Ensemble versus Individual System in Quantum Optics. G. C. Hegerfeldt in Fortschr. Phys. <595::aid-prop595>3.0.co;2-m 46:595 (1998).  What the paper says!?

This is an article self-described as a "lecture", although I could not identify in which context (see the volume as a whole, which contains other interesting articles such as Ref. «shi98a»). It addresses the interesting question of ensemble vs single-object averages, which connects to my Quantum-state ergodic hypothesis.

The discussion is oriented around the three-level intermittent fluorescence (here attributed to a much-cited 1975 paper by Dehmelt but which cannot be traced) and much of the actual development of the text is on deriving the quantum jump approach of the author. The general problem, however, is well paused:

With the advent of atom traps, in particular the Paul trap, and with laser cooling it became possible to store a single atom (ion) — or two, three or more — in a trap for hours or days and to experiment with it, e.g. study its interaction with light, microwave radiation or with other atoms. For a single system the statistical interpretation of quantum mechanics, based on ensembles, is not so readily applicable as in the case of beam or a gas. The question we want to address here is the following.

"Does quantum mechanics allow statements for a single system?"

The answer will be: "Yes, to some extent."

The answer—«to an extent»—is however less convincingly conveyed.

If the underlying stochastic process is ergodic then

time average over a single trajectory = ensemble average

and this equality allows easy calculation. In many cases, e.g. for a renewal process, ergodicity is easy to see. We believe that it is probably true in general for the quantum trajectories [18].

And note [18] states:

[18] It would indeed be very interesting if there were non-ergodic quantum trajectories. Then there could be different time averages for different trajectories.

This comment is interesting:

Hence for observables such as time averages quantum mechanics allows predictions for single systems.

As is this one:

There is a word of caution, however. For the observable "frequency spectrum of fluorescent radiation" from a single atom the above trajectories are not (directly) applicable. This has to do with the time-energy uncertainty relation. If all photon detection times were known by measurements then the spectrum would be broadened and deformed. This shows that the above quantum trajectories are not ªrealisticº and should therefore not be overinterpreted. They are just a useful quantum mechanical tool in certain situations. There is also a relation with the consistent-histories approach to quantum mechanics [20].

There is a list of references (his [4, 6, 9, 10, 11, 12, 13])

The text seems moderately well written, discussing suddenly things out of nowhere, e.g., after pausing the main question, the authors writes «Of course this is trivial if the probability in question is 0 or 1», although there never was a mention of any probability to that point. There also seems to be a typo there: