Abstract

We propose augmenting dipole-dipole interactions with the use of epsilon-and-mu-near-zero (EMNZ) supercoupling. In particular, we demonstrate via numerical simulations that this tunneling effect enables the coupling of two distant (arbitrarily located and oriented) quantum emitters as if they were located in the proximity of each other and within a straight waveguide. This effect is empowered by the zero phase advance and geometry independent nature of EMNZ supercoupling. We expect that this mechanism might be exploited in the development of waveguide QED setups, for example, by relaxing the requirements in positioning the emitters, and/or by facilitating the design and integration of complex waveguide networks with intricate geometry. To finalize, we also provide proposals for the design of several potential realizations of this concept, including a simple microwave waveguide setup, of interest for future proof-of-concept experiments, as well as an all-dielectric platform, suitable for operation at optical frequencies and integration on a chip.

© 2017 Optical Society of America

Corrections

A. M. Mahmoud, I. Liberal, and N. Engheta, "Dipole-dipole interactions mediated by epsilon-and-mu-near-zero waveguide supercoupling: publisher’s note," Opt. Mater. Express 7, 1096-1096 (2017)
https://www.osapublishing.org/ome/abstract.cfm?uri=ome-7-3-1096

16 February 2017: A correction was made to the title.

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 ×E=0and ×H=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 d1and d2, and equal transition frequency ω0, 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]

Γ12=2ω02ε0c2d1Im{G(r1,r2,ω0)}d2
Ω12=ω02ε0c2d1Re{G(r1,r2,ω0)}d2
Here, Γ12 is the dissipative coupling coefficient that describes the variation on the radiative decay, while Ω12 is the coherent coupling coefficient describing the frequency shift of atomic transitions, both due to dipole-dipole interaction. We remark that Eq. (2) is typically considered an approximation of a more general and complex expression that include the contribution from off-resonant terms [37]. We also note that since Eqs. (1) and (2) refer to coupling coefficients between different emitters, the dyadic Green’s functions are never evaluated at equal space points avoiding the need to treat their singularities. G is the dyadic Green’s function on the environment through which the emitters interact. In general, the phase and magnitude of the elements of G strongly depend on the positions of the quantum emitters [10,24,36]. However, as anticipated, by exploiting the ability of EMNZ media to “open up” and “stretch the space”, it is possible to mitigate the impact of the turns and bends of the waveguides connecting the emitters, while keeping an accurate control of the phase progression. This effect is depicted schematically in Fig. 1(b) in which two randomly oriented waveguides are connected via a region of arbitrary shape. Interestingly, if this region is filled with an EMNZ medium, the coupling parameters characterizing the dipole-dipole interaction would be exactly the same interaction as in the straight waveguide case shown in Fig. 1(a). Naturally, this concept could be extended to a complex quantum network in which the coupling between different qubits is powered, with zero-phase advance, by EMNZ supercoupling.

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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ΔΓ12=Γ12EMNZΓ120Γ120
ΔΩ12=Ω12EMNZΩ120Ω120

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., εh/ε0=εd(π/k0d)2, where εd 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 εr = 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, Ez, 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, Ez, 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 ∼38λ02), 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.

Funding

This work is supported in part by the US Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiatives (MURI) on Quantum Metaphotonics & Metamaterials, Award No. FA9550-12-1-0488.

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47. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110(1), 013902 (2013). [CrossRef]   [PubMed]  

48. C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93(19), 195152 (2016). [CrossRef]  

49. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011). [CrossRef]   [PubMed]  

50. Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015). [CrossRef]  

51. Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006). [CrossRef]  

