Quantum Transport of Rydberg Excitons with Synthetic Spin-Exchange Interactions

TL;DR

This paper proposes a method to control quantum transport of Rydberg excitons using synthetic spin-exchange interactions, enabling exploration of novel transport phenomena and topological effects in engineered quantum systems.

Contribution

It introduces a new scheme for quantum transport in Rydberg atoms utilizing synthetic interactions, avoiding the need for resonant dipole-dipole interactions, and is experimentally feasible.

Findings

Demonstrates topological exciton pumping with quantized entanglement transfer

Reveals nonlocal correlated transport phenomena resilient to dephasing

Provides a tunable, long-range interaction scheme for quantum transport

Abstract

We present a scheme for engineering quantum transport dynamics of spin excitations in a chain of laser-dressed Rydberg atoms, mediated by synthetic spin-exchange arising from diagonal van der Waals interaction. The dynamic tunability and long-range interaction feature of our scheme allows for the exploration of transport physics unattainable in conventional spin systems. As two concrete examples, we first demonstrate a topological exciton pumping protocol that facilitates quantized entanglement transfer, and secondly we discuss a highly nonlocal correlated transport phenomenon which persists even in the presence of dephasing. Unlike previous schemes, our proposal requires neither resonant dipole-dipole interaction nor off-diagonal van der Waals interaction. It can be readily implemented in existing experimental systems.

Equations

$$\hat{H}=\sum _i\frac{{\Omega }_i}{2}{\hat{\sigma }}_x^i+\sum _i{\Delta }_i{\hat{\sigma }}_{rr}^i+\sum _{i<j}V({\mathbf{r}}_{ij}){\hat{\sigma }}_{rr}^i{\hat{\sigma }}_{rr}^j,$$

$${\hat{H}}_{\mathrm{eff}}=\sum _i\left({\Delta }_i+\frac{{\Omega }_i^2}{2{\Delta }_i}\right){\hat{\sigma }}_{rr}^i+\sum _{i\ne j}{I}_{ij}{\hat{\sigma }}_{rr}^i{\hat{\sigma }}_{gg}^j+J_{ij}{\hat{\sigma }}_{+}^i{\hat{\sigma }}_{-}^j,$$

$$I_{ij}=\frac{{\Omega }_j^{2}V({\mathbf{r}}_{ij})}{4{\Delta }_j[{\Delta }_j+V({\mathbf{r}}_{ij})]},J_{ij}=\sum _{\beta =i,j}\frac{{\Omega }_i{\Omega }_{j}V({\mathbf{r}}_{ij})}{8{\Delta }_{\beta }[{\Delta }_{\beta }+V({\mathbf{r}}_{ij})]}.$$

$$J_i^{(2)}\approx \frac{{\Omega }_i{\Omega }_{i+2}V({\mathbf{r}}_{i,i+2})}{4[{\Delta }_i+V({\mathbf{r}}_{i,i+1})][{\Delta }_i+V({\mathbf{r}}_{i,i+1})+V({\mathbf{r}}_{i,i+2})]}.$$

$${\hat{H}}_{\mathrm{eff}}=$$

$$+\sum _i\left({\mu }_A{\hat{a}}_i^{†}{\hat{a}}_i+{\mu }_B{\hat{b}}_i^{†}{\hat{b}}_i+{\mu }_C{\hat{c}}_i^{†}{\hat{c}}_i\right),$$

$${\mathcal{C}}_n=\frac{1}{2\pi }{\int }_{\mathrm{BZ}}{\mathcal{B}}_n(\mathbf{k})d^2\mathbf{k},$$

Full text

††preprint: APS/123-QED

Developing controlled large-scale quantum systems constitutes a central goal of quantum simulation and quantum computation Cirac and Zoller (2012); Georgescu et al. (2014). Among the variety of physical realizations, neutral atoms present several unique advantages Weiss and Saffman (2017), their inherent qubit identity, long coherence time, flexible state maneuverability, as well as tunable qubit-qubit interactions, for instance mediated by Rydberg states Saffman et al. (2010). The continued progresses in Rydberg-atom studies offer great potential for probing many-body dynamics Labuhn et al. (2016); Bernien et al. (2017); Marcuzzi et al. (2017); Kim et al. (2018); Guardado-Sanchez et al. (2018). With improved operation fidelity Levine et al. (2018) and increased system size Barredo et al. (2018), quantum simulation on the Rydberg atom based platform Kim et al. (2018); Takei et al. (2016) is becoming increasingly attractive.

