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Engineering topology in waveguide arrays

Lavi K. Upreti

Abstract

The topological classification of a system depends on the discrete symmetries of its Hamiltonian. In Floquet photonic waveguide arrays, the abstract symmetries of the Altland--Zirnbauer (AZ) scheme -- chiral, particle-hole, and time-reversal (for photonics, $z$-reversal) -- arise from structural properties of the lattice, yet a systematic correspondence has not been established. Here, we illustrate this correspondence for a simpler system of one-dimensional waveguide arrays with real coupling coefficients, showing how bipartite structure and $z$-reflection symmetry alone determine the whole AZ class. We further demonstrate that non-bipartite networks -- lacking conventional particle-hole symmetry, chiral symmetry, and $z$-reversal symmetry -- can nonetheless support topologically protected boundary states at quasienergy $\varepsilon = π$, even in one dimension. The protecting symmetry -- \textit{shifted}-particle-hole symmetry -- applies equally to higher-dimensional Floquet waveguides.

Engineering topology in waveguide arrays

Abstract

The topological classification of a system depends on the discrete symmetries of its Hamiltonian. In Floquet photonic waveguide arrays, the abstract symmetries of the Altland--Zirnbauer (AZ) scheme -- chiral, particle-hole, and time-reversal (for photonics, -reversal) -- arise from structural properties of the lattice, yet a systematic correspondence has not been established. Here, we illustrate this correspondence for a simpler system of one-dimensional waveguide arrays with real coupling coefficients, showing how bipartite structure and -reflection symmetry alone determine the whole AZ class. We further demonstrate that non-bipartite networks -- lacking conventional particle-hole symmetry, chiral symmetry, and -reversal symmetry -- can nonetheless support topologically protected boundary states at quasienergy , even in one dimension. The protecting symmetry -- \textit{shifted}-particle-hole symmetry -- applies equally to higher-dimensional Floquet waveguides.
Paper Structure (20 sections, 46 equations, 10 figures, 1 table)

This paper contains 20 sections, 46 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Symmetries in photonic waveguide array
  3. Structural properties
  4. Bipartite structure (BpS)
  5. z-Reflection symmetry ($z-$Ref)
  6. Discrete symmetries in photonic systems
  7. Chiral symmetry (CS)
  8. $z$-Reversal symmetry ($z$-RS)
  9. Particle-hole symmetry (PHS)
  10. Symmetry compatibility
  11. Engineering symmetries in photonic waveguide arrays
  12. Chiral Symmetry
  13. Particle Hole Symmetry
  14. shifted-Particle Hole Symmetry
  15. $z$-Reversal Symmetry
  16. ...and 5 more sections

Figures (10)

  • Figure 1: One-dimensional waveguide array with (a) $z$-independent coupling, realizing a static effective Hamiltonian, and (b) $z$-periodic coupling with period $Z$, realizing a Floquet-type driven system.
  • Figure 2: Bipartite lattice structure: coupling exists only between the $A$ and $B$ families, with no intra-family connections.
  • Figure 3: Reflection symmetry points within one period: the coupling configuration is symmetric about both $z = 0$ and $z = Z/2$.
  • Figure 4: Symmetry relations in $z$-dependent photonic waveguide arrays. Each region represents a fundamental symmetry: CS, $z$-RS, PHS, and $s$-PHS. Overlapping regions indicate cases where multiple symmetries coexist. In the photonic case, the BpS implies both CS and PHS, and $z$-Ref symmetry implies $z$-RS.
  • Figure 5: In a one dimensional waveguide arrays with two waveguides per unit cell and period $Z$, there are two cases to preserve CS: (a) zero onsite potential preserves BpS and z-Ref, and (b) a non-zero time-varying onsite potential (indicated by color variation along the $z$-axis) breaks both BpS and z-Ref.
  • ...and 5 more figures