  We investigate the onset of quantum thermalization in a system governed by
the Jahn-Teller Hamiltonian which describes the interaction between a single
spin and two bosonic modes. We find that the Jahn-Teller model exhibits a
finite-size quantum phase transition between the normal phase and two types of
super-radiant phase when the ratios of spin-level splitting to each of the two
bosonic frequencies grow to infinity. We test the prediction of the Eigenstate
Thermalization Hypothesis in the Jahn-Teller model. We show that the
expectation value of the spin observable quickly approaches its long-time
average value. We find that the distance between the diagonal ensemble average
and the microcanonical ensemble average of the spin observable decreases with
the effective thermodynamic parameter. Furthermore, we show that the mean-time
fluctuations of the spin observable are small and are inversely proportional to
the effective system dimension.