52. S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

References

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  1. A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
    [Crossref] [PubMed]
  2. C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).
  3. Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A 80(1), 011810 (2009).
    [Crossref]
  4. A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
    [Crossref] [PubMed]
  5. M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
    [Crossref] [PubMed]
  6. A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
    [Crossref] [PubMed]
  7. A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
    [Crossref] [PubMed]
  8. J. S. Douglas, H. Habibian, C.-L. Hung, V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9(5), 326–331 (2015).
    [Crossref]
  9. V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces,” New J. Phys. 18(4), 4043041 (2016).
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  10. A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
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  11. J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
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  12. H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
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  16. M. Silveirinha and N. Engheta, “Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
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    [Crossref]
  18. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  22. R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4 Pt 2), 046608 (2004).
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  23. N. Engheta, “Materials science. Pursuing near-zero response,” Science 340(6130), 286–287 (2013).
    [Crossref] [PubMed]
  24. D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82(7), 075427 (2010).
    [Crossref]
  25. C. Gonzalez-Ballestero, A. González-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92(15), 155304 (2015).
    [Crossref]
  26. E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87(3), 033831 (2013).
    [Crossref]
  27. R. Sokhoyan and H. A. Atwater, “Quantum optical properties of a dipole emitter coupled to an ɛ-near-zero nanoscale waveguide,” Opt. Express 21(26), 32279–32290 (2013).
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  28. R. Sokhoyan and H. A. Atwater, “Cooperative behavior of quantum dipole emitters coupled to a zero-index nanoscale waveguide,” Prepress available http//arxiv.org/abs/1510.07071 (2015).
  29. R. Fleury and A. Alù, “Enhanced superradiance in epsilon-near-zero plasmonic channels,” Phys. Rev. B 87(20), 201101 (2013).
    [Crossref]
  30. P. A. Huidobro, A. Y. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. J. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012).
    [Crossref]
  31. I. Liberal and N. Engheta, “Nonradiating and radiating modes excited by quantum emitters in open epsilon-near-zero cavities,” Sci. Adv. 2(10), e1600987 (2016).
    [Crossref] [PubMed]
  32. C. L. Cortes and Z. Jacob, “Super-Coulombic atom-atom interactions in hyperbolic media,” http://arxiv.org/abs/1601.04013 1–5 (2016).
  33. J. Kästel and M. Fleischhauer, “Suppression of spontaneous emission and superradiance over macroscopic distances in media with negative refraction,” Phys. Rev. A 71(1), 011804 (2005).
    [Crossref]
  34. Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
    [Crossref] [PubMed]
  35. G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
    [Crossref]
  36. H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66(6), 063810 (2002).
    [Crossref]
  37. W. Vogel and D.-G. Welsch, Quantum Optics (John Wiley & Sons, 2006).
  38. W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
    [Crossref]
  39. D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468 (2006).
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  40. J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
    [Crossref]
  41. J. Kim, A. Dutta, G. V. Naik, A. J. Giles, F. J. Bezares, C. T. Ellis, J. G. Tischler, A. M. Mahmoud, H. Caglayan, O. J. Glembocki, A. V. Kildishev, J. D. Caldwell, A. Boltasseva, and N. Engheta, “Role of epsilon-near-zero substrates in the optical response of plasmonic antennas,” Optica 3(3), 339 (2016).
    [Crossref]
  42. G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
    [Crossref]
  43. G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
    [Crossref] [PubMed]
  44. N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616–622 (2015).
    [Crossref]
  45. J. Brown, “Artificial dielectrics having refractive indices less than unity,” Proc. IEEE100, 51–62 (1953).
  46. W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” Antennas Propagation, IRE Trans. 10(1), 17–19 (1962).
    [Crossref]
  47. E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110(1), 013902 (2013).
    [Crossref] [PubMed]
  48. C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93(19), 195152 (2016).
    [Crossref]
  49. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
    [Crossref] [PubMed]
  50. Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
    [Crossref]
  51. Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
    [Crossref]
  52. S. G. Joannopoulos, J. D. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2008).

2016 (6)

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces,” New J. Phys. 18(4), 4043041 (2016).
[Crossref]

I. Liberal, A. M. Mahmoud, and N. Engheta, “Geometry-Invariant Resonant Cavities,” Nat. Commun. 7, 10989 (2016).
[Crossref] [PubMed]

I. Liberal and N. Engheta, “Zero-Index Platforms: where light defies geometry,” Opt. Photonics News 27(7), 26–33 (2016).
[Crossref]

I. Liberal and N. Engheta, “Nonradiating and radiating modes excited by quantum emitters in open epsilon-near-zero cavities,” Sci. Adv. 2(10), e1600987 (2016).
[Crossref] [PubMed]

J. Kim, A. Dutta, G. V. Naik, A. J. Giles, F. J. Bezares, C. T. Ellis, J. G. Tischler, A. M. Mahmoud, H. Caglayan, O. J. Glembocki, A. V. Kildishev, J. D. Caldwell, A. Boltasseva, and N. Engheta, “Role of epsilon-near-zero substrates in the optical response of plasmonic antennas,” Optica 3(3), 339 (2016).
[Crossref]

