Exciton-polaritons in flatland: Controlling flatband properties in a Lieb lattice

In recent years, novel two-dimensional materials such as graphene, bismuthene, and transition-metal dichalcogenides have attracted considerable interest due to their unique physical properties. However, certain lattice geometries, such as the Lieb lattice, do not exist as atomic monolayers. Fortunately, a range of physical effects can be transferred to the realms of photonics by creating artificial photonic lattices emulating these two-dimensional materials. Here, exciton-polaritons in semiconductor microcavities offer an exciting opportunity to study a part-light, part-matter quantum fluid of light in a complex lattice potential. In this Rapid Communication, we study exciton-polaritons in a two-dimensional Lieb lattice of buried optical traps. The $S$ and $P_{xy}$ photonic orbitals of such a Lieb lattice give rise to the formation of two flatbands which are of greatest interest for the distortion-free storage of compact localized states. By using a well controlled etch-and-overgrowth technique, we manage to control the trapping as well as the site couplings with great precision. This allows us to spectroscopically monitor the flatness of the flatbands across the full Brillouin zone. Furthermore, we demonstrate experimentally that these flatbands can be directly populated by condensation under nonresonant laser excitation. Finally, using this advanced device approach we demonstrate resonant and deterministic excitation of flatband modes in transmission geometry. Our findings establish the exciton-polariton systems as a highly controllable, optical many-body system to study flatband effects and for distortion-free storage of compact localized states.

Figure 1

(a) Schematic drawing of the investigated etch-and-overgrowth Lieb lattice. $A$, $B$, and $C$ denote the three sites of the lattice unit cell, and $t$ and $t^{\prime }$ describe the nearest- and next-nearest-neighbor coupling, respectively. (b) Atomic force microscope image of the etched spacer layer. (c) Realspace and (d) first Brillouin zone representation of the Lieb lattice with $\mathrm{\Gamma }$, $X$, and $M$ denoting the high-symmetry points and $a$ being the center-to-center distance. (e) Zoom on two coupled nearest-neighbor sites in the etch-and-overgrowth geometry. Pronounced coupling and mode hybridization is possible even for traps that do not overlap. (f) Fourier space energy-resolved photoluminescence of the $S$ band of the investigated Lieb lattice. The blue data points are extracted from the measured dispersions, accurately revealing the four Dirac cones at the $M$ points. In red (flatband) and gray (dispersive bands) a tight-binding model is plotted, agreeing very well with the measured data. See Supplemental Material [34] for a video of the rotating figure. (g) Photoluminescence measurements of the $S$ and $P$ bands in the reduced Brillouin zone representation. The $S$ and $P$ flatbands are visible at around 1.603 and 1.606 eV, respectively.

Figure 2

Dispersion measurements below threshold for reduced trap distances of $v=0.80$, $v=0.90$, and $v=1.05$ along the (a)–(c) $M\text{−}X\text{−}M$ and (d)–(f) $X\text{−}\mathrm{\Gamma }\text{−}X$ directions, respectively. Particularly in the $X\text{−}\mathrm{\Gamma }\text{−}X$ direction, the next-nearest neighbor coupling ${\mathit{t}}^{\prime }$ leads to the strongest deviation from the tight-binding model and thus to a curvature of the flatbands. A full Bloch mode calculation accurately describes the band structure (gray lines) as well as the respective flatbands (red lines). (g) Measured spectral bandwidth of the $S$ and $P$ flatbands (symbols) compared to the theoretically predicted bandwidths, extracted from the calculations (solid lines).

Figure 3

Overview of the different spectroscopic methods used to populate the hybrid light-matter flatband states, using the $S$ flatband as an example. (a),(b) Nonresonant cw laser excitation with a large spot, leading to a uniformly occupied $S$-flatband condensate across several tens of unit cells. (c),(d) Nonresonant cw laser excitation of a polariton condensate in a single compact flatband site using a spiral phase plate. (e),(f) Resonant cw laser excitation of the $S$ flatband in transmission, using a backside polished sample and polarization optics to suppress the excitation laser. (b),(d),(f) Real-space images of the respective PL emission from the Lieb flatband. All methods show the distinct diamond-shaped, real-space signature of the Lieb flatband.