Topological Polaritons

The interaction between light and matter can give rise to novel topological states. This principle was recently exemplified in Floquet topological insulators, where classical light was used to induce a topological electronic band structure. Here, in contrast, we show that mixing single photons with excitons can result in new topological polaritonic states—or “topolaritons.” Taken separately, the underlying photons and excitons are topologically trivial. Combined appropriately, however, they give rise to nontrivial polaritonic bands with chiral edge modes allowing for unidirectional polariton propagation. The main ingredient in our construction is an exciton-photon coupling with a phase that winds in momentum space. We demonstrate how this winding emerges from the finite-momentum mixing between $s$-type and $p$-type bands in the electronic system and an applied Zeeman field. We discuss the requirements for obtaining a sizable topological gap in the polariton spectrum and propose practical ways to realize topolaritons in semiconductor quantum wells and monolayer transition metal dichalcogenides.

Popular Summary

The fact that light can propagate forward or backward (i.e., reciprocity) is part of our daily life experience: When we see someone, we can also be seen. Recently, scientists have started to explore ways of breaking this seemingly universal paradigm with the goal of creating one-way channels for light that could have applications, for example, as diodes in photonic circuits. Here, we propose a scheme to generate one-way photons by coupling ordinary ones to semiconductor excitons (bound pairs formed from a conduction-band electron and a valence-band hole). We describe how mixed exciton-photon states (or polaritons) can have a nontrivial topology, which can be thought of as “knots” encoded in their gapped band structure.

Our proposal to turn ordinary photons and excitons into topological polaritons relies on a winding in the exciton-photon coupling that provides a nontrivial mixing or “knotting” of exciton and photon bands. These knots have to be undone at the edge of the system, which leads to the presence of special edge modes. Remarkably, these modes are very robust and propagate in a single direction set by the winding of the knots (i.e., they are protected against backscattering). Similar chiral edge modes are well known in electronic systems, a prime example being quantum Hall states. Previous schemes to create analogs of quantum Hall states for light mostly relied on magneto-optical effects, however, limiting their applicability to the microwave regime.

Our scheme is not limited to excitons and photons, but can in principle also be used to generate nontrivial topology by mixing other types of bosonic bands. We expect that our results will be applicable to semiconductor quantum wells and monolayer transition metal dichalcogenides.

Figure 1

Schematic view of a system with topological polaritons. A chiral polaritonic mode consisting of a mixture of bound particle-hole pairs (excitons) and photons is found at the edge of the system.

Figure 2

Schematic polariton spectrum ${\omega }_q^{P\pm }$. The upper and lower polariton bands are drawn in red and blue, while the uncoupled bands ($g_q=0$) are shown as thin black lines for comparison. Because of the winding structure of the coupling, the two polariton bands acquire a nontrivial topology with finite Chern numbers $C_{\pm }=\pm 1$ in the case of $m=1$ [see Eq. (2)]. The associated edge modes are indicated by a dashed purple line.

Figure 3

Topological polariton spectrum. Spectrum obtained for a tight-binding lattice analog of the Hamiltonian Eq. (1) on a square lattice with periodic (open) boundary conditions in the $x$ direction ($y$ direction). Two edge modes are found within the gap, each localized at one of the system boundaries. The color scale encodes the edge localization, where red (blue) indicates a large weight of the wave function at the edge at $y=0$ ($y=L$). For the construction of the lattice model, we choose quadratic dispersions $q^2/2m_X+{\omega }_0^X$ and $q^2/2m_C$ for the exciton and photon, respectively, replacing $q_{x,y}^2\to 2-2\cos (q_{x,y})$ (note that a quadratic photon dispersion with $m_C=q_{\text{res}}/2c$ correctly reproduces the original linear dispersion close to resonance at $q_{\text{res}}$). Similarly, we model the winding coupling by $g_{\mathbf{q}}=g[\sin (q_x)+i\sin (q_y)]$. The system size is $40\times 40$ (with lattice constant $a=1$), and the other relevant parameters are given by $m_X=-500$, $m_C=1$, ${\omega }_0^X=0.25$, and $g=0.01$.

Figure 4

Schematic experimental setup for realizing topolaritons. A quantum well (in blue) is embedded in a photonic waveguide (or cavity). The boundary of the light blue disks represent equipotential lines of the periodic exciton potential with lattice constant $a$ (see Sec. 3b).

Figure 5

Scheme for opening a topological gap using a periodic potential. The solid (dashed) blue and red lines correspond to the dispersion of the lower and upper polaritons with (without) a periodic potential in a simplified one-dimensional scenario. As long as $\pi /a>q_{\text{res}}$, the lowest polariton band includes the winding around the resonance and the gap opened at the Brillouin-zone boundary is of topological nature.

Figure 6

Topological polariton spectrum with a periodic potential. The spectrum is obtained for a finite-size lattice version of the Hamiltonian Eq. (1) with an additional periodic potential of period $a$ and exhibits in-gap chiral edge modes (colored). The color scale encodes the edge localization, where red (blue) indicates a large weight of the wave function at the edge at $y=0$ ($y=L$). The system size is $40\times 40$ (with lattice constant $a=1$) and the other relevant parameters are given by $m_X=1\times {10}^5$, $m_C=1$, ${\omega }_0^X=0.9$, $g=0.1$,$V_X=0.05$.

Figure 7

Topolariton edge modes. The figure depicts the typical intensities of the exciton and photon fields obtained when pumping a finite square system optically at the center of its lower edge, shown here at time $t=4\times {10}^3$ after switching on the laser for a triangular-lattice exciton potential of strength $V_X=-0.05$ and lattice constant $a=3.86$. Units are defined here by setting ${\omega }_0^X$ and the speed of light to unity (other relevant parameters are $m_X=1\times {10}^3$, $g_q=0.1$, the pumping frequency ${\omega }_p=0.856$, and a polariton decay rate $\gamma =5\times {10}^{-4}$). A counterclockwise chiral edge mode is clearly visible, with an exciton component strongly peaked at the minima of the periodic potential. Exciton and photon fields both travel together as a unique polaritonic chiral edge mode.