  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.
