Topological Floquet edge states in periodically curved waveguides

Abstract

We study the Floquet edge states in arrays of periodically curved optical waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the bulk-edge correspondence, our study explores the interplay between band topology and periodic modulations. By analysing the quasienergy spectra and Zak phase, we reveal that, although topological and non-topological edge states can exist for the same parameters, they can not appear in the same spectral gap. In the highfrequency limit, we find analytically all boundaries between the different phases and study the coexistence of topological and non-topological edge states. In contrast to unmodulated systems, the edge states appear due to either band topology or modulation-induced defects. This means that periodic modulations may not only tune the parametric regions with nontrivial topology, but may also support novel edge states. may coexist, how to distinguish between topological and non-topological edge states.In this work, we study the Floquet edge states (FESs) in arrays of periodically curved optical waveguides described by a periodically modulated Su-Schrieffer-Heeger (SSH) model [33]. We analyse, for the first time to our knowledge, the interplay between band topology and periodic modulations, and describe the coexistence of both topological and non-topological edge states supported by the same parameters. Our results show that, for a specific gap, the Zak phase Z Gm is either 0 or π, so that the topological edge states appear only in the gap of Z Gm = π. Through controlling both modulation frequency and amplitude, we may drive the system from non-topological to topological regime, and vice versa. We demonstrate analytically that periodic modulations induce a virtual defect at the boundary, being the key mechanism for the formation of non-topological edge states.The paper is organized as follows. In Sec. II, we introduce our physical model and derive its coupled-mode equations. In Sec. III, we calculate the quasi-energy spectra under OBC. In Sec. IV, by employing the multi-scale perturbation analysis, we give the effective coupled-mode equations and demonstrate the periodic modulations can induce virtual defects at boundaries. The FESs include defect-free surface states and Shockley-like surface states, which induced by virtual defects and the alternating strong and weak couplings between waveguides, respectively. In Sec. V, we analytically obtain the asymptotic phase boundary and numerically give the phase digram of appear FESs, respectively. We explore the topological nature of all FESs via calculate the bulk topological invariant Zak phase. We find that Shockley-like surface

Gauge‐Induced Floquet Topological States in Photonic Waveguides

“…As such, our gauge‐induced interface modes are different from other modulation‐induced defect states. [ 54–56 ]…”

Modulation-induced long-range magnon bound states in one-dimensional optical lattices

“…Considering the periodic boundary condition, we have ψ l1,l2+Lt = ψ l1+Lt,l2 = ψ l1,l2 . Due to the time periodicity, we can express the probability amplitudes ψ l1l2 (t) by the Fourier components [11,24]…”

“…Periodic modulation not only provides a versatile tool to manipulate the quantum particles, but also brings novel states of matter into the quantum systems [9]. It has already been utilized to control the hopping [10,11], band structure [12,13], and quantized transport of a single particle [14][15][16][17][18][19]. Remarkably, artificial gauge field [20] and Floquet topological insulator [21][22][23] have been realized by well-designed modulation protocols.…”

Floquet Engineering of Two Dimensional Photonic Waveguide Arrays with Cornered $π$ Mode

“…By arraying the waveguides with alternating interval, the Floquet Su-Schrieffer-Heeger (SSH) model can be mimicked. Theoretical study found that zero modes and π modes, which are localized at the end of the one dimensional lattice, can be generated by the Floquet engineering [37][38][39][40][41][42]. The localized modes have been observed in experiment [43,44].…”