Controllable single-photon scattering in a Rydberg-atom-waveguide system via van der Waals interactions

Recently, there has been significant interest in light-matter interactions within waveguide structures, leading to the emergence of waveguide quantum electrodynamics (waveguide QED) [1, 2, 3, 4]. In quantum networks [5, 6], waveguides are typically regarded as quantum channels for photons, while atoms act as quantum nodes. By adjusting the properties of these quantum nodes, the transmission of photons within the waveguide can be controlled. Typically, atoms are three to four orders of magnitude smaller than the wavelength of photons traveling through the waveguide, allowing them to be effectively modeled as point-like dipoles. However, giant atoms [7, 8] have dimensions comparable to or even larger than the wavelength of photons propagating in the waveguide, rendering the traditional model inapplicable. Compared to traditional atomic systems, giant atoms offer several potential advantages, such as frequency-dependent relaxation rates and Lamb shifts [8], chiral and oscillating bound states [9, 10, 11, 12], anomalous single-photon scattering spectra [13, 14, 15, 16, 17, 18, 19, 20], decoherence-free interactions (DFIs) [21, 22, 23, 24, 25], nonexponential decay [26, 27, 28, 29, 30]. Currently, systems suitable for giant atoms include superconducting quantum circuits [22, 27, 31, 32], coupled waveguide arrays [28], synthetic frequency dimension [12], spin ensembles [33], and matter waves in optical lattices [34].

To explore new features and broaden the range of applications for giant atoms, it is crucial to develop additional methods for constructing them. In the search for approaches to build giant atoms, the goal is to find a controllable building block that exhibits the properties of giant atoms under specific conditions, while demonstrating distinct characteristics under others, and enabling the integration of multiple systems. Fortunately, Rydberg atoms precisely meet these requirements due to their controllability and compatibility, as they possess strong long-range dipole-dipole interactions and are highly sensitive to external fields [35]. Strong van der Waals interactions cause dipole blockade or antiblockade effects depending on the choice of detuning between the laser field and the atomic transition frequency. Both dipole blockade and antiblockade mechanisms have been successfully employed in various applications, including quantum information processing, quantum computation, and quantum simulation [36, 37, 38, 39, 40].

In pioneering work, a phase-dependent decay of the double Rydberg excitation, an archetype of the giant-atom effect, was observed using the Rydberg antiblockade effect [41, 42]. To the best of our knowledge, this represents the first real atoms realization of giant atoms in the optical regime. Rydberg-atom-assisted giant atoms open the door to exploring richer physics and unlocking more potential applications in quantum optics. Inspired by the core ideas in there pioneering works [41, 42], we investigate single-photon scattering in a coupled-resonator waveguide (CRW) system coupled to a pair of Rydberg atoms. Unlike the studies in [41, 42], our approach enables switching between the dipole blockade regime and the antiblockade regime [43, 44, 45] by directly changing the interatomic distance. In contrast, their systems are consistently operated within the antiblockade regime. Additionally, we focus on the single-photon scattering behaviors manifested through reflection spectrum in different regimes. This approach allows us to characterize the blockade and the antiblockade distances without measuring two-photon correlation function.

In Section II, we present the model and the equations governing the system. In Sections III and IV, we analyze the single-photon scattering phenomena arising from the coupling of a point-like Rydberg superatom and a giant atom with waveguide, respectively. In Section V, we discuss single-photon scattering during the transition between blockade and the antiblockade regimes by varying the interatomic distance. And obtain the effective criteria for Rydberg blockade and Rydberg antiblockade. By adjusting the ratio between the coupling strengths of the two atoms to their respective coupled resonators, we further verify our results. Finally, we conclude in Section VI.

As illustrated in Fig. 1(a), the system under consideration comprises an infinitely long coupled-resonator waveguide (CRW) with a pair of two-level Rydberg atoms trapped in the $0$th and $N$th resonators, respectively. The two Rydberg atoms, sharing a resonant frequency $\omega_{e}$, are driven both by the single photon propagating in the CRW (characterized by coupling frequency $\omega_{k}$ and coupling strengths $J_{0}$ and $J_{N}$) and by an external coherent field with frequency $\omega_{c}$ and Rabi frequency $\Omega_{c}$. When both atoms are excited into the Rydberg state, they interact via the van der Waals (vdW) potential $V_{d}=C_{6}/d^{6}$, where $C_{6}$ is the vdW coefficient and $d$ is the interatomic distance. It should be emphasized that $d$ and $N$ are mutually independent here. As shown in Figs. 1(b) and  1(c1), when $\omega_{e}$ lies within CRW band and $\omega_{c}$ resides outside it, the single photon in the CRW drives the lower transitions from $|g_{1}g_{2}\rangle$ to $|e_{1}g_{2}\rangle$ or $|g_{1}e_{2}\rangle$ with detuning $\Delta_{k}=\omega_{k}-\omega_{e}$, while the external field drives the upper transitions from $|e_{1}g_{2}\rangle$ or $|g_{1}e_{2}\rangle$ to $|e_{1}e_{2}\rangle$ with detuning $\Delta_{c}=\Delta+V_{d}$, where $\Delta=\omega_{e}-\omega_{c}$. The energy levels are represented as an effective four-level configuration in the two-atom basis (Fig. 1(c1)). By redefining the collective ground state as $|G\rangle=|g_{1}g_{2}\rangle$, the single-excitation state as $|E^{(1)}\rangle=(|{e_{1}}{g_{2}}\rangle+|{g_{1}}{e_{2}}\rangle)/{\sqrt{2}}$, and the double-excitation state as $|E^{(2)}\rangle=|e_{1}e_{2}\rangle$, we further obtain a superatomic configuration (Fig. 1(c2)). Varying the interatomic distance $d$ to be less than the blockade distance (such that $\Delta_{c}\gg\Omega_{c}$) enforces a strict dipole blockade effect. This closes the double-excitation channels, effectively truncating the configuration in Fig. 1(c2) to a two-level Rydberg superatom (Fig. 1(c3)). In the antiblockade regime, where the double Rydberg excitation occurs, the energy levels reduce approximately to a two-level system comprising $|G\rangle$ and $|E^{(2)}\rangle$, as the single-excitation state $|E^{(1)}\rangle$ becomes negligibly populated. During the dynamics of the excitation and de-excitation driven by the external coherent field, an accumulated phase of $\pi$ is obtained (Fig. 1(c4)).

In a rotating frame, the Hamiltonian of the entire system can be written explicitly as three terms (see Appendix A)

The Hamiltonian of the CRW reads ($\hbar\equiv 1$ hereafter)

where $\omega_{0}$ and $\omega_{e}$ denote the central frequency of the resonators and the atomic transition frequency, respectively. $a_{j}$ ($a^{\dagger}_{j}$) represents the bosonic annihilation (creation) operator on site $j$, and $\xi$ is the hopping strength between the nearest-neighbor resonators.

The interaction between a pair of atoms and their respective resonators is described by a Jaynes-Cummings Hamiltonian

The interaction between a pair of atoms and the external coherent field, as well as the vdW interaction between two Rydberg atoms, is described as follows:

If the frequency $\omega_{k}$ of anincident single photon lies within the CRW band, while the frequency $\omega_{c}$ of the coherent field lies outside it. The two atoms by the incident single photon with wave vector $k$ and energy $\omega_{k}=\omega_{0}-2\xi\cos(k)$, straightforwardly excites can be excited by the collective ground state $|G\rangle$ to the single-excitation state $|E^{(1)}\rangle$. Meanwhile, the external coherent field exclusively drives the transition from $|E^{(1)}\rangle$ to the double-excitation state $|E^{(2)}\rangle$ (see Fig. 1(c2)). Due to the conservation of the number of excitations in the system, the stationary eigenstate of the entire system in the superatom basis can be written as

where $c_{j}$ represents the probability amplitude for finding a photonic excitation in resonator $j$. $u_{e_{1}g_{2}}(u_{g_{1}e_{2}})$ denotes the probability amplitude of a single Rydberg atom being in the excited state $|e_{1}g_{2}\rangle$ ($|g_{1}e_{2}\rangle$), while $u_{e_{1}e_{2}}$ represents the probability amplitude of both atoms being in the double-excitation state $|e_{1}e_{2}\rangle$. In addition, $|0\rangle$ indicates that all of the resonators are in the vacuum state. For a 1D scattering problem, the probability amplitude $c_{j}$ can be written as:

