Floquet Quadrupole Photonic Crystals Protected by Space-Time Symmetry

High-order topological phases, such as those with nontrivial quadrupole moments [1,2], protect edge states that are themselves topological insulators in lower dimensions. So far, most quadrupole phases of light are explored in linear optical systems, which are protected by spatial symmetries [3] or synthetic symmetries [1,2,4–7]. Here we present Floquet quadrupole phases in driven nonlinear photonic crystals that are protected by space-time screw symmetries [8]. We start by illustrating space-time symmetries by tracking the trajectory of instantaneous optical axes of the driven media. Our Floquet quadrupole phase is then confirmed in two independent ways: symmetry indices at high-symmetry momentum points and calculations of the nested Wannier bands. Our Letter presents a general framework to analyze symmetries in driven optical materials and paves the way to further exploring symmetry-protected topological phases in Floquet systems and their optoelectronic applications.

Figure 1

Space-time screw symmetry in a driven nonlinear medium. (a) Schematic of a uniform slab of GaAs driven by a circularly polarized field ${\mathbf{E}}_d$ incident from the normal direction. (b) All three instantaneous optical axes of the driven medium, ${\widehat{e}}_{1-3}$, spin around the $z$ axis. (c) Screw symmetry can be found in each of the optical axes, defined through a combination of rotation in space ($x$, $y$) and translation in time ($t$).

Figure 2

Space-time symmetry indices of photonic bands dressed by optical nonlinearity. (a) Schematic drawing of a square GaAs PhC unit cell with periodicity $Na$ ($N=8$), consisting of veins of width $w$ and spacing $a$ as well as 4 disks of diameter $d$. (b) The PhC band structure, including both TE (blue lines) and TM bands (red). The ${\tilde{S}}_4$ symmetry indices are labeled at the high-symmetry momentum points ($\mathrm{\Gamma }$ and $M$) of each band. If driven by an external field, Floquet basis sharing the same ${\tilde{S}}_4$ index, such as $|4,-1⟩$ and $|2,0⟩$ at $\mathrm{\Gamma }$, will couple to each other, which leads to an energy splitting of $\mathrm{\Delta }\omega$ and opens a new gap in the Floquet spectrum. (c) The energy splitting $\mathrm{\Delta }\omega$ increases linearly with the driving field strength $E$.

Figure 3

Confirmation of quadrupole phases through Wannier band calculations. (a) Calculated Wannier bands and nested Wannier bands for the first two Floquet bands. The results confirm the vanishing dipole moments $p_x=p_y=0$ and the nontrivial quadrupole moment $q_{xy}=1/2$. (b) The gap between the two Wannier bands is opened by the external driving field.

Figure 4

Physical consequences of corner states and filling anomaly. (a) Schematic of a $10\times 10$ supercell of Floquet quadrupole PhCs. One unit cell is presented in the inset. (b) The eigenvalue spectrum confirms the existence of corner states and the filling anomaly in our system. (c) Accumulative time-averaged energy density profile of the first 200 eigenstates, showing fractional occupations ($2\pm 0.5$) at the four corners.