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arXiv:quant-ph/0504115 · International Journal of Quantum Information

LOGICAL ANALYSIS OF THE BOHR COMPLEMENTARITY PRINCIPLE IN AFSHAR'S EXPERIMENT UNDER THE NAFL INTERPRETATION

R. Srinivasan

Published Apr 1, 2010• 4 citations
arXivHTMLPDF

Abstract

The NAFL (non-Aristotelian finitary logic) interpretation of quantum mechanics requires that no "physical" reality can be ascribed to the wave nature of the photon. The NAFL theory QM, formalizing quantum mechanics, treats the superposed state (S) of a single photon taking two or more different paths at the same time as a logical contradiction that is formally unprovable in QM. Nevertheless, in a nonclassical NAFL model for QM in which the law of noncontradiction fails, S has a meaningful metamathematical interpretation that the classical path information for the photon is not available. It is argued that the existence of an interference pattern does not logically amount to a proof of the self-interference of a single photon. This fact, when coupled with the temporal nature of NAFL truth, implies the logical validity of the retroactive assertion of the path information (and the logical superfluousness of the grid) in Afshar's experiment. The Bohr complementarity principle, when properly interpreted with the time dependence of logical truth taken into account, holds in Afshar's experiment. NAFL supports, but not demands, a metalogical reality for the particle nature of the photon even when the semantics of QM requires the state S.

