Topological States and Adiabatic Pumping in Quasicrystals

The unrelated discoveries of quasicrystals and topological insulators have in turn challenged prevailing paradigms in condensed-matter physics. We find a surprising connection between quasicrystals and topological phases of matter: (i) quasicrystals exhibit nontrivial topological properties and (ii) these properties are attributed to dimensions higher than that of the quasicrystal. Specifically, we show, both theoretically and experimentally, that one-dimensional quasicrystals are assigned two-dimensional Chern numbers and, respectively, exhibit topologically protected boundary states equivalent to the edge states of a two-dimensional quantum Hall system. We harness the topological nature of these states to adiabatically pump light across the quasicrystal. We generalize our results to higher-dimensional systems and other topological indices. Hence, quasicrystals offer a new platform for the study of topological phases while their topology may better explain their surface properties.

Figure 1

The numerically calculated spectrum of Eq. (1) as a function of the phase $\varphi$ for $t=1$, $\lambda =0.5$, $b=(\sqrt{5}+1)/2$ (the golden mean), and $n=-49\dots 49$. The bulk of the spectrum remains fixed, whereas few modes, localized at the boundaries, sweep across the gaps. The insets depict the spatial density of typical eigenstates as a function of position along the 1D lattice: (1) a left boundary state, (2) a right boundary state, and (3) an extended state within the band.

Figure 2

Observation of topological boundary states in an Aubry-André photonic quasicrystal. (a) A sketch of the experimental setup. (b) An illustration of the conducted experiment. Light is injected into one of the waveguides and tunnels to neighboring waveguides as it propagates. (c)–(e) Experimental observation of the left boundary state for $\varphi =\pi /2$. Light was initially injected into a single waveguide (red arrows). The measured outgoing intensity is plotted versus the injection position along the lattice. (c),(d) An excitation at the middle of the lattice (site 0) and at the rightmost site (site 49) results in a significant spread. (e) For an excitation at the leftmost site (site $-49$), the light remains tightly localized at the boundary, marking the existence of a boundary state.

Figure 3

Experimental observation of adiabatic pumping via topologically protected boundary states in a photonic quasicrystal. (a) An illustration of the adiabatically modulated photonic quasicrystal, constructed by slowly varying the spacing between the waveguides along the propagation axis $z$. Consequently, the injected light experiences an adiabatically modulated Hamiltonian, $H_{\mathrm{off}}[\varphi (z)]$, as it propagates and is pumped across the sample. (b) The spectrum of the model as a function of the phase $\varphi$ for $t=40/75$, $\lambda =0.6$, $b=(\sqrt{5}+1)/2$, and $n=1\dots 21$. In the experiment, $\varphi$ was scanned between $0.35\pi$ and $1.75\pi$, marked by arrows (and red dots). The insets depict the spatial density of a boundary eigenstate as a function of the position at three different stages of the evolution: At $\varphi =0.35\pi$, the eigenstate is localized on the right boundary. At $\varphi =\pi$, it is delocalized across the system, while at $\varphi =1.75\pi$ the state is again localized, but on the left boundary. (c) Experimental results: Light was injected into the rightmost waveguide (site 1) at $z=0$ ($\varphi =0.35\pi$). The measured intensity distributions as a function of the position are presented at different stages of the adiabatic evolution, i.e., different propagation distances. It is evident that along the adiabatic evolution the light crossed the lattice from right to left.