  Quantum time dilation occurs when a clock moves in a superposition of
relativistic momentum wave packets. We utilize the lifetime of an excited
hydrogen-like atom as a clock to demonstrate how quantum time dilation
manifests in a spontaneous emission process. The resulting emission rate
differs when compared to the emission rate of an atom prepared in a mixture of
momentum wave packets at order $v^2/c^2$. This effect is accompanied by a
quantum correction to the Doppler shift due to the coherence between momentum
wave packets. This quantum Doppler shift affects the spectral line shape at
order $v/c$. However, its effect on the decay rate is suppressed when compared
to the effect of quantum time dilation. We argue that spectroscopic experiments
offer a technologically feasible platform to explore the effects of quantum
time dilation.
