Topological pumping over a photonic Fibonacci quasicrystal

Quasiperiodic lattices have recently been shown to be a nontrivial topological phase of matter. Charge pumping—one of the hallmarks of topological states of matter—was recently realized for photons in a one-dimensional off-diagonal Harper model implemented in a photonic waveguide array. However, if the relationship between topological pumps and quasiperiodic systems is generic, one might wonder how to observe it in the canonical and most studied quasicrystalline system in one dimension—the Fibonacci chain. This chain is expected to facilitate a similar phenomenon, yet its discrete nature hinders the experimental study of such topological effects. Here, we overcome this obstacle by utilizing the topological equivalence of a family of quasiperiodic models which ranges from the Fibonacci chain to the Harper model. Implemented in photonic waveguide arrays, we observe the topological properties of this family, and perform a topological pumping of photons across a Fibonacci chain.

Figure 1

Experimental setup. (a) Pulses from a femtosecond fiber laser (Raydiance Smart Light) with a central wavelength of $1552\text{nm}$ are focused inside the glass, resulting in a permanent change to the index of refraction. (b) Microscope image of a waveguide array used in one of the experiments. One waveguide runs from one side of the glass slide to the other and is used for input. The rest of the waveguides start at some distance from input facet of the glass, with the distance determining the propagation length along the array. Inset marker is $10\mu \text{m}$ long, and the distances between the waveguides range from 8 to $12\mu \text{m}$. (c) A $808\text{-nm}$-wavelength beam from a continuous-wave (cw) diode laser is injected into the waveguide array, allowed to propagate along it, and measured at the output using a CCD camera.

Figure 2

Experimental observation of the right boundary state for 21-waveguide-long arrays with $\beta =0.01$ (a Harper chain), $\beta =2.5$, and $\beta =200$ (a Fibonacci chain), with $\lambda =0.6$, $\overline{b}=(1+\sqrt{5})/2$, and $\varphi =0.7\pi$. Light was initially injected into a single waveguide (red arrows). The measured outgoing intensity is plotted versus the waveguide position. (a) An excitation at the middle of the array (site 11) results in a significant spread of the wave function. (b) Regardless of the value of $\beta$, for an excitation at the rightmost waveguide (site 21) the light remains tightly localized at the boundary, marking the existence of a boundary state.

Figure 3

Spectrum and wave-function localization vs $\varphi$. (a) Energy spectrum of a 28-site-long Fibonacci chain $(\beta =100)$ vs $\varphi$, with $\lambda =0.25$ and $\overline{b}=(1+\sqrt{5})/2$. The two boundary states that traverse the largest energy gaps as a function of $\varphi$ are marked in green. Blue, purple, and red dashed lines mark three values of $\varphi$ ($0.292\pi$, $0.8\pi$, and $1.39\pi$, correspondingly) for which experimental results are presented in the inset. (b) Output intensity for light inserted into the (blue) leftmost waveguide, exciting the left-hand-side boundary state; (purple) middle waveguide, when the state is found within the energy band; and (red) rightmost waveguide, exciting the right-hand-side boundary state. (c) The amount of light remaining at the two outermost waveguides vs $\varphi$ on the (blue) left-hand and (red) right-hand boundaries.

Figure 4

Comparison of spectra. Energy spectra as a function of $\varphi$, of a 13-waveguide-long (a) Fibonacci chain $(\beta =200)$ and (b) Harper chain $(\beta =0.01)$ with $\lambda =0.6$.

Figure 5

Two-parameter topological pumping. (a) Energy spectrum of a 13-waveguide-long array as a function of the propagation axis $z$, with scanned parameters $\varphi$ (green) and $\beta$ (orange). At $z=0$ the array starts as a Fibonacci chain $(\beta =25)$. An adiabatic process lowers $\beta$ to $0.01$, resulting in a Harper chain at $z=5\text{mm}$. $\varphi$ is then scanned from $-0.48\pi$ to $0.7\pi$ to pump light across the structure. Finally, $\beta$ is increased back to 25 between $z=70\text{mm}$ and $z=75\text{mm}$, ending with a Fibonacci chain. (b) Experimental results of topological pumping. The measured intensity distributions as a function of the position are presented at different stages of the adiabatic evolution, i.e., different propagation distances. Light was injected into the rightmost waveguide (site 13), pumped across the array from right to left, and ended up localized at the two leftmost waveguides (sites 1 and 2).