The transport of particle or spin via quantum state-changing interactions is essential for understanding energy or information flow. Emulating such problems on a quantum simulator constitutes a focused thrust within the broad quantum physics community Žutić et al. (2004); Najafov et al. (2010); Collini (2013); Jurcevic et al. (2014); Akselrod et al. (2014). Earlier efforts based on Rydberg-atom systems have provided first insights Barredo et al. (2015); Schönleber et al. (2015); Orioli et al. (2018); Günter et al. (2013); Schempp et al. (2015); Letscher and Petrosyan (2018); Wüster et al. (2011), where transport of spin excitation is facilitated typically by resonant dipole-dipole interaction (DDI) or by off-diagonal van der Waals (vdW) flip-flop interaction between Rydberg states. They include direct spin-exchange between different Rydberg states Barredo et al. (2015); Schönleber et al. (2015); Orioli et al. (2018), second-order exchange between ground state and Rydberg state Günter et al. (2013); Schempp et al. (2015); Letscher and Petrosyan (2018), and a fourth-order process inside ground internal state manifolds Wüster et al. (2011); Glaetzle et al. (2015); Van Bijnen and Pohl (2015).

In this Letter, we propose a simpler yet as effective method for engineering exciton transport dynamics in a Rydberg-atom system. The use of resonant DDI or flip-flop vdW interaction is avoided. Instead, our main idea relies on a perturbative spin-exchange process by off-resonantly dressing the ground state to a Rydberg state. Capitalizing on the diagonal vdW interaction-induced Rydberg level shift, perturbations from different pathways collectively contribute to a net exchange interaction between the ground and the Rydberg states. When exciton-exciton interaction as well as dephasing are included, our model system is shown to be capable of simulating various transport dynamics unattainable in conventional spin systems. In the first example, we establish an interesting topological pumping protocol, whereby the exciton experiences a quantized center-of-mass motion. In the second example, we show that the long-range interaction between excitons permits the formation of high-order magnon bound state, which exhibits nonlocal correlations even when ballistic transport turns into classical diffusion due to dephasing.

Model.—The system we study is an array of individually trapped cold atoms, dressed by laser fields that couple the ground state $|g\rangle$ to a Rydberg state $|r\rangle$ Marcuzzi et al. (2017); Levine et al. (2018). It is modeled by the Hamiltonian

[TABLE]

where $\Omega_{i}$ and $\Delta_{i}$ are Rabi frequencies and detunings of the dressing field [Fig. 1(a)], $\hat{\sigma}_{x}^{i}=|r_{i}\rangle\langle g_{i}|+|g_{i}\rangle\langle r_{i}|$ and $\hat{\sigma}_{\alpha\alpha}^{i}=|\alpha_{i}\rangle\langle\alpha_{i}|$ ($\alpha=g,r$) are spin-flip and projection operators for the $i$-th atom (located at $\mathbf{r}_{i}$), and $V(\mathbf{r}_{ij})=C_{6}/|\mathbf{r}_{i}-\mathbf{r}_{j}|^{6}$ is the diagonal vdW interaction between atoms in the Rydberg state Bernien et al. (2017) (we take $C_{6}>0$).

