1. Introduction

Dipole-dipole interactions mediated by photons propagating in a waveguide are a key element in basic research on light-matter interactions and quantum information processing [1–9]. Indeed, waveguide QED setups are particularly convenient as they facilitate inherently long-range and efficient interactions. However, the development of waveguide QED systems also confronts several challenges. For instance, the location of and the separation between quantum emitters (QEs) must be accurately controlled, since, due to the phase progression associated with propagation, the nature of the dipole-dipole interaction periodically changes from dissipative to coherent as a function of the position of and separation between the emitters [10]. This difficulty is likely to become even more acute when scaling preliminary waveguide QED setups into a complete quantum information network [11–13], particularly if it must be compact and integrated into a chip. In fact, although most studies so far have addressed dipole-dipole interactions occurring on a straight waveguide, a complete and compact waveguide network must necessarily include turns, bends, splitters, etc. In such an intricate geometry, keeping an accurate tracking of the phase progression might become a cumbersome task [14].

Here, we propose the use of epsilon-and-mu-near-zero (EMNZ) supercoupling [15] as a strategy to overcome this difficulty and, in general, as a tool to enhance dipole-dipole interactions and to build quantum networks with accurate phase tracking. This work is motivated by early theoretical and experimental efforts [15–19], which demonstrated that structures with near-zero parameters enable tunneling/supercoupling, i.e., full transmission with zero-phase advance through distorted waveguides with arbitrary cross-section. As schematically depicted in Fig. 1, the benefits of supercoupling to tailor dipole-dipole interactions are at least twofold: First, the lack of phase progression alleviates the negative impact of waveguide turns and bends in the coupling between the emitters. Second, the geometry-invariant [20,21] nature of this effect simplifies the integration of complex waveguide architectures with intricate and compact geometries.

 

Fig. 1 (a), (b) Schematic of an epsilon-and-mu-near-zero (EMNZ) medium emulating opening up or stretching the space at the dotted line between the two sections of an air-filled parallel-plate waveguide, while keeping the region as a “single point” electromagnetically. This keeps the interaction between two quantum emitters placed in the two sections of the waveguide the same as for a straight waveguide regardless from their relative orientation.

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In essence, within a EMNZ medium, from a macroscopic viewpoint we have $\nabla \times \overrightarrow{E}=\overrightarrow{0}$and $\nabla \times \overrightarrow{H}=\overrightarrow{0}$ simultaneously [22], which leads to a scenario where the electric and magnetic phenomena become decoupled and spatially distributed statically, while still temporally dynamic [22,23]. On the one hand we can still observe optical phenomena due to the dynamic nature of the field, while on the other hand the field distributions are spatially of static-like nature. As studied in Ref [15] and as schematically depicted in Fig. 1, by having such an EMNZ region we effectively “open up” or “stretch” the space, without affecting external electromagnetic entities and quantities, implying that we can have an electromagnetically large physical volume, that would have otherwise influenced the electromagnetic wave propagation externally, but now owing to the EMNZ effect it behaves as if it is a ‘single point’ electromagnetically as viewed from the external world. As we will show, this effect facilitates the coupling to distant, arbitrarily located and oriented, quantum emitters as if they were in the proximity of each other and placed within a straight waveguide.

Previous waveguide-based works have addressed dipole-dipole interactions in plasmonic wires [10,24], photonic crystals [6–8], chiral waveguides [25], metallic rectangular waveguides at cut-off [26–29] and superconducting waveguides [4]. Dipole-dipole interactions have also been studied in more exotic environments, including graphene [30], epsilon-near-zero (ENZ) [31], hyperbolic [32] and left-handed [33–35] media. We believe that these structures, although of great scientific and technological interest on its own, do not provide the flexibility in turning, bending, and deforming waveguides with zero phase advance offered by EMNZ supercoupling. Therefore, in this work we shed the light on the ability of EMNZ media to introduce a new platform within which long-range interactions occur between QEs. In addition, we propose and design several realizations for future experimentation of this concept including a microwave waveguide setup (which could serve for an experimental verification of the overall wave dynamics) as well as an all-dielectric structure (which could be the basis for operation in the optical domain and on-chip integration).

