Two coils in an anti-Helmholtz configuration are used to generate a weak quadrupolar magnetic field; here, we will consider the coils as being separated along the formula_2-axis. In the proximity of the field zero, located halfway between the two coils along the formula_2-direction, the field gradient is uniform and the field itself varies linearly with position. For this discussion, consider an atom with ground and excited states with formula_4 and formula_5, respectively, where formula_6 is the magnitude of the total angular momentum vector. Due to the Zeeman effect, these states will each be split into formula_7 sublevels with associated values of formula_8, denoted by formula_9 (note that the Zeeman shift for the ground state is zero and that it will not be split into sublevels by the field). This results in spatially-dependent energy shifts of the excited-state sublevels, as the Zeeman shift is proportional to the field strength and in this configuration the field strength is linear in position. As a note, the Maxwell equation formula_10 implies that the field gradient is twice as strong along the formula_2-direction than in the formula_12 and formula_13-directions, and thus the trapping force along the formula_2-direction is twice as strong.