An analytical theory of the time-orbiting-potential quadrupole magnetic trap for cold atoms is developed. It is shown that the rotating magnetic field used to create the time-averaged harmonic potential is responsible for the formation of quasienergy states of an atom in the trap. It is found that the motion of an atom near the origin of the trap can be represented as consisting of slow motion in the time-averaged potential and fast oscillations with small amplitude. Eigenstates and eigenfunctions for the motion are found. The eigenfunctions are used to calculate the coordinate and momentum distributions for a single atom. It is concluded that at low temperature the quantum-statistical momentum distribution for a single atom exhibits a ring shaped structure due to the fast oscillations in the atomic linear momentum.

• Received 16 April 1998

DOI:https://doi.org/10.1103/PhysRevA.58.3138