High-fidelity, deterministic interactions between individual quantum emitters and photons, as well as photonmediated interactions between emitters, are central to quantum science and engineering [1][2][3][4][5][6]. In free space, these interactions are limited by the emitter's scattering cross section, which is typically bounded by a small geometrical limit [7]. To circumvent these limits, optical cavities and waveguides can be used to enhance interaction probabilities between atomic emitters [8][9][10][11][12]. Recent research has shown that photonic crystals can also engineer such atom-photon interactions [13][14][15][16]. While substantial experimental progress toward these goals has been made [17][18][19][20][21][22], widespread applications remain limited by multiple obstacles. For instance, many of these techniques require exquisite control of emitter arrays near or at nanostructured surfaces, which is experimentally challenging.
Coherent control of quantum emitters with 2D arrays furnishes inherent advantages over solid state architectures or classical dipole arrays such as dynamic reconfigurability [23], substantially larger coherent coupling strengths [24], and environments devoid of surface imperfections [25]. Additionally, these systems feature intrinsic quantum nonlinearities and fermionlike behavior of interacting photons [26,27]. While 1D atomic chains have been studied for coupling impurity atoms of broad and delocalized linewidth [28] and similar investigations have focused on other quantum emitters such as superconducting qubits [29][30][31], only recently has it been shown that 2D optical atomic lattices, in contrast to their 1D counterparts, can interact strongly with individual photons [32][33][34]. In particular, the rich, two-dimensional mode structure of 2D arrays can provide a highly coherent interface capable of directional photon transfer [35] and a variety of other quantum information applications not feasible in 1D geometries [26,32,[36][37][38][39].
In this Letter, we demonstrate that 2D atom arrays can be used to engineer emitter-photon interactions and to enable high-fidelity, long-range interactions between emitters. We consider impurity atom emitters [Fig. 1 dipole-dipole interactions with other impurity atoms. This allows one to confine and guide impurity emission within the 2D surface and to engineer impurity-photon bound states that generate strong and coherent interaction between distant impurities. We extend the analogy between atom arrays and cavities by defining quality factors of impurity coupling to far-field light Q ð1Þ and photon transfer between two impurities Q ð2Þ , which describe the number of coherent photon exchanges obtainable during the system's relaxation time [40].
We first summarize the formalism of an isolated array in the absence of impurities [26,32]. We consider a square 2D atom array in the xy plane with lattice spacing a ≲ λ, where ω L ¼ 2πc=λ is the resonance frequency of the lattice atoms [Fig. 1(a)]. Subwavelength spacing is obtainable, e.g., using ultracold atoms in optical lattices [33,41,42]. Taking ℏ ¼ 1 and lowering operators σ i , the non-Hermitian Hamiltonian for the isolated N-atom array is
J ij and Γ ij are the coherent and dissipative parts of the free-space dipole-dipole interactions between the ith and jth lattice atoms separated by displacement vector
for real-space Green's tensor Gðr; ωÞ with normalized transition dipole vector di of atom i [43][44][45][46]. This formalism holds while the retardation of light within the spatial scale of our system is negligible [47]. In the limit of large lattices, Eq. ( 1) is diagonal in momentum space, yielding eigenstates that are collective surface modes of xy-plane quasimomentum k, lowering operator
where Gðk;
, there exist guided modes jkj > ω L such that k z is imaginary and the excitation is an evanescent mode, propagating along the lattice without loss [26,32]. Conversely, modes for which jkj < ω L decay into the far field and are said to be within the light cone [Fig. 1(b)].
We now introduce an impurity atom into the array interstitially [24]. Provided that jω I -ω L j ≪ ω I , ω L , the responses of both atomic species are narrow peaks around ω L , and the Green's tensor formalism holds [43][44][45][46]. This condition could also be fulfilled by a single atomic species by shifting one or more of the resonant frequencies using, e.g., an ac shift produced by optical tweezers. The coupling of an impurity with lowering operator s and position vector r s to any surface excitation σ k is described by the non-Hermitian Hamiltonian under the rotating wave approximation
where
As seen from Eq. ( 4), the coupling between the impurity and lattice atoms depends on their relative polarization. We assume that all atoms have either right or left-handed circular polarization in the xy plane and identify the two polarization configurations key to this work: (1) the identical configuration, where both the lattice and impurity atoms have the same polarization (e.g., both right-handed) and (2) the orthogonal configuration, where the lattice and impurity atoms have the opposite polarization (e.g., right and left-handed; see Fig. S2). These polarizations could be individually addressed by inducing Zeeman shifts with a z-axis magnetic field. The orthogonal configuration still leads to impurity-lattice interaction, as these polarizations are only orthogonal for light emitted along the z axis, not within the xy plane.
