Abstract :
[en] This PhD thesis investigates scrambling in near-integrable and chaotic systems. First, we present a quasiclassical formalism of out-of-time-ordered correlators (OTOCs), the main chaos probe that we use, using the semiclassical van Vleck-Gutzwiller propagator and the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner-Moyal formalism, i.e. an exponential growth in a chaotic regime. For long times, for which the semiclassical framework is, in principle, still valid, the diagonal approximation yields an asymptotic saturation value for the quasiclassical OTOC under the assumption of fully developed classical chaos. Numerical simulations in Bose-Hubbard systems show that this estimation strongly underestimates the actual threshold value of the quantum OTOCs. This indicates that genuinely quantum contributions are primarily responsible for describing OTOCs beyond the short-time regime. The thesis then shifts its focus to near-integrable systems, where chaos is present only in the form of a thin chaotic layer. In a fully integrable system, around a fixed point, the dynamics is dictated by a stability exponent, and in a fully chaotic system by a maximal Lyapunov exponent. The question remains open as to how the two compete in near-integrable regimes. We describe this integrability-chaos transition by establishing a description of the dynamics inside the chaotic layer in a universal system: the periodically driven pendulum. We support our claim for the universality of this mechanism by studying two paradigmatic examples of the integrability-to-chaos transition, namely the kicked rotor and the kicked Bose-Hubbard dimer. Finally, we study quantum scars in single- and many-particle systems. For the former, we find that the short-time behaviour of indicators such as the OTOC, computed in the kicked rotor, is not sensitive to the presence of a scar, i.e. to the fulfilment of Heller's criterion. Nonetheless, we show that the OTOC can serve as a probe of unstable periodic orbits. For the latter, we look at the persistent oscillations that manifest in the presence of scars in different observables computed for staggered dimer states in Bose-Hubbard models. As the interaction strength increases, one expects scarring to be a less prevalent phenomenon. This is corroborated by the oscillations that progressively vanish.
