How do atomic clocks work? Do they have to do with Zeeman?
In broad outline, they use radiation from a specific atomic transition of precisely known frequency to drive an oscillator (in practice, there are many atomic physics and technical details). To get a precise frequency, one actually should avoid the Zeeman effect as much as possible: if there are stray magnetic fields around, that would change the atomic energy levels and hence shift the frequency. So levels with should be used. The November 2007 issue of Physics Today has an up-to-date article about atomic clocks.