Theory of localization and resonance phenomena in the quantum kicked rotor

Abstract

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 $\he$. We here show that for rational values of $\he/(4π)=p/q$, 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 $E(\tilde{t})\sim \Delta {\tilde{t}}^2$, where $\Delta =2\pi /q$ 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 $\sim \exp (-q/\xi )$), thus reflecting essentially localized behavior.

27 pages, 3 figures

References in corpus (5)

• Quantum resonances and decoherence for delta-kicked atoms
• Stable Quantum Resonances in Atom Optics
• Ehrenfest time in the weak dynamical localization
• Weak dynamical localization in periodically kicked cold atomic gases
• Fractional $\hslash$-scaling for quantum kicked rotors without cantori

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• Quantum phase transitions in the quasi-periodic kicked rotor
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• Sigma models for quantum chaotic dynamics
• Experimental observation of time singularity in classical-to-quantum chaos transition
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• Floquet simulators for topological surface states in isolation
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• Localization properties and high-fidelity state transfer in electronic hopping models with correlated disorder
• Unconventional quantum criticality in the kicked rotor
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• Quantum dynamics of a planar rotor driven by suddenly switched combined aligning and orienting interactions
• Enhancing Quantum Metrology by Quantum Resonance Dynamics
• Effective field theory of random quantum circuits
• Symmetry and dynamics universality of supermetal in quantum chaos
• Spacing ratios in mixed-type systems
• Interscale entanglement production in a quantum system simulating classical chaos
• Mapping atomic trapping in an optical superlattice onto the libration of a planar rotor in electric fields