Floquet Topological Phases in One-Dimensional Nonlinear Photonic Crystals

Jian Lu, Li He, Zachariah Addison, Eugene J. Mele, and Bo Zhen

Phys. Rev. Lett. 126, 113901 – Published 15 March 2021

Abstract

We report on a theoretical analysis of the Floquet topological crystalline phases in driven one-dimensional photonic crystals mediated by second-order optical nonlinearity. We define the photonic Berry connection and photonic polarization in such systems using different methods and prove their equivalence. We present two examples of topological phase transitions in which two Floquet bands cross and open new gaps under the driving field. Finally, we analyze the physical consequences of each topological phase transition by examining edge states and filling anomalies. Our study presents routes toward the realization of robust reconfigurable photonic cavities with topologically protected light confinement.

Received 18 July 2020
Accepted 28 January 2021

DOI:https://doi.org/10.1103/PhysRevLett.126.113901

Physics Subject Headings (PhySH)

Nonlinear optics Symmetry protected topological states Topological materials

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Jian Lu , Li He, Zachariah Addison , Eugene J. Mele , and Bo Zhen ^*

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

^*Corresponding author. bozhen@sas.upenn.edu

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Issue

Vol. 126, Iss. 11 — 19 March 2021

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Images

Figure 1
Floquet topological phase transition in a driven 1D PhC. (a) Schematic of a 1D nonlinear PhC that consists of alternating layers of Si and ${LiNb O}_{3}$ . An external driving field $E (Ω)$ couples different TM modes via the electro-optic effect of ${LiNb O}_{3}$ . (b) The dispersion of the four lowest TM bands is calculated. The $C_{2}^{y}$ eigenvalues ( $\pm 1$ ) of the mode profiles at the BZ center ( $Γ$ ) and edge ( $X$ ) are evaluated. The second (red) and third (blue) static bands, both topologically trivial, are coupled via the external driving field. (c) Under the driving field, Floquet bands are created and a band inversion is induced between two Floquet bands, i.e., band $| 3, - 1 ⟩$ and $| 2, 0 ⟩$ at $Γ$ . After inversion, a Floquet gap is opened (yellow) and both Floquet bands have topologically nontrivial bulk polarizations.
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Figure 2
Trivial static to polarized Floquet phase transition. (a) Schematic drawing of drive-induced edge states (red and blue circles) and filling anomalies that do not exist in the static system. (b) The finite system consists of a PhC with 100 unit cells terminated by PECs, shown in brown. The width of the air gaps $g$ can be modified. (c) Numerical simulation results of the energy spectrum of the trivial static finite system, where both band continua consist of 100 states, corresponding to bands $| 2 ⟩$ and $| 3 ⟩$ in Fig. 1. (d) Under an external driving field, two edge states (blue and red circles) emerge in the Floquet gap (shaded in yellow) corresponding to states 100 and 101. The mode profiles confirm that these two states are localized at the top and bottom boundaries.
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Figure 3
Polarized static to anomalous Floquet phase transition. (a) Schematic drawings of static edge states (triangles), drive-induced edge states (circles), and energy spectra of the polarized static phase and the anomalous Floquet phase. (b) Numerical simulation results of the energy spectrum of a static finite system with 100 unit cells in the polarized phase, exhibiting edge states (triangles) in the static energy gap (shaded in gray). (c) Under a driving field, two new edge states (red and blue circles), corresponding to states 99 and 100, emerge in the Floquet gap (shaded in yellow), while the static edge states (states 199 and 200), are preserved (triangles). The upper panel shows an enlarged view of the spectrum within the dashed box. (d) Mode profiles of the static edge states (199 and 200), shown in black lines, and drive-induced edge states (99 and 100), shown in red and blue lines, which are all localized at the boundaries.
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