We present a quantum theory of a magneto-optical trap (MOT) from first principles based on the quantum kinetic equation for the atomic density matrix, taking into account the recoil effects caused by the interaction of atoms with the laser field. An efficient method for solving the quantum kinetic equation is proposed. It is shown that the steady-state solution describing the atoms in the MOT has a significantly nonequilibrium nature and can be described within the framework of a two-temperature distribution. The momentum distribution of cold atoms in the MOT depends on the magnetic field gradient and, in general, significantly differs from the momentum distribution of atoms in the optical molasses, which is usually used as an approximation to describe the MOT. We have also shown that with an increase in the magnetic field gradient, a spatial two-component distribution of atoms in the trap is formed even for a single-particle approximation when interatomic interactions are neglected.

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• Cooling & trapping