2. Results and discussion

A conceptual sketch of the proposed system is depicted in Fig. 1(a): Two two-dimensional (2D) emitters, with electric dipole moments $d_1$and $d_2$, and equal transition frequency ${\omega }_0$, are placed inside a waveguide and separated by a distance S. Their dipole-dipole interaction can be described in terms of the coupling parameters [36,37]

We start by discussing potential implementations of this concept. For instance, it was proposed in Refs [15,17]. that using a host medium that has a near zero permittivity (ENZ), - for example, a dispersive continuous media near its plasma frequency [38–44] or a waveguide section with the TE₁₀ mode near its cut-off frequency [45–47] - and loading it with a properly designed dielectric rod that generates enough magnetic field inside the rod mostly in the opposite direction to the magnetic field in the ENZ region (i.e., outside the rod) in order to make the total flux of to reach zero, we can achieve an effective near-zero permeability as well, and consequently achieve an effectively EMNZ medium [15,17]. It was also discussed that owing to the ENZ nature of the host medium the shape of the EMNZ structure as well as the location of the rod within the structure does not influence the EMNZ behavior, and that for a fixed cross-sectional area and dielectric rod properties (its radius and relative permittivity), the performance would be preserved regardless of how arbitrary the external shape of the region is or how arbitrary the location of the rod is within the structure.

The structure to be studied is shown in Fig. 2(a): an arbitrarily shaped two-dimensional (2D) ENZ region of a cross-sectional area (1.37 λ₀)² (where λ₀ is the free space wavelength) is loaded by a dielectric rod with circular cross section of radius 0.137 λ₀ and relative permittivity of 8, such that the overall structure exhibits an EMNZ behavior at the frequency of operation. The dielectric rod is then moved within the structure and at every location the fields within the structure is numerically calculated and consequently the electric field dyadic Green function is evaluated. In order to quantify the deviation of those quantities from their values for the case of two emitters in a single straight waveguide, separated by a distance S, we define the following two figures of merit:

 

Fig. 2 (a) Schematic of proposed structure for interaction between two two-dimensional (2D) quantum emitters separated by an epsilon-and-mu-near-zero (EMNZ) medium emulating opening up or stretching the space by being embedded between two air-filled waveguides. (b), (c) Colormap of ∆Γ₁₂ and ∆Ω₁₂, respectively, as a function of the location of the center of the rod C within the proposed EMNZ region. The interaction between two 2D emitters placed in the two sections of the waveguide is effectively the same as for a straight waveguide regardless of their relative orientation and the location of the rod.

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Color-maps for ∆Γ₁₂ and ∆Ω₁₂ are generated versus the location of the rod’s center C within the EMNZ region. It is clear from Figs. 2(b) and (c) that for ∆Γ₁₂ and ∆Ω₁₂ are almost zero everywhere showing that our proposed structure, which is indeed able to maintain the coupling between emitters over large distance regardless of the shape of the structure as well as the location of the rod within it.

In practice, even the need of a natural ENZ medium can be avoided by exploiting the structural dispersion in metallic waveguides. In fact, it has been shown that parallel metallic plates can simulate a two-dimensional artificial plasma when the TE₁₀ mode is considered [45,46]. Specifically, the effective permittivity of the such waveguide structures follows a Drude-like model [45,46], i.e., ${\epsilon }_h/{\epsilon }_0={\epsilon }_d-{(\pi /k_{0}d)}^2$, where ε_d is the relative permittivity of the dielectric between the metallic plates, d is the separation between the metallic plates, and k₀ is the wave number in free space. Here we investigate the ability of that structure to exhibit the phenomenon of long-range interaction between arbitrarily oriented emitters in a similar manner to the previously discussed 2D scenario. Note, however, that in this case a fully three-dimensional (3D) structure is considered.