First, we consider the dynamics of a single Rydberg exciton. In the limit of large detuning with $\Omega_{i}\ll|\Delta_{i}|$, $|\Delta_{i}+V(\mathbf{r}_{ij})|$; and $|\Delta_{i}-\Delta_{j}|\ll|\Delta_{i/j}|$, the singly excited state set $\{|\Psi_{i}\rangle=|g_{1}g_{2}\cdots r_{i}\cdots g_{N}\rangle,i=1,2,\cdots N\}$ forms a quasi-degenerate subspace $\Pi_{1}$. As a result, the perturbative coupling with the rest of the Hilbert space can induce strong state mixing inside $\Pi_{1}$, giving rise to coherent position exchanges of the exciton. To clarify the basic physics, we take the example of $N=2$. As shown in Fig. 1(b), the degenerate states $|r_{1}g_{2}\rangle$ and $|g_{1}r_{2}\rangle$ are off-resonantly coupled to $|g_{1}g_{2}\rangle$ and $|r_{1}r_{2}\rangle$, which can be approximately treated as two Raman pathways. For the non-interacting case ($V=0$), the contributions of these two paths cancel out. In the presence of the vdW interaction, the level shift $V$ for $|rr\rangle$ causes the two pathways collectively to yield a nonvanishing spin exchange interaction $J=\Omega^{2}/4\Delta-\Omega^{2}/4(\Delta+V)$. For a many-body system, applying second-order van Vleck perturbation theory Shavitt and Redmon (1980) to the original model [Eq. (1)] and dropping the constant $-\sum_{j}\Omega_{j}^{2}/4\Delta_{j}$, we arrive at an effective Hamiltonian sup

[TABLE]

where $\hat{\sigma}_{+}^{i}=|r_{i}\rangle\langle g_{i}|$ and $\hat{\sigma}_{-}^{i}=|g_{i}\rangle\langle r_{i}|$ are spin raising and 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

[TABLE]

In contrast to earlier dressing schemes Glaetzle et al. (2015); Van Bijnen and Pohl (2015), the spin-exchange interaction we find constitutes a pure synthetic interaction as the initial Hamiltonian Eq. (1) contains only diagonal vdW interactions. It exhibits different $r$-dependence compared with previous schemes [Fig. 1(c)] and is highly tunable in terms of $\Omega_{i}$ and $\Delta_{i}$. This effective model is not restricted to any particular type of lattice, and this work considers the simplest one-dimensional periodic chain with a spacing $d$.

The exciton transport is conveniently described by mapping spins to hard-core bosons with $\hat{\sigma}_{+}^{i}=\hat{a}_{i}^{\dagger}$ and $\hat{\sigma}_{-}^{i}=\hat{a}_{i}$, where $\hat{a}^{\dagger}_{i}$ ($\hat{a}_{i}$) creates (annihilates) a Rydberg exciton at site $i$. The effective Hamiltonian for a single exciton can then be expressed in a tight-binding form with $\hat{H}_{\mathrm{eff}}=\sum_{i}\mu_{i}\hat{a}^{\dagger}_{i}\hat{a}_{i}+\sum_{i<j}J_{ij}\left(\hat{a}^{\dagger}_{i}\hat{a}_{j}+\hat{a}^{\dagger}_{j}\hat{a}_{i}\right)$, where $\mu_{i}=\Delta_{i}+\Omega_{i}^{2}/2\Delta_{i}+\sum_{j\neq i}I_{ij}$ is the on-site potential. To benchmark the effective model, we focus on an entanglement distribution protocol, in which the entangled state $(|r_{0}g_{1}\rangle+|g_{0}r_{1}\rangle)/\sqrt{2}$ is transferred to $(|r_{0}g_{N}\rangle+e^{i\phi}|g_{0}r_{N}\rangle)/\sqrt{2}$ over a chain of $N+1$ atoms. For systems dominated by nearest-neighbor (NN) hopping, perfect entanglement transfer can be achieved when the conditions $\mu_{i}=\mu$, $J_{i,i+1}=J\sqrt{i(N-i)}$ Christandl et al. (2004) are satisfied. The initial entangled state can be set up via Rydberg blockade, and the perfect transfer condition can be met by tuning local parameters $\Omega_{i}$ and $\Delta_{i}$. As verified by numerical results ($N=6$) shown in Fig. 1(d), entanglement between the two end nodes gradually establishes and approaches the maximal value eventually, calibrated by their concurrence Wootters (1998) and state fidelity. The potential existence of disorder in atomic positions is also taken into account in the calculation Marcuzzi et al. (2017). As long as the disorder-induced interaction fluctuation $\delta V_{ij}$ is much smaller than $V_{ij}$ itself ($\delta V_{ij}\ll V_{ij}$), the transport efficiency remains high sup . We note that all numerical results presented in this work are based on solving the exact model Eq. (1).

