is conveniently described bymapping spins to hard-core bosons with ˆ σ i + = ˆ a † i and ˆ σ i − = ˆ a i , where ˆ a † i (ˆ a i ) creates (annihilates) a Ryd-berg exciton at site i . The effective Hamiltonian for asingle exciton can then be expressed in a tight-bindingform with ˆ H eff = P i µ i ˆ a † i ˆ a i + P i 75 for ∆ / Ω = 10 and N = 50). Example 1 .—To illustrate the dynamical tunability ofour scheme, we consider an implementation for topolog-ical exicton pumping, for which a time-dependent andsite-dependent exchange interaction is required [36–41].A periodic system with broken parity symmetry is as-sumed [42], with three lattice sites (labeled as A , B , C , and separated by d ) forming a unit cell (with theperiod l = 3 d ), dressed by control fields of three in-tensities [Fig. 2(a)] with corresponding Rabi frequenciesΩ A , Ω B , Ω C = Ω × { sin ( φ + π/ , sin ( φ ) , sin ( φ − π/ } and φ a time-dependent control parameter. Such a dress-ing scheme can be realized by using three independentlycontrolled acousto-optic deflectors. Retaining the NN in-teraction, the system can be described by the generalizedRice-Mele Hamiltonian [43]ˆ H eff = X i (cid:16) J A ˆ a † i ˆ b i + J B ˆ b † i ˆ c i + J C ˆ c † i ˆ a i +1 + H . c . (cid:17) E ne r g y EffectiveRigorous (a)(b) (c)(d) A B C FIG. 2. (a) Illustration of the dressing scheme for topologicalexciton pumping. (b) Energy band and Berry curvature ofthe effective model. (c) Illustration of the pumping sequence.(d) Mean displacement h x i of the exciton at different time t .The blue dots are calculated by the refined density matrixusing the exact model Eq. (1), and the red lines are obtainedwith the effective Hamiltonian Eq. (3). The inset shows themodulation detail φ/π = + tanh[5 . t/T − / . . The simula-tions are performed with Ω / π = 5 MHz, ∆ / π = 20 MHz, V ( d ) = 3∆, N = 12, and T = 27 . µ s. + X i (cid:16) µ A ˆ a † i ˆ a i + µ B ˆ b † i ˆ b i + µ C ˆ c † i ˆ c i (cid:17) , (3)where ˆ a i , ˆ b i , and ˆ c i are exciton annihilation operatorsfor site A , B , and C of the i -th unit, respectively. Ac-cording to the Bloch theorem, this system can be de-scribed in the quasi-momentum k space with a single-particle Hamiltonian ˆ H ( k, φ ) [28], which is also periodicin φ . Thus, we can define the energy band in the syn-thetic space k = ( k, φ ) with the first Brillouin zone (BZ) k ∈ ( − π/l, π/l ] and φ ∈ ( − π/ , π/ C n = 12 π Z BZ B n ( k ) d k , (4)where B n ( k ) = i ( h ∂ φ u n | ∂ k u n i − c . c . ) is the Berry cur-vature of the n -th band, and | u n i is the eigen state ofˆ H ( k, φ ). For the system considered above, we find threegapped bands with respective nontrivial topological num-bers C , C , C = { , − , } , as shown in Fig. 2(b).In this case, we can implement Thouless pumping [36],while the parameter φ ( t ) is slowly modulated in time t . After a pumping cycle in which φ changes by π , theHamiltonian returns to its initial form. If an energy bandis filled or homogeneously populated, the mean displace-ment h x i of the exciton after one pumping cycle is quan-tized in units of lattice constant, i.e., h x i /l = C n . Forour system, the energy gap Ω V ( d ) / 8∆ [∆ + V ( d )] be-tween the upper and the middle band is about two or-ders of magnitude larger than the gap between the mid-dle and the lower band. Thus, to achieve better adia-baticity, we