Interference, distinguishability, and apparent contradiction in an experiment on induced coherence

Abstract

It has been shown in experiments on induced coherence without induced emission using two coherently pumped nonlinear crystals that (a) no second-order interference occurs as long as it is possible, even if only in principle, to determine which of the two crystals emitted the photon pair; (b) second-order interference does occur whenever it is intrinsically impossible to determine which of the two crystals emitted the photon pair. The present paper discusses a variant of the previous experiments in which, apparently, (c) each photon of the pair is emitted by a different crystal. It is also shown that interference is not necessarily reduced when there is no intrinsic indistinguishability.The experiment provides a proof of Bell's theorem for two particles without using an inequality and, at least in principle, could be used to test local realism against quantum mechanics.However, as in the case of the other experiments so far performed, some additional assumption would have to be introduced. It is then shown that the introduction of a reasonable assumption reduces the detectors's efficiencies needed to test local realism, even if nonenhancement is not assumed.PACS number(s): 03.65. Bz, 42.50.Dv Experiments in which interference is observed in the superposition of signal photons from two coherently pumped parametric down-converters have been reported by Mandel and co-workers [I]. It has been shown that no interference occurs as long as it is possible to determine which of the two crystals emitted the photon pair. It does not matter whether we have or do not have the practical means to distinguish which of the two crystals emitted the photons. If it is possible in principle, no interference will be observed. It has also been shown that interference does occur whenever it is intrinsically impossible to determine which of the two crystals emitted the photon pair. In the present paper, I wish to discuss a variant of these previous experiments. In this experiment, apparently, each photon of the pair is emitted by a different crystal, and it is possible to determine which of the two crystals emitted the idler and which emitted the signal. A situation is also discussed in which interference is not reduced even though there is no intrinsic indistinguishability.That is, in some cases it is possible to determine which of the two crystals emitted the photon pair. Although at first sight this should correspond to a reduction in the interference, it in fact does not. As we will also see, there is a sort of conservation of information. The information gained about the photons emitted by one of the crystals is compensated by a loss of information about the photons emitted by the other crystal. In other words: whenever it is possible to determine which of the two crystals emitted the photon pair, one possibility is actualized, so to speak, and no interference is observed. On the other hand, whenever it is not possible to determine which of the two crystals emitted the photon pair, two possibilities are actualized, and interferenc...

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“…Reciprocally, distinguishability in principle, that is, the possibility (even if only in principle) of obtaining which-path information, would be a sufficient condition for reducing two-photon interference (in this case the indistinguishability would be only accidental). That this is not necessarily so has been outlined in a previous paper [5]. Here, this point is made clear by discussing a variant of an experiment that has already been performed many times [2].…”

“…Then, introducing the third interferometer, we obtain (5). From (5) we see that we can only have coincident detections at sites 2 and 3 when γ 1 follows paths S 1 and L 3 and γ 2 follows path L 2 . The possibility in which γ 1 follows paths S 1 and S 3 , and γ 2 follows path S 2 , does not exist in (5).…”

“…From (5) we see that we can only have coincident detections at sites 2 and 3 when γ 1 follows paths S 1 and L 3 and γ 2 follows path L 2 . The possibility in which γ 1 follows paths S 1 and S 3 , and γ 2 follows path S 2 , does not exist in (5). Thus, using (4) and (5), we see that the probability of coincident detections occurring at sites 2 and 3 is given by…”

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“…The influence of the SPDC spatial modes overlap [62,69], the SPDC temporal properties [61], and the losses in the system [59] on the visibility of the interference fringes have been widely studied since the first demonstration of the phenomenon. However, our present understanding of the influence of polarization properties of the down-converted photons on the nonlinear interference is rather limited [123][124][125][126][127].…”