Line Shapes of High Resolution Photoassociation Spectra of Optically Cooled Atoms

High resolution photoassociation spectra of colliding ultracold trapped atoms contain detailed information about ground and excited state interactions. We calculate the asymmetric line shapes for ultracold ${\mathrm{Na}}_2$ absorption using a resonance scattering expression, proportional to a free-bound Franck-Condon factor, that is justified by full quantum scattering calculations of a collision in a radiation field. The line shapes illustrate Wigner threshold law behavior, which is characteristic of the quantum limit as $T\to 0$. Using an adiabatic hyperfine analysis, we calculate and compare a model spectra for $J=1,2,3, \mathrm{and} 4$ features of the ${\mathrm{Na}}_2$ ${}_{}{}^1{}_{}{}^{}\Pi _g^{}{}_{}{}^{}(\nu =48)$ state to recent high resolution experimental data.