Localized gap modes in nonlinear dimerized Lieb lattices

Abstract

Compact localized modes of ring type exist in many two-dimensional lattices with a flat linear band, such as the Lieb lattice. The uniform Lieb lattice is gapless, but gaps surrounding the flat band can be induced by various types of bond alternations (dimerizations) without destroying the compact linear eigenmodes. Here, we investigate the conditions under which such diffractionless modes can be formed and propagated also in the presence of a cubic on-site (Kerr) nonlinearity. For the simplest type of dimerization with a three-site unit cell, nonlinearity destroys the exact compactness, but strongly localized modes with frequencies inside the gap are still found to propagate stably for certain regimes of system parameters. By contrast, introducing a dimerization with a 12-site unit cell, compact (diffractionless) gap modes are found to exist as exact nonlinear solutions in continuation of flat band linear eigenmodes. These modes appear to be generally weakly unstable, but dynamical simulations show parameter regimes where localization would persist for propagation lengths much larger than the size of typical experimental waveguide array configurations. Our findings represent an attempt to realize conditions for full control of light propagation in photonic environments.

Compact discrete breathers on flat-band networks

“…(6) of the saw-tooth in presence of onsite nonlinearity read iȧ n = −b n − b n+1 + γa n |a n | 2 iḃ n = −b n − b n−1 − b n+1 − a n−1 − a n + γb n |b n | 2 (36) The CLSs of the linear regime can be continued as compact discrete breathers written Eq. (4,5) with frequency Ω = E F B + γA 2 C n,n0 (t) = A 1 0 δ n,n0−1 + 1 −1 δ n,n0 e −iΩt (37) Comparing to the U = 1 case of the cross-stitch lattice, the new feature is the non-orthogonality of neighboring CLSs at the linear limit. While the flat band is gapped away from the dispersive band, at weak nonlinearities we can expect a resonant interaction between neighboring CLSs, which may -or may not -lead to model dependent linear local instability.…”

“…The existence of compact discrete breathers in nonlin-ear flat band networks was observed in Ref. [36,37]. Furthermore, the coexistence between nonlinear terms and spin-orbit coupling has been discussed in the framework of ultra-cold atoms in a diamond chain [38].…”

Photonic flat band dynamics

“…The dimensional reduction decreases the degrees of freedom and prevents transport of localized modes across the lattice, due to the impossibility of radiating energy to the rest of the system. For a nonlinear Lieb lattice [58,59], we found a tendency to have stability inversion regions but not as clear or complete as in kagome. As a consequence, we observed good mobility only for narrow regions in parameter space, what changes abruptly depending on the system size.…”

Observation of quincunx-shaped and dipole-like flatband states in photonic rhombic lattices without band-touching

“…Flatband geometries [1][2][3][4][5][6][7][8][9][10][11][12] have attracted great interest in recent years due to the existence of at least one completely dispersionless band in their energy spectrum which bring new perspectives to the study of various fascinating phenomena, including fractional quantum Hall effect [13][14][15][16] , inverse Anderson localization [17][18][19][20][21][22] , conservative PT-symmetric compact solutions [23][24][25][26][27][28] , and nonlinear compact breathers [29][30][31][32] . Destructive interference is the essence of a flatband existence, and the associated eigenmodes are compact in real space -hence dubbed compact localized states (CLSs).…”