Observation of Topological Phase Transitions in Photonic Quasicrystals

Topological insulators and topological superconductors are distinguished by their bulk phase transitions and gapless states at a sharp boundary with the vacuum. Quasicrystals have recently been found to be topologically nontrivial. In quasicrystals, the bulk phase transitions occur in the same manner as standard topological materials, but their boundary phenomena are more subtle. In this Letter we directly observe bulk phase transitions, using photonic quasicrystals, by constructing a smooth boundary between topologically distinct one-dimensional quasicrystals. Moreover, we use the same method to experimentally confirm the topological equivalence between the Harper and Fibonacci quasicrystals.

Figure 1

Experimental methods. (a) Illustration of a photonic waveguide array implementing a deformation between two QCs [cf. Eq. (2)]. (b) A schematics of the experimental setup. We focus a coherent light beam into a waveguide in the array, allow it to propagate along the structure, and image the output intensity using a CCD camera. (c) Illustration of the relation between the localization parameter ${\xi }_n$ and the measured intensity distribution $|{\psi }_n{|}^2$ for two injection sites, $n=66$ and $n=121$. The shaded intervals denote the $\Delta$-distance neighborhood around the insertion point. When more light remains within this neighborhood, the value of ${\xi }_n$ increases.

Figure 2

Summary of results for a smooth deformation between (a)–(c) two topologically inequivalent Harper QCs, and (d)–(f) topologically equivalent Harper and Fibonacci QCs. In both experiments (from left to right), $L_I=84$ (blue), $L_D=51$ (purple hues), and $L_{II}=84$ (red). (a) The hopping amplitude $t_n$ as a function of the lattice site $n$, for modulation frequencies $b_{\mathrm{I}}=2/(1+\sqrt{5})$ and $b_{\mathrm{II}}=2/(1+\sqrt{6.5})$ of the Harper QCs. (b) Experimentally measured ${\xi }_n$ for $\Delta =7$, as a function of the injection site $n$. The two peaks within the deformation region imply the existence of localized states. (c) Numerically obtained LDOS of the structure, $D_n(E)$. The energy bands are composed of extended states, while localized states (roughly 15 sites wide) traverse the gaps in the deformation region. These states manifest the transition between the inequivalent QCs. (d)–(f) Same as (a)–(c), but with a Harper QC deformed into a Fibonacci QC with $b_{\mathrm{I}}=({\tau }_{\mathrm{II}}+1)/{\tau }_{\mathrm{II}}=2/(1+\sqrt{5})$. Here, ${\xi }_n$ shows no sign of localized states. Accordingly, though the distribution of the bands changes along the structure, the energy gaps appear to remain open. This confirms the equivalence between the two QCs.