Quantum precursors to Kolmogorov-Arnold-Moser theorem in Floquet spin-$J$ systems

The Kolmogorov-Arnold-Moser (KAM) theorem proves that the resonant tori of classical integrable Hamiltonians are broken when a non-integrable perturbation is introduced, whereas non-resonant tori only get deformed for up to a finite value of the perturbative parameter. In this letter, we identify quantum precursors to the KAM theorem in one-degree-of-freedom spin Hamiltonians periodically perturbed by instantaneous kicks. After recognizing quantum signatures of resonances in the Floquet eigenstates of the perturbed Hamiltonian, we reveal a differentiated sensitivity to the perturbation of the eigenstates of the unperturbed Hamiltonian, depending on whether the states satisfy a resonant condition or not. It is also shown that this differentiated sensitivity becomes more pronounced as the system size increases, leading to the KAM theorem in the classical limit $J\rightarrow\infty$. Numerical and analytical results obtained from unitary perturbation theory strongly support these findings. Although specific to kicked models, our results can be easily extended to more general scenarios, allowing the identification of the quantum mechanism that corresponds to the KAM theorem in the classical limit.