To gain intuition for the distinct effects of these two polarization configurations, we study a toy model: an impurity in a 2 × 2 atom array, with details derived in the Supplemental Material (SM) [24]. The impurity only couples to two of four array modes: vk , the lowest momentum mode with the largest linewidth, and v⊥ , the highest momentum mode with the narrowest linewidth. In vk , all atoms oscillate in phase, whereas in v⊥ they oscillate π out of phase in a checkerboard pattern. These modes form the symmetry points of the Brillouin zone [Fig. 1(b)], with vk at the center (Γ) and v⊥ at the corner (M). An impurity in the orthogonal configuration only couples to v⊥ and an impurity in the identical configuration only couples to vk . In this latter combination, the impurity and array oscillate π out of phase, forming a state comparable to a dark state in V-type electromagnetically induced transparency [48]. The orthogonal configuration impurity couples to v⊥ , forming a bright state. The effect of the array on the impurity converges with relatively few atoms [24].
Provided that γ I ≪ γ L , the array's dynamics occur on a timescale much shorter than that of the impurity s, rendering it a Markovian bath. To simultaneously this condition and the resonance frequency requirement, different isotopes of the same element could be used, e.g., 87 Sr and 88 Sr [28,49]. Alternatively, tightly focused beams on select atoms could induce two-photon transitions by coupling them to metastable states, selectively tuning both the target atoms' resonance frequency and linewidth. The impurity exchanges photons, both real and virtual, with the array, giving rise to the so-called self-energy term Σ SE through which the impurity is influenced by its own presence, reducing Eq. (3) to effective Hamiltonian H Eff ¼ ðΣ SEiγ I =2Þs † s. Formally, the self-energy is computed from Eq. ( 3) by calculating the impurity's Schrodinger equation of motion and eliminating the array degrees of freedom by solving for the lattice modes in steady state (Markovian bath) [24], yielding
The self-energy is key to understanding impurity-lattice interactions as it modifies the effective frequency and decay rate of the impurity to ω Eff ¼ ω I þ Re½Σ SE and Γ Eff ¼ γ I -2Im½Σ SE , respectively. These equations are valid as long as Σ SE varies little on the interval δ LI þ Re½Σ SE AE Γ Eff , such that the electromagnetic response of the lattice atoms with respect to δ LI is approximately constant compared to that of the impurity atom. Under these same conditions, ω Eff -ω L ≈ δ LI . For broad Γ Eff , Σ SE can vary considerably, and non-Markovian analysis is valuable [15,50]. Figure 2(a) displays Γ Eff in the identical configuration. Below ω BE (red curve), Γ Eff is enhanced as the impurity couples to resonant lattice modes, particularly those in the light cone. Above ω BE , however, the linewidth of these states is suppressed by destructive interference between impurity radiation and off-resonant coupling with these modes. We can maximize the impurity lifetime (creating the "dark" state explained above) due to a particular σ k by minimizing the corresponding term in Γ Eff [maximizing Eq. ( 5)] with respect to δ LI . As we place s at a plaquette center, Js ðkÞ, Γs ðkÞ are real, and we obtain the optimized lattice detuning
This quantity is plotted in black in Fig. 2(a) for k ¼ 0 and corresponds to the curve of smallest Γ Eff and largest excitation probability. The correspondence of k ¼ 0 demonstrates that light cone coupling dominates identical configuration dynamics. In the SM, we show that linewidth suppression is lattice spacing limited, as Γ Eff → 0 in the limit a=λ ≪ 1, while δ D LI ∝ 1=a 3 [24]. Figure 2(b) depicts Γ Eff in the orthogonal configuration. Like in the identical configuration, Γ Eff is enhanced by impurity coupling to resonantly driven lattice modes for δ LI < ω BE . However, orthogonal configuration Γ Eff has greater enhancement that occurs near resonance with the band edge, not the light cone.
If we add an incident driving field 3), the impurity will experience both direct Rabi drive Ω I and an array-mediated driving response. Assuming that Ω L ðkÞ=γ L ≪ 1, we eliminate the array degrees of freedom and find the effective Rabi frequency
The resultant single-impurity quality factor Q ð1Þ ¼ Ω Eff =Γ Eff can be very large, indicating many coherent oscillations [40]. Ω Eff =Γ Eff ≥ Ω I =γ I for the identical polarization case with a weak, perpendicularly incident drive [24].
We now focus on lattice-mediated interactions between two impurities, s and q, which exchange photons via dipole-dipole interactions. This exchange has a latticeindependent component ϕ, which is simply the free-space dipole-dipole interaction between the impurities [51], and a lattice-mediated component, which represents the modification of the interimpurity dipole-dipole interactions by PHYSICAL REVIEW LETTERS 126, 223602 (2021) 223602-3 lattice interactions. Eliminating the lattice degrees of freedom [24], the effective dipole-dipole interaction between s and q is H ð2Þ
Eff s † q þ c:c. [24], where
The quantity Φ sq Eff is a key metric because it describes the lattice-mediated photon transfer between impurities, analogous to Eq. ( 7), but with the driving field replaced by that of the second impurity. Thus, Φ sq Eff depends on both the distance between impurities d and the placement of the impurities within their respective plaquettes. In regimes of large dissipative Φ sq Eff , the system experiences gain that can be interpreted as parity-time symmetry breaking [52].
