Analytical solutions of the time-evolution of the populations for D1 transition line of the optically-pumped alkali-metal atoms with I = 3/2

We present an analytical calculation of the time-evolution of the populations of the D1 transition line in optically pumped alkali-metal atoms with a nuclear spin of I = 3/2 such as 7Li, 27Na, 39K, 41K, or 87Rb. We consider the simple case of a circularly (or linearly) polarized laser beam with weak intensity. The results we obtained for the populations can be used in calculating the analytical form of the line shapes in velocity-dependent pump-probe laser spectroscopy.