There are different types of atomic clocks, but in most common types, the output comes from a crystal oscillator, or more generally frequency synthesizer, which is then slaved to some spectral feature in the "physics package". That is to say, the phase noise is as good as that of the crystal in the unit, but the longer term frequency stability is much improved by the slaving.

An exception is an active hydrogen maser, which directly outputs the frequency of atomic transition. It has very good phase noise, but is a rare beast, which is only used where it is absolutely necessary.