Anapole-Plasmon Strong Coupling Induced Large Rabi Splitting in Dielectric-Metallic Hybrid Nanostructures

Manipulation of light-matter interaction is critical in modern physics, especially in the strong coupling regime. Based on the highly localized field enhancement in the near field, we demonstrate that strong coupling can be achieved in the dielectric-metallic hybrid structure consisting of a perforated Si nanoblock filled with a slotted Ag nanodisk (nanoprism) in its center. Large Rabi splitting of 275 (262) meV is achieved due to the generation of new hybrid energy states via the strong anapole-plasmon coupling. Our results pave a feasible way to realize large Rabi splitting and offer an attractive platform to explore strong coupling regime in the light-matter interaction.


I. INTRODUCTION
I NTERACTION between light and matter has attracted considerable attention and has become the heart of modern optics due to its important role in optoelectronics, quantum optics and nanophotonic devices and so on [1], [2], [3]. Strong light-matter interaction might create new hybrid energy states with excellent application prospects in tunable low-threshold semiconductor lasers [4], low-energy switching [5] and cutting -edge nanophotonic devices, etc [6], [7], [8].
A specific regime of light-matter interaction can be realized when the intrinsic dissipation rate becomes slower than the rate of coherent energy exchange between matter and optical excitation [9]. Such a regime is called strong coupling, which can promote the formation of hybrid states of part-light and part-matter [10]. The half-light and half-matter bosonic quasiparticles are greatly important in quantum science, optoelectronics and nonlinear optics and so on [11], [12], [13], [14]. Strong coupling also opens up a lot of possibilities for fascinating advances such as Bose-Einstein condensation [15], [16], [17] and optical spin switching [18]. Therefore, manipulation of The light-matter interaction in the strong coupling regime is critical in modern physics. Exciton-polariton coupling in microcavities can serve as an excellent platform to control exciton conductance [19] and strong optical nonlinearity [20]. Initially, the observation of strong coupling was limited to experiments at cryogenic temperatures [21], [22], [23], [24]. With the development of nanoscience, various photonic and plasmonic nanostructures were used to achieve strong coupling at room temperature [25], [26], [27]. Hybridized states of part-light and part-matter characteristics formed in these structures were manifested by the mode splitting in the optical spectrum [22], [28]. As a result, two entirely hybrid energy states were produced, i.e., Rabi splitting. This so-called Rabi splitting has a wide application range [29], including control and modification of chemical reactions [30], [31], [32], low-threshold lasers [33], optical switching [34] and quantum information processing [35].
In this work, we take an alternative route by utilizing the anapole modes in the cube nano-pixel geometry [47] to acquire large Rabi splitting. Strong coupling appears between the anapole and localized surface plasmon modes in the Si nanoblock-Ag nanodisk (nanoprism) hybrid structures and considerable Rabi splitting of 275 (262) meV is achieved. The obtained Rabi splitting in the Si nanoblock-Ag nanodisk (nanoprism) hybrid nanostructure is larger than those reported in Refs. [40], [41], [42], [43]. Our findings open new horizons to develop strong coupling effect for superior optoelectronic applications at nanoscales. Fig. 1 shows the schematics of the perforated Si nanoblock (a), the perforated Si nanoblock filled with the slotted metallic nanodisk (b) and nanoprism (c). The shape of perforation in the Si nanoblock corresponds to the shape of metal component in the hybrid structure. Au and Ag are employed to build the slotted nanodisk and nanoprism. The length and width of the air hole in the metal constituent both are 5 nm. The radius of Au or Ag nanodisk is 13 nm. The Au or Ag nanoprism is rhombus with the two diagonals of 50 nm and 140 nm. The heights of metallic nanodisk (nanoprism), Si nanoblock and air hole are all 50 nm. The length and width of the perforated Si nanoblock are the same and are denoted with L. A three-dimensional (3D) finite-difference time-domain (FDTD) method is used to execute the calculations. A total-field/scattered-field (TFSF) source with a wavelength ranging from 550 nm to 950 nm is used to act as the light source with the polarization along the x-axis direction. The permittivities of Si [48] and Au (Ag) [49] are taken from the published data. The mesh volume of Si nanoblock is 5 × 5 × 5 nm 3 . The mesh size is set to be 1 nm around the metallic constituent. The refractive index of surrounding medium is 1.0. Perfect matching layers are adopted in all directions.  of corresponding perforated Si nanoblocks with L = 355 and 390 nm are also calculated and shown in Fig. 2(a), marked with red solid and dashed lines, respectively. Prominent scattering dips are observed here, indicating the excitation of anapole modes of perforated Si nanoblocks. The red-shift of scattering dip with L indicates the tunable anapole mode. Moreover, the scattering dip at 715 nm of the perforated Si nanoblock with L = 355 nm overlaps with that of the absorption peak of the slotted Ag nanodisk. For the perforated Si nanoblock with L = 390 nm, the scattering dip at 758 nm overlaps with that of the slotted Au nanodisk. The overlapping of the scattering dip of perforated Si nanoblock with the absorption peak of slotted metallic nanodisk, contributing to strong couple of localized surface plasmon and anapole when these two components are integrated together. Similar phenomena are found in the metal nanoprisms and corresponding perforated Si nanoblocks ( Fig. 2(b)). As L increases from 435 to 460 nm, the scattering dip of perforated Si nanoblock firstly overlaps with the absorption peak of Ag nanoprism and then with that of Au nanoprism.

