The first problem is to produce low energy muons. There are only few facilities in the world which can do that. The second problem is getting the muons slow eno

The first problem is to produce low energy muons. There are only few facilities in the world which can do that. The second problem is getting the muons slow enough to bind with the protons, and making sure the muonic atoms are at the right levels. The third is to get a strong enough signal from the setup. But there are other problems too. For example, you need to measure the frequency of the light. Since the (quite recent) invention of frequency combs, this is possible with astounding precision. While this experiment is not the most complicated in this regard, it is still a challenge. Also, even after you have measured the level difference, you need quite a lot of theory to get the radius out, which had to be developed first, as you need to know where to look.