Photonic realization of a transition to a strongly driven Floquet topological phase

Floquet systems provide a platform with significant potential for generating and controlling topological phases of matter. By introducing an external driving field, a topologically trivial system can be driven to topological phases possessing nonzero bulk invariants and associated gapless surface modes. One rich feature of Floquet systems is that, as one moves away from the weak-field regime by increasing the amplitude of the driving field, one can encounter a series of topological transitions which place the system in distinct topological phases. Here we experimentally demonstrate this phenomenon in a photonic system consisting of an array of evanescently coupled helical waveguides. We show that, by moving between the weakly and strongly driven regimes, we can induce a transition in which the bulk topological invariant changes sign and the associated topological edge mode reverses its propagation direction. These two topological phases are part of a larger phase diagram and serve to demonstrate both the rich topological physics present in Floquet systems as well as the accessibility of the strongly driven regime—a regime typically associated with the difficulties of large radiative losses for photonic systems and significant heating for their condensed matter counterparts.

Figure 1

(a) Illustration of our honeycomb array of helical waveguides. Light is injected at the front of the structure and evolves in the transverse plane as it propagates into the page along the $z$ direction. (b) Isolated waveguides highlighting the parameters $(R,Z)$, which specify the helix radius and helix period, respectively. By varying these parameters, we can place the system in a variety of topological phases. The upper and lower panels, respectively, illustrate the waveguides used in realizing the weakly and strongly driven topological phases discussed in the text.

Figure 2

Floquet topological phase diagram showing the winding number associated with the $\beta =0$ gap as a function of the dimensionless parameters $\mathrm{\Omega }/c$ and $A$. Each region has been colored according to the value of the winding number, and corresponding numerical labels have been added to the key regions. The inset shows a theoretical estimate of the bending loss plotted on axes identical to those of the phase diagram. The regions of the phase diagram probed in our experiment are labeled “sample I” and “sample II,” respectively, residing in the weakly driven and strongly driven regimes. Shown alongside the labels is an arrow that indicates the path taken by a wavelength sweep from 1480 to 1600 nm. The three points highlighted along the path correspond to the wavelengths of Fig. 4.

Figure 3

Band structures for samples taken to be finite along the $y$ direction and periodic along the $x$ direction. The $W=+1$ phase is shown in (a), and the $W=-1$ phase is shown in (b). These band structures are evaluated at the points in the phase diagram that correspond to the locations of the samples used in the experiment when operating at a wavelength of 1550 nm. Edge modes localized on the bottom (top) of the sample are labeled B (T) and are highlighted in blue (orange).

Figure 4

Light arriving at the output facet after 14 cm of propagation. Panels (a) and (b) correspond, respectively, to samples that have been placed in the $W=+1$ (weakly driven, sample I) and $W=-1$ (strongly driven, sample II) regions of the phase diagram. Light is injected at the corner indicated by the arrow: The first (second) row corresponds to injection at the left (right) corner. The dashed white lines have been overlaid on the images to indicate the sample boundaries. By sweeping the wavelength, we effectively observe the light at different propagation distances along the sample (see the text). A clear transition from counterclockwise to clockwise circulation is observed, consistent with the sign change of the bulk topological invariant.