-In classical mechanics the theory of non-linear dynamics provides a detailed framework for the distinction between near-integrable and chaotic systems. Quite in opposition, in quantum many-body theory no generic microscopic principle at the origin of complex dynamics is known. Here we show that the non-equilibrium dynamics of homogeneous Gaudin models can be fully described by underlying classical Hamiltonian equations of motion. The original Gaudin system remains fully quantum and thus cannot exhibit chaos, but the underlying classical description can be analyzed using the powerful tools of the classical theory of motion. We specifically apply this strategy to the Tavis-Cummings model for quantum photons interacting with an ensemble of two-level systems. We show that scattering in the classical phase space can drive the quantum model close to thermal equilibrium. Interestingly, this happens in the fully quantum regime, where physical observables do not show any dynamic chaotic behavior.Many aspects of the transition from regular dynamics of an integrable system to erratic behavior of a complex system are understood in classical mechanics. On the one hand, there is the Kolmogorov-Arnold-Moser (KAM) theorem [1], that proves the stability of weakly perturbed integrable systems. On the other hand, a variety of mechanisms leading to chaos and eventually to the ergodic exploration of phase space have been found (see, e.g., [2]). For quantum systems, the main paradigms for the description of quantum chaotic phenomena are quasiclassical [3] and random matrix [3,4] theories. Moreover, in a number of specific model studies thermalization processes have been observed (e.g., [5]). However, there remains an important conceptual gap between regular and complex behaviors. In this letter we investigate a non-trivial integrable quantum system without going to the quasiclassical limit and gain microscopic insight into the emergence of irregularity when breaking integrability by driving an internal parameter.A good starting point to approach regular dynamics of non-trivial quantum systems are Bethe ansatz (BA) integrable models, which possess a complete set of integrals of motion. The exact solutions of time-independent BA many-body solvable systems played a crucial role in the understanding of various fundamental phenomena and concepts in physics. Famous examples are the solutions for the Ising model, the Heisenberg spin chain, the onedimensional Hubbard model or the Lieb-Liniger gas [6,7]. Also certain aspects of quantum chromodynamics can be described by the integrable quantum spin chain with complex spin [8]. However, the non-equilibrium dynamics of these models are rich [9][10][11][12][13][14][15][16][17][18] and much more difficult to be calculated within the BA than the static properties. Formulating a theory of integrability breaking for timedependent problems is thus not only a conceptual, but also a technical challenge.In this letter we restrict ourself to a certain class of Gaudin-type models for which we can...