Edge solitons in Lieb topological Floquet insulator

Abstract

We describe topological edge solitons in a continuous dislocated Lieb array of helical waveguides. The linear Floquet spectrum of this structure is characterized by the presence of two topological gaps with edge states residing in them. A focusing nonlinearity enables families of topological edge solitons bifurcating from the linear edge states. Such solitons are localized both along and across the edge of the array. Due to the non-monotonic dependence of the propagation constant of the edge states on the Bloch momentum, one can construct topological edge solitons that either propagate in different directions along the same boundary or do not move. This allows us to study collisions of edge solitons moving in the opposite directions. Such solitons always interpenetrate each other without noticeable radiative losses; however, they exhibit a spatial shift that depends on the initial phase difference.Topological insulation [1, 2] is a fundamental phenomenon that spans across several areas of physics. Topological photonics, initiated by the seminal paper [3], is one of such areas attracting nowadays growing attention [4,5]. Floquet insulators [6,7] are a particular form of topological insulators, which are characterized by special topological invariants [8]. In such systems, a periodic modulation in the evolution coordinate breaks time-reversal symmetry, resulting in the appearance of the in-gap unidirectional edge states. Photonic Floquet topological insulators have been realized with helical waveguide arrays [9] and were also explored in more complex modulated structures [10-12], including quasicrystals [13].Topological effects in optical systems of helical waveguides can be combined with nonlinear self-action, enabling a plethora of phenomena including modulational instabilities [14,15], and the existence of topological edge solitons. Edge solitons have been obtained numerically in continuous [15], discrete [16][17][18], and Dirac [19] models. They are essentially two-dimensional objects, propagating along the boundary of the topological insulator with the velocity imposed by the group velocity of the Floquet edge state on which soliton is constructed. In previously considered Floquet insulators such solitons, obtained for a given interface, were always co-propagating and suffering from considerable radiative losses due to strong localization [15]. In such setting it was practically impossible to implement interactions of edge solitons with appreciably different Bloch momenta of the carrier waves on finite distances in finite samples.In this Letter we show that it is possible to obtain edge solitons that move, along the same edge, in the opposite directions or even remain immobile in specially designed Floquet insulators with non-monotonous topological branches of the spectrum. We obtain them in different gaps of real-world continuous system -dislocated Lieb array of helical waveguides. Counter-propagating Floquet edge solitons interact almost elastically, but acquire phase-dependent spatial shift after collision....

Topological dipole Floquet solitons

“…Nonlinearity allows the formation of edge solitons -unique states that exhibit topological protection and simultaneously feature a rich variety of shapes and interactions. Edge solitons were predicted in photonic Floquet insulators in continuous [26,34,39,40] and discrete [41][42][43][44] models, and in polariton microcavities [28,45,46]. Their counterparts in nontopological photonic graphene were observed in [47].…”

Floquet Edge Multicolor Solitons

“…For instance, topological edge solitons have been introduced theoretically, and in some cases demonstrated experimentally in mechanical systems [36,37], in nonlinear topological electric circuits [38,39], in Bose-Einstein condensates with spin-orbit interactions [40], and in topological systems governed by Dirac equation [41]. In photonic systems, topological edge solitons have been shown to form in helical waveguide arrays [24,[42][43][44][45][46][47], and were very recently observed in anomalous topological insulator [48], in optically induced Su-Schrieffer-Heeger [49,50] lattices, and studied in polaritonic systems [26,51,52]. Edge solitons in optically induced lattices have been also observed in resonant atomic vapors [53] and topological edge solitons and frequency combs were also predicted to form in driven two-dimensional arrays of coupled ring resonators [54].…”

Topological dipole Floquet solitons

“…Nonlinearity allows the formation of edge solitonsunique states that exhibit topological protection and simultaneously feature a rich variety of shapes and interactions. Edge solitons were predicted in photonic Floquet insulators in continuous [26,34,39,40] and discrete [41][42][43][44] models, and in polariton microcavities [28,45,46]. Their counterparts in nontopological photonic graphene were observed in [47].…”