The proposed configuration is as shown in Fig. 3(a), where the two waveguides that are required to emulate an air-filled parallel plate waveguide with the TEM mode are mimicked using TE₁₀ mode in rectangular waveguides filled with a dielectric of relative permittivity of 2. The EMNZ section is formed of an air-filled waveguide of a surface area 1.5 λ₀ x 1.5 λ₀ (where λ₀ is the free space wavelength) loaded by a dielectric rod of radius 0.1 λ₀ and relative permittivity of about 15.74. Two scenarios are then considered in order to investigate the ability of such structure to mimic the features discussed above. The dielectric rod is placed at two different locations within the proposed EMNZ section, at the center and at one of its corners. In both cases ∆Γ₁₂ and ∆Ω₁₂ are evaluated in a similar fashion than in the previous section. However, it is clear from Figs. 3(b) and (c) that unlike the 2D implementation, the performance of the structure is not as robust with respect to the rod location within the EMNZ region. In other words, moving the rod from the center to the corner of the structure leads to a non-zero ∆Γ₁₂ and ∆Ω₁₂. This effect is due to mode coupling from the desired TE₁₀ mode into an undesired TM₁₀ mode which is not accounted for while designing the EMNZ structure [48]. This difficulty can be circumvented by surrounding the perimeter of the dielectric rod with 8 thin PEC wires to prohibit the excitation of the TM mode in the vicinity of the rod [48], as schematically depicted in Fig. 3(a). Indeed, the numerical simulations reported in Figs. 3(b) and 3(c) illustrate how adding a wire mesh generally mitigates the impact of the positions of the rod in the coupling coefficients. However, the system is not optimized and some small mode coupling might still persist. We expect that further optimization of the wire mesh (number, location and radius of the wires) could results in ∆Γ₁₂ and ∆Ω₁₂ being almost zero for arbitrary rod locations.

 

Fig. 3 (a) Schematic of two emitters separated by the proposed 3D structure for an effective epsilon-and-mu-near-zero (EMNZ) medium by being embedded between two waveguides. This keeps the interaction between two 3D emitters placed in the two sections of the waveguide effectively the same as for a straight waveguide regardless of their relative orientation. (b), (c) ∆Γ₁₂ and ∆Ω₁₂, respectively, as a function of normalized frequency. Addition of PEC wires around the dielectric rod mitigates the impact of the location of the rod within the structure.

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This result suggests that EMNZ-mediated dipole-dipole interactions could be recreated with a simple microwave waveguide experiment. This setup might be valuable for a future proof-of-concept experiment aiming to validate the overall wave dynamics of the process. Furthermore, ENZ behavior of waveguides at cut-off has been validated even for visible light [47]. However, this configuration might not be suitable for high-performance operation at optical frequencies and/or integration on a chip. Therefore, we next investigate the possibility of applying our concept of coupling two emitters over large distances within an all-dielectric platform. Following Ref [49], our design is based on a 2D photonic crystal (PC) exhibiting a ‘Dirac cone at k = 0’. In fact, it was verified by numerical simulations and through experiments [49,50] that an effective medium theory [51] can indeed link such ‘Dirac cone at k = 0’ PCs with reasonable dielectric constants to an EMNZ system, and that purely dielectric PCs can behave as if they acquire near-zero constitutive parameters (both permittivity and permeability) for the TE mode (whose electric field is along the axis of the PC). In addition, it is well known that a photonic crystal with a complete bandgap would not allow any light mode to propagate within it [52]. Moreover, if one creates defects into the photonic crystals light can be guided in such chain of defects [52].

Here, we numerically show that combining a 2D PC with a complete photonic band gap and a PC with a ‘Dirac cone at k = 0’ PCs (as in Ref [49].) we are able to create an all-dielectric platform within which our concept can be applied with low losses in the optical domain domains. The geometry of the proposed configuration is depicted in Fig. 4(a). Region 1 is a 2D PC with a complete band gap at the operation frequency constituted with a square lattice of dielectric pillars of a lattice constant a₁ = 0.28 λ₀, radius r₁ = 0.0847 λ₀ and ε_r = 12.5 in a host medium that is air, where λ₀ is the free-space operating wavelength. Two waveguides are obtained by removing one row of pillars from the PC and creating a defect waveguide. We then consider a ‘Dirac cone at k = 0’ PCs region (Region 2, following Ref [49].) between the two waveguides within the 2D PC environment that is designed using the design flow proposed in Ref [49]. for achieving the zero index condition in a PC. The host medium is kept the same as in region 1 and the ‘Dirac cone at k = 0’ medium is also formed using a square lattice of dielectric pillars of a lattice constant a₂ = 0.56 λ₀, radius r₂ = 0.112 λ₀. The ‘Dirac cone at k = 0’ region has an area of 6.16 λ₀ x 6.16 λ₀. Figure 4(b) shows the band structure of both the complete bandgap PC (blue line) and the ‘Dirac cone at k = 0’ PC (red line). The dotted horizontal line shows the operating frequency. As depicted, at this frequency Region 1 indeed exhibits a complete bandgap behavior, while Region 2 exhibits a Dirac cone with accidental degeneracy [49] that leads to EMNZ performance. It is worth mentioning here that this system is designed to work at slightly higher frequency than the exact Dirac cone point as shown in Fig. 4(b) in order to avoid exciting the longitudinal mode by the higher order K-vector components appearing at the interface of the waveguides and the ‘Dirac cone at k = 0’ PC.