consider motion of the exciton within theupper band. As indicated by the pumping sequenceshown in Fig. 2(c), we first shine a resonant field onsites C j and A j +1 to produce an entangled state | ψ j i =(1 / √ c † j + ˆ a † j +1 ) | i using Rydberg blockade. With suchan initialization and φ (0) = 0, we create an equallyweighted Bloch states for the upper band. Then, we adi-abatically ramp φ from 0 to π , and observe the positionof the exciton. As shown in Fig. 2(d), the mean displace-ment of the exciton after one pumping cycle is indeed h x i ≈ d = l , in agreement with the topology of the up-per band. Since this energy band is almost flat in the k -dimension, such a quantized motion indicates a high-efficiency entangled state transfer from | ψ j i to | ψ j +1 i .It is worth pointing out that during the long pump-ing cycle T , the NNN interaction also comes into play.In fact, the long-range interaction induced NNN hop-pings P i (cid:16) J ′ A ˆ a † i ˆ c i + J ′ B ˆ b † i ˆ a i +1 + J ′ C ˆ c † i ˆ b i +1 + H . c . (cid:17) andthe modifications to on-site potential can be viewed asperturbations to ˆ H eff . We find that although these per-turbations can significantly modify the spread h x i of theexciton, they do not change the mean displacement h x i [28]. Such a robust center-of-mass (COM) motion is pro-tected by the topology of the band, which is invariant un-der continuous deformation of the Hamiltonian [44, 45]. Example 2 .—The strong and nonlocal exciton-excitoninteraction in the proposed system also makes it feasiblefor studying correlated transport [46, 47]. Here, we con-sider the dynamics of two excitons in a homogeneouslydressed (∆ i = ∆ and Ω i = Ω) chain with V (2 d ) ≪ | ∆ | .We first consider the dynamics of the dimer state | Ψ i,i +1 i , which can be prepared via anti-blockade exci-tation satisfying 2∆ + V ( d ) = 0 [48]. As explained pre-viously, such a tightly bound state can migrate throughNNN hopping [see the upper panel of Fig. 3(a)], the hop-ping rate of which can be significant near the facilita-tion [7] region ∆ + V ( d ) = 0. This correlated transportcan be measured by the second-order correlation function g (2) i,j = h ˆ a † i ˆ a † j ˆ a j ˆ a i i . As shown in Fig. 3(a), we find g (2) i,j rapidly spreads on the diagonals j = i ± H ′ eff . Unlike spin systems re-ported earlier [23, 46], the long-range interaction U ij canbe tuned much larger than the exchange rate J ij here,which results in a highly anisotropic XXZ model andpermits the existence of high-order bound states. If theNNN interaction U i,i +2 is sufficiently larger than the NNhopping rate J i,i +1 , excitons separated by one lattice sitealso forms bound state and exhibits correlated motion.Different from the dimer state | Ψ i,i +1 i , transport of thehigh-order bound state | Ψ i,i +2 i relies on a second-orderprocess with hopping rate ∼ J i,i +1 /U i,i +2 [see the up-per panel of Fig. 3(b)]. As verified by numerical resultsshown in Fig. 3(b), g (2) i,j also localizes on the skew di- Site index i S i t e i nde x j S i t e i nde x j -6 -6 -6 -6 -6 -5 -5 -5 -5 -5 (b)(a)(c) S i t e i nde x j Site index i (d) P r obab ili t y -6 Center of mass position -6 -3 3-6 FIG. 3. (a) Evolution of the density-density correlation g (2) i,j for the dimer state | Ψ i,i +1 i . (b) Evolution of the density-density correlation for the high-order bound state | Ψ i,i +2 i .