  The classical Kolmogorov-Arnold-Moser (KAM) theorem provides the underlying
mechanism for the stability of the solar system under some small chaotic
perturbations. Despite many previous efforts, any quantum version of the KAM
theorem remains elusive In this work, we provide a quantum KAM theorem in the
context of the anisotropic Dicke model which is the most important quantum
optics model. It describes a single mode of photons coupled to $ N $ qubits
with both a rotating wave (RW) term and a counter-RW (CRW) term. As the ratio
of the CRW over the RW term increases from zero to one, the systems evolves
from quantum integrable to quantum chaotic. We establish a quantum KAM theorem
to characterize such a evolution quantitatively by both large $ N $ expansion
and Random Matrix Theory and find agreement from the two complementary
approaches. Connections and differences between the Dicke models and
Sachdev-Ye-Kitaev (SYK) or hybrid SYK models are examined. Possible Quantum KAM
theorem in terms of other quantum chaos criterion such as quantum Lyapunov
exponent is also discussed.
