  A plausible mechanism of thermalization in isolated quantum systems is based
on the strong version of the eigenstate thermalization hypothesis (ETH), which
states that all the energy eigenstates in the microcanonical energy shell have
thermal properties. We numerically investigate the ETH by focusing on the large
deviation property, which directly evaluates the ratio of athermal energy
eigenstates in the energy shell. As a consequence, we have systematically
confirmed that the strong ETH is indeed true even for near-integrable systems,
where we found that the finite-size scaling of the ratio of athermal
eigenstates is double exponential. Our result illuminates universal behavior of
quantum chaos, and suggests that large deviation analysis would serve as a
powerful method to investigate thermalization in the presence of the large
finite-size effect.