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93(19), 195152 (2016).
[Crossref]

2015 (8)

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616–622 (2015).
[Crossref]

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
[Crossref]

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

J. S. Marcos, M. G. Silveirinha, and N. Engheta, “µ-near-zero supercoupling,” Phys. Rev. B 91(19), 195112 (2015).
[Crossref]

C. Gonzalez-Ballestero, A. González-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92(15), 155304 (2015).
[Crossref]

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87(4), 1379–1418 (2015).
[Crossref]

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
[Crossref] [PubMed]

J. S. Douglas, H. Habibian, C.-L. Hung, V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9(5), 326–331 (2015).
[Crossref]

2014 (4)

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

A. M. Mahmoud and N. Engheta, “Wave-matter interactions in epsilon-and-mu-near-zero structures,” Nat. Commun. 5, 5638 (2014).
[Crossref] [PubMed]

G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
[Crossref]

2013 (7)

N. Engheta, “Materials science. Pursuing near-zero response,” Science 340(6130), 286–287 (2013).
[Crossref] [PubMed]

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87(3), 033831 (2013).
[Crossref]

R. Sokhoyan and H. A. Atwater, “Quantum optical properties of a dipole emitter coupled to an ɛ-near-zero nanoscale waveguide,” Opt. Express 21(26), 32279–32290 (2013).
[Crossref] [PubMed]

R. Fleury and A. Alù, “Enhanced superradiance in epsilon-near-zero plasmonic channels,” Phys. Rev. B 87(20), 201101 (2013).
[Crossref]

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
[Crossref] [PubMed]

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110(1), 013902 (2013).
[Crossref] [PubMed]

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

2012 (1)

P. A. Huidobro, A. Y. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. J. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012).
[Crossref]

2011 (3)

A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref] [PubMed]

G. V. Naik, J. Kim, and A. Boltasseva, “Oxides and nitrides as alternative plasmonic materials in the optical range,” Opt. Mater. Express 1(6), 1090–1099 (2011).
[Crossref]

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

2010 (1)

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82(7), 075427 (2010).
[Crossref]

2009 (2)

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A 80(1), 011810 (2009).
[Crossref]

2008 (3)

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

2007 (2)

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[Crossref]

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

2006 (3)

M. Silveirinha and N. Engheta, “Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B 23(3), 468 (2006).
[Crossref]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

2005 (1)

J. Kästel and M. Fleischhauer, “Suppression of spontaneous emission and superradiance over macroscopic distances in media with negative refraction,” Phys. Rev. A 71(1), 011804 (2005).
[Crossref]

2004 (2)

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4 Pt 2), 046608 (2004).
[Crossref] [PubMed]

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 degrees bend obtained utilizing topology optimization,” Opt. Express 12(24), 5916–5921 (2004).
[Crossref] [PubMed]

2002 (1)

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66(6), 063810 (2002).
[Crossref]

1997 (1)

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
[Crossref]

1962 (1)

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” Antennas Propagation, IRE Trans. 10(1), 17–19 (1962).
[Crossref]

1959 (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[Crossref]

Akimov, A. V.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

Alù, A.

R. Fleury and A. Alù, “Enhanced superradiance in epsilon-near-zero plasmonic channels,” Phys. Rev. B 87(20), 201101 (2013).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

Arcari, M.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Atwater, H. A.

Bezares, F. J.

Blais, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
[Crossref] [PubMed]

Boltasseva, A.

Borel, P.

Bose, R.

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Brown, J.

J. Brown, “Artificial dielectrics having refractive indices less than unity,” Proc. IEEE100, 51–62 (1953).

Bulu, I.

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A 80(1), 011810 (2009).
[Crossref]

Caglayan, H.

Caldwell, J. D.

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A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref] [PubMed]

Martín-Moreno, L.

P. A. Huidobro, A. Y. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. J. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012).
[Crossref]

Mazur, E.

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

McClung, A. C.

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

McMillan, J. F.

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Moreno, E.

C. Gonzalez-Ballestero, A. González-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92(15), 155304 (2015).
[Crossref]

A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref] [PubMed]

Mukherjee, A.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

Muniz, J. A.