(c) Correlation function at 9 µ s, with the same initial statein (b) and γ = 0 . / π = 5 MHz,∆ / π = − 400 MHz, V ( d ) = − . N = 12 in (a), and∆ / π = 30 MHz, V ( d ) = 3∆, N = 13 in (b)-(d). The corre-lation functions are normalized to the maximal value. agonals, but spreads on the j = i ± t > /γ , with γ thedephasing rate introduced previously. Interestingly, thestrong bunching of g (2) i,j along i = j ± g (2) i,j for | i − j | > U i,i +1 forbids the diffusion into | i − j | = 1 region. Togain a deeper insight into such a correlated transport, weinvestigate the COM motion of two excitons. For coher-ent transport, this motion is characterized by a quantumrandom walk [49], with a density distribution describedby the Bessel function [see the upper panel of Fig. 3(d)].In contrast, the COM density distribution at t > /γ iswell fitted by a Gaussian function [see the lower panelof Fig. 3(d)]. This indicates that the high-order boundstate we study exhibits diffusive expansion as a compos-ite, which does not reach equilibrium due to its reduceddiffusion rate compared to free excitons.In conclusion, we propose a Rydberg-atom systemfor studying quantum transport dynamics, utilizing syn-thetic spin-exchange induced by vdW interaction. Ourscheme does not require resonant DDI or off-diagonalvdW interaction, and thus avoids the complicated exci-tation schemes in multi-Rydberg-level systems. For thestate-of-the-art experimental setup, its typical Rydberglifetime ∼ µ s [10] will not hinder the exciton trans-port we discuss, and its influence on the dynamics canbe eliminated by using projective measurement [28]. Inaddition to simulating quantum transport phenomena,this work opens up an avenue towards constructing exoticspin models with Rydberg atoms, which for instance canfacilitate the study of many-body localization [50, 51].This work is supported by the National Key R&DProgram of China (Grant No. 2018YFA0306504) andby NSFC (Grant No. 11804181). F. Yang acknowledgesvaluable discussions with Prof. K. Ohmori, Prof. S. Sug-awa, Dr. J. Yu and Dr. F. Reiter. ∗ [email protected] † [email protected][1] J. I. Cirac and P. Zoller, Nat. Phys. , 264 (2012).[2] I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod.Phys. , 153 (2014).[3] D. S. Weiss and M. Saffman, Phys. Today , 44 (2017).[4] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. , 2313 (2010).[5] H. Labuhn, D. Barredo, S. Ravets, S. De L´es´eleuc,T. Macr`ı, T. Lahaye, and A. Browaeys, Nature ,667 (2016).[6] H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Om-ran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres,M. 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MANY-BODY QUASI-DEGENERATE PERTURBATION ANALYSIS We first consider the effective Hamiltonian for a single Rydberg exciton. The initial Hamiltonian of the modelsystem considered [see Eq. (1) in the main text] can be decomposed as ˆ H = ˆ H + ˆ V , whereˆ H = X i ∆ i | Ψ i ih Ψ i | + X i In this section, we consider the effective exciton dynamics in a realistic system, where the dephasing, the spontaneousdecay of the Rydberg state, and the atomic positional disorder could influence the exciton transport model establishedin the previous section. Site index i Time ballistic diffusive EffectiveRigorous (a) (b) (c) Site index i T i m e T i m e FIG. 1. (a) Evolution of the mean