What's done in such an experiment is that we initialize a quantum state in a particular state (that this is possible is not actually obvious, but let's assume it's true) and then we make a lot of repeated measurements after different amounts of times.

So we're not talking about a single system, but instead about statistics about a set of measurements with systems that have been set up in the same initial state. (Each system only being measured once after a certain time after it's being set up).