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I refer to this Wikipedia article http://en.wikipedia.org/wiki/Resolved_sideband_cooling

I understand how everything works except for one detail. When the atom is moving towards the laser, the laser frequency is Doppler shifted such that its frequency matches the energy gap [itex]\omega_{0}[/itex]. So when it absorbs that photon, it goes from its ground state to excited state. I also understand that the process of absorbing a photon must reduce the atom's momentum by the same amount as the momentum of the incoming photon.

My question is, how do we know that this absorption is able to exactly bring the oscillating atom from [itex]|n+1>[/itex] to [itex]|n>[/itex]? The energy gap of the harmonic oscillator has nothing to do with [itex]\omega_{0}[/itex]. So how does that work out so nicely?

As always, thank you for all the help.