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The non-integrability of quantum systems, often associated with chaotic behavior, is a concept typically applied to cases with a high-dimensional Hilbert space Among different indicators signaling this behavior, the study of the long-time oscillations of the out-of-time-ordered correlator (OTOC) appears as a versatile tool, that can be adapted to the case of systems with a small number of degrees of freedom. Using such an approach, we consider the oscillations observed after the scrambling time in the measurement of OTOCs of local operators for an Ising spin chain on a nuclear magnetic resonance quantum simulator [J. Li,et al, Phys. Rev. X 7, 031011 (2017)]. We show that the systematic of the OTOC oscillations describes qualitatively well, in a chain with only 4 spins, the integrability-to-chaos transition inherited from the infinite chain., Comment: 6+ pages, 5 figures. Comments welcome

Out-of-time-order correlators (OTOCs) have been proposed as a probe of chaos in quantum mechanics, on the basis of their short-time exponential growth found in some particular set-ups. However, it has been seen that this behavior is not universal. Therefore, we query other quantum chaos manifestations arising from the OTOCs and we thus study their long-time behavior in systems of completely different nature: quantum maps, which are the simplest chaotic one-body system and spin chains, which are many-body systems without a classical limit. It is shown that studying the long-time regime of the OTOCs it is possible to detect and gauge the transition between integrability and chaos, and we benchmark the transition with other indicators of quantum chaos based on the spectra and the eigenstates of the systems considered. For systems with classical analogue, we show that the proposed OTOC indicators have a very high accuracy that allow to detect subtle features along the integrability-to-chaos transition., Comment: Closest to published version. Included changes suggested by referees, grammar changes according to editors, enhanced quality and resolution of figures, etc