Generation of Photon-Plasmon Quantum States in Nonlinear Hyperbolic Metamaterials

We develop a general theoretical framework of integrated paired photon-plasmon generation through spontaneous wave mixing in nonlinear plasmonic and metamaterial nanostructures, rigorously accounting for material dispersion and losses in the quantum regime through the electromagnetic Green function. We identify photon-plasmon correlations in layered metal-dielectric structures with 70% internal heralding quantum efficiency and reveal a novel mechanism of broadband generation enhancement due to topological transition in hyperbolic metamaterials.

Figure 1

(a) Scheme of the photon-pair generation from a nonlinear metamaterial. (b) Diagrammatic representation of the two-photon interference according to Eq. (1).

Figure 2

(a)–(c) Scheme of SFWM generation of a pair of (a) entangled plasmons, (b) entangled photons, and (c) a photon entangled with a plasmon in the gold or nonlinear dielectric structure. (d) Color map of the two-photon detection probability $|T({\mathit{k}}_i,{\mathit{k}}_s){|}^2$ in the reciprocal space vs the in-plane wave vector components (arbitrary units) in transverse magnetic (TM) polarization (${\mathit{d}}_{i,s}\propto \mathit{k}\times \mathit{k}\times \widehat{\mathit{z}}$) at $z_i=z_s=100\mathrm{nm}$. The signal (solid circle) and idler (dashed circle) light lines are plotted in white. The letters a–d mark the near- and far-field signal and idler generation regimes. (e) Efficiency of signal heralding by far-field idler photons; see Eq. (2). For all plots $\hslash {\omega }_i\approx \hslash {\omega }_s\approx \hslash {\omega }_p\approx 3\mathrm{eV}$, ${ϵ}_{\mathrm{diel}}=2$, $d_{\text{silver}}=20\mathrm{nm}$, the pump is TM polarized, $k_{p,x}=0.5{\omega }_p/c$, and silver permittivity is from Ref. [40].

Figure 3

(a) Schematic illustration of two-photon generation from a nonlinear hyperbolic metamaterial. (b) Top view of $|T_{is}({\mathit{k}}_i,{\mathit{k}}_s){|}^2$ for $z=-100\mathrm{nm}$. (c) Side view of $\mathrm{Re}{T}_{is}(k_{s,x},z_i)$. (d) Side view of $|T_{is}(k_{s,x},k_{i,z}){|}^2$. Calculated for TM polarizations of signal and idler detectors, normal pump incidence is polarized along $x$, $\hslash {\omega }_s=3.46\mathrm{eV}$, $\hslash {\omega }_p=3.6\mathrm{eV}$, $d_{\text{silver}}=12.5\mathrm{nm}$, and $d_{\mathrm{diel}}=25\mathrm{nm}$.

Figure 4

(a) Phase matching map vs the signal energy and in-plane wave vector. The color corresponds to $|\mathrm{Re}(k_{s,z}+k_{i,z}-2k_{\mathrm{pump},z})(d_1+d_2)|$. Horizonal black lines show the boundaries between the spectral regions with topologically different dispersion of signal and idler photons (shown in the insets). (b) Photon-pair spectrum in the TM polarization for signal and idler, integrated over the wave vectors inside the signal light cone, calculated for $\hslash {\omega }_{\mathrm{pump}}=3.6\mathrm{eV}$ (indicated by a horizontal line) and the same other parameters as in Fig. 3.