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 z -axis. In the proximity of the field zero, located halfway between the two coils along the z -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 J = 0 and J = 1 , respectively, where J is the magnitude of the total angular momentum vector. Due to the Zeeman effect, these states will each be split into 2 J + 1 sublevels with associated values of m J , denoted by | J , m J ⟩ ⟩ (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 ∇ ∇ ⋅ ⋅ B = 0 implies that the field gradient is twice as strong along the z -direction than in the x and y -directions, and thus the trapping force along the z -direction is twice as strong.