where $r$ and $t$ are the reflection and transmission amplitudes, respectively. In the range $0<j<N$ between the two Rydberg atoms, the photon propagates forward with a probability amplitude $\alpha$ and backward with a probability amplitude $\beta$ between the resonators. By substituting $j=0$ and $j=N$ into Eq. 6, we obtain the continuity boundary conditions $1+r=\alpha+\beta$ and $\alpha e^{ikN}+\beta e^{-ikN}=te^{ikN}$ at the respective sites. Together with the two continuity boundary conditions and solving the Schrödinger equation $H|E\rangle=(\omega_{k}-\omega_{e})|E\rangle$. In a rotating frame, the real-space Hamiltonian $H_{1}$ transforms to $H$ (see Appendix A), while simultaneously, the energy of the incident photon changes to $\omega_{k}-\omega_{e}$. The reflection rate $R=|r|^{2}$ can be obtained with (see Appendix B):

where $\eta=\Delta_{k}-\Delta_{c}$, and $J_{0}=J_{N}=J$.

In this paper, we select realistic parameters of cold ${}^{\mathrm{87}}\mathrm{Rb}$ atoms, such as the transition frequency $\omega_{e}=2\pi\times 1009\times 10^{12}$Hz and the van der Waals coefficient $C_{6}=2\pi\times 2.8\times 10^{12}\mathrm{s}^{-1}\mu m^{6}$ with the ground state $|g_{1,2}\rangle=|5S_{1/2},F=2,m_{F}=2\rangle$ and the Rydberg state $|e_{1,2}\rangle=\mid 75P_{3/2},m_{J}=3/2\rangle$. To ensure that the frequency $\omega_{c}$ lies outside the CRW band while the atomic transition frequency $\omega_{e}$ remains within it, we set $\Delta\neq 0$ throughout this paper. Other parameters are given in the captions of the figures.

When the distance between a pair of Rydberg atoms is $d=3.1\mu\mathrm{m}$, they experience a significant vdW potential of $V_{d}=19.8$GHz. In this case, since $V_{d}\gg|\Delta|$, the pair of Rydberg atoms behaves as a truncated superatom, excluding the double-excitation state due to the blockade effect. The blockade effect on single-photon scattering can be demonstrated through the single-photon reflection spectrum. In Fig. 2, we compare two types of single-photon reflection spectrum: one from a system with a small atom coupled to the CRW  [46], and the other from our system consisting of a pair of Rydberg atoms coupled to the CRW. For the case of $N=0$, we observe in Fig. 2(a) that their behaviors match perfectly, indicating that the effect produced by the pair of Rydberg atoms is equivalent to that of a small atom. The reason is that for $N=0$, the pair of Rydberg atoms is located in the same resonator, and therefore can be regarded as a truncated superatom without a double-excitation state due to the blockade effect, regardless of the size of the resonator.

We now consider the case of $N\neq 0$ while keeping the distance d constant. When N takes a relatively large value (e.g. $N=21$), the accumulated phase $\phi=kN$ becomes significant, leading to a pronounced non-Markovian retardation effect. This phenomenon is manifested in Fig. 2(b) through strong oscillations in the single-photon reflection spectrum [19]. However, the envelope of the oscillating lines remains identical to that shown in Fig. 2(a). In Figs. 2(c) and  2(d), for smaller $N$, it is clearly observed that the width of the reflection window decreases with increasing $N$ for even $N$, while it remains nearly constant regardless of $N$ for odd $N$. Furthermore, the reflection window in the latter case is narrower than that in the former. From the fully consistent envelope of all oscillating lines in Fig. 2, we conclude that the underlying physics is governed by the regime in which the CRW is coupled to the truncated superatom (blockade), exhibiting point-like behavior akin to a small atom, independent of the internal structure of the CRW ($N\neq 0$). In this case, when the incident photon resonates with the atomic transition, i.e., $\Delta_{k}=0$, it is completely reflected with $R=1$.

When the interatomic distance is $d=9.5\mu$m, the two atoms experience a vdW potential of $V_{d}=24$MHz. In this case, Rydberg antiblockade occurs because the vdW interaction-induced shift is well compensated by the detuning [43, 44, 45], i.e., $\Delta+V_{d}=0$. In the antiblockade regime, as shown in Fig. 3, we observe completely different behaviors in the single-photon scattering arising from the Rydberg double-excitation (see Fig. 1(c4)). These characteristics align perfectly with the single-photon scattering of a giant atom coupled to the CRW [13], apart from a $\pi$-phase accumulation. This phenomenon can be clearly explained by the effective relaxation rate of the giant atom. Under the Markovian limit, the effective relaxation rate of a giant atom can be expressed as  [8, 22, 47, 48]

