Emergence of correlations in the process of thermalization of interacting bosons

We address the question of the relevance of thermalization and scrambling to the increase of correlations in the quench dynamics of an isolated system with a finite number of interacting bosons. Specifically, we study how, in the process of thermalization, the correlations between occupation numbers increase in time, resulting in the emergence of the Bose-Einstein distribution. Despite the exponential increase of the number of principal components of the wave function, we show, both analytically and numerically, that the two-point correlation function before saturation increases quadratically in time according to perturbation theory. In contrast, we find that the out-of-time-order correlator increases algebraically and not exponentially in time after the perturbative regime and before the saturation. Our results can be confirmed experimentally in traps with interacting bosons and they may be relevant to the problem of black hole scrambling.