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A probabilistic view of wave-particle duality for single photons

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Andrea Aiello

Max Planck Institute for the Science of Light, Staudtstrasse 2, 91058 Erlangen, Germany

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Abstract

One of the most puzzling consequences of interpreting quantum mechanics in terms of concepts borrowed from classical physics, is the so-called wave-particle duality. Usually, wave-particle duality is illustrated in terms of complementarity between path distinguishability and fringe visibility in interference experiments. In this work, we instead propose a new type of complementarity, that between the continuous nature of waves and the discrete character of particles. Using the probabilistic methods of quantum field theory, we show that the simultaneous measurement of the wave amplitude and the number of photons in the same beam of light is, under certain circumstances, prohibited by the laws of quantum mechanics. Our results suggest that the concept of “interferometric duality” could be eventually replaced by the more general one of “continuous-discrete duality”.

To describe the world as it appears to our senses, we often resort to the concepts of waves and of particles. Thus, we talk about the waves that appear on the surface of a pond when we throw a stone into it, or about the Brownian motion of pollen particles suspended in water. In such a world described by the laws of classical physics, waves and particles represent different aspects of reality: a physical object can be described as either a wave or a particle, and these two descriptions are mutually exclusive.

Things are essentially different in the quantum world, where wave and particle aspects are both present in the description of the same physical system, such as an electron or a photon. This fact of nature is called wave-particle duality and manifests itself, for example, in double-slit interference experiments. As Richard Feynman wrote in his celebrated lectures on physics, wave-particle duality is “[…] a phenomenon which is impossible, $absolutely$ impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the $only$ mystery.”

In this paper we study the problem of wave-particle duality by exploiting the fundamental fact that the amplitudes of waves vary continuously, whereas particles can be counted one by one. For this study, we consider a light beam excited in a single-photon state, which impinges upon two distinct detectors. The first detector measures the continuous values of a quadrature of the electromagnetic field. The second detector simultaneously counts the discrete number of photons that fall on it. We find that the outcomes of these two simultaneous measurements are linearly uncorrelated but statistically dependent. In other words, the wave (quadrature) and particle (photon number) aspects of light coexist and affect each other in a nontrivial way.

These results also suggests that linear correlations, which are commonplace in many applications of quantum mechanics, may in some cases not provide a complete picture of a quantum system. The use of probability distributions describing the results of measurements of the physical observables of the system, becomes mandatory to obtain an accurate description of the latter.

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