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
[Crossref] [PubMed]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

Muñoz, P.

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

Naik, G. V.

Nikitin, A. Y.

P. A. Huidobro, A. Y. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. J. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012).
[Crossref]

Painter, O.

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
[Crossref] [PubMed]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

Park, H.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

Paulisch, V.

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces,” New J. Phys. 18(4), 4043041 (2016).
[Crossref]

Polman, A.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110(1), 013902 (2013).
[Crossref] [PubMed]

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Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A 80(1), 011810 (2009).
[Crossref]

Reinecke, T. L.

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
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A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87(4), 1379–1418 (2015).
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Rempe, G.

A. Reiserer and G. Rempe, “Cavity-based quantum networks with single atoms and optical photons,” Rev. Mod. Phys. 87(4), 1379–1418 (2015).
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Reshef, O.

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

Rotman, W.

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” Antennas Propagation, IRE Trans. 10(1), 17–19 (1962).
[Crossref]

Sanders, B. C.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
[Crossref] [PubMed]

Shahmoon, E.

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87(3), 033831 (2013).
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Shalaev, V. M.

N. Kinsey, C. DeVault, J. Kim, M. Ferrera, V. M. Shalaev, and A. Boltasseva, “Epsilon-near-zero Al-doped ZnO for ultrafast switching at telecom wavelengths,” Optica 2(7), 616–622 (2015).
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G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

Shvets, G.

Sigmund, O.

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

Silveirinha, M. G.

J. S. Marcos, M. G. Silveirinha, and N. Engheta, “µ-near-zero supercoupling,” Phys. Rev. B 91(19), 195112 (2015).
[Crossref]

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[Crossref]

Sokhoyan, R.

Söllner, I.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Song, G.

G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
[Crossref]

Song, J. D.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Sørensen, A. S.

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82(7), 075427 (2010).
[Crossref]

Spitzer, W. G.

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[Crossref]

Stobbe, S.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Sun, F. W.

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Tejedor, C.

A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref] [PubMed]

Thyrrestrup, H.

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

Tischler, J. G.

Urzhumov, Y.

van Loo, A. F.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
[Crossref] [PubMed]

Vesseur, E. J. R.

E. J. R. Vesseur, T. Coenen, H. Caglayan, N. Engheta, and A. Polman, “Experimental verification of n = 0 structures for visible light,” Phys. Rev. Lett. 110(1), 013902 (2013).
[Crossref] [PubMed]

Vulis, D. I.

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

Vurgaftman, I.

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
[Crossref]

Wallraff, A.

A. F. van Loo, A. Fedorov, K. Lalumière, B. C. Sanders, A. Blais, and A. Wallraff, “Photon-Mediated Interactions Between Distant Artificial Atoms,” Science 342(6165), 1494–1496 (2013).
[Crossref] [PubMed]

Walsh, D.

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[Crossref]

Welsch, D.-G.

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66(6), 063810 (2002).
[Crossref]

Wong, C. W.

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Wu, Y.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

Xu, J.

G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
[Crossref]

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Yang, Y.

G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
[Crossref]

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Yin, M.

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

Young, M. E.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

Yu, C. L.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

Yu, S.-P.

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
[Crossref] [PubMed]

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

Zhang, Z. Q.

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Zhu, S.

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

Zibrov, A. S.

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

Ziolkowski, R. W.

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4 Pt 2), 046608 (2004).
[Crossref] [PubMed]

Zoller, P.

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
[Crossref]

Adv. Mater. (1)

G. V. Naik, V. M. Shalaev, and A. Boltasseva, “Alternative plasmonic materials: Beyond gold and silver,” Adv. Mater. 25(24), 3264–3294 (2013).
[Crossref] [PubMed]

Antennas Propagation, IRE Trans. (1)

W. Rotman, “Plasma simulation by artificial dielectrics and parallel-plate media,” Antennas Propagation, IRE Trans. 10(1), 17–19 (1962).
[Crossref]

J. Opt. Soc. Am. B (1)

Nanophotonics (1)

J. D. Caldwell, L. Lindsay, V. Giannini, I. Vurgaftman, T. L. Reinecke, S. A. Maier, and O. J. Glembocki, “Low-loss, infrared and terahertz nanophotonics using surface phonon polaritons,” Nanophotonics 4(1), 44–68 (2015).
[Crossref]

