Optical pumping effect in absorption imaging of F=1 atomic gases

We report our study of the optical pumping effect in absorption imaging of $^{23}$Na atoms in the $F=1$ hyperfine spin states. Solving a set of rate equations for the spin populations in the presence of a probe beam, we obtain an analytic expression for the optical signal of the $F=1$ absorption imaging. Furthermore, we verify the result by measuring the absorption spectra of $^{23}$Na Bose-Einstein condensates prepared in various spin states with different probe beam pulse durations. The analytic result can be used in the quantitative analysis of $F=1$ spinor condensate imaging and readily applied to other alkali atoms with $I=3/2$ nuclear spin such as $^{87}$Rb.