III. RESULTS AND DISCUSSIONS
To obtain the correct Rabi splitting, extinction cross-section curves of these two kinds of hybrid nanostructures were calculated [36]. Fig. 2(c) shows the extinction cross-section curves of the integrated Si nanoblock-Au (Ag) nanodisk hybrid nanostructures. Different from the only one dip found in the scattering cross-section curve of the perforated Si nanoblock and only one peak in the absorption spectrum of slotted metal nanodisk, there are two identifiable dips, denoted as the upper (λ 3 = 668 nm) and lower (λ 4 = 784 nm) hybrid energy modes, appearing in the Si nanoblock-metal nanodisk hybrid nanostructure. The lower and upper hybrid energy modes represent an anti-crossing behavior, indicating the resonance coupling effect of anapole and plasmon and large Rabi splitting [50], [51]. The two dips both shift to the longer wavelength region (corresponding 714 nm and 814 nm) when the Ag nanodisk is replaced by the Au nanodisk in the hybrid nanostructure. This again indicates the lower and upper hybrid energy modes derivation from the resonance coupling of plasmon and anapole. Moreover, the two dips both become shallow when the Ag nanodisk is replaced by the Au nanodisk with a decreased interval (Δλ changing from 119 nm to 100 nm), indicating that larger Rabi splitting might be acquired in the hybrid nanostructure with Ag nanodisk. This can be explained by the fact that the resonance energy of Au nanodisk is larger but its coupling strength is lower than that of Ag nanodisk [52], [53].
When the slotted metallic nanoprism is inserted in the perforated Si nanoblock, similar phenomena are also observed in the extinction cross-section curves (Fig. 2(d)). The upper and lower hybrid energy modes both shift to the longer wavelength region with the increased L. Moreover, the dip in the short wavelength region is shallower than that observed in Fig. 2(c). While for the dip in the longer wavelength region, opposite phenomenon is obtained. These differences can be ascribed to the shape change of metal constituent in the hybrid structures. Moreover, for the hybrid nanostructure with a Ag nanoprism, the interval between the two distinct dips λ 7 (722 nm) and λ 8 (852 nm) is close to that obtained in the hybrid nanostructure with the Au nanoprism but the depths of these two dips both become shallower, again indicating the weakened coupling strength in the hybrid nanostructure with the Au nanoprism.
To understand the physical mechanisms of the phenomena observed in Fig. 2, corresponding electric field distributions in the x-y and y-z planes are calculated. Considering the similar optical properties of Au and Ag constituents with the same configurations, only the electric field distributions of the structures with Ag components are shown in Fig. 3. The electric field distribution pattern of the Ag nanodisk at λ 1 = 715 nm (Fig. 3(a)) verifies the excitation of localized surface plasmon. The electric field distribution and electric field energy flow directions of the perforated Si nanoblock at λ 2 = 715 nm (Fig. 3(b)) demonstrate the excitation of anapole mode in such a structure [41], [54], [55]. When the slotted Ag nanodisk is placed in the perforated Si nanoblock, the electric field energies for the dips at λ 3 = 668 nm and λ 4 = 784 nm both are mainly gathered in the center as shown in Fig. 3(c) and (d). These indicate the excitation of stronger localized surface plasmons of the Ag nanodisk. Strong electric energies in the neighborhoods, with similar distribution patterns to that of the perforated Si nanoblock, are also observed here, implying the excitation of anapole mode simultaneously. That is, strong coupling of anapole and plasmon occurs and then promotes the appearance of the upper and lower energy modes in such hybrid nanostructures. Moreover, the energies of these two dips both are stronger than those observed in Fig. 3(a) and (b), demonstrating that the anapole mode can enhance the light-matter interaction via the strong coupling effect. Fig. 3(e)-(h) displays the electric field distributions for the slotted Ag nanoprism, perforated Si nanoblock and their integrated structure at resonance modes, respectively. Similar electric field distribution patterns again verify the excited localized surface plasmon of Ag nanoprism, anapole of perforated Si nanoblock and strong coupling of anapole and plasmon in the integrated structure at λ 7 = 722 nm and λ 8 = 852 nm. For the dip at λ 7 = 722 nm, although the coupling of anapole and plasmon can also be found here, the surrounding electric field energy is very weaker than that in the center, implying the domination role of localized surface plasmon on this dip. This might be attributed to the particular shape of Ag nanoprism with two unequal diagonals. These findings again imply that the Rabi splitting can be ascribed to the excited plasmon and anapole modes and their strong resonance couple effect.
