Scattering-state theory of open Floquet lattices: transfer matrices, branch openness, and robust asymmetry

We establish a scattering-state theory for open one-dimensional Floquet lattices based on a frequency-domain transfer-matrix formulation. For real quasienergy, the conjugate-symplectic structure of the transfer matrix separates bulk Floquet–Bloch modes into propagating and evanescent sectors, enabling a consistent treatment of interface matching and the shrinking-window smoothing required for long-sample transport. By tracking how incoming states populate deep-bulk propagating branches, we define branch-resolved weights $p_{\mu \alpha }$ and total branch weights $p_{\mu }$. We prove that $p_{\mu }$ equals the escape probability of a wave packet initialized on the corresponding branch. In the open geometries considered here, true bound trapping of propagating branches is nongeneric, yielding $p_{\mu }=1$ for generic parameters. This generic openness implies that long-sample transport is governed by deep-bulk branch populations rather than by boundary-sensitive interference. Consequently, the integrated left–right transmission asymmetry reduces to the net chirality, and hence the winding contribution, of an isolated Floquet band. The robust topological observable is therefore the accumulated asymmetry plateau, not the detailed transmission line shape, which remains strongly reshaped by nonadiabatic boundaries. A spatially adiabatic boundary serves only as a transparent benchmark for resolving the branch structure, not as the origin of the topological response.