squared displacement. The blue dots are calculated by the exact model followed by projectivemeasurements, and the red line represent the analytic solution. (b) and (c) show the evolution of exciton density distributionsin the ballistic and diffusive regime, respectively, where a brighter color indicates a larger probability. The parameters used areΩ / π = 5 MHz, ∆ / π = 50 MHz, V ( d ) = 3∆ ( d = 4 . µ m), N = 11, and γ = 0.8, 0.1, 1 MHz for (a), (b), (c). A. Effects of dephasing For the state-of-the-art experimental system [2], the main decoherence channel is the pure dephasing caused byatomic Doppler effect and phase noises of the dressing laser. Thus, we first consider the open-system dynamics withlocal dephasings. In this case, the evolution of system density matrix ˆ ρ can be described by the master equation ∂ t ˆ ρ = − i [ ˆ H, ˆ ρ ] + P i L [ √ γ ˆ σ irr ]ˆ ρ with the Lindblad operator L [ˆ σ ]ˆ ρ = ˆ σ ˆ ρ ˆ σ † − (ˆ σ † ˆ σ ˆ ρ + ˆ ρ ˆ σ † ˆ σ ) [3]. Before discussingexciton transport, we discuss the influence of dephasing on a single atom. In the weak-dephasing regime ( | ∆ | ≫ γ ),the equation of motion for ρ rr = h r | ˆ ρ | r i is governed by ddt ρ rr = − Γ (cid:18) ρ rr − (cid:19) , with Γ = Ω γ/ + ( γ/ ≈ Ω γ , (12)which indicates that for an excitation-free initial state | g i [ ρ rr (0) = 0], the Rydberg exciton grows with a rate Γ / | r i has a loss rate Γ / 2. In multi-atom systems, such an exciton growth (loss) canbe suppressed by vdW interactions [3], but the growth (loss) rate can still be roughly estimated as Γ / t c = min (cid:8) ∆ i /γ Ω i (cid:9) , during which the exciton growth (loss)event rarely occurs and can be subtracted out by projective measurement. In this regime, the dynamics can beeffectively described by ∂ t ˆ ρ = − i [ ˆ H ′ eff , ˆ ρ ] + P i L [ √ γ ˆ a † i ˆ a i ]ˆ ρ , which conserves the total exciton number P i ˆ a † i ˆ a i .To see the crossover between coherent and incoherent exciton motion, we consider a simple case: transport of asingle excitation in a homogenously dressed infinite chain described by ˆ H eff = P i,d J d (cid:16) ˆ a † i ˆ a i + d + ˆ a † i + d ˆ a i (cid:17) , where thehopping rate J d = J i,i + d only depends on the distance d > 0. The density matrix for a single exciton can be expressedas ˆ ρ = P m,n ρ m,n ˆ a † m | ih | ˆ a n , and the resulting equation of motion is equivalent to the Haken-Reineker-Strobl modelwith cohorent hoppings and on-site dephasings:˙ ρ m,n = − i X d> J d ( ρ m + d,n + ρ m − d,n − ρ m,n − d − ρ m,n + d ) − γ (1 − δ mn ) ρ m,n . (13)For the initial state ˆ ρ = ˆ a † | ih | ˆ a , the equation of motion for the mean square displacement h x i = P n n ρ n,n is d dt h x i + γ ddt h x i = 4 X d ( dJ d ) , with d h x i dt (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 , h x i (cid:12)(cid:12) t =0 = 0 . (14)The solution of Eq. (14) is given by h x ( t ) i = 4 P d ( dJ d ) γ (cid:0) γt + e − γt − (cid:1) ≈ (cid:2) P d ( dJ d ) (cid:3) t , for t ≪ γ − , (cid:2) P d ( dJ d ) /γ (cid:3) t, for t ≫ γ − , (15)which indicates that the transport experiences a crossover from ballistic spreading ( ∼ t ) to diffusive expansion ( ∼ t ) full modeldephasing model full modeldephasing model F i de li t y (c) Time Time (a) (b) without projective measurement with projective measurement FIG. 2. (a) and (b) show the evolution of h x i , where (b) and (a) are calculated with and without projective measurement,respectively. The parameters are γ = 0 . κ = 0 . 