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Physical Review Letters

Exponential Localization of Photons

Iwo Białynicki‐Birula

Published Jun 15, 1998• 103 citations
Publisher

Abstract

It is shown that photons can be localized in space with an exponential falloff of the energy density and photodetection rates. The limits of localization are determined by the fundamental Paley-Wiener theorem. A direct mathematical connection between the spatial localization of photons and the decay in time of quantum mechanical systems is established.

DOI
10.1103/physrevlett.80.5247

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It is shown that photons can be localized in space with an exponential falloff of the energy density and photodetection rates. The limits of localization are determined by the fundamental Paley-Wiener theorem. A direct mathematical connection between the spatial localization of photons and the decay in time of quantum mechanical systems is established.
In contradistinction not only to the seventy-year-old notion of poor localizability of a photon but also to a recent result of principal revisions of the notion, we demonstrate that N -photon pulsed wave functions of a specific type exhibit faster than an exponential falloff of energy and photodetection rate with distance from the pulse center. The wave functions are cylindrical and exhibit an exceptionally strong localization in two dimensions out of three. Our approach involves a “technology transfer” in the sense that, in order to tackle the problem belonging to quantum optics, we make use of results obtained recently in the study of the so-called nondiffracting localized solutions to the classical wave equation.
Using the theory of generalized imprimitivities for the description of weakly localized single-photon states, the spatial dependence of the photodetection of such states is calculated and an ${x}^{\mathrm{\ensuremath{-}}7}$ falloff is found. A general technique for determining the falloff of densities associated with weakly localized states is given and in particular the same spatial dependence is found for the falloff of energy density in agreement with the previous work of Amrein.
Photons cannot be localized in a sharply defined region. The expectation value of their energy density and the photon number density can only be approximately localized, leaving an exponential tail. We show that one may sharply localize either electric or magnetic (but not both) footprints of photons and only momentarily. In the course of time evolution this localization is immediately destroyed. However, the coherent states, like their classical counterparts, can be localized without any limitations. The main tool in our analysis is a set of space-dependent photon creation and annihilation operators defined without any reference to the mode decomposition.
The theoretical concept of photons is very important for the interpretation of quantum optics experiments. Since Eugene P. Wigner published the important work on the concept of particle localization in relativistic quantum mechanics, it has long been argued in the theoretical literature that photons cannot be exactly located in free space.1 The argument claims that localized photons necessarily have an energy density that drops off as a power law of distance. If this is true, one should seriously examine the experimental consequences of such a claim.