Polariton $\mathbb{Z}$ Topological Insulator

We demonstrate that honeycomb arrays of microcavity pillars behave as an optical-frequency two-dimensional photonic topological insulator. We show that the interplay between the photonic spin-orbit coupling natively present in this system and the Zeeman splitting of exciton polaritons in external magnetic fields leads to the opening of a nontrivial gap characterized by a $C=\pm 2$ set of band Chern numbers and to the formation of topologically protected one-way edge states.

Figure 1

(a) Topologically protected light propagation through an edge of polariton topological insulator. The polariton graphene considered is based on an etched planar microcavity. The cavity is constituted by two distributed Bragg reflectors (DBRs) sandwiching a cavity with embedded QWs. The energy splitting existing between TE and TM polarized modes provides the photonic SOI. The application of a real magnetic field perpendicular to the $x\text{−}y$ plane of the structure opens the nontrivial gap. Edge modes are one-way propagative modes that cannot elastically scatter to the bulk states. In a stripe geometry, normally incident light is guided either clockwise or anticlockwise, depending on the external magnetic field sign. (b) A zoom view of two coupled microcavity pillars $A$ and $B$; $L$ and $T$ localized polariton modes are linearly polarized along and transversely to the $AB$ axis between the centers of the pillars.

Figure 2

Nontrivial band structure of the polariton graphene stripe in an external magnetic field. (a) Bulk energy dispersion without (dashed lines) and with (solid lines) magnetic field. (b) Illustration of degeneracy lifting in $K$ and $K^{\prime }$ points. Because of coupling between sublattice and polarization, states localized on one sublattice go up in energy at one point and down in the other. Chern numbers at each point are shown in green. (c) Numerical calculation of eigenstates: edge states are marked with color. Direction of their propagation is given by the sign of the product $H_z{g}_X$ and is protected from both backscattering and scattering into the bulk.

Figure 3

Propagation of light in the conducting $(\mathrm{\Delta }=0)$ and topological insulator phase $(\mathrm{\Delta }\ne 0)$. Calculated spatial distribution of emission intensity. (a) Rapid expansion of the bulk propagative states after 20 ps at $\mathrm{\Delta }=0$; (b) surface states after 100 ps at $\mathrm{\Delta }=0.1\mathrm{meV}$. White circles show the pumping spots located at ${\mathbf{r}}_j$. The contours of the potential are traced in black. The parameters are $\beta ={\hslash }^2(m_l^{-1}-m_t^{-1})/4m$ where $m_{l,t}$ are the effective masses of TM and TE polarized particles, respectively, and $m=2(m_t-m_l)/m_t{m}_l$; $m_t=5\times {10}^{-5}{m}_0$, $m_l=0.95m_t$, where $m_0$ is the free electron mass; ${\tau }_0=35\mathrm{ps}$, $\sigma =1\mu \mathrm{m}$, $\omega =1.6\mathrm{meV}$, $\tau =25\mathrm{ps}$. Pumping $P$ was circular polarized.