Topological polaritons from photonic Dirac cones coupled to excitons in a magnetic field

We introduce an alternative scheme for creating topological polaritons (topolaritons) by exploiting the presence of photonic Dirac cones in photonic crystals with triangular lattice symmetry. As recently proposed, topolariton states can emerge from a coupling between photons and excitons combined with a periodic exciton potential and a magnetic field to open up a topological gap. We show that in photonic crystals the opening of the gap can be substantially simplified close to photonic Dirac points. Coupling to Zeeman-split excitons breaks time reversal symmetry and allows to gap out the Dirac cones in a nontrival way, leading to a topological gap similar to the strength of the periodic exciton potential. Compared to the original topolariton proposal [T. Karzig et al., Phys. Rev. X 5, 031001 (2015)], this scheme significantly increases the size of the topological gap over a wide range of parameters. Moreover, the gap opening mechanism highlights an interesting connection between topolaritons and the scheme of [F. D. M. Haldane and S. Raghu, Phys. Rev. Lett. 100, 013904 (2008)] to create topological photons in magneto-optically active materials.

Figure 1

Schematic view of the setup. A triangular photonic crystal (red) is coupled to an exciton-carrying quantum well (blue). Time reversal symmetry is broken due to a Zeeman splitting of the excitons in the external magnetic field.

Figure 2

Schematic dispersion close to the photonic Dirac cone at $q=K$ without (inset) and with a finite exciton-photon coupling. The main panel shows the splitting of the photonic Dirac cone into two polaritonic Dirac cones. The latter are gapped out when the exciton-photon coupling has a winding and an additional periodic exciton potential removes accidental degeneracies (dashed lines). The color coding indicates the strength of the excitonic (blue lines) and photonic (red lines) components in the polariton wave function. Inset: Position (here on resonance, $\delta =0$) of the bare exciton dispersion (blue line) relative to the photonic Dirac cone (red lines).

Figure 3

Schematic energy spectrum close to the photonic Dirac point with (right) and without (left) the presence of an exciton-photon coupling $g_K$. All energies are measured relative to the free exciton energy ${\omega }_0^X$ indicated by the dashed black line. We choose a periodic exciton potential strength $u_X<0$ and neglect the higher energy photonic mode ${|0\rangle }_P \left(u_P\gg g\right)$. The dashed blue lines at $-2u_X$ and $6u_X$ indicate the energy range of the high-momentum excitons corresponding to the maximum and minimum of the excitonic potential defined in Eq. (7).

Figure 4

Spectrum of the tight-binding model for a cylinder geometry. Gray lines denote bulk bands, while the coloring indicates localization at one of the edges of the system, with red (blue) coloring corresponding to the edge at $y=0\left(L\right)$. Parameters used: $m_P=1,m_X=10000,{\omega }_0^X=1.2,g_0=0.1,u_P=0.25,u_X=-0.025,L=35$.

Figure 5

Size of the (lowest) topological gap for different parameters. The red curve shows the gap of the scheme presented in this paper $\left(u_P=1.0\right)$ with $g_K=0.05$. Left inset: $g_K=0.01$. Right inset: $g_K=0.1$. Other parameters are given by $m_X=5000$ and $u_X=-0.005$. For small $g_K$ the largest gaps can be obtained when the excitons are tuned close to the photonic Dirac point $\left(\delta \approx 0\right)$. The blue curve shows the original scheme [20] (no isolated Dirac points) for comparison using the same parameters except that $u_P=0$. As discussed in the text the gap opening scheme in this paper allows for much larger gaps when the exciton-photon coupling is not too large.

Figure 6

Comparison of the (schematic) band structures of different topological photon/polariton proposals. The corresponding topological gaps are represented by gray regions. (a) Haldane and Raghu [5, 6] proposal. Photonic Dirac cones of TE character (dashed red lines) are gapped out by a finite magneto-optical effect, thus creating bands with nontrivial topology (solid red lines). (b) The proposal presented in this paper (see Fig. 2). TE photonic Dirac cones (dashed red lines) are resonantly coupled to a single (say, $J_z=+1$) exciton species (dashed blue line). Hybridized modes (solid purple lines) exhibit gapped-out Dirac cones, which lead to topological bands similar to those in (a). (c) Original topological polariton proposal [20]. The nontrivial topology of the polariton modes (solid purple lines) emerges from a winding coupling of TE photons (dashed red line) and a single (say, $J_z=+1$) exciton species (dashed blue line), without the need for any Dirac cones. Since the band gaps that form at the edge of the (here quadratic) Brillouin zone (defined by the periodic exciton potential) are shifted between the $M=(\pi /a_s,\pi /a_s)$ and the $X=(\pi /a_s,0)$ points, the size of the overall topological gap is reduced.