Analytical populations of a multilevel atom in a weak linearly-polarized light

We present analytical solutions of the populations of an alkali-metal atom in the presence of a weak linearly-polarized laser beam. After a general method for calculating rate equations is described, the analytical solutions of the populations for the transition Fg = 1 → Fe = 0, 1, 2 are obtained. Provided that the laser intensity is weak, the time-dependent analytical populations are exact. As examples, explicit results for the D2 and D1 transitions of an alkali-metal atom with a nuclear-spin angular momentum of I = 3/2, such as 87Rb, 23Na, 7Li, or 39K, are presented.