Anomalous and Quantum Hall Effects in Lossy Photonic Lattices

We theoretically discuss analogues of the anomalous and the integer quantum Hall effect in driven-dissipative two-dimensional photonic lattices in the presence of a synthetic gauge field. Photons are coherently injected by a spatially localized pump, and the transverse shift of the in-plane light distribution under the effect of an additional uniform force is considered. Depending on pumping parameters, the transverse shift turns out to be proportional either to the global Chern number (integer quantum Hall effect) or to the local Berry curvature (anomalous Hall effect). This suggests a viable route to experimentally measure these quantities in photonic lattices.

Figure 1

(a)–(d) Photon amplitude distribution $|a_{m,n}|$ on a $41\times 41$ square lattice with $\alpha =1/5$. The central sites are pumped. The force $F$ is zero for (a)–(c) and $F=0.1J$ for (d). In (a),(b),(d), the pump frequency is tuned to ${\omega }_0/J=-2.95$ within the lowest energy band; in (c), it is tuned to ${\omega }_0/J=-2.85$ within a band gap. The loss rate is $\gamma =0.01J$ for (a),(c),(d) and $\gamma =0.02J$ for (b). The bright regions have higher intensity than the dark regions. (e) Displacement $⟨x⟩$ as a function of $F$, in units of $J$, for a pump frequency ${\omega }_0/J=-2.95$ with $\alpha =1/5$. The solid (blue) line is for $\gamma /J=0.05$, and the dashed (green) line is for $\gamma /J=0.08$.

Figure 2

The estimated Berry curvature $\mathrm{\Omega }$ as a function of ${\omega }_0$, in units of $J$ (dots) as compared to the exact value (solid lines) for $\gamma /{\mathrm{\Delta }}_{\text{width}}=1/30$ on $201\times 201$ lattices. (a) Lowest band of square lattice at $\alpha =1/3$ and (b) middle band of square lattice at $\alpha =1/5$.

Figure 3

(a) Photonic honeycomb lattice. Sites belonging to the sublattices $A$ and $B$ are denoted by empty and filled circles, respectively. (b) The Berry curvature of the upper band of a honeycomb lattice with $\mathrm{\Delta }=0.3J$. The Dirac points are the vertices of the hexagon. (c) The estimated Berry curvature $\mathrm{\Omega }$ as a function of ${\omega }_0$, in units of $J$ (dots) as compared to the exact value (solid line), for the honeycomb lattice without a synthetic magnetic field. System parameters are $\gamma /J=0.04$ and $\sigma =3$ on a $200\times 200$ unit cell honeycomb lattice with site energy mismatch $\mathrm{\Delta }/J=0.3$. The shaded region indicates the band gap.