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.
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.
Abstract Out-of-time-ordered correlators (OTOCs), defined via the squared commutator of a time-evolving and a stationary operator, represent observables that provide useful indicators for chaos and the scrambling of information in complex quantum systems. Here we present a quasiclassical formalism of OTOCs, which is obtained from the semiclassical van Vleck–Gutzwiller propagator through the application of the diagonal approximation. For short evolution times, this quasiclassical approach yields the same result as the Wigner–Moyal formalism, i.e. OTOCs are classically described via the square of the Poisson bracket between the two involved observables, thus giving rise to 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. However, numerical simulations, carried out within chaotic few-site Bose–Hubbard systems in the absence and presence of periodic driving, demonstrate that this saturation value strongly underestimates the actual threshold value of the quantum OTOC, which is normally attained after the Ehrenfest time. This indicates that nondiagonal and hence genuinely quantum contributions, thus exceeding the framework of the quasiclassical description, are primarily responsible for describing OTOCs beyond the short-time regime.
Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.
Fast scrambling of quantum correlations, reflected by the exponential growth of out-of-time-order correlators (OTOCs) on short pre-Ehrenfest time scales, is commonly considered as a major quantum signature of unstable dynamics in quantum systems with a classical limit. In two recent works [Phys. Rev. Lett. 123, 160401 (2019)0031-900710.1103/PhysRevLett.123.160401] and [Phys. Rev. Lett. 124, 140602 (2020)10.1103/PhysRevLett.124.140602], a significant difference in the scrambling rate of integrable (many-body) systems was observed, depending on the initial state being semiclassically localized around unstable fixed points or fully delocalized (infinite temperature). Specifically, the quantum Lyapunov exponent λ_{q} quantifying the OTOC growth is given, respectively, by λ_{q}=2λ_{s} or λ_{q}=λ_{s} in terms of the stability exponent λ_{s} of the hyperbolic fixed point. Here we show that a wave packet, initially localized around this fixed point, features a distinct dynamical transition between these two regions. We present an analytical semiclassical approach providing a physical picture of this phenomenon, and support our findings by extensive numerical simulations in the whole parameter range of locally unstable dynamics of a Bose-Hubbard dimer. Our results suggest that the existence of this crossover is a hallmark of unstable separatrix dynamics in integrable systems, thus opening the possibility to distinguish the latter, on the basis of this particular observable, from genuine chaotic dynamics generally featuring uniform exponential growth of the OTOC.
Out-of-time-order correlators (OTOCs) have proven to be a useful tool for studying thermalization in quantum systems. In particular, the exponential growth of OTOCS, or scrambling, is sometimes taken as an indicator of chaos in quantum systems, despite the fact that saddle points in integrable systems can also drive rapid growth in OTOCs. By analyzing the Dicke model and a driven Bose-Hubbard dimer, we demonstrate that the OTOC growth driven by chaos can, nonetheless, be distinguished from that driven by saddle points through the long-term behavior. Besides quantitative differences in the long-term average, the saddle point gives rise to large oscillations not observed in the chaotic case. The differences are also highlighted by entanglement entropy, which in the chaotic-driven dimer matches a Page curve prediction. These results illustrate additional markers that can be used to distinguish chaotic behavior in quantum systems, beyond the initial exponential growth in OTOCs.
Out-of-time-order correlators (OTOCs) have been proposed as sensitive probes for chaos in interacting quantum systems. They exhibit a characteristic classical exponential growth, but saturate beyond the so-called scrambling or Ehrenfest time τ_{E} in the quantum correlated regime. Here we present a path-integral approach for the entire time evolution of OTOCs for bosonic N-particle systems. We first show how the growth of OTOCs up to τ_{E}=(1/λ)logN is related to the Lyapunov exponent λ of the corresponding chaotic mean-field dynamics in the semiclassical large-N limit. Beyond τ_{E}, where simple mean-field approaches break down, we identify the underlying quantum mechanism responsible for the saturation. To this end we express OTOCs by coherent sums over contributions from different mean-field solutions and compute the dominant many-body interference term amongst them. Our method further applies to the complementary semiclassical limit ℏ→0 for fixed N, including quantum-chaotic single- and few-particle systems.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate Λ as a function of the number of degrees of freedom, f, and inverse temperature, β.
We show that the scaled growth rate, Λ/f, can be described by a universal function of fβ and displays a cross-over from classical to quantum behavior as we increase f and/or lower the temperature. In the deep quantum regime of infinite f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.