This paper examines dynamical localization in periodically driven quantum systems through the lens of the thin-thick classification. The central finding is that dynamical localization—where quantum interference suppresses classical chaotic diffusion—is equivalent to a large Floquet thinness index under the natural momentum or principal quantum number projection. In systems like the quantum kicked rotor and microwave-driven hydrogen, the localization length precisely equals the diameter of the projected reachable set of Floquet eigenstates. Throughout this localized regime, the quantum coherence horizon exceeds the classical prediction horizon by a factor bounded below by a positive power of Planck's constant. This means quantum coherence persists far longer than classical chaos would predict. The system only becomes "thick"—with the thinness index collapsing to order unity—when the driving strength reaches the quantum resonance-overlap threshold, where multiphoton resonances in the quasi-energy spectrum begin to overlap. This threshold lies above the classical chaos border, creating an operational window where systems are classically chaotic yet remain dynamically localized. The results apply to Floquet systems with one-and-a-half or two degrees of freedom and hold for fixed positive Planck's constant. The practical implication is that keeping driving strengths below the quantum resonance-overlap threshold preserves coherence on timescales much longer than classical predictions.
This paper examines dynamical localization in periodically driven quantum systems through the lens of the thin-thick classification. The central finding is that dynamical localization—where quantum interference suppresses classical chaotic diffusion—is equivalent to a large Floquet thinness index under the natural momentum or principal quantum number projection. In systems like the quantum kicked rotor and microwave-driven hydrogen, the localization length precisely equals the diameter of the projected reachable set of Floquet eigenstates. Throughout this localized regime, the quantum coherence horizon exceeds the classical prediction horizon by a factor bounded below by a positive power of Planck's constant. This means quantum coherence persists far longer than classical chaos would predict. The system only becomes "thick"—with the thinness index collapsing to order unity—when the driving strength reaches the quantum resonance-overlap threshold, where multiphoton resonances in the quasi-energy spectrum begin to overlap. This threshold lies above the classical chaos border, creating an operational window where systems are classically chaotic yet remain dynamically localized. The results apply to Floquet systems with one-and-a-half or two degrees of freedom and hold for fixed positive Planck's constant. The practical implication is that keeping driving strengths below the quantum resonance-overlap threshold preserves coherence on timescales much longer than classical predictions.
This paper presents a fundamental reinterpretation of quantum chaos, arguing that the connection between quantum complexity and operational failure—such as decoherence, information loss, or leakage from protected quantum states—has been poorly understood. The author proposes that quantum chaos should be viewed through the Trojan framework of thin and thick chaos, where projection onto operational variables plays the central role. The core insight is that a quantum system may display all the traditional signatures of chaos—Wigner-Dyson spectral statistics, rapid operator growth, entanglement spreading—yet remain operationally harmless. The key distinction is whether the projected quantum state, rather than the full quantum state, reaches a failure region. This is analogous to classical systems where local instability may remain confined (thin chaos) or may connect to failure (thick chaos). The paper establishes a Quantum Projection-Containment Theorem, which provides a rigorous sufficient condition for avoiding failure: as long as the projected quantum displacement remains smaller than the distance to the failure set, the system remains quantum-thin regardless of its spectral complexity. The Quantum Thinness Ratio is introduced as a quantitative diagnostic, comparing actual projected motion with the available safety margin. Several quantum protection mechanisms are identified that keep chaos thin: spectral gaps that suppress transitions from protected subspaces, symmetries and selection rules that create invariant sectors, localization that prevents delocalization failure, and quantum error-correcting codes that protect logical information even when physical degrees of freedom are chaotic. Quantum thickening occurs when these protections fail—through gap collapse, symmetry breaking, delocalization, or code failure—allowing quantum chaos to become operational failure. The paper provides a practical protocol for diagnosing quantum thinness and emphasizes that quantum chaos is not automatically decoherence. The difference between harmless quantum complexity and operational failure is reachability: chaos becomes thick only when information spreading or leakage reaches protected variables. This framework unifies classical and quantum understandings of stability while respecting the fundamental differences between trajectories and quantum states.
Periodically driven systems are nowadays a very powerful tool for the study of condensed quantum matter: indeed, they allow the observation of phenomena in huge contrast with the expected behavior of their classical counterparts. An outstanding example is dynamical localization: this phenomenon consists in the prevention of heating despite an external periodic perturbation. It has been defined for the first time in a single particle model, the kicked rotor. This chaotic system undergoes an unbounded heating in time in its classical limit; on the opposite, in the quantum regime, the kinetic energy grows until a saturation value is reached and the system stops heating. In my thesis I consider a chain of coupled quantum kicked rotors in order to investigate the fate of dynamical localization in presence of interactions. I remarkably show that a dynamically localized phase persists in the quantum system also in presence of interactions. This is an unexpected behavior since periodically driven, interacting, non-integrable quantum systems heat up to an infinite temperature state. Moreover, I find a genuine quantum dynamics also in the delocalized phase: the heating is not diffusive, as it happens in the classical system, but it follows a sub-diffusive power law. A focus on the properties of the Floquet eigenstates and operators matrix properties gives interesting hints, still under investigation, for a possible justification of the above mentioned slow heating.
