Parity Anomaly and Landau-Level Lasing in Strained Photonic Honeycomb Lattices

We describe the formation of highly degenerate, Landau-level-like amplified states in a strained photonic honeycomb lattice in which amplification breaks the sublattice symmetry. As a consequence of the parity anomaly, the zeroth Landau level is localized on a single sublattice and possesses an enhanced or reduced amplification rate. The selection of the sublattice depends on the strain orientation but is independent of the valley. The spectral properties of the higher Landau levels are constrained by a generalized time-reversal symmetry. In the setting of two-dimensional photonic crystal lasers, the anomaly affects the mode selection and lasing threshold while in three-dimensional photonic lattices it can be probed via the beam dynamics.

Figure 1

(a) Segment of a honeycomb lattice, with vertices representing states in a two-dimensional photonic crystal or weakly coupled optical fibers in a three-dimensional setting. (b) Sketch of a deformed arrangement which results in a constant pseudomagnetic field. (c) We investigate the interplay of this field with amplification and absorption that breaks the sublattice symmetry. The two sublattices $A$ and $B$ have amplification rates ${\gamma }_A$ and ${\gamma }_B$, respectively (negative values correspond to absorption). The pseudomagnetic field resulting from the strain is modeled via smooth coupling functions $t_l$ whose definition (5) involves the bond vectors ${\mathit{\rho }}_l$, $l=1$, 2, 3.

Figure 2

Dependence of the Landau level spectrum (8, 9) on the amplification imbalance $\gamma =({\gamma }_A-{\gamma }_B)/2$, for strain leading to a pseudomagnetic field of strength $\beta$. (a) Real part, which vanishes for the zeroth Landau level, as well as for the other Landau levels beyond their bifurcation thresholds ${\gamma }_n$, Eq. (10). (b) Imaginary part, which becomes finite beyond the bifurcation. Because of the parity anomaly, the zeroth Landau level breaks the symmetry of the spectrum as it is located on the $A$ sublattice for $\beta >0$ (thick solid line) while it is located on the $B$ sublattice for $\beta <0$ (thin dashed line). The circles indicate the lasing threshold when $|\gamma |$ is increased at fixed average absorption $\overline{\gamma }=-{\gamma }_1$. For $\gamma >0$ this threshold is set by the zeroth Landau level, ${\gamma }_L={\gamma }_1$ (solid circle), while for $\gamma <0$ it is set by the first Landau level, ${\gamma }_L=-\sqrt{2}{\gamma }_1$ (open circle).