Bosonic Condensation and Disorder-Induced Localization in a Flat Band

We report on the engineering of a nondispersive (flat) energy band in a geometrically frustrated lattice of micropillar optical cavities. By taking advantage of the non-Hermitian nature of our system, we achieve bosonic condensation of exciton polaritons into the flat band. Because of the infinite effective mass in such a band, the condensate is highly sensitive to disorder and fragments into localized modes reflecting the elementary eigenstates produced by geometric frustration. This realization offers a novel approach to studying coherent phases of light and matter under the controlled interplay of frustration, interactions, and dissipation.

Figure 1

(a) Scanning electron micrograph of a 1D Lieb (or Stub) lattice of coupled micropillars etched out of a cavity. The enlargement shows schematically the pillar structure, with two distributed Bragg reflectors (DBRs) forming the cavity, and quantum wells (QWs) inserted at the antinodes of the optical field. (b) Geometric structure of the lattice: it contains three types of pillars ($A,B,C$) linked by couplings $t_1$, $t_2$ to nearest neighbors and $t^{\prime }$ to next-nearest neighbors. An example of plaquette state (see text) is highlighted in gray. (c) Band structure calculated in the tight-binding model with $t^{\prime }=0$, at zero and finite spectral detuning between $A$ and $C$ sites (solid and dotted lines, respectively). (d) Experimental real space emission at the flat-band energy [conditions of Fig. 2]. (e) Calculated density profile of a typical flat-band mode (only one unit cell is shown).

Figure 2

(b),(h) Energy-resolved far field emission of the structure at low excitation power, for polarization parallel (TM) and perpendicular (TE) to the lattice. Black lines are fits using the tight-binding Hamiltonian ${\mathcal{H}}_{\text{Lieb}}$ [see Eq. (1)]. When the pump excites equally all type of pillars (a), condensation occurs in the upper dispersive band and is TM polarized: panels (b)–(d) monitor the far field emission when increasing the pump power from $P=0.05P_{\mathrm{th}}$ to $P=2P_{\mathrm{th}}$, where $P_{\mathrm{th}}$ is the threshold power ($P_{\mathrm{th}}\simeq 4\mathrm{mW}$ for this condensation process). In (d) condensation is achieved at the top of the upper dispersive band. (e) Corresponding real space image of the condensate. (f) Calculated trajectory of the eigenvalues of Eq. (1) in the complex plane, as the power is increased (see arrows). (g) When the pump is centered on the line of pillars $A$, condensation occurs in the flat band and is TE polarized: panels (h)–(j) monitor the far field emission with increasing pump power ($P_{\mathrm{th}}\simeq 5\mathrm{mW}$ for this condensation process). (k) Corresponding real space image of the flat-band condensate. (l) Same calculation as (f) but for the pump configuration (g), reproducing condensation in the flat band.

Figure 3

(a)–(c) Interferograms obtained by superposing two images of the flat-band condensate (at $P=2P_{\mathrm{th}}$) shifted by $\mathrm{\Delta }x=0$, 2, and 4 unit cells. (d)–(f) Same experiment realized with the upper dispersive band condensate. (g) First-order spatial coherence $g^{(1)}$ deduced from the visibility of the interferograms, as a function of $\mathrm{\Delta }x$. (h) Solid lines: half participation ratio $\mathrm{H}\mathrm{P}\mathrm{R}$ calculated from ${\mathcal{H}}_{\text{Lieb}}$ as a function of (diagonal) disorder strength, in the flat band and at the top ($k=0$) of the upper dispersive band. Shaded areas denote the standard deviation, and the vertical line indicates the estimated disorder strength in our sample. Dots: $\mathrm{H}\mathrm{P}\mathrm{R}$ in the flat band calculated from the full Hamiltonian $\mathcal{H}(P)$ [Eq. (1)].

Figure 4

(a) Energy-resolved emission of the flat-band condensate in real space [see Fig. 2] along the line of pillars $A$. We color code in (b) the emission energy along the lattice: several localized modes are visible, emitting at distinct energies. (c) Energy-resolved emission of the upper dispersive band condensate in real space [see Fig. 2], along the line of pillars $B$ and $C$. (d) Corresponding spatial map of the emission energy, showing monochromatic emission over the whole condensate.