Cross-damping effects in 1S-3S spectroscopy of hydrogen and deuterium

We calculate the cross-damping frequency shift of a laser-induced two-photon transition monitored through decay fluorescence, by adapting the analogy with Raman scattering developed by Amaro et al. [P. Amaro et al., PRA 92, 022514 (2015)]. We apply this method to estimate the frequency shift of the 1S-3S transition in hydrogen and deuterium. Taking into account our experimental conditions, we find a frequency shift of less than 1 kHz, that is smaller than our current statistical uncertainty.