Neither in the article, nor in the comments, is it clear to me that the time-resolution of these clocks is sufficient to track relative variations in the rate of time between spatially displaced clocks. This would speak to the sort of periods involved in measuring gravitational waves. Alternatively, one could sample the readouts at varying rates and signal-average the readout instances, and look for spikes in the sample-rate spectrum.
Ie, one could sample the readouts 1000 times at 0.1 ms, at 0.2, 0.4.... 6.4ms ... etc.
If I'm not clear, I need my morning second cuppa, sorry. Maybe someone can repair the above so it makes more sense.