Measuring topological invariants from generalized edge states in polaritonic quasicrystals

We investigate the topological properties of Fibonacci quasicrystals using cavity polaritons. Composite structures made of the concatenation of two Fibonacci sequences allow one to investigate generalized edge states forming in the gaps of the fractal energy spectrum. We employ these generalized edge states to determine the topological invariants of the quasicrystal. When varying a structural degree of freedom (phason) of the Fibonacci sequence, the edge states spectrally traverse the gaps, while their spatial symmetry switches: The periodicity of this spectral and spatial evolution yields direct measurements of the gap topological numbers. The topological invariants that we determine coincide with those assigned by the gap-labeling theorem, illustrating the direct connection between the fractal and topological properties of Fibonacci quasicrystals.

Figure 1

(a) Nominal potential energy corresponding to a laterally modulated 1D cavity. (b) SEM image of a portion of a 1D cavity reproducing the Fibonacci sequence (top view). The letters A and B correspond to two different widths of the cavity. (c) SEM image showing the full view of three fabricated Fibonacci structures, corresponding to three different values of the phason $\varphi$. Each structure consists of the concatenation $\overleftarrow{F_N}\overrightarrow{F_N}$ of a Fibonacci sequence $\overrightarrow{F_N}$ and its mirror symmetric $\overleftarrow{F_N}$. Vertical white arrows indicate the position of local changes in the sequence introduced when scanning $\varphi$. (d) Schematics of the experimental setup.

Figure 2

(a), (b) Energy-resolved emission of a Fibonacci structure (a) in real space and (b) in momentum space, for a given value of the phason ($\varphi =0.62\pi$). Edge states are visible in the two lowest main energy gaps, characterized by $q=+2$ and $q=-1$. These states are localized at the interface ($x=0$) between the $\overleftarrow{F_N}$ and $\overrightarrow{F_N}$ Fibonacci sequences. (c) Spatial profile of the edge state of gap $q=+2$ measured in a series of structures of the same A-letter width $w_A=3.5\mu \mathrm{m}$ but various B-letter widths $w_B$, yielding different contrasts for the Fibonacci potential.

Figure 3

(a) Measured energy of the edge states of gaps $q=+2$ and $q=-1$ as a function of the phason $\varphi$. $E_0$ denotes the energy of the lowest bulk mode, and the solid lines indicate the gap boundaries. (b) Corresponding spatial symmetry of the edge states. When scanning $\varphi$, the wave functions evolve from symmetric (S) to antisymmetric (AS) with respect to the interface ($x=0$). (c) Relative spectral position within the gaps of the two considered edge states, obtained from scattering matrix calculations. (d) Calculated symmetry of the two considered edge states.

Figure 4

Measured (left) and calculated (right) spatial profile of the $q=+2$ and $q=-1$ edge states for values of $\varphi$ taken in four different quadrants: $[0,\pi /2], [\pi /2,\pi ], [\pi ,3\pi /2]$, and $[3\pi /2,2\pi ]$. The mode spatial structure switches from symmetric (S) to antisymmetric (AS) with respect to the interface ($x=0$), with a direction and periodicity yielding a direct measurement of the topological invariants of the quasicrystal. The blue lines are shifted vertically to improve their visibility.