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Physical Review A

Effects of detector size and position on a test of Born's rule using a three-slit experiment

Etienne Gagnon, Christopher D. Brown, Amy L. Lytle

Published Jul 23, 2014• 9 citations
Publisher

Abstract

Recent high-precision measurements in a three-slit diffraction experiment [Sinha et al., Science 329, 418 (2010)] have been performed as an explicit test of the validity of Born's rule for quantum probabilities. This experiment aims to establish an upper limit to the possibility of higher-order interference, which, if observed, could support generalization of quantum probability theory. We reproduce this three-slit experiment using position-resolved detection, compare our results to a computational model, and find significant limitations to the normalization scheme proposed by Sinha et al. that influence interpretation. We further show that the dependence of the measurements on detector size and position must be taken into account for proper interpretation of results and meaningful comparison with other experimental schemes.

DOI
10.1103/physreva.90.013832

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Recent high-precision measurements in a three-slit diffraction experiment [Sinha et al., Science 329, 418 (2010)] have been performed as an explicit test of the validity of Born's rule for quantum probabilities. This experiment aims to establish an upper limit to the possibility of higher-order interference, which, if observed, could support generalization of quantum probability theory. We reproduce this three-slit experiment using position-resolved detection, compare our results to a computational model, and find significant limitations to the normalization scheme proposed by Sinha et al. that influence interpretation. We further show that the dependence of the measurements on detector size and position must be taken into account for proper interpretation of results and meaningful comparison with other experimental schemes.
In Mod. Phys. Lett. A 9, 3119 (1994), one of us (R.D.S) investigated a formulation of quantum mechanics as a generalized measure theory. Quantum mechanics computes probabilities from the absolute squares of complex amplitudes, and the resulting interference violates the (Kolmogorov) sum rule expressing the additivity of probabilities of mutually exclusive events. However, there is a higher order sum rule that quantum mechanics does obey, involving the probabilities of three mutually exclusive possibilities. We could imagine a yet more general theory by assuming that it violates the next higher sum rule. In this paper, we report results from an ongoing experiment that sets out to test the validity of this second sum rule by measuring the interference patterns produced by three slits and all the possible combinations of those slits being open or closed. We use attenuated laser light combined with single photon counting to confirm the particle character of the measured light.
This paper is based on work published in [1]. It describes a triple slit experiment using single photons that has been used to provide a bound on one of the most fundamental axioms of quantum mechanics i.e. Born’s rule for probabilities [2]. In spite of being one of the most successful theories which describes various natural phenomena, quantum mechanics has enough intricacies and “weirdness” associated with it which makes many physicists believe that it may not be the final theory and hints towards the possibility of more generalized versions. Quantum interference as shown by a double slit diffraction experiment only occurs from pairs of paths. Even in multi‐slit versions, interference can only occur between pairs of possibilities and increasing the number of slits does not increase the complexity of the theory that still remains second‐order. However, more generalized versions of quantum mechanics may allow for multi‐path i.e. higher than second order interference. This experiment also provides a bound on the magnitude of such higher order interference. We have been able to bound the magnitude of three‐path interference to less than 10−2 of the expected two‐path interference, thus ruling out third and higher order interference and providing a bound on the accuracy of Born’s rule.
Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text C. J. Brown, E. Gagnon, and A. L. Lytle, "Effect of Detector Size and Position on a Test of Born's Rule Using a Three-Slit Experiment," in Frontiers in Optics 2013, I. Kang, D. Reitze, N. Alic, and D. Hagan, eds., OSA Technical Digest (online) (Optica Publishing Group, 2013), paper FTh1C.4. Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article
Quantum mechanics predicts that interference arising from multiple slits can be reduced to a sum of outcomes from the simpler one- and two-slit counterparts. Testing deviations from these results has been discussed as testing the validity of Born's rule. Here an experiment that considers the evolution between three levels of an atom as the quantum paths is devised, providing precision tools that can be used to test universality of Born's and to constrain theories designed to supersede quantum mechanics.
In which-way double-slit experiments with perfect detectors, it is assumed that having a second detector at the slits is redundant because it will not change the interference pattern. We, however, show that if higher-order or nonclassical paths are accounted for, the presence of the second detector will have an effect on the interference pattern. Accounting for these nonclassical paths also means that the Sorkin parameter in triple-slit experiments is only an approximate measure of Born-rule violation. Using the difference between single and double which-way detectors, we give an alternative parameter which is an exact measure of Born-rule violation.
It has been suggested by Sorkin that a three-slit Young experiment could reveal the validity a fundamental ingredient in the foundations of one of the cornerstones in modern physics namely quantum mechanics. In terms of a certain parameter κSκ_SκS​, it was argued that a non-zero value could imply a breakdown of the fundamental Born's rule as well as the superposition principle. Here we argue that a physical realization of such arguments could lead to an erroneous conclusion and contradict the basic rules of quantum mechanics. In fact, we argue that a proper interpretation of the procedures involved in a physical determination of κSκ_SκS​ does not necessarily lead to κS=0κ_S=0κS​=0. In order to show this we consider a mono-chromatic source of photons prepared in an {\it arbitrary} quantum state and a simple version of the well-established photon detection theory of Glauber which, by construction, obeys all the rules of quantum mechanics. It is, however, also argued that after a proper identification of the relevant quantum-mechanical probability amplitudes one can be reach κS=0κ_S=0κS​=0. As long as one only consider a single photon detector, it is verified that, in this context, there is no fundamental difference between quantum-mechanical interference and interference as expressed in terms of classical electro-magnetic waves.