In addition to simulating coherent dynamics, the system we consider also provides a platform for probing the crossover between coherent and incoherent transport. In the presence of dephasing Lesanovsky and Garrahan (2013); Lienhard et al. (2018), the evolution of the density matrix $\hat{\rho}$ is governed by the master equation $\partial_{t}{\hat{\rho}}=-i[\hat{H},\hat{\rho}]+\sum_{i}\mathcal{L}[\sqrt{\gamma}\hat{\sigma}_{rr}^{i}]\hat{\rho}$, where the Lindblad operator $\mathcal{L}$ gives $\mathcal{L}[\hat{\sigma}]\hat{\rho}=\hat{\sigma}\hat{\rho}\hat{\sigma}^{\dagger}-\frac{1}{2}(\hat{\sigma}^{\dagger}\hat{\sigma}\hat{\rho}+\hat{\rho}\hat{\sigma}^{\dagger}\hat{\sigma})$. For weak dephasing ($\gamma\ll|\Delta_{i}|$), the dynamics remain confined within the subspace $\Pi_{1}$ for times smaller than $t_{c}=\min\left\{\Delta_{i}^{2}/\gamma\Omega_{i}^{2}\right\}$, while for $t>t_{c}$ incoherent spin flips and exciton growth takes over Lesanovsky and Garrahan (2013). In the transport regime ($t<t_{c}$), the dynamics can be effectively described by $\partial_{t}{\hat{\rho}}=-i[\hat{H}_{\mathrm{eff}},\hat{\rho}]+\sum_{i}\mathcal{L}[\sqrt{\gamma}\hat{a}_{i}^{\dagger}\hat{a}_{i}]\hat{\rho}$, equivalent to the Haken-Reineker-Strobl (HRS) model with coherent hopping and on-site dephasing Haken and Reineker (1972); Haken and Strobl (1973); Reineker (1973). The exciton motion remains coherent for $t<1/\gamma$, and exhibits incoherent features as it enters the diffusion region $t>1/\gamma$ sup .

Next we discuss the case involving multi-excitons. With $n$ Rydberg excitons, the quasi-degenerate perturbation analysis can no longer be simply applied to the subspace $\Pi_{n}$ spanned by the states $\{|\Psi_{i_{1},\cdots,i_{n}}\rangle=|g_{1}\cdots r_{i_{1}}\cdots r_{i_{n}}\cdots g_{N}\rangle,i_{1}<\cdots<i_{n}\}$, since the large vdW interaction between excitons removes some of the degeneracy. For example, in the $n=2$ case, the doubly excited subspace $\Pi_{2}$ can be decomposed into $\Pi_{2}=\Pi^{\prime}_{2}\bigcup\Pi^{\prime\prime}_{2}$, with $\Pi^{\prime}_{2}$ spanned by the dimer states $\{|\Psi_{i,i+1}\rangle=|g_{1}\cdots r_{i}r_{i+1}\cdots g_{N}\rangle,i=1,\cdots,N\}$ and $\Pi^{\prime\prime}_{2}$ the complementary set of $\Pi^{\prime}_{2}$. If $V(\mathbf{r}_{i,i+1})$ is of the same order as $|\Delta_{i}|$, $\Pi^{\prime}_{2}$ and $\Pi^{\prime\prime}_{2}$ forms two decoupled quasi-degenerate subspaces. Inside $\Pi^{\prime}_{2}$, the dimer states are coupled to each other through next-nearest-neighbor (NNN) hoppings $\sum_{i}J^{(2)}_{i}\hat{\sigma}^{i}_{+}\hat{\sigma}^{i+1}_{rr}\hat{\sigma}^{i+2}_{-}+\mathrm{H.c.}$, with an effective three-body exchange interaction

