Chiral States in Coupled-Lasers Lattice by On-Site Complex Potential

The ability to control the chirality of physical devices is of great scientific and technological importance, from investigations of topologically protected edge states in condensed matter systems to wavefront engineering, isolation, and unidirectional communication. When dealing with large networks of oscillators, the control over the chirality of the bulk states becomes significantly more complicated and requires complex apparatus for generating asymmetric coupling or artificial gauge fields. Here we present a new approach for a precise control over the chirality of the bulk state of a triangular array of hundreds of symmetrically coupled lasers, by introducing a weak non-Hermitian complex potential, requiring only local on-site control of loss and frequency. In the unperturbed network, lasing supermodes with opposite chirality (staggered vortex and staggered antivortex) are equally probable. We show that by tuning the complex potential to an exceptional point, a nearly pure chiral lasing supermode is achieved. While our approach is applicable to any oscillators network, we demonstrate how the inherent nonlinearity of the lasers effectively pulls the network to the exceptional point, making the chirality extremely resilient against noise and imperfections.

Figure 1

Experimental arrangement and lasing modes. (a) Schematic arrangement of the digital degenerate cavity laser. (b) The laser network geometry and complex potential that were applied by the SLM. In each triangle, loss of $\mathrm{\Delta }\alpha$ was applied to one laser, and a detuning of $\pm (\mathrm{\Delta }\mathrm{\Omega }/2)$ was applied to the other two lasers. The dashed purple line and curve represent the direct and indirect links between the frequency-detuned lasers. (c) Detected far-field image of the triangular laser lattice without a complex potential ($\mathrm{\Delta }\mathrm{\Omega }=0$, $\mathrm{\Delta }\alpha =0$). The spots marked by V (AV) correspond to staggered vortex (antivortex) lasing supermode. The presence of both the V and AV spots is due to the coexistence of multiple uncoupled longitudinal modes, which act as independent realizations of the same experiment. (d) A detected near-field image of the lasers indicating local uniformity of their intensity. (e) Illustration of the two loss-degenerate lasing supermodes of a triangular lattice of negatively coupled lasers, the staggered vortex and staggered antivortex. The circles include the lasers’ phases in the different lattice sites, the edges connect nearest neighbors. The orange and green filling colors indicate the triangles’ vorticity.

Figure 2

Experimentally measured chirality, induced by complex potential. (a) The measured chirality $c$ of the lasing mode as a function of the applied frequency detuning $\mathrm{\Delta }\mathrm{\Omega }$ and relative loss $\mathrm{\Delta }\alpha$. Maximal chirality is obtained along the lines $\mathrm{\Delta }\mathrm{\Omega }=\mathrm{\Delta }\alpha$, marked by the black dashed lines. (b) Cross sections of the chirality for varying $\mathrm{\Delta }\mathrm{\Omega }$ at fixed $\mathrm{\Delta }\alpha$ values [marked by the dotted colored lines in (a)]. The measurement points are connected by lines. Insets show typical far-field images at selected points.

Figure 3

Theoretical analysis. (a) and (b) The real and imaginary parts of the eigenvalues of the linear system’s minimal loss eigenmodes as a function of $\mathrm{\Delta }\mathrm{\Omega }$ and fixed $\mathrm{\Delta }\alpha$. Exceptional points, marked by the vertical purple dashed lines, emerge at $\mathrm{\Delta }{\mathrm{\Omega }}_{\mathrm{EP}}\approx 1.1\mathrm{\Delta }\alpha$ (for $\mathrm{\Delta }\alpha =0.08\kappa$), where a square-root splitting is apparent. The blue and orange curves correspond to the two eigenvalues of $\mathcal{H}$ with the lowest imaginary part. (c) Chirality of a three-laser system. The solid green curve presents the chirality of the minimal-loss eigenmode of the linear system. The red circles display the chirality of the laser network, obtained from simulation of the nonlinear laser rate equations. (d) and (e) The calculated averaged chirality of a triangular lattice with a noisy complex potential. The chirality of the minimal-loss cold-cavity linear mode (green circles) and the chirality of the nonlinear laser network (red circles) are displayed as a function of the standard deviation of the normal distribution from which the complex potential is drawn. In (d) $\mathrm{\Delta }\alpha$ is held constant while in (e) $\mathrm{\Delta }\mathrm{\Omega }$ is constant. The mean values are displayed, and correspond to an exceptional point. The data points in (c)–(e) are connected by a solid line.

Figure 4

Simulation results for the effective loss $\mathrm{\Delta }{\alpha }_{\mathrm{eff}}$ (red circles) as a function of the applied loss $\mathrm{\Delta }\alpha$ for a fixed $\mathrm{\Delta }\mathrm{\Omega }=0.085|\kappa |$ and unsaturated gain of $g_0=1.2g_{0,\mathrm{th}}$, where $g_{0,\mathrm{th}}$ is the unsaturated gain at the lasing threshold. The circles’ diameter represents the prevalence of $\mathrm{\Delta }{\alpha }_{\mathrm{eff}}$ among the 1000 realizations. The dashed line shows an ensemble average $⟨\mathrm{\Delta }{\alpha }_{\mathrm{eff}}⟩$. It is evident that $\mathrm{\Delta }{\alpha }_{\mathrm{eff}}$ approaches the values of $\mathrm{\Delta }{\alpha }_{\mathrm{EP}}$ (horizontal lines). The lasers’ nonlinearity pulls the system to the EPs. Similar results obtained for other values of $\mathrm{\Delta }\mathrm{\Omega }$ and $g_0$ [30].