Calculations of collisions between cold alkaline-earth-metal atoms in a weak laser field

We calculate the light-induced collisional loss of laser-cooled and trapped magnesium atoms for detunings up to 50 atomic linewidths to the red of the ${}^1{S}_0{-}^1{P}_1$ cooling transition. We evaluate loss rate coefficients due to both radiative and nonradiative state-changing mechanisms for temperatures at and below the Doppler-cooling temperature. We solve the Schrödinger equation with a complex potential to represent spontaneous decay, but also give analytic models for various limits. Vibrational structure due to molecular photoassociation is present in the trap loss spectrum. Relatively broad structure due to absorption to the ${\mathrm{Mg}}_2{\hspace{0.5em}}^1{\Sigma }_u$ state occurs for detunings larger than about ten atomic linewidths. A much sharper structure, especially evident at low temperature, occurs even at smaller detunings due to ${\mathrm{Mg}}_2{\hspace{0.5em}}^1{\Pi }_g$ absorption, which is weakly allowed due to relativistic retardation corrections to the forbidden dipole transition strength. We also perform model studies for the other alkaline-earth-metal species Ca, Sr, and Ba, and for Yb, and we find similar qualitative behavior as for Mg.