Angle-resolved extinction cross-section spectra of Si nanoblock-Ag nanodisk and Si nanoblock-Ag nanoprism hybrid nanostructures are shown in Fig. 4(a) and (b), respectively. It can be clearly seen that the positions of the two dips in these two structures are almost invariable but the depths both become shallow and shallow when the incident angle increases from 0, 15, 30, and 45°in intervals of 15°. These illustrate that the incidence angle effect on the Rabi splitting can be ignored when it is less than 45°. Fig. 4(c) and (d) respectively shows the extinction crosssection curves of Si nanoblock-Ag nanodisk and Si nanoblock-Ag nanoprism nanostructures at different polarization angles. No distinct change can be found in the curve of the Si nanoblock- Ag nanodisk hybrid nanostructure with the polarization angle increasing from 0 to 90°with a step of 30°(from TM to TE polarization), indicating the polarization-insensitivity of Rabi splitting. Differentially, opposite change trends in depths and positions of these two dips in the Si nanoblock-Ag nanoprism hybrid nanostructure imply the polarization-dependence of Rabi splitting, which can be ascribed to the specific shape of Ag nanorpism with unequal diagonals.
In order to identify the anapole-plasmon coupling behavior, dispersion curves of perforated Si nanoblock, slotted Au nanodisk and their hybrid nanostructure are calculated by changing the L of the perforated Si nanoblock. The results are depicted in Fig. 5(a). An obvious anti-crossing behavior is exhibited in the energy of the lower branch (LB) and upper branch (UB), ensuring the Rabi splitting. When the energies of the uncoupled plasmon mode (E P ) and anapole mode (E A ) are equal (zero detuning, δ = E P -E A = 0), large Rabi splitting can be extracted as the minimal splitting occurs between the LB and UB [3], [53]. Hopfield coefficients of UB and LB that illustrate the intermixing behavior between the anapole and plasmon are displayed in Fig. 5(b). The contribution of uncoupled states in the hybrid structure can be obtained via the energy fraction of plasmon and anapole [53], [56]. The large Rabi splitting Ω, proportional to the coupling strength g ( Ω = 2 g) [40], [41], [42], [43], [53], can be obtained as δ = 0. Here, it can easily obtain that the Rabi splitting value is Ω AP = 214 meV when L = 390 nm, satisfying the strong coupling criterion [57], [58]: Ω AP > ( Γ A + Γ p )/2 between the plasmon ( Γ p = 169 meV) and anapole ( Γ A = 179 meV) modes. Negative detuning (δ < 0) implies a larger anapole fraction in UB and smaller anapole fraction in LB. Positive detuning (δ > 0) indicates a larger plasmon fraction in UB and smaller plasmon fraction in LB. In this work, the detuning is modulated by the anapole mode to obtain larger Rabi splitting with δ = 0.
When the Au nanodisk is replaced by the Ag nanodisk, corresponding dispersion curves and Hopfield coefficients for anapole and plasmon contributions to UB and LB are shown in Fig. 5(c) and (d), respectively. The calculated Rabi splitting is as high as 275 meV when L = 355 nm, larger than that of the Si nanoblock-Au nanodisk nanostructure. Large Rabi splitting of 262 meV can also be achieved in the Si nanodisk-Ag nanoprism hybrid nanostructure with L = 435 nm, also larger than 240 meV obtained in the Si nanodisk-Au nanoprism hybrid nanostructure with L = 460 nm. These results again demonstrate the appearance of strong anapole-plasmon hybrid energy states and superior Rabi splitting in such structures. Table I shows a brief comparison of coupling related nanostructures and Rabi splitting values in previous works (including calculation (cal.) and experiment (exp.)) and our work. Obviously, our proposed nanostructures are with the Rabi splitting value larger than those observed in the conventional nanostructures [40], [41], [42], [43], [50] and are simpler than those with close Rabi splitting values reported in Refs. [44], [45], [46]. Our findings open a new way for realizing strong anapole-plasmon coupling and large Rabi splitting in simple nanostructures.

IV. CONCLUSION
In summary, we have successfully demonstrated that strong coupling between the anapole and plasmon modes can be realized in the Si nanoblock-Au (Ag) nanodisk (nanoprism) hybrid structures. Two anapole-plasmon hybrid states are generated in such structures. Large Rabi splitting value reaching to 275 meV is achieved when the slotted Ag nanodisk is inserted in the perforated Si nanoblock, larger than those obtained in the anapole-plasmon systems reported previously [41], [42]. Furthermore, these hybrid nanostructures are simpler than those reported in Refs. [44], [45], [46]. Our findings open a new way to realize strong coupling of anapole-plasmon and pave an excellent platform to manipulate the light-matter interaction in optoelectronic field at nanoscales.