Beyond Ginibre statistics in open Floquet chaotic systems with localized leaks

We show that the spectral properties of driven quantum systems with a classically chaotic counterpart and spatially localized openness, such as optical or microwave billiards with leaks, deviate from predictions of Ginibre ensembles. Our analysis focuses on the leaky quantum standard map (QSM) of the kicked rotor. We compare its complex resonance spectrum with both Ginibre and truncated circular orthogonal ensembles (TCOEs). We find that the long-lived resonances follow TCOE statistics, reproducing the density of states and level spacing correlations, but depart from Ginibre predictions. Short-lived resonances, however, do not show a clear correspondence with either random-matrix ensemble. We also demonstrate that increasing the leak size takes the density of states of the TCOE toward the Ginibre limit, yet their spectral correlations remain distinct.