Extended Range of Dipole-Dipole Interactions in Periodically Structured Photonic Media

The interaction between quantum two-level systems is typically short range in free space and in most photonic environments. We show that diminishing momentum isosurfaces with equal frequencies can create a significantly extended range of interaction between distant quantum systems. The extended range is robust and does not rely on a specific location or orientation of the transition dipoles. A general relation between the interaction range and properties of the isosurface is described for structured photonic media. It provides a new way to mediate long-range quantum behavior.

Figure 1

Schematics of interactions between two TLSs mediated by (a) evanescent near-field modes and (b) propagating far-field modes. (c) The momentum isosurface $S_{\omega (\mathbf{k})={\omega }_0}$ with equal frequencies ${\omega }_0$ and $d{S}_{\mathbf{k}}$ is a small surface element.

Figure 2

(a) Two dipolar TLSs spaced by a distance $R=10\lambda$ in free space, where $\lambda =2\pi c/\omega$. (Right panel) Isosurface for the transition frequency in momentum space. The real part of ${\rho }_{\mathbf{k}}{e}^{i\mathbf{k}·\mathbf{R}}$ is plotted on the isosurface. Red and blue colors indicate positive and negative maxima, respectively. (b) Similar to (a) but with a shorter distance $R=0.3\lambda$ and thus slow oscillation on the isosurface. (c) Two TLSs are placed in a specific photonic environment, such as a Weyl photonic crystal, where the isosurface radius, $q=|\mathbf{k}-{\mathbf{k}}_c|$, can be very small. Here $R=10\lambda$. $\widehat{\mathbf{R}}$ in (a)–(c) is fixed as $(1,0,1)/\sqrt{2}$. (d),(e) The real part of radiative interaction, normalized by ${\mathrm{\Gamma }}_{\mathrm{Re}}(R=0)$, as a function of distance between two TLSs in free space and the Weyl photonic crystal, respectively. Red dots correspond to the cases in (a)–(c), respectively.

Figure 3

(a) Structure of the Weyl photonic crystal. A double-gyroid dielectric unit cell with four air spherical defects is the same as in Ref. [39]. (b) Dispersion relation on the plane of $k_z=0$. $k_{x,y}$ are normalized by $2\pi /a$, where $a$ is the lattice constant. (c) The real part of the radiative interaction ${\mathrm{\Gamma }}_{\mathrm{Re}}$ [normalized by ${\mathrm{\Gamma }}_{\mathrm{Re}}(R=0)$] as a function of distance for TLS transition frequencies (upper panel) $\omega =0.5545$, (middle panel) 0.5520, and (lower panel) $0.5512[2\pi c/a]$, which are marked with white contours (i), (ii), and (iii) in (b), respectively. The inter-TLS direction is $\widehat{\mathbf{R}}=(-1,1,1)/\sqrt{3}$. The dipole orientations are ${\widehat{\mathbf{\mu }}}_{1,2}=(-1,1,1)/\sqrt{3}$, and ${\mathbf{\mu }}_1$ is fixed at a central point of the unit cell. Green dashed curves are the envelopes of the solid curves. Insets (i)–(iii) are the isosurfaces in momentum space. (d) The linear relationship between the decay length ${\ell }_D$ “and the inverse of the isosurface size $1/\overline{q}$.

Figure 4

(a) Real part of the integrand in Eq. (2) on an elliptical isosurface with (left panel) $\widehat{\mathbf{R}}=(0,1,0)$, (middle panel) $\widehat{\mathbf{R}}=(0,1,1)/\sqrt{2}$, and (right panel) $\widehat{\mathbf{R}}=(0,0,1)$. Unit vectors $\widehat{\mathit{s}}$ and $\widehat{\mathit{l}}$ represent the short and long axes of the anisotropic isosurface. The dipole orientation is fixed as ${\widehat{\mathbf{\mu }}}_{1,2}=(0,0,1)$. (b) The same as (a), except that the isosurface is in the Weyl photonic crystal in Fig. 3 at frequency $\omega =0.555[2\pi c/a]$ and the dipole orientation is fixed at ${\widehat{\mathbf{\mu }}}_{1,2}=(0,1,0)$. (c) The absolute value of ${\mathrm{\Gamma }}_{\mathrm{Re}}$ as a function of distance $R$. Light green, blue, and red curves, respectively, correspond to $\widehat{\mathbf{R}}$ in the left, middle, and right cases of (b).