So these are the rotation matrices inside the cavity (U1) and for free flight (U2), and the total action of the clock is then the triple matrix product U1.U2.U1. I plot the magnitude squared of the off-diagonal element which stands for the transition from lower to higher state which we want to count. In the first picture I chose a shorter gap time, in the second picture I chose a longer gap time.
As you can see, the function oscillates much quicker and these so-called fringes become narrower for a longer gap time. Now your clock oscillator sits on the central one, tunes a bit to the left and a bit to the right, and makes sure that it sits on the spot where the change is symmetric. Thanks to these narrow fringes, this is super sensitive to small changes in the frequency.
(the x axis shows the de-tuning, with the zero in the center meaning we are exactly on the cesium hyperfine frequency)
Anyway, I thought that was neat.