When δ LI is above the band edge (Markovian regime) and the impurities are the photon transfer dynamics form an iSWAP gate between the impurities with interaction strength described by modified excitation transfer rate Φ sq Eff and decay rate Γ Eff . As iSWAP gates have the necessary nonlinearity to be universal for quantum computation [53,54], they serve as a basis for quantum computing architectures. This interaction results in coherent oscillations with large two-impurity quality factors (number of coherent excitation transfers [40])
Eff =Γ Eff (Fig. 3). We now characterize Q ð2Þ for both polarization configurations, highlighting the distinct advantages of each case.
Figure 3(a) shows time-dependent transfer of excitation probability (Q ð2Þ ∼ 10 2 ) between impurities s and q in the orthogonal configuration with d ¼ 0.4λ and a ¼ 0.1λ. The high frequency, small amplitude modulations are induced by lattice mode interactions, especially those near the band edge. As this coupling leads to impurity-lattice states outside of our Markovian approximation, the analytic value for orthogonal configuration Q ð2Þ in Figs. 3(b) and 4 are slight overestimates [15], whereas the oscillations of Fig. 3(a) are exact numerical solutions. Fig. 3(b) is restricted to δ LI > 1.05ω BE and δ LI ¼ 1.05ω BE in Fig. 3(a) to limit this error. The yellow regions show the strong coupling regimes near the band edge (Q ð2Þ ∼ 10 2 ), while the dark blue lines represent regions of vanishing Q ð2Þ occurring when the free-space and lattice-mediated components of Φ sq Eff destructively interfere. Strong impurity-impurity coupling also occurs in the identical configuration. , where the two impurities in a Markovian bath approximation hold nearly exactly. This configuration also exhibits regions of low Q ð2Þ due to vanishing Φ sq Eff . For each polarization configuration, we examine the effect of impurity distance d and lattice spacing a on Q ð2Þ . Larger d weakens both free-space and array-mediated dipole-dipole interactions, reducing Φ sq Eff . In the orthogonal configuration, Q ð2Þ is proportional to e -d=ξ for some parameter dependent length scale ξ [Fig. 4(a)]. This scaling is consistent with the width of exponentially localized impurity-array bound states [14,15,28] and holds until Q ð2Þ approaches its free-space limit (dashed, light-blue curve for a=λ ¼ 0.2). The behavior of the identical configuration is similar but demonstrates larger Q ð2Þ for small d. However, as the identical configuration stems from few atom dark states, its Q ð2Þ decreases more rapidly with growing d, rendering it preferable for nearby impurities [24].
Maximum coupling occurs between impurities in adjacent plaquettes. Figure 4(b) displays Q ð2Þ max ≡ Q ð2Þ ðd ¼ aÞ. In the identical configuration, Q ð2Þ max diverges as 1=a 6 for small a, which is consistent with the 1=a 3 dipole-dipole interaction strength that mediates the coupling enhancement and linewidth suppression of the dark state. Similarly, the orthogonal and free-space configurations exhibit a 1=a 3 scaling, which is consistent with coupling enhancement in a system of relatively static linewidth [Γ Eff ≈ γ L for ω > ω BE ; see Fig. network would be limited by the maximum number of lattice spacings d=a at which a desired Q ð2Þ could be achieved. Figure 4(c) shows approximately logarithmic scaling in λ=a for Q ð2Þ > 1.
Overall, lattice-mediated coupling improves Q ð2Þ by several orders of magnitude and extends nonlinear impurity-impurity coupling to tens of lattice sites. While both configurations achieve these effects, we reemphasize that the identical and orthogonal configurations yield larger Q ð2Þ for small and large d, respectively.
In conclusion, we have demonstrated that 2D atom arrays can effectively mediate between single photons and impurity atoms and have elucidated the role of polarization in these interactions. We find that 2D arrays feature superior performance to that of their 1D counterparts, as their rich, planar mode structure not only leads to the higher fidelity photon transfer required for quantum information processing but also deterministically guides photons within the 2D plane [35], creating quantum network geometries that are infeasible in one dimension. As the optimal detuning for these coherent interactions is above the lattice band edge, the excitation can be localized near the impurity. This allows for nonlinear impurityimpurity interactions that are substantially stronger and reach farther than those in free space, resulting in highly coherent two-atom interactions of large two-impurity quality factors Q ð2Þ . Such predictions could be experimentally detected by absorption measurement of individual impurity atoms or by probing time-resolved coherent dynamics of impurity atom pairs [33,55,56]. These results provide a framework for a multilevel treatment [57] that could extend to both array and impurity atoms and devise coherent switching, quantum gates, and guided mode excitation [26]. As the system's strong coherence and controllable dissipation display parity-time symmetry breaking [52], they apply to studies of exceptional points [58]. Finally, we note that similar effects can be explored in solid state systems, such as transition metal dichalcogenides [59], where excitons could mediate interactions between localized impurities.