05 MHz, Ω / π = 5 MHz, ∆ = 10Ω and V ( d ) = 3∆. (c) Fidelity of theentanglement transfer for different uncertainties σ , with other parameters the same as Fig. 1(d) in the main text. The datapoints are averaged over 500 calculations, and the error bars mark one standard deviation intervals. as the system evolves, with the characteristic time being 1 /γ . Such a crossover is verified by numerical simulationsshown in Fig. 1(a). The transport process is coherent in the ballistic spreading regime [Fig. 1(b)], as characterized bythe interference fringes during the spreading. In the diffusive expansion regime, the exciton motion exhibits incoherentfeatures, as indicated by the Gaussian type density distributions [Fig. 1(c)]. B. Effects of spontaneous radiative decay While the dephasing can significantly modify the transport dynamics, the influence of finite Rydberg lifetime isnegligible. This is because for the current experimental system [2], the Rydberg state lifetime τ r ∼ µ s is sufficientlylong for observing exciton transport at a rate ∼ ∂ t ˆ ρ = − i [ ˆ H, ˆ ρ ] + P i L [ √ γ ˆ σ irr ]ˆ ρ + P i L [ √ κ ˆ σ i − ]ˆ ρ , where κ is the spontaneous decay rate. In this case,the density matrix can be written as ˆ ρ = P m,n ρ m,n ˆ a † m | ih | ˆ a n + ˆ ρ ′ , where ˆ ρ ′ represents the density matrix whenspontaneous decay occurs. The evolution of the matrix element ρ m,n is then given by˙ ρ m,n = − κρ m,n − i X d> J d ( ρ m + d,n + ρ m − d,n − ρ m,n − d − ρ m,n + d ) − γ (1 − δ mn ) ρ m,n . (16)Comparing Eq. (16) with Eq. (13), we find that the spontaneous decay simply contributes an exponential decay factor e − κt to the elements ρ m,n . With the projective measurement, the events described by ˆ ρ ′ are discarded, and the refineddensity matrix is just the same as that described by the dephasing model. This is verified by numerical results shownin Fig. 2, where we calculate the mean squared displacement h x i for the exciton initially localized at the center of aperiodic chain containing 7 sites. As shown in Fig. 2(a), the only difference between the dephasing model and the fullmodel is simply an exponential factor. By using the projective measurement, the full model has the same predictionas the dephasing model, as proved by Fig. 2(b). C. Influences of positional disorders For the current experiment, the finite temprature of the atom induces an uncertainty of its position, resulting in anuncertainty of the vdW interaction V ij . In the facilitation regime discussed in Ref. [4], the exciton transport requiresthe fluctuation of the NN interaction much smaller than the Rabi frequency, i.e., δV i,i +1 ≪ Ω, otherwise the facilitationcondition ∆+ V i,i +1 = 0 is broken and the initial excitation becomes highly localized. In our dressing scheme, both theon-site potential and the exchange rate are determined by the effective interaction strength J ij , such that the influenceof the disorder is determined by its fluctuation δJ ij . If the ratio δJ ij /J ij = ∆( δV ij )(∆+ V ij ) V ij ∼ δV ij /V ij ≤ δV i,i +1 /V i,i +1 ismuch smaller than unity, the positional disorder will not hinder the transport process. Due to the fact V i,i +1 ≫ Ω,the condition δV i,i +1 ≪ V i,i +1 is much looser than δV i,i +1 ≪ Ω, which suggests