Dynamical Localization as Quantum Thinning: A Thinness Ratio for Microwave Hydrogen and the Kicked Rotator

Abstract

This paper asks a narrower question than whether a quantum system is chaotic. It asks whether quantum transport reaches a stated failure scale over a stated horizon. The setting is dynamical localization, where momentum or photon-number diffusion in a classically chaotic Floquet system stops after a break time because of interference. The main example is microwave-driven hydrogen, where the failure scale is the photon number needed for ionization. The companion example is the quantum kicked rotator, where the failure scale is an imposed momentum window and must be labeled as such. The author adapts the thinness ratio by using a quantile of the quantum probability distribution rather than a root-mean-square width, since failure is a threshold crossing and exponential tails need not rank states like variances do. Under an assumed exponential-localization picture, the quantile remains below the failure scale for all times if the localization length is smaller than that scale by a logarithmic factor set by the tolerance. This gives a precise sense in which a classically unstable system can be quantum-thin: local classical instability coexists with confined quantum transport. The paper emphasizes that classical thickness is independent of this quantum bound. Classical diffusion eventually exceeds any fixed window, but that does not imply the quantum state follows. The localization bound can fail in several cases: before the break time, at quantum resonances with ballistic growth, under decoherence that restores diffusion, and in the semiclassical limit where the localization length grows as the effective Planck constant shrinks. The paper also notes that accelerator modes can produce transport without local instability and should not be confused with thick chaos. It proposes comparing a quantum contour where the ratio equals unity against measured ionization thresholds, but does not perform that fit. It distinguishes localization from spectral statistics, arguing that random-matrix tests do not measure transport. Limitations include the assumed localization shape, the one-dimensional hydrogen model, qualitative treatment of decoherence, and the dependence of the classification on tolerance, horizon, and ensemble.