Nat. Commun. (3)

A. Goban, C.-L. Hung, S.-P. Yu, J. D. Hood, J. A. Muniz, J. H. Lee, M. J. Martin, A. C. McClung, K. S. Choi, D. E. Chang, O. Painter, and H. J. Kimble, “Atom-light interactions in photonic crystals,” Nat. Commun. 5, 3808 (2014).
[Crossref] [PubMed]

A. M. Mahmoud and N. Engheta, “Wave-matter interactions in epsilon-and-mu-near-zero structures,” Nat. Commun. 5, 5638 (2014).
[Crossref] [PubMed]

I. Liberal, A. M. Mahmoud, and N. Engheta, “Geometry-Invariant Resonant Cavities,” Nat. Commun. 7, 10989 (2016).
[Crossref] [PubMed]

Nat. Mater. (1)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10(8), 582–586 (2011).
[Crossref] [PubMed]

Nat. Photonics (2)

Y. Li, S. Kita, P. Muñoz, O. Reshef, D. I. Vulis, M. Yin, M. Lončar, and E. Mazur, “On-chip zero-index metamaterials,” Nat. Photonics 9(11), 738–742 (2015).
[Crossref]

J. S. Douglas, H. Habibian, C.-L. Hung, V. Gorshkov, H. J. Kimble, and D. E. Chang, “Quantum many-body models with cold atoms coupled to photonic crystals,” Nat. Photonics 9(5), 326–331 (2015).
[Crossref]

Nature (2)

A. V. Akimov, A. Mukherjee, C. L. Yu, D. E. Chang, A. S. Zibrov, P. R. Hemmer, H. Park, and M. D. Lukin, “Generation of single optical plasmons in metallic nanowires coupled to quantum dots,” Nature 450(7168), 402–406 (2007).
[Crossref] [PubMed]

H. J. Kimble, “The quantum internet,” Nature 453(7198), 1023–1030 (2008).
[Crossref] [PubMed]

New J. Phys. (1)

V. Paulisch, H. J. Kimble, and A. González-Tudela, “Universal Quantum Computation in Waveguide QED using Decoherence Free Subspaces,” New J. Phys. 18(4), 4043041 (2016).
[Crossref]

Opt. Express (2)

Opt. Mater. Express (1)

Opt. Photonics News (1)

I. Liberal and N. Engheta, “Zero-Index Platforms: where light defies geometry,” Opt. Photonics News 27(7), 26–33 (2016).
[Crossref]

Optica (2)

Photonics Nanostructures – Fundam. Appl. (1)

C. W. Wong, J. Gao, J. F. McMillan, F. W. Sun, and R. Bose, “Quantum information processing through quantum dots in slow-light photonic crystal waveguides,” Photonics Nanostructures – Fundam. Appl. 7, 47–55 (2009).

Phys. Rev. (1)

W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared properties of hexagonal silicon carbide,” Phys. Rev. 113(1), 127–132 (1959).
[Crossref]

Phys. Rev. A (5)

E. Shahmoon and G. Kurizki, “Nonradiative interaction and entanglement between distant atoms,” Phys. Rev. A 87(3), 033831 (2013).
[Crossref]

G. Song, J. Xu, and Y. Yang, “Quantum interference between Zeeman levels near structures made of left-handed materials and matched zero-index metamaterials,” Phys. Rev. A 89(5), 053830 (2014).
[Crossref]

H. T. Dung, L. Knöll, and D.-G. Welsch, “Resonant dipole-dipole interaction in the presence of dispersing and absorbing surroundings,” Phys. Rev. A 66(6), 063810 (2002).
[Crossref]

Q. Quan, I. Bulu, and M. Lončar, “Broadband waveguide QED system on a chip,” Phys. Rev. A 80(1), 011810 (2009).
[Crossref]

J. Kästel and M. Fleischhauer, “Suppression of spontaneous emission and superradiance over macroscopic distances in media with negative refraction,” Phys. Rev. A 71(1), 011804 (2005).
[Crossref]

Phys. Rev. B (8)

R. Fleury and A. Alù, “Enhanced superradiance in epsilon-near-zero plasmonic channels,” Phys. Rev. B 87(20), 201101 (2013).
[Crossref]