where $k_{a}$ and $D(\omega_{a})$ are, respectively, the wave vector and the density of states of the field at the atomic frequency $\omega_{a}$. $\phi$ denotes the accumulated phase arising from the coherent driving, i.e., $\phi=\pi$ in this context. $\Gamma_{\mathrm{eff}}$ indicates that the single-photon reflection rate $R$ is generally depends on $N$ with $k_{a}/\pi=k/\pi=-0.5$ and $k_{a}/\pi=0.5$, respectively (see Fig. 1(b)). Specifically, as shown in Fig. 3(a), when $N=2,6,\dots,4m+2,\dots$ ($m=0,1,\dots$), $kN$ becomes an odd multiple of $\pi$, leading to maximum dissipation for the giant atom. This results in the complete reflection of the single photon with $R=1$; When $N=4,8,\dots,4m,\dots$ ($m=1,2,\dots$), $kN$ becomes an even multiple of $\pi$, rendering the giant atom dissipationless. In this case, the giant atom and the resonator are effectively decoupled, allowing a single photon to travel through the waveguide without exciting any atoms, leading to $R=0$. As shown in Fig. 3(b), when $N=1,5,\dots,4m+1,\dots$ or $N=3,7,\dots,4m+3,\dots$ ($m=0,1,\dots$), $kN$ becomes a odd multiple of $\pi/2$, resulting in normal dissipation for the giant atom, with $R=0.5$. However, the reflection windows for these cases are separated and symmetrically distributed on both sides of $\Delta_{k}=0$. Based on the analysis above, we conclude that a pair of Rydberg atoms in the antiblockade regime, coupled to the CRW, exhibits the giant-atom effect as described in Refs. [41, 42].

In Fig. 4(a), we plot the single-photon reflection rate $R$ as a function of the interatomic distance $d$ with detuning $\Delta_{k}=0$ for different values of $N$. Clearly, as the interatomic distance $d$ varies, three distinct regimes emerge: the blockade regime, the transition regime, and the antiblockade regime. The most characteristic feature of the blockade and antiblockade effects is that the behavior of the single-photon scattering remains almost unchanged as the distance varies. In contrast, in the transition regime, the scattering behavior changes significantly, except for $N=4m+2$ ($m=0,1,\dots$). Here, we focus on the blockade and antiblockade regimes. Specifically, in the blockade regime, for all values of $N$, the incident photon is completely reflected ($R=1$) up to the blockade distance $d_{b}=3.5\mu$m due to the small-atom characteristics of the pair of Rydberg atoms, as demonstrated in Fig. 2. In the antiblockade regime, when $N=4m$ ($m=1,2,\dots$) and $N=4m+2$ ($m=0,1,\dots$), the incident photon remains completely reflected ($R=1$) and completely transmitted ($R=0$), respectively. However, when $N=4m+1$ and $N=4m+3$, $R=0.5$ emerges at the antiblockade distance $d_{ab}$, and beyond $d_{ab}$, $R$ is slightly above $0.5$ and slightly below $0.5$, respectively, due to the imperfect antiblockade effect ($\Delta+V_{d}\neq 0$). These behaviors are consistent with the single-photon scattering characteristics in the antiblockade regime, as shown in Fig. 3. Based on the results, the effective criteria for Rydberg excitation blockade and antiblockade are as follows: Blockade: $R=1$ for all $N$. Antiblockade: $R=0.5$ for odd $N$; $R=0$ for $N=4m$ ($m=1,2,\dots$); and $R=1.0$ for $N=4m+2$ ($m=0,1,\dots$). Using these criteria, the maximally entangled state $(|GE^{(1)}\rangle+|E^{(1)}G\rangle)/\sqrt{2}$ (blockade regime) and non-entangled state $|E^{(2)}\rangle$ (antiblockade regime), can be accurately identified via the single-photon scattering.

In Fig. 4(b), we plot the blockade distance $d_{b}$ and the antiblockade distance $d_{ab}$ as functions of the detuning $\Delta$ with $\Delta_{k}=0$. As $\Delta$ varies, the blockade distance $d_{b}$ remains unchanged. This is due to two factors: first and foremost, the resonant excitation condition $\Delta_{k}=0$ is consistently maintained, and second the single-excitation Rydberg state is unaffected by the external field. In contrast, the antiblockade distance $d_{ab}$ decreases as the detuning $\Delta$ increases, following $d_{ab}=(-C_{6}/\Delta)^{1/6}$.

