I am trying to understand cesium atomic clocks better.
I am not getting HOW the cesium oscillation is actually being counted.
So from my understanding of an older atomic clock:
Cesium gets heated.
Heated cesium goes through magnets.
Excited cesium atoms get discarded and non-excited cesium atoms get hit with microwave radiation at about $9,192,631,770 \, \mathrm{Hz} .$
This excites electrons to the hyperfine splitting upper energy level of the $6\text{s}$ orbital.
Electrons that get excited hit a detector, and the ratio between these and the total number entering the microwave chamber is calculated, subsequently the frequency of the microwave generator is adjusted to reach the highest ratio. To this point to me all we did was measure that cesium has a two energy levels separated by some energy,$$ {E}_{\text{photon}} ~=~ h f ~=~ h \times {9,192,631,770} \, \mathrm{Hz} \,,$$where $h$ is Planck's constant and $f$ is a photon's frequency.
So great – but how is this actually being counted? Is there a device somewhere that says – ah ok, maximum cesium atoms are being excited, let's start counting... start counting what though?!? And what device is doing the actual counting, as in: $1,$ $2,$ $3,$ $4,$ ..., $9,192,631,770,$ and then saying, BOOM, that's a second.