We consider cooling of a trapped particle in the limit when the position and velocity variables do not change adiabatically. The slowly varying quantity in a harmonic trap is then the energy, and a representation based on oscillator eigenstates is used. For large excitation, fast particles, a Fokker-Planck expansion is obtained which is valid when our adiabatic description breaks down. Estimates of the initial cooling rate are given and compared with earlier results. Our treatment requires a large number of oscillator states to be coupled by each one-photon process, and in this limit the diffusion will make the expansion invalid towards the end of the cooling; our physical interpretation is that particles leak out of the trap. The opposite case, the Lamb-Dicke regime, is advantageous for cooling experiments, and then a difference equation replaces the Fokker-Planck equation of the present paper.
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Javanainen, J., Stenholm, S. Laser cooling of trapped particles II. Appl. Phys. 24, 71–84 (1981). https://doi.org/10.1007/BF00900402