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We study quench dynamics in the many-body Hilbert space using two isolatedsystems with a finite number of interacting particles: a paradigmatic model ofrandomly interacting bosons and a dynamical (clean) model of interactingspins-$1/2$. For both systems in the region of strong quantum chaos, the numberof components of the evolving wave function, defined through the number ofprincipal components $N_{pc}$ (or participation ratio), was recently found toincrease exponentially fast in time [Phys. Rev. E 99, 010101R (2019)]. Here, weask whether the out-of-time ordered correlator (OTOC), which is nowadays widelyused to quantify instability in quantum systems, can manifest analogoustime-dependence. We show that $N_{pc}$ can be formally expressed as the inverseof the sum of all OTOC's for projection operators. While none of the individualprojection-OTOC's shows an exponential behavior, their sum decreasesexponentially fast in time. The comparison between the behavior of the OTOCwith that of the $N_{pc}$ helps us better understand wave packet dynamics inthe many-body Hilbert space, in close connection with the problems ofthermalization and information scrambling.

Timescales in the quench dynamics of many-body quantum systems: Participation ratio vs out-of-time ordered correlator