Abstract
In this paper we consider the laser cooling of a trapped particle when the discrete nature of the quantum levels must be taken into account. We particularize the equations to the Lamb-Dicke limit when the oscillational, amplitude is much less than the optical wavelength. Then the quantum levels are coupled only to the neighbouring ones, and we can also find a small parameter to use for adiabatic elimination of rapidly varying terms. There emerges one rate equation for the probability distribution over the quantum states. The ultimate distribution is found to be of the Planckian type independently of the initial one. We solve the equation, generally and obtain the time evolution for an arbitrary initial distribution. In particular we look at an initially thermal, Poisson, or pure state distribution. For the thermal one we find that its shape is preserved during the cooling. The results are discussed.
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