Theory of localization and resonance phenomena in the quantum kicked rotor

We present an analytic theory of quantum interference and Anderson localization in the quantum kicked rotor (QKR). The behavior of the system is known to depend sensitively on the value of its effective Planck’s constant $\text{he}$. We here show that for rational values of $\text{he}\mathrm{/}(4\pi )=p\mathrm{/}q$, it bears similarity to a disordered metallic ring of circumference $q$ and threaded by an Aharonov-Bohm flux. Building on that correspondence, we obtain quantitative results for the time–dependent behavior of the QKR kinetic energy, $E(\tilde t)$ (this is an observable which sensitively probes the system’s localization properties). For values of $q$ smaller than the localization length $\xi$, we obtain scaling $E(\tilde{t})\sim \mathrm{\Delta }{\tilde{t}}^2$, where $\mathrm{\Delta }=2\pi \mathrm{/}q$ is the quasi–energy level spacing on the ring. This scaling is indicative of a long time dynamics that is neither localized nor diffusive. For larger values $q\gg \xi$, the functions $E(\tilde{t})\to {\xi }^2$ saturates (up to exponentially small corrections $\sim \exp (-q\mathrm{/}\xi )$), thus reflecting essentially localized behavior.