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According to Ian Stewart's 2013 Symmetry: A Very Short Introduction (pp. 119-120),

Experiment and theory suggest that superposed states should not be observable as such; only individual eigenfunctions can be observed. More precisely, observing a superposition is delicate and only possible in unusual circumstances; until recently it was believed to be impossible. [My emphasis.]

1) What experimental and (or?) theoretical breakthrough (presumably) is he referring to in the second sentence?

I initially guessed that he refers to the "Quantum Microphone", but I'm not so sure. However, if he is, then 2) how does that count as "observing a superposition" and how wasn't that theoretically expected? ("Yup, quantum mechanics still works.")

Edit. I'm not sure whether it helps, but I'll quote the continuation (and add two tags).

Associated with this suggestion is the Copenhagen interpretation, in which any observation somehow 'collapses' the state to an eigenfunction. This proposal led to quasi-philosophical ideas such as Schrödinger's cat and the many-worlds interpretation of quantum mechanics. All we need here, however, is the underlying mathematics, which tells us that observable states correspond to irreducible representations of the symmetry group of the equation. In particle physics, observable states are particles. So symmetry groups and their representations are a basic feature of particle physics.

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Superposed states of what observable? Every state is simultaneously a superposition of many eigenstates of an observable AND a single eigenstate of another observable. To be a superposition or not for a given state depends on the considered observable. – Valter Moretti Apr 8 '14 at 17:54
This is why citations are a good thing. – BMS Apr 8 '14 at 18:04

I would guess, and it can only be a guess, that Stewart is referring to weak measurement.

There is a rather vague description of this in New scientist. Annoyingly I can't track down the original paper, but if Stewart's book was written in 2013 the timing fits.

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In my opinion, Stewart is speaking of experiments like Cavity Quantum Electrodynamics ones for instance, where quasi-classical states superposition can be achieved and observed. A lot of work has been done by Serge HAROCHE (2012 Nobel Prize) and his co-workers at Laboratoire Kastler Brossel concerning mesoscopic superposition states of the electromagnetic field stored in the mode of a high Quality superconducting cavity.

Have a look for instance on the 2004 PHD thesis of Alexia AUFFEVES GARNIER, Rabi oscillation at the quantum-classical boundary and generation of Schrödinger cat states.

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