that the transport in our scheme ismore robust against positional disorders compared with the facilitation regime.The robustness of our scheme is verified by the entanglement transfer protocol shown in Fig. 1(d) of the maintext. In the calculation, we consider the Gaussian distribution of the atom position along the chain direction withan uncertainty σ . For vdW interaction V ij = C /r ij , we have δV ij /V ij = 6 δr ij /r ij ≤ √ σ/d . The fidelities of theentanglement transfer for different values of σ/d are shown in Fig. 2(c). For a realistic σ = 0 . µ m [4], the ratio σ/d ≈ 2% ( δV ij /V ij < . 2) is small enough to ensure a high transport efficiency. III. TOPOLOGICAL EXCITON TRANSPORT With the NN interacting approximation, the system for implementing topological exciton pumping can be mappedto the generalized Rice-Mele model [see Eq. (3) in the main text] withˆ H eff ( φ ) = X i h J A ( φ )ˆ a † i ˆ b i + J B ( φ )ˆ b † i ˆ c i + J C ( φ )ˆ c † i ˆ a i +1 + H . c . i + X i h µ A ( φ )ˆ a † i ˆ a i + µ B ( φ )ˆ b † i ˆ b i + µ C ( φ )ˆ c † i ˆ c i i . (17)The hopping rates and the on-site potentials (after dropping the constant term ∆) are given by J A ( φ ) = U sin ( φ + π/ 4) sin ( φ ) , µ A ( φ ) = E sin ( φ + π/ 4) + U (cid:2) sin ( φ ) + sin ( φ − π/ (cid:3) , (18) J B ( φ ) = U sin ( φ ) sin ( φ − π/ , µ B ( φ ) = E sin ( φ ) + U (cid:2) sin ( φ − π/ 4) + sin ( φ + π/ (cid:3) , (19) J C ( φ ) = U sin ( φ − π/ 4) sin ( φ + π/ , µ C ( φ ) = E sin ( φ − π/ 4) + U (cid:2) sin ( φ + π/ 4) + sin ( φ ) (cid:3) , (20)with E = Ω / 2∆ and U = Ω V ( d ) / V ( d )]. Transforming real-space exciton operators into quasi-momentum k -space with k ∈ ( − π/l, π/l ] andˆ a † k = 1 √ N N X j =1 ˆ a † j e ikjl , ˆ b † k = 1 √ N N X j =1 ˆ b † j e ikjl , ˆ c † k = 1 √ N N X j =1 ˆ c † j e ikjl , (21)the effective Hamiltonian can be decomposed into ˆ H eff ( φ ) = P k ˆ H ( k, φ ) withˆ H ( k, φ ) = (cid:0) ˆ a † k ˆ b † k ˆ c † k (cid:1) µ A ( φ ) J A ( φ ) J C ( φ ) e − ikl J A ( φ ) µ B ( φ ) J B ( φ ) J C ( φ ) e ikl J B ( φ ) µ C ( φ ) ˆ a k ˆ b k ˆ c k . (22)The single-particle eigen state of ˆ H ( k, φ ) satisfying ˆ H ( k, φ ) | u n ( k, φ ) i = E n ( k, φ ) | u n ( k, φ ) i can be expressed as | u n ( k, φ ) i = (cid:0) ˆ a † k | i ˆ b † k | i ˆ c † k | i (cid:1) u n,a ( k, φ ) u n,b ( k, φ ) u n,c ( k, φ ) , (23)where n denotes the band index. Since ˆ H ( k, φ ) is periodic in both k and φ , we can introduce the energy band in thetwo-dimensional space spanned by k = ( k, φ ) and calculate its topologic number according to Eq. (4) of the main text.For an initial state | ψ (0) i = (1 / √ N ) P k e iξ ( k ) | u n ( k, φ ) i that homogeneously populates [ ξ ( k ) ∈ R ] each k componentof the n -th band, the state evolution in the adiabatic limit is given by | ψ ( t ) i = (1 / √ N ) P k | ψ k ( t ) i , with | ψ k ( t ) i = exp ( iξ ( k ) − i Z t dt ′ E n [ k, φ ( t )] − Z φ ( t ) φ dφ h u n ( k, φ ) | ∂ φ | u n ( k, φ ) i ) | u n ( k, φ ) i . (24)The mean position of the exciton in the continuous limit is expressed as h X ( t ) i = l π Z π/l − π/l dk h ψ k ( t ) | i∂ k | ψ k ( t ) i + dP b ( t ) + 2 dP c ( t ) , (25)where P b and P c denote the exciton distribution probability at site B and C , respectively. After one pumping cycle T with φ ( T ) = φ + π , the mean displacement h x ( t ) i = h X ( T ) i − h X (0) i is