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Golfing with a Single Photon

Rebecca M Slayton

Published Aug 1, 2001• 0 citations
Publisher

Abstract

The probability of finding a photon in a narrow range of both position and momentum simultaneously appears to be negative, thanks to the uncertainty principle.
DOI
10.1103/physrevfocus.8.7

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The uncertainty principle suggests a quantitative trade-off between the control of position and the control of momentum in particle propagation. However, a superposition of two states with very different uncertainty trade-offs introduces an interference term that seems to combine precise statements about position and about momentum, allowing us to study how quantum mechanics describes the propagation of individual particles in free space. Here, we present a detailed experimental study of photons prepared in a superposition of position and momentum generated in a Sagnac interferometer. The transverse distribution of photons was obtained with three different measurement settings at the output port of the interferometer, corresponding to the initial position distribution, the initial momentum distribution, and an intermediate propagation time at which the contributions of initial position and momentum uncertainties are approximately equal to each other. We show that the interference effect localizes the photons in narrow intervals of position and momentum, resulting in a quantitative violation of Newton's first law as the interference pattern spreads out at the intermediate position. The data obtained can be used to demonstrate the negativity of the Wigner function in regions outside the position and momentum intervals in which the position and momentum contributions are confined.
In radar analysis there exists an analogue of the Heisenberg uncertainty principle of quantum mechanics. These uncertainty principles say that it is impossible to determine simultaneously the range (resp. position) and range rate (resp. momentum) of a moving target (resp. non-relativistic quantum-mechanical particle) with an arbitrarily high accuracy at a given instant of time. This analogy suggests that there should be a common mathematical structure behind that explains both of them. The concept of real Heisenberg nilpotent group lies at the foundations of signal theory as well as of quantum mechanics. Adopting the view point of nilpotent harmonic analysis it becomes obvious that the radar auto-ambiguity functions are of pure positive type on the symplectic time-frequency plane which will be identified with the tangent space to the Schrödinger coad-joint orbit of the Heisenberg group. Combining the classical Bochner theorem with Hudson’s characterization of the state vectors that give rise to non-negative Wigner quasi-probability densities we identify the Gabor functions as pulse envelopes of those radar auto-ambiguity functions on the time-frequency plane that are of positive type simultaneously with respect to its usual vector group structure as well as to its symplectic structure.