Summary form only given. Dynamical localization is one of the signatures of quantum chaos most intensely studied in recent years. These studies rely on the experimental realization of a kicked rotor by placing cold atoms in a far detuned, periodically driven, stationary wave. For a set of parameters such that the classical limit of such a system shows a chaotic diffusive motion in the momentum space, quantum interferences have been shown to suppress the chaotic diffusion in the quantum case. The momentum distribution then presents a characteristic double exponential form. We have performed an experiment in which dynamical localization is studied in the presence of a stationary wave driven by two series of pulses that can have commensurable or incommensurable frequencies, which can be shown to be formally equivalent to perform the experiment in two dimensions. We have thus shown that localization is still observed when the two frequencies are commensurable, but that the classical diffusion in momentum space persists for a much longer time if the frequencies are incommensurable.
Control over the quantum dynamics of chaotic kicked rotor systems is demonstrated. Specifically, control over a number of quantum coherent phenomena is achieved by a simple modification of the kicking field. These include the enhancement of the dynamical localization length, the introduction of classical anomalous diffusion assisted control for systems far from the semiclassical regime, and the observation of a variety of strongly nonexponential line shapes for dynamical localization. The results provide excellent examples of controlled quantum dynamics in a system that is classically chaotic and offer opportunities to explore quantum fluctuations and correlations in quantum chaos.
New experimental results are presented for short-pulse microwave ionization of highly excited hydrogen atoms. A comparison of these results with quantum numerical computations and analytical predictions provides for the first time experimentally grounded evidence of the localization phenomenon that leads to the suppression of the quantum version of the chaotic diffusion in action space occurring in the classical limit. The onset of chaotic motion in externally driven classical systems is a key problem in nonlinear dynamics that is now well understood in its essential aspects. In many physically interesting cases, as the perturbation strength increases beyond some critical value, the system starts absorbing energy in a diffusive way. The question whether “diffusive” excitation processes can take place also in externally driven quantum systems is then a very interesting one for the physics of atoms and molecules in external electromagnetic fields. This is a deep question involving the nature and the validity of the quasiclassical approximation when the underlying classical dynamics is chaotic. The question of “quantum diffusion” was first addressed in a simple model system, the kicked rotator. It was chosen because the features of classical chaos in it were relatively clean and well understood. Besides that, the numerical simulation of its quantum dynamics could be easily accomplished. The major indication of this model was that quantum mechanics suppresses the classical chaotic diffusion in action via a destructive interference effect similar to that responsible for the Anderson localization well known in condensed matter physics.
This thesis investigates quantum transport in the energy space of two paradigm systems of quantum chaos theory. These are highly excited hydrogen atoms subject to a microwave field, and kicked atoms which mimic the delta-kicked rotor model. Both of these systems show a complex dynamical evolution arising from the interaction with an external time-periodic driving force. In particular two quantum phenomena, which have no counterpart on the classical level, are studied: the suppression of classical diffusion, known as dynamical localisation, and quantum resonances as a regime of enhanced transport for the delta-kicked rotor. The first part of the thesis provides new support for the quantitative analogy between energy transport in strongly driven highly excited atoms and particle transport in Anderson-localised solids. A comprehensive numerical analysis of the atomic ionisation rates shows that they obey a universal power-law distribution, in agreement with Anderson localisation theory. This is demonstrated for a one-dimensional model as well as for the real three-dimensional atom. We also discuss the implications of the universal decay-rate distributions for the asymptotic time-decay of the survival probability of the atoms. The second part of the thesis clarifies the effect of decoherence, induced by spontaneous emission, on the quantum resonances which have been observed in a recent experiment with delta-kicked atoms. Scaling laws are derived, based on a quasi-classical approximation of the quantum evolution. These laws describe the shape of the resonance peaks in the mean energy of an experimental ensemble of kicked atoms. Our analytical results match perfectly numerical computations and explain the initially surprising experimental observations. Furthermore, they open the door to the study of the competing effects of decoherence and chaos on the stability of the time evolution of kicked atoms. This stability may be characterised by the overlap of two identical initial states which are subject to different time evolutions. This overlap, called fidelity, is investigated in an experimentally accessible situation.