Physical limitations on quantum nonlocality in the detection of gamma photons emitted from positron/electron annihilation

Recent experimental measurements of the time interval between detection of the two photons emitted in positron/electron annihilation have indicated that collapse of the spatial part of the photon’s wavefunction, due to detection of the other photon, does not occur. Although quantum nonlocality actually occurs in photons produced through parametric down-conversion, the recent experiments give strong evidence against measurement-induced instantaneous spatial-localization of high-energy gamma photons. A new quantummechanical analysis of the EPR problem is presented which may help to explain the observed differences between photons produced through parametric down-conversion and photons produced through positron/electron annihilation. The results are found to concur with the recent experiments involving gamma photons. 03.65.Ud Typeset using REVTEX 1 Quantum non-locality in measurements involving polarization and two-photon interference of correlated photons has been experimentally confirmed at many independent laboratories [1-4]. It has been generally postulated that non-local effects may also occur in regard to the spatial wavefunctions of the emitted photons. As an example, detection of one of the photons produced in parametric down conversion is predicted to cause “instantaneous” localization of the other photon, subsequently eliminating any uncertainty in the time of arrival of the second photon at a second detector. Experimental support of this prediction has been reported by Hong et al. [4]. The two-photon interference method utilized in Ref. [4] indicates that the minimum time uncertainty, in the time interval between detection of the two down-converted photons, is less than 100fs. This uncertainty in time is much less than the coherence time of the initial pump photons, which subsequently gives strong indication of nonlocal collapse of the photon wave-function. One may expect to observe similar nonlocal effects involving photons emitted from positron/electron annihilation. Recent high-resolution measurements of the time interval between the two photons emitted in positron/electron annihilation have been carried out by Irby [5]. The results of the measurements indicate that the absolute minimum uncertainty in detection time between arrival of the two photons is ∆tQM = 117 ± 9 ps, which surprisingly, agrees with the lifetime of positrons in bulk sodium (119 ps) predicted by quantum electrodynamics [6] [7]. Although nonlocal effects are observed to occur in the case of down converted photons, the experimental results give strong evidence against the instantaneous spatial-localization of gamma photons emitted from annihilation events. In this paper, we present a quantum-mechanical analysis of the time interval between detection of correlated photons. The analysis is basically the same as that first presented by Einstein, Podolsky, and Rosen (EPR) in 1935 [8]. The main difference, however, is that we include time dependence in the quantum wavefunctions and also take into account restrictions on photon momenta due to energy conservation. As in the original EPR paper, we assume that the total momentum before the particles interact (or are emitted) is zero. In addition, we assume that the particles interact at times 2 t < 0. The total wavefunction can then be written (for t ≥ 0) Ψ(x1, x2, t) = ∫ ∞ −∞ ψp(x2, t) up(x1, t) dp , (1) where up(x1, t) are eigenstates of particle one’s momentum and energy up(x1, t) = e ipx1/h̄ e . (2) In order to conserve momentum, let us also assume ψp(x2, t) = e −ip(x2 + xo)/h̄ e , (3) which are eigenstates of particle two’s momentum and energy. (xo is an arbitrary constant introduced in the original EPR paper. In this case, however, since we are including the explicit time dependence, we will set xo = 0). Note that if a measurement of particle one’s momentum yields a value of p, the total wavefunction collapses to Ψ(x1, x2, t) = ψp(x2, t) up(x1, t) , (4) which has momentum eigenvalues of p and −p respectively for particles one and two. Before any measurement takes place, the total wavefunction is thus given by