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I was at a journal club some time around 2020 where we discussed a paper that I am trying to find again. The claim was that it had experimentally shown that wave function collapse was not "instant" by doing quantum tomography on a system being measured, and seeing the system go continuously from a superposition to an eigenstate of the measured basis. It probably involved some kind of weak measurement.

Can you point me to any literature performing such an experiment?

Candidates I have looked at so far, which do not seem to be the paper I am looking for, are

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You may be thinking of this paper, which concerns not measurement per se but quantum "jumps" between energy levels in an artificial atom. The jump is inferred from a break in the fluorescence of the three-level "atom", which I think would count as, as you say, some kind of weak measurement. https://www.nature.com/articles/s41586-019-1287-z

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