Topological Pumping Through a Localized Bulk in a Photonic Hofstadter System

Abstract

Photonic systems provide a highly tunable platform for emulating quantum Hall physics. This tunability enables probing of the interplay between strong disorder and robust topological transport that remains difficult to access in solid-state systems. Here we realize a photonic version of the Harper-Hofstadter and Aubry-André models using a one-dimensional multilayer photonic crystal (Bragg stack) with a synthetic dimension encoded in its geometry. By modulating the layer thicknesses, we observe the Hofstadter butterfly and its chiral edge states from a family of one-dimensional multilayer structures, consistent with the Thouless pump picture. Exploiting the quasiperiodicity in this model, we show that increasing quasiperiodic modulation induces a wavelength-selective localization transition: specific Chern bands become fully localized along one dimension, while chiral edge states persist and continue to wind across the gap. We confirm this behavior through numerical simulations and experiments, and eigenmode analysis reveals that edge transport in this regime proceeds via a sequence of Landau-Zener transitions between localized states. These results demonstrate a crossover from adiabatic Thouless pumping under weak quasiperiodic modulation to a Landau-Zener-mediated topological pump at strong modulation, realized in a a compact and easily tunable photonic system.

Figures (6)

• Figure 1: (a) A schematic of a 1D multilayer PhC consisting of alternating dielectric materials of Si and SiO$_2$. An effective unit cell $n$ is defined as a pair of neighboring Si and SiO$_2$ layers. Within a unit cell, the layers share the same thickness $t_n$. (b) The Hofstadter butterfly plotted by normalized wavelength and colored by $\log_{10}$(IPR) realized in a family of 1D PhCs of $N=144$ with $\beta$ as a tuning parameter. Fractal butterflies can be seen for $\beta$ away from the middle ($\beta = 0.50$). A red dashed line indicates the slice of $\beta$ fixed to realize: (c) The chiral edge states colored by $\log_{10}$(IPR) calculated from a family of 1D PhCs of $N=144$ with $\phi$ as a tuning parameter. This chosen slice of $\beta$ slices through three extended bulk bands, and localized edge states traverse the gaps and wind once within a period of $\phi$. Near-flat cladding modes pad the top and bottom of the spectra.
• Figure 2: (a) SEM image of an $N=8$ 1D PhC including cladding layers used for characterizing chiral edge states. The multilayer stack is composed of Si (dark grey) and SiO$_2$ (light grey) deposited onto a borosilicate glass cover slide as the substrate. Cladding layers on each side of the PhC extend the edge states' lifetime. (b) Transfer-matrix simulation of chiral edge states within the bandgap at $\beta=1/3$, $A = 0.2$, for an $N=8$ system. An edge branch crosses the top bandgap. (c) Experimentally measured transmission spectrum showing the same edge state. (d) Transfer-matrix simulation of Hofstadter butterfly in an $N=13$ system. (e) Experimentally measured transmission showing the butterfly.
• Figure 3: (a) MPB simulation of the photonic Hofstadter butterfly at $N=144$ and high modulation $A=0.5$. The simulation shows pockets of localization that are wavelength- and $\beta$-dependent. The red dashed line indicates the inverse golden ratio, $\beta \approx 2/(\sqrt{5}+1)$, which is the slice chosen to reproduce the chiral edge states in (b) and (c). (b) MPB simulation of the chiral edge states at $N=21$, $\beta = 34/21$, and at low modulation $A = 0.2$. Nearly all modes are extended except the chiral edge states, which are localized at the edges of the 1D PhQC. An edge branch can be smoothly tracked as it enters the bulk and wraps around to the other branch within a period of $\phi$, indicated by the red arrows from Edge 1 that wrap onto the blue arrows at Edge 2. (c) The same as (b) except at higher modulation $A=0.5$. The bulk bands located near $\lambda/\braket{a} \sim 3.5$, indicated by the black arrow, are localized, while the other bands remain extended. Chiral edge states continue to cross the gap in this case. (d) Center-of-mass calculations in real space of the energy density for eigenstates highlighted in the red dashed box in (b). A possible eigenstate's evolution starts at the right edge (Edge 1) and traverses multiple avoided crossings indicated by the color. (e) Transfer-matrix simulations of the logarithmic transmission of the same region of (d), showing avoided crossings. (f) Experimental results for $\phi = 1.25$, showing extended bulk bands (at the edges of the spectrum) and the localizing bulk bands and chiral edge states (highlighted by the grey shadow). Faster attenuation of the edge states and bulk bands with increasing system size, relative to the extended bulk bands, is a signature of localization.
• Figure S1: Transfer-matrix simulation of logarithmic transmission at $N=89$ and $\beta = 144/89$ of the chiral edge states at (a) low modulation $A=0.2$ and (b) high modulation $A=0.5$. The bulk bands indicated by the white arrow disappear upon high quasiperiodic modulation.
• Figure S2: The transfer-matrix simulations at: (a) $N=5$ and $\beta = 8/5$ (b) $N=8$ and $\beta = 13/8$ (c) $N=13$ and $\beta =21/13$ (d) $N=21$ and $\beta = 34/21$, which shows progressively less transmissive bulk bands and edge states as the system size increases, implying localization. The white dashed line indicates the slice $\phi = 1.25$ which was used in the experiment in the main text to diagnose the localization transition.