text: 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 -axis. In the proximity of the field zero, located halfway between the two coils along the -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  and , respectively, where  is the magnitude of the total angular momentum vector. Due to the Zeeman effect, these states will each be split into  sublevels with associated values of , denoted by  (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  implies that the field gradient is twice as strong along the -direction than in the  and -directions, and thus the trapping force along the -direction is twice as strong.
  In combination with the magnetic field, pairs of counter-propagating circularly-polarized laser beams are sent in along three orthogonal axes, for a total of six MOT beams (there are exceptions to this, but a minimum of five beams is required to make a 3D MOT). The beams are red-detuned from the  transition by an amount  such that , or equivalently, , where  is the frequency of the laser beams and  is the frequency of the transition. The beams must be circularly polarized to ensure that photon absorption can only occur for certain transitions between the ground state   and the sublevels of the excited state , where . In other words, the circularly-polarized beams enforce selection rules on the allowed electric dipole transitions between states.
  At the center of the trap, the magnetic field is zero and atoms are "dark" to incident red-detuned photons. That is, at the center of the trap, the Zeeman shift is zero for all states and so the transition frequency  from  remains unchanged. The detuning of the photons from this frequency means that there will not be an appreciable amount of absorption (and therefore emission) by atoms in the center of the trap, hence the term "dark". Thus, the coldest, slowest moving atoms accumulate in the center of the MOT where they scatter very few photons.
  Now consider an atom which is moving in the -direction. The Zeeman effect shifts the energy of the  state lower in energy, decreasing the energy gap between it and the  state; that is, the frequency associated with the transition decreases. Red-detuned photons, which only drive  transitions, propagating in the -direction thus become closer to resonance as the atom travels further from the center of the trap, increasing the scattering rate and scattering force. When an atom absorbs a photon, it is excited to the  state and gets a "kick" of one photon recoil momentum, , in the direction opposite to its motion, where . The atom, now in an excited state, will then spontaneously emit a photon in a random direction and after many absorption-spontaneous emission events, the atom will have, on average, been "pushed" back towards the field-zero of the trap. This trapping process will also occur for an atom moving in the -direction if photons are traveling in the -direction, the only difference being that the excitation will be from  to <math>|J=1,m_J
  =+1\rangle. Since the magnetic field gradient near the trap center is uniform, the same phenomenon of trapping and cooling occurs along the  and -directions as well.
  Mathematically, the radiation pressure force that atoms experience in a MOT is given by:
  where  is the damping coefficient,  is the Landé g-factor and  is the Bohr magneton.