Topological Properties of Floquet Winding Bands in a Photonic Lattice

The engineering of synthetic materials characterized by more than one class of topological invariants is one of the current challenges of solid-state based and synthetic materials. Using a synthetic photonic lattice implemented in a two-coupled ring system we engineer an anomalous Floquet metal that is gapless in the bulk and shows simultaneously two different topological properties. On the one hand, this synthetic lattice presents bands characterized by a winding number. The winding emerges from the breakup of inversion symmetry, and it directly relates to the appearance of Bloch suboscillations within its bulk. On the other hand, the Floquet nature of the lattice results in well-known anomalous insulating phases with topological edge states. The combination of broken inversion symmetry and periodic time modulation studied here enriches the variety of topological phases available in lattices subject to Floquet driving and suggests the possible emergence of novel phases when periodic modulation is combined with the breakup of spatial symmetries.

Figure 1

Floquet winding metals. (a) The experimental platform consists of two 40 m long fiber rings with a 0.55 m difference of length coupled via a variable beamsplitter (VBS). One ring contains a PM that controls the phase of light pulses. (b) The dynamics in the rings can be mapped onto a lattice: propagation in the left (right) fiber ring is represented with orange (blue) lines, and lattice sites are shown with circles. $T_F$ represents one Floquet driving period. (c)–(d) Calculated band structure of a Floquet insulator (c) with $K=0$ ($c_1=1$, $c_2=-1$) and a Floquet winding metal (d) with $K=-1$ ($c_1=-2$, $c_2=0$). (e) Selected band cuts for different values of $\phi$ corresponding to the red band in (d).

Figure 2

Tomography of the quasienergy bands. (a) Experimentally observed split-step coherent walk for $\phi =0$. (b) Reconstructed band structure of the system. (c) Measured band tomographies for $K=0$ ($c_1=1$, $c_2=-1$), $K=1$ ($c_1=2$, $c_2=0$), and $K=2$ ($c_1=3$, $c_2=1$) integrated over the quasimomentum $k$. (d) Band structure from simulations of Eq. (1) for the same parameters as in (c).

Figure 3

Topological Bloch suboscillations. (a)–(c) Measured real-space evolution of a wave packet injected close to $k=0$ into one of the bands and evolved under an adiabatic increase of $\phi$ for (a) $K=1$ ($c_1=2$, $c_2=0$), (b) $K=2$ ($c_1=5$, $c_2=-1$), and (c) $K=3$ ($c_1=8$, $c_2=-2$). Note that in (a) and (b), the initial wave packet has a momentum slightly smaller and larger, respectively, than $k=0$ due to the experimental injection technique. Dashed orange lines show analytical curves. The increase rate $d\phi /dt$ is $2\pi ·0.008\mathrm{rad}/\text{turn}$ for (a) and (b), and $2\pi ·0.012\mathrm{rad}/\text{turn}$ for (c). (d) Dots, measured evolution of the center-of-mass of the wave packet in (c); solid line, analytic solution. Error bars represent $1\sigma$ confidence intervals and generally are smaller than the dot size. Gray areas emphasize the difference between the maximal and the minimal achievable amplitudes of suboscillations in the analytic curve. (e) Illustration of the experimental procedure.

Figure 4

Topological edge states. (a) Phase diagram of anomalous Floquet phases as a function of the coupling amplitudes in the first ${\theta }_1$ and second step ${\theta }_2$. (b) Measured dynamics when exciting the lattice at a single site located at the interface between two lattices belonging to two different phases [triangles in (a)], showing a localized interface state. Both lattices are prepared with $K=-1$ ($c_1=1$, $c_2=-3$) and $\phi =\pi /2$ but different values of ${\theta }_1$ and ${\theta }_2$. (c) Measured dispersion showing a flat band in one gap (shown with the arrow), corresponding to the interface state. (d) Experimental and (e) theoretical band tomography of the winding metal for all values of $\phi$ confirming the presence of edge state bands in each of the gaps.