DOI
10.1142/s021974991000640x

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The NAFL (non-Aristotelian finitary logic) interpretation of quantum mechanics requires that no "physical" reality can be ascribed to the wave nature of the photon. The NAFL theory QM, formalizing quantum mechanics, treats the superposed state (S) of a single photon taking two or more different paths at the same time as a logical contradiction that is formally unprovable in QM. Nevertheless, in a nonclassical NAFL model for QM in which the law of noncontradiction fails, S has a meaningful metamathematical interpretation that the classical path information for the photon is not available. It is argued that the existence of an interference pattern does not logically amount to a proof of the self-interference of a single photon. This fact, when coupled with the temporal nature of NAFL truth, implies the logical validity of the retroactive assertion of the path information (and the logical superfluousness of the grid) in Afshar's experiment. The Bohr complementarity principle, when properly interpreted with the time dependence of logical truth taken into account, holds in Afshar's experiment. NAFL supports, but not demands, a metalogical reality for the particle nature of the photon even when the semantics of QM requires the state S.
In an interesting series of articles published few years ago Afshar and coworkers [1,2] reported an optical experiment in which they claimed to refute the well known N. Bohr principle of complementarity [3, 4]. Obviously this result, if justified, would constitute a serious attack against the orthodox interpretation of quantum mechanics (known as the Copenhagen interpretation). This work stirred much debate in different journals (see for examples references [5–12]). We think however that there are still some important misunderstandings concerning the interpretation of this experiment. In a preprint written originally in 2004 [5] (and following some early discussions with Afshar) we claimed already that the interpretation by Afshar et al. can be easily stated if we stay as close as possible from the texts written by Bohr. The aim of the present article (which was initially written in 2005 to precise a bit the thought developed in [5]) is to comment the interpretation discussed in [1]. We will in the following analyze the meaning of Bohr principle and show that far from disproving its content the experiment [1] is actually a complete confirmation of its general validity. The difficulties associated with the understanding of this principle are not new and actually complementarity created troubles even in Einstein mind [3] so that we are here in good company. To summarize a bit emphatically Bohr’s complementarity we here remind that this principle states that if one of a pair of non commuting observables of a quantum object is known for sure, then information about the second (complementary) is lost [3, 4, 15, 16]. This can be equivalently expressed as a kind of duality between different descriptions of the quantum system associated with different experimental arrangements which mutually exclude each other (read in particular [3,4]). Later in the discussion we will try to precise this definition but for the moment it is enough to illustrate the concepts by examples Consider for instance the well known Young double-pinholes interference experiment made with photons. The discrete nature of light precludes the simultaneous observation of a same photon in the aperture plane and in the interference pattern: the photon cannot be absorbed twice. This is already a trivial manifestation of the principle of Bohr. Here it implies that the two statistical patterns associated with the wave in the aperture plane and its Fourier (i. e., momentum) transform require necessarily different photons for their recording. It is in that sense that each experiment excludes and completes reciprocally the other. In the case considered before the photon is absorbed during the first detection (this clearly precludes any other detection). However even a non-destructive solution for detection implying entanglement with other quantum systems has a radical effect of the same nature: the complementarity principle is still valid. For example, during their debate Bohr and Einstein [3] discussed an ideal which-way experiment in which the recoil of the slits is correlated to the motion of the photon. Momentum conservation added to arguments based on the uncertainty relations are sufficient to explain how such entanglement photon-slits can erase fringes [15–19]. It is also important for the present discussion to remind that the principle of complementarity has a perfidious consequence on the experimental meaning of trajectory and path followed by a particle. Indeed the unavoidable interactions existing between photons and detectors imply that a trajectory existing independently of any measurement process cannot be unambiguously defined. This sounds even like a tragedy when we consider once again the two-holes experiment. Indeed for Bohr this kind of experiments shows definitely the essential element of ambiguity which is involved in ascribing conventional physical attributes to quantum systems. Intuitively (i. e., from the point of view of classical particle dynamic) one would expect that a photon detected in the focal plane of the lens must have crossed only one of the hole 1 or 2 before to reach its final destination. However, if this is true, one can not intuitively understand how the presence of the second hole (through which the photon evidently did not go) forces the photon to participate to an interference pattern (which obviously needs an influence coming from both holes). Explanations to solve this paradox have been proposed by de Broglie, Bohm, and others using concepts such as empty waves or quantum potentials [20, 21]. However all these explanations are in agreement with Bohr principle (since they fully reproduce quantum predictions) and can not be experimentally distinguished. Bohr and Heisenberg proposed for all needed purposes a much more pragmatic and simpler answer: don’t bother, the complementarity principle precludes the simultaneous observation of a photon trajectory and of an interference pattern. For Bohr [3]: This point is of great logical consequence, since it is only the circumstance that we are presented with a choice of either tracing the path
A recent experiment performed by Afshar et al. has been interpreted as a violation of Bohr's complementarity principle between interference visibility and which-path information a two-path interferometer. The authors have reproduced the same experiment, using true single-photon pulses emitted by a single nitrogen-vacancy (NV) color center in a diamond crystal and propagating in a two-path wavefront-splitting interferometer realized with a Fresnels' biprism, followed by a grating with transmitting slits of adjustable width. The experimental setup is illustrated. The results go far beyond the reported results in the previous works of Afshar et al. and the subtle flaws in their analysis are then pointed out. The measured values of the interference visibility V and the which-path information, characterized by the distinguishability parameter D, are found, with values close to the ideal-case upper bound. This result demonstrates that the experiment can be perfectly explained by the Copenhagen interpretation of quantum mechanics, without any violation of Bohr's complementarity principle.
This work is about Bohmian mechanics, a non-relativistic quantum theory about the motion of particles and their trajectories, named after its inventor David Bohm (Bohm,1952). This mechanics resolves all paradoxes associated with the measurement problem in nonrelativistic quantum mechanics. It accounts for quantum randomness, absolute uncertainty, the meaning of the wave function of a system, collapse of the wave function, and familiar (macroscopic) reality. We review the purpose for which Bohmian trajectories were invented: to serve as the foundation of quantum mechanics, i.e., to explain quantum mechanics in terms of a theory that is free of paradoxes and allows an understanding that is as clear as that of classical mechanics. To achieve this we analyse an optical interferometry experiment devised and carried out 2005 by Shahriar Afshar (Afshar,2005). The radical claim of Afshar implies in his own words the 'observation of physical reality in the classical sense' for both 'which path (particle-like)' and 'interference (wave-like)' properties of photons in the same experimental setup through the violation of the Englert-Greenberger duality relation (Englert,1996) that according to Englert can be regarded as quantifying of the 'principle of complementarity'.
A recent experiment performed by Afshar et al (2007 Found. Phys. 37 295–305) has been interpreted as a violation of Bohr's complementarity principle between interference visibility and which-path information (WPI) in a two-path interferometer. We have reproduced this experiment, using true single-photon pulses propagating in a two-path wavefront-splitting interferometer realized with a Fresnel's biprism, and followed by a grating with adjustable transmitting slits. The measured values of interference visibility V and WPI, characterized by the distinguishability parameter D , are found to obey the complementarity relation V 2 + D 2 ⩽1. This result demonstrates that the experiment can be perfectly explained by the standard interpretation of quantum mechanics.