## Quantum States of Motion

The motion of atoms undergoing laser cooling is usually treated classically.
This means that their motion is not considered when calculating their interaction with the cooling light using the Schrödinger equation,
but is separately described by Newton's laws.
However, when very cold atoms are confined in a region of size comparable to their deBroglie wavelength, their motion must be considered
quantum mechanically, within the Hamiltonian, using the operator p^{2}/2M.

One important example is motion in a magnetic trap. Atoms can be cold enough to satisfy the condition above, and thus their motion needs to be described in terms of a wave function bound by the trap potential as discussed in 1989 and its reference 6. If the classical orbit of a trapped atom passes near a zero-field point in the trap, there can be a spin flip and the atom is ejected. The quantum analog of this is the amplitude of the wave function near such a point, resulting in a Majorana transition that cause ejection.

Another example is confinement in the optical potential resulting from the light shifts in an inhomogeneous field. This was first observed in single-beam trapping by Ashkin and colleagues at Bell Labs in 1986, and the next year in Paris. when the optical potential was produced in a 1-D standing wave. This latter case was the forerunner of multitudinous experiments in 3-D optical lattices where spectroscopy was performed among the quantum levels at NIST in 1992.

Still another example arises when the velocity change from a single absorption, ℏk/M = recoil velocity, is important in the description. The case of velocity selective coherent population trapping is discussed elswhere, but recoil-induced resonance is a different kind of example. In this case, two laser beams of different k-vectors cause a stimulated Raman transition between two momentum states of the same atomic sublevel, thus amplifying one beam at the expense of the other (see Ref's. [4] and [6]). These different momentum states arise from including the atomic motion in the Hamiltonian. The momentum difference of the light beams is imparted to the atoms, and also must satisfy the resonance condition for energy conservation, thus allowing velocimetry at the recoil velocity scale.