[TABLE]

For exciton dynamics inside $\Pi^{\prime\prime}_{2}$, the perturbation analysis yields an effective Hamiltonian $\hat{H}_{\mathrm{eff}}^{\prime}=\hat{H}_{\mathrm{eff}}+\hat{H}_{\mathrm{int}}$ sup , where $\hat{H}_{\mathrm{eff}}$ is the single-exciton effective Hamiltonian, while $\hat{H}_{\mathrm{int}}=\sum_{i<j}U_{ij}\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{j}\hat{a}_{i}$ describes exiciton-exciton interactions, with strength $U_{ij}=V(\mathbf{r}_{ij})-2(I_{ij}+I_{ji})$. For the case of $n$ excitons, the dynamics of the system can still be approximately described by $\hat{H}_{\mathrm{eff}}^{\prime}$, as long as the initial separations between excitons are large enough to ensure that their mutual interactions are much smaller than the detuning. If some of the excitons are close to each other initially, they will form tightly bound state (such as the dimer state described above), whose transport property needs further elaborations.

When simulating transport physics, we focus on the case where the total exciton number $\hat{N}_{e}=\sum_{i}\hat{\sigma}_{rr}^{i}=\sum_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}$ is conserved. However, $\hat{N}_{e}$ is not strictly conserved due to a finite spin-flip probability $|\Omega_{i}/\Delta_{i}|^{2}$ for each ($i$-th) atom. This could significantly influence the quality of our simulations especially as the number of atoms increases. However, for observables whose expectation values only depend on the diagonal elements of $\hat{\rho}$ (e.g., density correlations), the simulation results can be refined via post selection based on projective measurements. If the dynamics of $n$ excitons are of interest, the expectation values of the observables are calculated by the refined density matrix $\hat{\rho}_{p}=p^{-1}\sum_{k}p_{k}\hat{P}_{k}$, where $\hat{P}_{k}=|\phi_{k}\rangle\langle\phi_{k}|$ denotes the projection operator of state $|\phi_{k}\rangle\in\Pi_{n}$, and $p_{k}=\mathrm{Tr}(\hat{\rho}\hat{P}_{k})$ is the probability of the measurement. The post-selection probability $p=\sum_{k}p_{k}$ scales as $1-N\Omega^{2}/2\Delta^{2}$, which is acceptable for a reasonable sized system ($p\approx 0.75$ for $\Delta/\Omega=10$ and $N=50$).

Example 1.—To illustrate the dynamical tunability of our scheme, we consider an implementation for topological exicton pumping, for which a time-dependent and site-dependent exchange interaction is required Thouless (1983); Qian et al. (2011); Wang et al. (2013); Lohse et al. (2016); Nakajima et al. (2016); Lu et al. (2016). A periodic system with broken parity symmetry is assumed Mueller (2004), with three lattice sites (labeled as $A$, $B$, $C$, and separated by $d$) forming a unit cell (with the period $l=3d$), dressed by control fields of three intensities [Fig. 2(a)] with corresponding Rabi frequencies $\Omega_{A},\Omega_{B},\Omega_{C}=\Omega\times\{\sin^{2}(\phi+\pi/4),\sin^{2}(\phi),\sin^{2}(\phi-\pi/4)\}$ and $\phi$ a time-dependent control parameter. Such a dressing scheme can be realized by using three independently controlled acousto-optic deflectors. Retaining the NN interaction, the system can be described by the generalized Rice-Mele Hamiltonian Rice and Mele (1982)