In Fig. 5, we plot the single-photon reflection rate $R$ as a function of detuning $\Delta_{k}$ and interatomic distance $d$ for different values of $N$. In each plot, a branch appears for $\Delta_{k}<0$, governed by the dependence of the blockade distance $d_{b}\propto|\Delta_{k}|^{1/6}$  [49]. Along this branch, except at discontinuity points ($R=0$) occurring at $\Delta_{k}=-2\xi\cos[(2n+1)\pi/N]$ (where $n$ is a natural number), the blockade effect dominates despite the presence of non-Markovian effect present at those discontinuities. For instance, at the left end of the branch, $R=1$ persists uniformly for all $N$. Here, the blockade distance $d_{b}$ spans from $d=3.5\mu\mathrm{m}$ at $\Delta_{k}=0$ to $d=7.6\mu\mathrm{m}$ at $\Delta_{k}=-2\xi$. For $\Delta_{k}>0$, the blockade branch does not appear because the incident single photon with $\Delta_{k}>0$ is decoupled from the resonator due to the external field’s detuning $\Delta<0$. This behavior is illustrated in Fig. 6, where increasing the detuning $\Delta$ gives rise to another blockade branch that gradually converges toward the initial branch. In the limit of $\Delta\rightarrow-\infty$ (here, $\Delta=-4.5~{}\text{GHz}$), corresponding to an effective Rabi frequency $\Omega_{c}^{2}/\Delta\rightarrow 0$, the two branches merge and exhibit symmetry with respect to $\Delta_{k}=0$, as the influence of the external field vanishes. Under these conditions, the blockade distance $d_{b}$ for $\Delta_{k}\neq 0$ diminishes as $\Delta$ increases, following $d_{b}\propto|\Delta|^{-1/6}$. In addition, Fig. 6(d) reveals a narrow gap corresponding to the antiblockade regime where $\Delta+V_{d}=0$ (see the white dashed line).

Finally, we plot the single-photon reflection rate $R$ as a function of the ratio $J_{N}/J_{0}$ with detuning $\Delta_{k}=0$ for different values of $N$. As shown in Fig. 7(a), in the blockade regime, $R$ at $\Delta_{k}$ is independent of $J_{N}/J_{0}$ and the coupling sites, due to the point-like nature of the system, which holds regardless of $N$. In the antiblockade regime, as shown in Fig. 7(b): for $N=4m+1$ and $N=4m+3$, the single-photon reflection rate starts at $R=1$, decreases rapidly to $R=0.5$ at $J_{N}/J_{0}=1$, and then gradually increases back to $R=1$ as $J_{N}/J_{0}\rightarrow\infty$. This indicates that the coupling site of the giant atom plays a critical role because the size of the giant atom becomes significant in the antiblockade regime. For $N=4m+2$, the single-photon reflection rate $R$ remains constant at $R=1$, independent of $J_{N}/J_{0}$. For $N=4m$, the single-photon reflection rate $R$ remains at $R=1$ except for $R=0$ at $J_{N}/J_{0}=1$. These results align well with the behaviors observed in Figs. 3 and 4.

We investigate single-photon scattering in system consisting of a waveguide coupled to a pair of Rydberg atoms, illuminated by an external coherent field. By tuning the van der Waals interactions between the Rydberg atoms through changes in the interatomic distance, a transition between the Rydberg blockade and antiblockade regime can be achieved, resulting in significant changes in single-photon scattering spectrum. In the blockade regime, a pair of Rydberg atoms behaves as a small atom, whereas in the antiblockade regime, it acts as a giant atom. Remarkably, both the blockade and the antiblockade distances can be estimated directly from the single-photon scattering spectrum without directly requiring calculations of two-photon correlation function. Our work enables the realization of controllable quantum networks with simplified architectures, offering new pathways for complex quantum-network integration. Furthermore, it advances precise quantum control and quantum information processing in Rydberg-atom-waveguide quantum electrodynamics systems, while also guiding the design of single-photon quantum devices.

We thank Dr. Lei Du, Prof. Jin-Hui Wu, and Prof. Zhi-Hai Wang for their fruitful discussions. This work is supported by the National Natural Science Foundation of China (Grant No.11874004, No.11204019, and No.12204137). It is also supported by the specific research fund of The Innovation Platform for Academicians of Hainan Province (Grant No.YSPTZX202207) and the Hainan Provincial Natural Science Foundation of China (Grant No.122QN302).