I have been reading about the Scully-Druhl experiments where the photons are directed to down-converters, which create a "signal" photon and an "idle" photon, and in which it appears that the idle-photon gives information about which path the signal photon took. This has made me curious about something. If I have a laser, and fire a photon into a beam-splitter, then my photon can go L or R through a slit towards my photo-sensitive paper. If I can detect the path, I understand, I will get no interference pattern, but if I cannot, then I will get an interference pattern as if the photon went along both paths. I place on each path a down-converter, splitting my photon into two lower-energy photons, one - the signal photon - which goes to the slit and the photographic paper, and another - the idler photon - which does not. If I place detectors at the ends of the path of the idler photons, I can tell whether my signal photon went L or R. However, I could also direct my idler photons to a single detector, making me unable to see which path my signal photon took. The first scenario would give me no interference pattern, but the second would. My idler photons travel down a huge length of optical fibre to a human settlement on another planet 10 light years away. At the end of their ten-year journey, they either all go into separate detectors, so that I can tell if they went L or R, or all into a single detector, so I cannot tell. The decision is made by my friend at the other end, whom I spoke to before conducting the experiment, and the decision is made 9.5 years after my photons hit the down-converters. I can look at the photographic paper 10 years before it is determined that the idler photons will be able to reveal any information about what path the signal photons took. What will I see?
Er, no it wouldn't. You'd get no interference in either case. The reason is that sending the photon from each path through a parametric downconverter produces an entangled state. If you call one of the paths \(|1 \rangle\) and the other \(| 2 \rangle\), then the parametric downconverters would take a superposition state like \(| 1 \rangle + | 2 \rangle\) to the entangled state \(| 1 \rangle_{s} | 1 \rangle_{i} + | 2 \rangle_{s} | 2 \rangle_{i}\) (s indicating the signal photon, i the idler). If you send the signal photons through a double slit but throw away the idler photons, you're trying to produce an interference pattern with only half an entangled state, which will never work. In this sort of case you can only produce an interference pattern in the collective statistics of both the signal and idler photons. If you look at actual delayed quantum erasure experiments you'll see this is what happens: the results of measurements performed on the idler photons is used to make a post-selection on the signal photons, and it's only after this post-selection is made that you exctract an interference pattern in the signal photon statistics. Still no interference in either case.
Right, but my confusion is that in the delayed quantum erasure experiments that I know of, there was always a 50% chance that the idler photon would give information about the path of the signal photon, so the post-selection would draw out an interference pattern from a more obvious non-interference pattern. I'm trying to think up an experiment where all the photons give path information, or none of the photons give path information, and not 50/50. For example, if 50% of the photons give out path information, then I can post-select the 50% that didn't and I would see an interference pattern. In the case where no photons gave path information, wouldn't that mean that all the remaining photons in my post-selection would give an interference pattern? Wouldn't this indicate that if I looked at the photosensitive paper I would see an interference pattern?
This won't work. In the type of setup you've suggested the signal photons will never exhibit an interference pattern if you take the full set of results. Your only chance of seeing any interference in this type of experiment will involve post-selection. No, not necessarily. The rule of thumb is that you can see an interfererence pattern as long as the paths are in principle indistinguishable, regardless of whether or not you actually try to distinguish them. In the setup you've suggested the paths are distinguishable - the idler photons travel along differen optical fibers until they're aborbed by detectors. It's just that in some cases you're choosing to throw the information away by using one photon detector instead of two.
So post-selection of a full set of results is not a special circumstance of post-selection but is something different altogether? Why is it so different if the data either totally belongs to the path-information set or totally doesn't belong rather than 50/50? Does post-selection work for 60/40 or 99/1? What happens when it reaches 100/0 that changes it? I thought that this set-up was effectively the same as the Scully-Druhl, except that the 50/50 chance of a photon giving out path-information was changed to 100/0. If Scully-Druhl had put a length of optical fibre between the downconverter and the next beam-splitter (which will send the photon to the indistinguishable detector or the distinguishable detector) would that give the same effect as my suggestion (i.e. the paths are distinguishable for a long period of time)? Thank you very much for your knowledge, by the way. Please Register or Log in to view the hidden image!