[TABLE]

where $\hat{a}_{i}$, $\hat{b}_{i}$, and $\hat{c}_{i}$ are exciton annihilation operators for site $A$, $B$, and $C$ of the $i$-th unit, respectively. According to the Bloch theorem, this system can be described in the quasi-momentum $k$ space with a single-particle Hamiltonian $\hat{\mathcal{H}}(k,\phi)$ sup , which is also periodic in $\phi$. Thus, we can define the energy band in the synthetic space $\mathbf{k}=(k,\phi)$ with the first Brillouin zone (BZ) $k\in(-\pi/l,\pi/l]$ and $\phi\in(-\pi/2,\pi/2]$. The topology of each band is characterized by the Chern number

[TABLE]

where $\mathcal{B}_{n}(\mathbf{k})=i\left(\langle\partial_{\phi}u_{n}|\partial_{k}u_{n}\rangle-\mathrm{c.c.}\right)$ is the Berry curvature of the $n$-th band, and $|u_{n}\rangle$ is the eigen state of $\hat{\mathcal{H}}(k,\phi)$. For the system considered above, we find three gapped bands with respective nontrivial topological numbers $\mathcal{C}_{1},\mathcal{C}_{2},\mathcal{C}_{3}=\{1,-2,1\}$, as shown in Fig. 2(b).

In this case, we can implement Thouless pumping Thouless (1983), while the parameter $\phi(t)$ is slowly modulated in time $t$. After a pumping cycle in which $\phi$ changes by $\pi$, the Hamiltonian returns to its initial form. If an energy band is filled or homogeneously populated, the mean displacement $\langle x\rangle$ of the exciton after one pumping cycle is quantized in units of lattice constant, i.e., $\langle x\rangle/l=\mathcal{C}_{n}$. For our system, the energy gap ${\Omega^{2}V(d)}/{8\Delta\left[\Delta+V(d)\right]}$ between the upper and the middle band is about two orders of magnitude larger than the gap between the middle and the lower band. Thus, to achieve better adiabaticity, we consider motion of the exciton within the upper band. As indicated by the pumping sequence shown in Fig. 2(c), we first shine a resonant field on sites $C_{j}$ and $A_{j+1}$ to produce an entangled state $|\psi_{j}\rangle=(1/\sqrt{2})(\hat{c}^{\dagger}_{j}+\hat{a}^{\dagger}_{j+1})|0\rangle$ using Rydberg blockade. With such an initialization and $\phi(0)=0$, we create an equally weighted Bloch states for the upper band. Then, we adiabatically ramp $\phi$ from [math] to $\pi$, and observe the position of the exciton. As shown in Fig. 2(d), the mean displacement of the exciton after one pumping cycle is indeed $\langle x\rangle\approx 3d=l$, in agreement with the topology of the upper band. Since this energy band is almost flat in the $k$-dimension, such a quantized motion indicates a high-efficiency entangled state transfer from $|\psi_{j}\rangle$ to $|\psi_{j+1}\rangle$. It is worth pointing out that during the long pumping cycle $T$, the NNN interaction also comes into play. In fact, the long-range interaction induced NNN hoppings $\sum_{i}\left(J_{A}^{\prime}\hat{a}^{\dagger}_{i}\hat{c}_{i}+J_{B}^{\prime}\hat{b}^{\dagger}_{i}\hat{a}_{i+1}+J_{C}^{\prime}\hat{c}^{\dagger}_{i}\hat{b}_{i+1}+\mathrm{H.c.}\right)$ and the modifications to on-site potential can be viewed as perturbations to $\hat{H}_{\mathrm{eff}}$. We find that although these perturbations can significantly modify the spread $\langle x^{2}\rangle$ of the exciton, they do not change the mean displacement $\langle x\rangle$ sup . Such a robust center-of-mass (COM) motion is protected by the topology of the band, which is invariant under continuous deformation of the Hamiltonian Niu and Thouless (1984); Mei et al. (2018).

Example 2.—The strong and nonlocal exciton-exciton interaction in the proposed system also makes it feasible for studying correlated transport Fukuhara et al. (2013); Preiss et al. (2015). Here, we consider the dynamics of two excitons in a homogeneously dressed ($\Delta_{i}=\Delta$ and $\Omega_{i}=\Omega$) chain with $V(2d)\ll|\Delta|$.

We first consider the dynamics of the dimer state $|\Psi_{i,i+1}\rangle$, which can be prepared via anti-blockade excitation satisfying $2\Delta+V(d)=0$ Amthor et al. (2010). As explained previously, such a tightly bound state can migrate through NNN hopping [see the upper panel of Fig. 3(a)], the hopping rate of which can be significant near the facilitation Marcuzzi et al. (2017) region $\Delta+V(d)=0$. This correlated transport can be measured by the second-order correlation function $g^{(2)}_{i,j}=\langle\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{j}\hat{a}_{i}\rangle$. As shown in Fig. 3(a), we find $g^{(2)}_{i,j}$ rapidly spreads on the diagonals $j=i\pm 1$ while remains localized on the orthogonal directions, which confirms the existence of such a mobile bound state.

For excitons separated by more than one site, the state evolution is governed by $\hat{H}^{\prime}_{\mathrm{eff}}$. Unlike spin systems reported earlier Fukuhara et al. (2013); Letscher and Petrosyan (2018), the long-range interaction $U_{ij}$ can be tuned much larger than the exchange rate $J_{ij}$ here, which results in a highly anisotropic $XXZ$ model and permits the existence of high-order bound states. If the NNN interaction $U_{i,i+2}$ is sufficiently larger than the NN hopping rate $J_{i,i+1}$, excitons separated by one lattice site also forms bound state and exhibits correlated motion. Different from the dimer state $|\Psi_{i,i+1}\rangle$, transport of the high-order bound state $|\Psi_{i,i+2}\rangle$ relies on a second-order process with hopping rate $\sim J_{i,i+1}^{2}/U_{i,i+2}$ [see the upper panel of Fig. 3(b)]. As verified by numerical results shown in Fig. 3(b), $g^{(2)}_{i,j}$ also localizes on the skew diagonals, but spreads on the $j=i\pm 2$ diagonals in this case. The influence of dephasing on this highly nonlocal correlated behavior is also investigated. In Fig. 3(c), we calculate the correlation function for $t>1/\gamma$, with $\gamma$ the dephasing rate introduced previously. Interestingly, the strong bunching of $g_{i,j}^{(2)}$ along $i=j\pm 2$ survives, although its distribution is different from the coherent case. The small and uniform distribution of $g_{i,j}^{(2)}$ for $|i-j|>2$ indicates the diffusion equilibrium for unpaired free excitons has been established, while the strong NN interaction $U_{i,i+1}$ forbids the diffusion into $|i-j|=1$ region. To gain a deeper insight into such a correlated transport, we investigate the COM motion of two excitons. For coherent transport, this motion is characterized by a quantum random walk Mülken and Blumen (2011), with a density distribution described by the Bessel function [see the upper panel of Fig. 3(d)]. In contrast, the COM density distribution at $t>1/\gamma$ is well fitted by a Gaussian function [see the lower panel of Fig. 3(d)]. This indicates that the high-order bound state we study exhibits diffusive expansion as a composite, which does not reach equilibrium due to its reduced diffusion rate compared to free excitons.

In conclusion, we propose a Rydberg-atom system for studying quantum transport dynamics, utilizing synthetic spin-exchange induced by vdW interaction. Our scheme does not require resonant DDI or off-diagonal vdW interaction, and thus avoids the complicated excitation schemes in multi-Rydberg-level systems. For the state-of-the-art experimental setup, its typical Rydberg lifetime $\sim~{}50~{}\mu\mathrm{s}$ Levine et al. (2018) will not hinder the exciton transport we discuss, and its influence on the dynamics can be eliminated by using projective measurement sup . In addition to simulating quantum transport phenomena, this work opens up an avenue towards constructing exotic spin models with Rydberg atoms, which for instance can facilitate the study of many-body localization Ponte et al. (2015); Smith et al. (2016).