It seems as though they avoid that problem by resetting the experiment after the measurement. From Li's paper:
> Then we can use a global probe laser which is only scattered by the mark ion (state dependent fluorescence) to measure its angular displacement after a time separation Δt. [...] After the measurement, we can cool the ions back to the ground state and repeat the experiment again.
No. The problem is that they are not measuring the ground state; they are measuring an exited state. It doesn't mater what they do with the system later.
It's my impression that the crystal arrangement really refers to the distribution of ions in the trap. Exciting one atom does add energy to the system, but in a near-eigenstate that's quite orthogonal to the crystal's structure.
Whether the distinguishability of one ion over the others changes the quantum statistics is an important and subtle problem.
You're right to be skeptical. When the time comes to interpret the results of this experiment, you'll find yourself in good company. Doubly so if the experiment resolves T violation.
It seems like you could easily fix this problem by substituting a different atom instead of exciting one of them. It would be interesting to see if the results are the same.
> Then we can use a global probe laser which is only scattered by the mark ion (state dependent fluorescence) to measure its angular displacement after a time separation Δt. [...] After the measurement, we can cool the ions back to the ground state and repeat the experiment again.