Scattering State Theory for One-dimensional Floquet Lattices

Abstract

We develop a Floquet transfer matrix method to solve scattering in extended 1D Floquet lattices, uncovering an underlying conjugate symplectic structure that enforces current conservation across sidebands. By engineering a spatial adiabatic boundary, we suppress multi-channel sideband interference, allowing us to establish a direct mapping between the bulk winding number $C$ and a rigid shift in the transmission energy windows--quantified as $C\hbarω$. We further propose two experimental realizations: cold-atom Bragg scattering to directly verify the transmission shift, and surface-acoustic-wave-induced transport demonstrating the quantized zero-bias current plateau.

Figures (8)

• Figure 1: Spectrum of a Floquet transfer matrix $F$. Dark blue (red) points represent left‑decaying (right‑decaying) evanescent waves; pale blue (red) points correspond to left‑ (right‑) moving Bloch waves.
• Figure 2: (a) Schematic of the scattering between two semi-infinite regions A (left gray part) and B (right yellow part). (b) Diagonalization of Floquet transfer matrix. (c) Schematic of three-segment scattering. (d) Schematic of the loop and path expansions for the total transmission matrix in three-segment scattering.
• Figure 3: (a) Potential profile with an adiabatic staircase gradient ($N_{\text{grad}}=N_s-1=3$, $L_{0}=1$), defined by $V(x,t;\xi)=8\xi\cos(2\pi x/d-\omega t)+2\xi$. (b) Schematic of the one-to-one mapping between incident waves and Floquet-Bloch states enabled by the adiabatic boundary. (c) Quasi-energy band structure, total transmittance $\bar{\mathcal{T}}=\sum_n \bar{\mathcal{T}}_{n0}$, and same-mode transmittance $\bar{\mathcal{T}}_{00}$ for left/right incidence with varying $N_{\text{grad}}$. The homogeneous lattice region ($\xi=1$) is sufficiently long $L \to \infty$.
• Figure 4: Schematic of the cold-atom setup. Bottom right plots transmission spectrum for $V(x,t;\xi) = 8\xi\cos(2\pi x - 2 t) + 2\xi$ with $\xi(x)=e^{-x^2/50}$.
• Figure 5: (a) Modeling and parameter selection for the surface acoustic wave transport experiment. (b) With the above parameters, temperature and top-gate boundary length are scanned to calculate the bias-free transport current as functions of driving strength and gate voltage, where the chemical potential $\mu$ is set to $10\;\text{meV}$. The red line plots the maximum current for a given SAW amplitude during a gate voltage scan.

Theorems & Definitions (8)

• Lemma 1: Spectrum reciprocal-conjugate pairing
• proof
• Lemma 2: $J$-inner product orthogonality
• proof
• Proposition 1: Extended Lemma 2 — blockwise biorthogonalization in the diagonalizable case
• proof
• Theorem 1: Exponential generation in the diagonalizable case
• proof