Floquet-Sambe Bottleneck and Frequency-Selective Localization in a Driven Synthetic Spin Chain

We study a finite Floquet chain in which a uniform nearest-neighbor hopping coexists with a periodically rotating, $\text{SU(2)}$-dictated spin-assisted hopping profile. The resulting coupling is spatially inhomogeneous – weakest at the chain boundaries and strongest in the bulk – and produces a frequency-dependent Floquet-Sambe bottleneck. In the closed system, the mean inverse participation ratio ($\text{MIPR}$) of the Floquet eigenstates exhibits a striking nonmonotonic dependence on the driving frequency $\omega$: the states remain extended at both low and high frequencies, but become maximally localized at an intermediate frequency. We demonstrate that this localization maximum occurs at ω_{\mathrm{peak}}\sim μ_{-s}=\sqrt{% 2s}, a scale controlled by the first boundary bottleneck. To connect these spectral properties to measurable transport, we construct an open-system Floquet-Sambe Green-function inverse participation ratio from the spatial density of the injected scattering state. This open-system diagnostic recovers the same nonmonotonic localization trend as its closed-system counterpart, with the peak shifted to higher frequencies by the static bandwidth and the lead self-energy. These findings establish the driven synthetic spin chain as a directly realizable, frequency-tunable platform for coherent information storage and retrieval, rooted in the interplay of Floquet-Sambe virtual channels, boundary-controlled localization, and frequency-selective transport in emerging multi-level superconducting circuit architectures.