determined by h x ( T ) i = l π Z φ + πφ dφ Z π/l − π/l dk i [ h ∂ φ u n ( k, φ ) | ∂ k u n ( k, φ ) i − c . c . ] = l C n , (26) D en s i t y d i s t r i bu t i on p r obab ili t y (a) Site index (b) (c) (d) (e)(g) (h) (i) (j) (k)(m) (n) (o) (p) (q) (f)(l)(r) FIG. 3. (a) Evolution of the exciton density distribution probability and the mean displacement h x ( t ) i . The parameters usedare the same as in Fig. 2 of the main text. which proves that the center of mass (COM) motion of the exciton is quantized in units of the lattice constant.Exact diagonalization of ˆ H ( k, φ ) reveals that the energy gap between the upper ( n = 3) and the middle band( n = 2) is U/ 2, while the gap between the middle and the lower band ( n = 1) is ∼ U sin ( π/ − U / [32( E − U ) sin (3 π/ ≈ . U for the parameter adopted in the main text [ V ( d ) = 3∆]. To achieve a better adiabaticity,we consider the transport of the exciton within the upper band. To this end, we consider to prepare an initial state | ψ j i = (1 / √ c † j + ˆ a † j +1 ) | i by using Rydberg blockade, and initialize the pumping with φ (0) = 0 where the eigenstates of ˆ H ( k, φ ) are | u ( k, i = ˆ b † k | i , | u ( k, i = (1 / √ e − ikl ˆ a † k − ˆ c † k ) | i , and | u ( k, i = (1 / √ e − ikl ˆ a † k + ˆ c † k ) | i ,such that | ψ j i = (1 / √ N ) P k e − ikjl | u ( k, i creates a homogeneous population of the upper band. Adiabaticallyramping φ ( t ) from 0 to π , we can observe the topological exciton transport within the upper band.In a more rigorous treatment, we need to take the vdW interaction between the NNN into consideration, whichcontributes a perturbation Hamiltonian∆ ˆ H eff = X i h J ′ A ˆ a † i ˆ c i + J ′ B ˆ b † i ˆ a i +1 + J ′ C ˆ c † i ˆ b i +1 + H . c . i + X i h ∆ µ A ˆ a † i ˆ a i + ∆ µ B ˆ b † i ˆ b i + ∆ µ C ˆ c † i ˆ c i i . (27)The NNN hopping strength J ′ A , J ′ B , J ′ C , and the on-site potential modifications ∆ µ A , ∆ µ B , ∆ µ C are of the orderas U ′ = Ω V (2 d ) / V (2 d )] ≈ . U . Although ∆ ˆ H eff is much smaller than ˆ H eff , it has significant impact onthe transport dynamics during the long pumping period T ≈ U − . As shown in Fig. 3, the evolution of excitondensity distributions obtained with the exact model [Figs. 3(a)-3(e)] can be described by the NN approximate model[Figs. 3(g)-3(k)] only at early times. At large times, the spreading of excitations are much larger than that predictedby the NN model, but can be well described by the effective model including the NNN perturbation term ∆ ˆ H eff [Figs. 3(m)-3(q)]. Such a spreading indicates that the NNN perturbation can significantly modify the mean squaredisplacement h x i of the exciton. Nevertheless, the mean displacement h x i is not influenced by this perturbation,which is almost the same for the exact model [Fig. 3(f)], the NN effective model [Fig. 3(l)], and the NNN effectivemodel [Fig. 3(r)]. Such a robust COM motion is protected by the topology of the corresponding energy band. [1] I. Shavitt and L. T. Redmon, J. Chem. Phys. , 5711 (1980).[2] H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner, V. Vuleti´c, and M. D.Lukin, Phys. Rev. Lett. , 123603 (2018).[3] I. Lesanovsky and J. P. Garrahan, Phys. Rev. Lett. , 215305 (2013).[4] M. Marcuzzi, J. Min´aˇr, D. Barredo, S. De L´es´eleuc, H. Labuhn, T. Lahaye, A. Browaeys, E. Levi, and I. Lesanovsky, Phys.Rev. Lett.118