In the experiment, they applied a potential by applying an intense laser pulse (what is more time-periodic than light?!). As solid state physicists we are familiar with what happens when one applies a periodic potential in space, and the mathematical structure here is identical. The analogy is made concrete below:
Periodicity in space:
Periodicity in time:
There is a beautiful visualization of this periodicity in the paper as seen below (click to enlarge):
One can see the repetition in (in A and B in the figure) as determined by the frequency of the incoming laser pulse. Furthermore, one can see the bands interacting with each other and causing gaps to open up as pictured in the schematic on the right and in the data in C and D in the figure.
A very simple idea, a very elegant experiment.