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Wikipedia shows a heating effect that causes minimum temperature achievable in doppler cooling as: "The atom performs a random walk in momentum space with steps equal to the photon momentum due to spontaneous emission and photon absorption. This constitutes a heating effect, which counteracts the cooling process and imposes a limit on the amount by which the atom can be cooled." What exactly is this heating process?

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Heat is random motion of atoms.

In doppler cooling, lasers are slightly below a transition frequency when viewed in the lab frame of reference. Atoms moving faster than average toward the beam see it blue shifted just enough to absorb a photon. These atoms all receive a kick that reduces their kinetic energy.

Now they are excited. They decay by emitting a photon in random directions. This kicks that atoms in random directions, adding heat.

Since the atoms are moving slower now, they emit a photon that is closer to the transition frequency when viewed in the lab frame of reference.

Atoms absorb a low frequency photon and emit a slightly higher frequency photon. This is the source of cooling.

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