Controlled Ordering of Topological Charges in an Exciton-Polariton Chain

We demonstrate, experimentally and theoretically, controlled loading of an exciton-polariton vortex chain into a 1D array of trapping potentials. Switching between two types of vortex chains, with topological charges of the same or alternating signs, is achieved by appropriately shaping an off-resonant pump beam that drives the system to the regime of bosonic condensation. In analogy to spin chains, these vortex sequences realize either a “ferromagnetic” or an “antiferromagnetic” order, whereby the role of spin is played by the orbital angular momentum. The ferromagnetic ordering of vortices is associated with the formation of a persistent chiral current. Our results pave the way for the controlled creation of nontrivial distributions of orbital angular momentum and topological order in a periodic exciton-polariton system.

Figure 1

(a) Schematic of a 1D mesa array microstructured in an $\mathrm{AlAs}/\mathrm{AlGaAs}$ microcavity formed between two distributed Bragg mirrors (BM) with embedded GaAs quantum wells (QW) illuminated by an off-resonant optical pump (red shaded area, not to scale). (b) Dispersion measurement of the polariton emission from the mesa traps below the condensation threshold. A $p$ energy band consists of two subbands and is formed by the hybridization of the $p$ modes of the individual mesas. Panel (c) shows schematics of the $\sigma$-bond and $\pi$-bond hybridization forming the $p$ band in panel (b).

Figure 2

Photoluminescence above the condensation threshold. (a) Dispersion of exciton polaritons excited by a large-area pump spot (inset, not to scale). Two distinct peaks at the edges of the Brillouin zone (BZ) are visible. The signal in the center of the BZ was detected for $k_y\ne 0$. The dotted white lines represent the calculated dispersion subbands, which are blueshifted with respect to the linear dispersion subbands shown by the red dotted lines. (b) The intensity profile of the antiferromagnetic vortex chain measured in real space and (c) the associated interferometry image. (d) Dispersion of exciton polaritons excited by a large-area pump spot with the mesas shielded by a mask (inset, not to scale). (e) The real-space intensity profile of the ferromagnetic vortex chain and (f) the associated interferometry image. The on-site topological charges $+1$ and $-1$ are represented by the up and down arrows, respectively. Lack of circular symmetry in (b) and (e) is due to unequal population of the Bloch modes forming the vortex chain (see text).

Figure 3

Calculated Bloch mode dispersions of the $p$ band (a) and (d) and their linear superpositions (b),(c) and (e),(f). (a) $\sigma$ and $\pi$ subbands of the chain. Insets in (a) and (d) schematically show the excitation conditions. The circles on the dispersion curves in (a) and (d) mark the Bloch states “BM1” and “BM2” populated by the respective pump spots, with the respective intensity profiles shown on the panels below. (b) Intensity and (c) phase profiles of the linear superposition of the two Bloch modes marked by the circles in panel (a) with the fixed phase difference of $\pi /2$. (e) Intensity and (f) phase profiles of the linear superposition of the two Bloch modes marked by the circles in panel (d) with the fixed phase difference of $\pi /2$.

Figure 4

Nonlinear dynamics of the Bloch mode amplitudes governed by the model in Eq. (2). Panels (a) and (b) show the temporal dynamics of the amplitude ratio and the phase difference between the two Bloch modes, respectively, for the cases of balanced pumping $P_a=P_b=35\mathrm{meV}/\mu {\mathrm{m}}^2$ and zero frequency detuning ${\mathrm{\Delta }}_{ab}={\mathrm{\Delta }}_b-{\mathrm{\Delta }}_a=0$. Panels (c) and (d) show the temporal dynamics of the amplitude ratio and the phase difference between the two Bloch modes, respectively, for the cases of imbalanced pumping $P_a<P_b$ ($P_a=32\mathrm{meV}/\mu {\mathrm{m}}^2$, $P_b=45\mathrm{meV}/\mu {\mathrm{m}}^2$) and nonzero frequency detuning ${\mathrm{\Delta }}_{ab}=0.63\mathrm{meV}$. For other parameters of the model, see the Supplemental Material [24].