P. A. Huidobro, A. Y. Nikitin, C. González-Ballestero, L. Martín-Moreno, and F. J. García-Vidal, “Superradiance mediated by graphene surface plasmons,” Phys. Rev. B 85(15), 155438 (2012).
[Crossref]

J. S. Marcos, M. G. Silveirinha, and N. Engheta, “µ-near-zero supercoupling,” Phys. Rev. B 91(19), 195112 (2015).
[Crossref]

D. Dzsotjan, A. S. Sørensen, and M. Fleischhauer, “Quantum emitters coupled to surface plasmons of a nanowire: A Green’s function approach,” Phys. Rev. B 82(7), 075427 (2010).
[Crossref]

C. Gonzalez-Ballestero, A. González-Tudela, F. J. Garcia-Vidal, and E. Moreno, “Chiral route to spontaneous entanglement generation,” Phys. Rev. B 92(15), 155304 (2015).
[Crossref]

M. G. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75(7), 075119 (2007).
[Crossref]

C. Della Giovampaola and N. Engheta, “Plasmonics without negative dielectrics,” Phys. Rev. B 93(19), 195152 (2016).
[Crossref]

Y. Wu, J. Li, Z. Q. Zhang, and C. T. Chan, “Effective medium theory for magnetodielectric composites: Beyond the long-wavelength limit,” Phys. Rev. B 74(8), 085111 (2006).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

R. W. Ziolkowski, “Propagation in and scattering from a matched metamaterial having a zero index of refraction,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70(4 Pt 2), 046608 (2004).
[Crossref] [PubMed]

Phys. Rev. Lett. (8)

Y. Yang, J. Xu, H. Chen, and S. Zhu, “Quantum interference enhancement with left-handed materials,” Phys. Rev. Lett. 100(4), 043601 (2008).
[Crossref] [PubMed]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100(3), 033903 (2008).
[Crossref] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of Electromagnetic Energy through Subwavelength Channels and Bends using ε-Near-Zero Materials,” Phys. Rev. Lett. 97(15), 157403 (2006).
[Crossref] [PubMed]

M. Arcari, I. Söllner, A. Javadi, S. Lindskov Hansen, S. Mahmoodian, J. Liu, H. Thyrrestrup, E. H. Lee, J. D. Song, S. Stobbe, and P. Lodahl, “Near-unity coupling efficiency of a quantum emitter to a photonic crystal waveguide,” Phys. Rev. Lett. 113(9), 093603 (2014).
[Crossref] [PubMed]

A. Goban, C.-L. Hung, J. D. Hood, S.-P. Yu, J. A. Muniz, O. Painter, and H. J. Kimble, “Superradiance for Atoms Trapped along a Photonic Crystal Waveguide,” Phys. Rev. Lett. 115(6), 063601 (2015).
[Crossref] [PubMed]

A. González-Tudela, D. Martin-Cano, E. Moreno, L. Martin-Moreno, C. Tejedor, and F. J. Garcia-Vidal, “Entanglement of two qubits mediated by one-dimensional plasmonic waveguides,” Phys. Rev. Lett. 106(2), 020501 (2011).
[Crossref] [PubMed]

J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, “Quantum State Transfer and Entanglement Distribution among Distant Nodes in a Quantum Network,” Phys. Rev. Lett. 78(16), 3221–3224 (1997).
[Crossref]

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Rev. Mod. Phys. (1)

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[Crossref]

Sci. Adv. (1)

I. Liberal and N. Engheta, “Nonradiating and radiating modes excited by quantum emitters in open epsilon-near-zero cavities,” Sci. Adv. 2(10), e1600987 (2016).
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Figures (4)

Fig. 1
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.
Fig. 2
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.
Fig. 3
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.
Fig. 4
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, Ez, 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.

Equations (4)

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Γ 12 = 2 ω 0 2 ε 0 c 2 d 1 Im { G ( r 1 , r 2 , ω 0 ) } d 2
Ω 12 = ω 0 2 ε 0 c 2 d 1 Re { G ( r 1 , r 2 , ω 0 ) } d 2
Δ Γ 12 = Γ 12 E M N Z Γ 12 0 Γ 12 0
Δ Ω 12 = Ω 12 E M N Z Ω 12 0 Ω 12 0

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