Anisotropic Dirac cone and slow edge states in a photonic Floquet lattice

Yuan Li and Xiankai Sun

Phys. Rev. B 105, 014306 – Published 12 January 2022

Abstract

Dirac cones with isotropic and linear dispersion, as in graphene, exhibit ultrahigh carrier mobility and offer an intriguing method to manipulate the behavior of particles. This concept has recently been extended to anisotropic Dirac cones with anisotropic dispersion in condensed matter, but their photonic counterparts are yet to be explored. Here, by introducing anisotropic coupling, we propose a Floquet lattice with a realistic photonic scheme that supports anisotropic Dirac cones and abundant topological phases. Under the highly anisotropic circumstances, the presence of anomalous Floquet insulators is demonstrated, where topological slow edge states are found along a specific direction. The group velocity of the edge states can be tailored continuously by adjusting the anisotropy of the lattice. These slow edge states are robust against disorders and can be harnessed for developing intriguing applications such as anisotropic devices, topological delay lines, and optical nonlinear devices.

Received 20 May 2021
Revised 17 August 2021
Accepted 10 September 2021

DOI:https://doi.org/10.1103/PhysRevB.105.014306

Physics Subject Headings (PhySH)

Topological effects in photonic systems Topological phase transition

Floquet systems

Atomic, Molecular & Optical

Authors & Affiliations

Yuan Li and Xiankai Sun ^*

Department of Electronic Engineering, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong SAR, China

^*Corresponding author: xksun@cuhk.edu.hk

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Vol. 105, Iss. 1 — 1 January 2022

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Images

Figure 1
Floquet system and photonic implementation. (a) Schematic representation of a reduced Floquet lattice composed of sublattice A-type (red dots) and B-type (blue dots) atoms, where the black dashed lines enclose a unit cell. The coupling between neighboring waveguides can be decomposed into four steps. In each step, the evolution of photons is governed by an evolution matrix $S_{j}$ ( $j = i, ii, iii$ , and iv). (b) Corresponding Brillouin zone, where the red dots represent high-symmetry points and the blue solid lines enclose the first Brillouin zone. (c) Schematic illustration of a bipartite square lattice composed of evanescently coupled elliptic-helical waveguides. This bipartite square lattice is composed of A-type (red) and B-type (blue) sublattices, twisting counterclockwise along the propagation direction ( $z$ axis) with a relative shift. (d) Cross sections of the waveguide lattice showing the coupling between neighboring waveguides in the four steps. In each step, the coupling between the neighboring waveguides occurs only along a specific direction.
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Figure 2
Schematic illustrations of the Dirac cones and phase diagram. (a) Schematic illustration of an isotropic two-dimensional Dirac cone with symmetric effective velocities ( $w_{x} = w_{y}$ ). (b) Schematic illustration of an anisotropic two-dimensional Dirac cone with asymmetric effective velocities ( $w_{x} \neq w_{y}$ ). (c) Phase diagram of the Floquet lattice in the case of zero phase shift $Δ = 0$ . TI, trivial insulator; AFI, anomalous Floquet insulator.
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Figure 3
Floquet band structures for different topological phases and equifrequency contours for the corresponding Dirac cones. (a) Band structure of a trivial insulator [ $\arcsin (κ_{x}) = \arcsin (κ_{y}) = 0.2 π, | M | = 0.2 π$ ] with a trivial band gap. (b) Band structure of an anomalous Floquet insulator [ $\arcsin (κ_{x}) = 0.25 π, \arcsin (κ_{y}) = 0.35 π, | M | = 0.2 π$ ] with a nontrivial band gap. (c) Band structure of a critical-phase-transition point ( $κ_{x} = κ_{y} = \sqrt{2} / 2$ ), which hosts an isotropic Dirac cone under the isotropic condition. (d) Band structure of another critical-phase-transition point [ $\arcsin (κ_{x}) = 0.15 π$ and $\arcsin (κ_{y}) = 0.35 π$ ], which hosts an anisotropic Dirac cone under the anisotropic condition. (e) Equifrequency contours for the isotropic Dirac cone, which exhibits a circular shape at the Dirac point Γ. (f) Equifrequency contours for the anisotropic Dirac cone, which exhibits an elliptic shape at the Dirac point Γ.
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Figure 4
Comparison of conical diffraction between isotropic and anisotropic Dirac cones. (a),(b) Initial excitation by a Gaussian beam (a) and single-unit-cell beam (b). (c),(e) Numerically simulated output intensity profile $| Ψ (x, y, L) |^{2}$ of the Floquet lattice for an isotropic Dirac cone (c) and for an anisotropic Dirac cone (e) under initial excitation of a Gaussian beam. (d),(f) Numerically simulated output intensity profile $| Ψ (x, y, L) |^{2}$ of the Floquet lattice for an isotropic Dirac cone (d) and for an anisotropic Dirac cone (f) under initial excitation of a single-unit-cell beam. Both lattices contain 12 × 12 unit cells, and the propagation distance is $L = 5 Z$ .
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Figure 5
Topological transition and edge-state band structures. (a) Schematic representation of a stripe structure composed of seven unit cells, where the orange dashed lines represent the periodic boundary. (b)–(d) Edge-state band structures β(k) along the $x$ (red) and $y$ (blue) directions with fixed $\arcsin (κ_{x}) = 0.15 π$ ( $R_{x} = 6.0 μ m$ ), for a trivial insulator with $\arcsin (κ_{y}) = 0.25 π$ and $| M | = 0.4 π$ ( $R_{y} = 6.45 μ m$ ) (b), for a critical phase with $\arcsin (κ_{y}) = 0.35 π$ and $| M | = 0$ ( $R_{y} = 6.80 μ m$ ) (c), and for an anomalous Floquet insulator with $\arcsin (κ_{y}) = 0.45 π$ and $| M | = 0.4 π$ ( $R_{y} = 7.10 μ m$ ) (d). In (d) the yellow (green) dots plot the fast (slow) edge states with a larger (smaller) group velocity. (e)–(g) Numerically simulated output intensity profile $| Ψ (x, y, L) |^{2}$ for different topological phases. The blue dashed boxes represent the boundary of the lattice, and the white dashed box indicates the initial excitation distribution for (e)–(g). With the majority of edge states excited initially around $k_{y} = 0$ , the output intensity profiles exhibit diverse propagation behaviors. Note that the fast edge states along the $y$ direction convert adiabatically into the slow edge states along the $x$ direction around the corner.
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Figure 6
Numerically simulated dynamic propagation of the edge states. (a),(b) Dynamic propagation of the edge states with a larger group velocity (a) and a smaller group velocity (b). The unit scale of the ruler is $2 a$ , and the blue dashed lines represent the boundary. (c),(d) Dynamic propagation of the edge states with a larger group velocity (c) and a smaller group velocity (d) when a defect is introduced on the edge. (e) Dynamic propagation of the edge states along an arbitrary edge.
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