Observation of a Localized Flat-Band State in a Photonic Lieb Lattice

We demonstrate the first experimental realization of a dispersionless state, in a photonic Lieb lattice formed by an array of optical waveguides. This engineered lattice supports three energy bands, including a perfectly flat middle band with an infinite effective mass. We analyze, both experimentally and theoretically, the evolution of well-prepared flat-band states, and show their remarkable robustness, even in the presence of disorder. The realization of flat-band states in photonic lattices opens an exciting door towards quantum simulation of flat-band models in a highly controllable environment.

Figure 1

(a) Edge-centered square (Lieb) lattice. The basis contains three sites ($A$, $B$, and $C$). To avoid edge effects and effects due to lattice inhomogeneity with depth, all measurements were performed near the circled $A$ site ($A_{7-7}$); see also (b). The lattice constant $a=44\mu \mathrm{m}$. (b) White-light transmission optical micrograph of the facet of a Lieb lattice with 323 waveguides fabricated by femtosecond laser writing. Each waveguide supports only a single fundamental mode at 780 nm. The next-nearest-neighbor coupling for a 7-cm-long glass chip was observed to be negligible. To minimize the difference in the next-nearest-neighbor coupling constants along the $x$ and $y$ axes, the lattice was fabricated such that the $x$ and $y$ axes of the lattice were at 45° relative to the vertical axis. (c) Representation of the three energy bands, including the flat band in the middle, for $\{k_{x}a,k_{y}a\}=[0,\pi ]$. (d) The femtosecond laser-writing technique.

Figure 2

Average diffraction patterns when (a) $A$, (b) $B$, and (c) $C$ sites surrounding $A_{7-7}$ were excited separately. The excited sites are circled. Each image is normalized such that the total output power is 1, and the field of view is approximately $225\mu \mathrm{m}\times 225\mu \mathrm{m}$. Simulations based on these averaged diffraction patterns indicate a waveguide-to-waveguide coupling constant of $0.01\pm 0.001{\mathrm{mm}}^{-1}$.

Figure 3

(a) Experimental setup for exciting the flat band. $L_1–L_8$ are convex lenses, $M_1–M_5$ are silver-coated mirrors with $M_5$ on a flipped mount, and BS is a beam splitter. $L_1$ collimated the 780-nm light emerging from a single-mode fiber. A zero-order nulled, binary-phase, square-checker-board diffractive optical element (DOE) generated a square array of diffraction-order “spots” at the focus of $L_2$. Using $L_3$ and $L_4$, these spots were relay imaged to the spatial filter, the transmission aperture of which was adjusted to pass only the four first orders (and a very weak 0th order). Using $L_5$ and $L_6$, the four spots were relay imaged to the input facet of the Lieb lattice. The output facet of the lattice was flood illuminated using $780\pm 10$-nm light, filtered from a broadband white-light source using a bandpass filter ($F$). This flood illumination excited the lattice modes, and enabled precise alignment of the four spots with the desired waveguide modes using camera 1. Once light had been coupled to the lattice, the light distribution at the output facet could be viewed using camera 2. A polarizer ($P$) passing only vertically polarized light was placed in front of camera 2, ensuring that measurements were not affected by polarization-dependent coupling in the lattice. (b) Intensity profile of the four spots that were coupled into the lattice (field of view approximately $80\mu \mathrm{m}\times 80\mu \mathrm{m}$). Note the very weak 0th order. (c)–(d) Fraunhofer diffraction patterns for the flat-band state and equal-phase state, respectively. The yellow dotted lines represent the positions of the optic axis.

Figure 4

(a)–(d) The nondiffracting states observed at the output of the lattice when the flat-band state was launched into the $B$ and $C$ sites of the (7–7), (6–7), (7–6), and (6–6) cells, respectively. When the equal-phase state was launched into the same cells, the output state is not localized, as shown by (e)–(h). (i) The average diffraction pattern of (a)–(d) and (j) is the average diffraction pattern of (e)–(h). Each image is normalized so that total intensity is 1. The field of view is approximately $210\mu \mathrm{m}\times 210\mu \mathrm{m}$.