  As a realistic model of a quantum system of matter, this paper investigates
the gravitational-wave effects on a hydrogen-like atom. By formulating the
tetrad formalism of linearized gravity, we naturally incorporate the
gravitational-wave effects through minimal coupling in the covariant Dirac
equation. The atomic electron transition rates induced by the gravitational
wave are calculated using the first-order perturbation theory, revealing a
distinctive selection rule along with Fermi's golden rule. This rule can be
elegantly understood in terms of gravitons as massless spin-2 particles. Our
results suggest the existence of gravitons and may lead to a novel approach to
probe Ultra-High-Frequency Gravitational Waves (UHF-GWs).
