nomadreid said:
one "films" a strontium ion in an electric field during the microsecond of its wave function "collapse"
This is a misleading description of what the experiment described in the paper is actually testing. A much better explanation of the point of the experiment is given in the Introduction to the paper:
What is an ideal measurement in quantum mechanics? What are its inner workings? How does the quantum state change because of it? These have been central questions in the development of quantum mechanics[1]. Notably, today’s accepted answer to the latter question is conceptually different from the one given in the first formalization of quantum mechanics by von Neumann[2]. Then, it was thought that an ideal measurement on aquantum system would inevitably destroy all quantum superpositions. Later, Lüders pointed out[3]that certain superpositions should survive, so that a sequence of ideal measurements would preserve quantum coherence. Lüders’s version is the one accepted today.
In other words, what this experiment is demonstrating is simply that a measurement on a quantum system only destroys superpositions (a better term would be "destroys quantum coherence") for the degrees of freedom involved in the measurement. In this case, only one of the three possible states of the trapped ion is involved in the measurement (i.e., has a nonzero interaction with whatever is being used to make the measurement), so the measurement only destroys quantum coherence that involves that one state--quantum coherence between the other two states is preserved. Or, to put it another way, since only one of the three states is "being measured", only quantum coherence involving that state is lost.
The "filming" part comes in because the measurement of the one state that is "being measured" (the state called ##|0\rangle## in the paper) only actually counts as a "measurement" if a fluorescence photon is detected; if no fluorescence photon is detected, then no "measurement" has taken place. Since the probability of detection of a fluorescence photon varies according to the interaction strength that is used, one can run the experiment multiple times with different interaction strengths to take what amount to "snapshots" of the process of loss of quantum coherence involving the state ##|0\rangle##. Very high interaction strength corresponds to the limit in which the probability of detection of a fluorescence photon approaches 1, and quantum coherence involving the state ##|0\rangle## is completely lost; very low interaction strength is the limit in which the probability of detection approaches 0, and quantum coherence is not affected at all. In between interaction strengths correspond to partial loss of quantum coherence.
Note, however, that this "partial" loss of quantum coherence is not a property of a single run of the experiment; it is only a property of an ensemble of runs all made with the same (intermediate) interaction strength. In any single run, either a fluorescence photon is detected or it is not. If it is, quantum coherence is lost; if it is not, quantum coherence is not lost. So there is no "partial" loss of coherence in any single run of the experiment, and there is also no "partial measurement" in any single run; no single run involves the measurement process being "caught" part way through. In any single run, either the measurement happens all the way, or it doesn't happen at all.