Two coils in an
anti-Helmholtz configuration are used to generate a weak quadrupolar magnetic field; by convention, the coils are 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 displacement from the field zero. 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, the J\neq0 states will each be split into 2J+1 sublevels with associated values of m_J, denoted by |J,m_J\rangle. This results in spatially-dependent energy shifts of the excited-state sublevels, as the Zeeman shift is linearly proportional to the field strength. As a note, the
Maxwell equation \nabla\cdot\mathbf{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. In combination with the magnetic field, three pairs of counter-propagating circularly-polarized laser beams are sent in along orthogonal axes, such that their intersection lies at the location of the magnetic field zero. The beams are red-detuned from the J=0\rightarrow J=1 transition by an amount \delta such that \delta\equiv \nu_0- \nu_L > 0, or equivalently, \nu_L=\nu_0-\delta, where \nu_L is the frequency of the laser beams and \nu_0 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 |0,0\rangle and the sublevels of the excited state |1,m_J\rangle, where m_J=-1,0,1. In other words, the circularly-polarized beams enforce selection rules on the allowed electric dipole transitions between states. Now consider an atom which is displaced from the field zero in the +z-direction. The Zeeman effect shifts the energy of the |J=1,m_J=-1\rangle state lower in energy, decreasing the energy gap between it and the |J=0,m_J=0\rangle state; that is, the frequency associated with the transition decreases. Red-detuned \sigma^-photons, which only drive \Delta m_J=-1 transitions, propagating in the -z-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 \sigma^-photon, it is excited to the |J=1,m_J=-1\rangle state and gets a "kick" of one photon recoil momentum, \hbar k, in the direction opposite to its motion, where k=2\pi\nu_0/c. The atom, now in an excited state, will then spontaneously emit a photon in a random direction as it returns to the ground state, which will result in another momentum "kick". Because this "kick" from the emitted photon occurs in a random direction, the net effect of many absorption-spontaneous emission events will result in the atom being "pushed" back towards the field-zero of the trap. This trapping process will also occur for an atom moving in the -z-direction if \sigma^+photons are traveling in the +z-direction, the only difference being that the excitation will be from |J=0,m_J=0\rangle to |J=1,m_J =+1\rangle since the magnetic field is negative for z. Since the magnetic field gradient near the trap center is uniform, the same phenomenon of trapping and cooling occurs along the x and y-directions as well. 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 \nu_0 from J=0\rightarrow J=1 remains unchanged. The detuning of the photons from this frequency means that there will not be an appreciable amount of absorption 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. Mathematically, the radiation pressure force that atoms experience in a MOT is given by: \mathbf{F}_\mathrm{MOT}=-\alpha\mathbf{v}-\frac{\alpha g \mu_B}{\hbar k}\mathbf{r}\nabla \|\mathbf{B}\|, where\alpha = 4\hbar k^2 \frac{I}{I_0} \frac{2\delta/\Gamma}{[1 + (2\delta/\Gamma)^2]^2}is the damping coefficient, g is the
Landé g-factor, \mu_B is the Bohr magneton, \hbar is the reduced Planck constant, I_0 is the saturation intensity, \delta is the laser detuning, \Gamma is the linewidth of the atom-cooling transition and k is the norm of its wavevector. ==Atomic structure necessary for magneto-optical trapping==