The two-photon interferometric experiment proposed by Franson [Phys. Rev. Lett. 62, 2205] is often treated as a "Bell test of local realism". However, it has been suggested that this is incorrect due to the 50% postselection performed even in the ideal gedanken version of the experiment. Here we present a simple local hidden variable model of the experiment that successfully explains the results obtained in usual realizations of the experiment, even with perfect detectors. Furthermore, we also show that there is no such model if the switching of the local phase settings is done at a rate determined by the internal geometry of the interferometers.The two-particle interferometer introduced by Franson [1] has been used in many two-photon interferometric experiments [2,3] that reveal complementarity between single and two-photon interference. The experiments cannot be described using standard methods involving classical electromagnetic fields [4]. However, the original paper was entitled Bell Inequality for Position and Time, and many subsequent papers claimed that the experiment constitutes a "Bell test of local realism involving time and energy". Some authors were more skeptical that a true, unambiguous test of a Bell inequality was possible with these experiments, even in principle, since even the ideal gedanken model of the experiment requires a post-selection procedure in which 50% of the events are discarded when computing the correlation functions [5,6]. If all events are taken into account the Bell inequalities are not violated. Thus, a local hidden-variable (LHV) model is not ruled out, but even so, no LHV model for the experiment has yet been constructed [7].The situation is further obscured by similar claims concerning certain other two-photon polarization experiments [8] where the problem of discarding 50% of the events also appears [5,9]. This was initially treated on equal footing with the problems of Franson-type experiments, but a recent analysis in [10] reestablishes the possibility of violating local realism. Unfortunately, that analysis cannot be adapted to the Franson experiment.Our aim is to resolve this uncertainty. First, we shall construct a simple local realistic model for the usual operational realization of the experiment. Second, we shall prove that under the additional condition that the random changes of the state of the local interferometers are at a rate dictated by the internal geometry of the interferometers, no local hidden variable model exists for the perfect gedanken version of this type of experiment. Even then, the usual Bell inequality will be inadequate. Let us briefly describe the idea behind the Fransontype experiments (Fig. 1). The source yields photon pairs, correlated to within their coherence times, and the two photons are fed into two identical unbalanced MachZehnder interferometers. The difference of the optical paths in those interferometers, ∆L, satisfies the relation ∆L ≫ cT coh , where c is the speed of light and T coh is the coherence time of the photons. Such o...
