Quantum theory of a resonant photonic crystal

We present a quantum model of two-level atoms localized in a three-dimensional lattice based on the Hopfield polariton theory. In addition to a polaritonic gap at the excitation energy, a photonic band gap opens up at the Brillouin zone boundary. Upon tuning the lattice period or angle of incidence to match the photonic gap with the excitation energy, one obtains a combined polaritonic and photonic gap as a generalization of Rabi splitting. For typical experimental parameters, the size of the combined gap is on the order of $25{\mathrm{cm}}^{-1}$, up to ${10}^5$ times the detuned gap size. The dispersion curve contains a branch supporting slow-light modes with vanishing probability density of atomic excitations.

Figure 1

(Color online) Single-polariton dispersion for a 3D cubic lattice along [100] in the extended zone scheme, with $ϵ=3\mathrm{eV}$ and three different coupling strengths, associated with the parameters $x_{01}=0$ (noninteracting), $1\mathrm{\AA }$ $(C_Q\sim 0.18\mathrm{meV})$, and $2\mathrm{\AA }$ $(C_Q\sim 0.35\mathrm{meV})$. The vertical dashed line indicates the BZ boundary at $\mid \vec{Q}\mid =1.00025ϵ∕\hslash c$. The graphs are generated numerically from Eq. (7), summing over 125 BZs.

Figure 2

(Color online) Single-polariton dispersion along [100], with $ϵ=3\mathrm{eV}$, $x_{01}=2\mathrm{\AA }$, and different lattice periods: (a) $\mid \vec{Q}\mid =1.00025ϵ∕\hslash c$ and (b) $\mid \vec{Q}\mid =ϵ∕\hslash c$. Plots (c) and (d) show the corresponding overlaps of the polariton with the bare excitation, $⟨0\mid b_{\vec{q}}{\alpha }_{\vec{q}}^{†}\mid 0⟩$, for the polaritons on the dispersion curve leading to the purely photonic state at $\vec{q}=\vec{Q}$ [indicated with arrows in (a) and (b)], which have no atomic component.

Figure 3

(Color online) Photonic gap at wave vectors $\vec{Q}=[k_x,\pi ∕\ell ,0]$ along the BZ boundary for $ϵ=3\mathrm{eV}$, $x_{01}=2\mathrm{\AA }$, and $\pi ∕\ell =0.9ϵ∕\hslash c=1.4\times {10}^5{\mathrm{cm}}^{-1}$ (red detuned). The dashed lines show $k_y$ vs $k_x$ for the surface $\mid \vec{k}\mid =ϵ∕\hslash c$ and the BZ boundary; here, the ordinate is not drawn to scale. The gap is largest at the intersection of the two surfaces, i.e., $\mid \vec{Q}\mid =ϵ∕\hslash c$.