 

Fig. 4 (a) Geometry of proposed structure: A photonic crystal (PC) with a Dirac point at k=0 (Region 2, following [48]) is surrounded by a photonic band gap (PBG) region (Region 1). Input and output waveguides from the internal PC are constructed by removing single rows of rods in the PBG region. (b) Band structure of the PBG (Region 1, blue line) and the Dirac cone PC (Region 2, red line), (c) Sketch, snapshot and phase of the electric field, E_z, for three different configurations corresponding to “laterally shifted” (left), “90deg bend”(center), “180deg bend” (right) waveguides. The simulations setup is excited with a 2D magnetic y-oriented dipole (shown as blue arrow). (d) ∆Γ₁₂ and ∆Ω₁₂, as a function of normalized frequency for all three aforementioned cases. The numerical simulations ratify the enhanced transmission with almost zero-phase advance in all studied configurations.

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Then, the system is excited numerically by placing a y-oriented 2D magnetic dipole within the upper-left waveguide (see 4(c), shown as blue arrow). Using this configuration, we numerically study three different examples in order to illustrate the flexibility of the proposed configuration in facilitating the integration of waveguide systems: a “laterally shifted” waveguide, a “90-deg bend” waveguide, and a “180-deg bend” waveguide. Figure 4(c) includes snapshots of the electric field, E_z, and its phase distribution, for these three different examples. It can be concluded that regardless of the orientation of the output waveguide, the emitted fields from the 2D magnetic dipole couple and propagate through the ‘Dirac cone at k = 0’ PC with almost no reflection and no phase progression. This effect is further ratified in Fig. 4(d), which depicts the figures of merit ∆Γ₁₂ and ∆Ω₁₂, as a function of frequency. As expected, both quantities approach zero at the frequency of operation, and in fact the coupling parameters for all “laterally shifted”, “90-deg bend” and “180-deg bend” cases are very similar across the studied bandwidth. We remark that the bandwidth of EMNZ tunneling is inversely proportional to the area of the EMNZ region, and thus the proposed configuration (with area ∼38${\lambda }_0^2$), presents a narrow bandwidth. Intuitively, the larger the area of the region required to exhibit a zero phase progression, the more accurate the material parameters must approach zero. Then, due to the compulsory dispersive nature of a near-zero response, the smaller the bandwidth is. Optimization of the bandwidth and additional aspects of the system performance should be carried out for each specific technological implementation and it is left for future efforts. Nonetheless, this preliminary design supports the possibility of exploiting EMNZ supercoupling for the development of rather flexible platforms for long-range emitter interaction within an all-dielectric PC environment.

3. Conclusions

We proposed EMNZ media as a rather flexible platform to engineer long-range dipole-dipole interactions. Specifically, we theoretically demonstrated that EMNZ supercoupling facilitates the interaction of two distant quantum emitters as if they were closely placed and connected via a straight waveguide. This configuration might be of interest in the development of waveguide QED setups for basic research of light-matter interactions and/or quantum information processing. By using numerical simulations, the performance of the proposed structures were shown to be robust in various aspects, including shape, orientation with respect to the connected waveguides and the location of the dielectric inclusion placed inside. Furthermore, we proposed some designs for future experimental realizations based on exploiting structural dispersion in metallic waveguides and/or arrays of dielectric rods. We expect that these designs will catalyze future experimental verifications either in the form of proof-of-concept experiments at microwave frequencies, and/or all-dielectric platforms operating at optical frequencies. The configurations presented here could be extended to more complex waveguide architectures and/or multiple dipole-dipole interactions.