Exponential Localization of Photons

The purpose of this Letter is to prove that, contrary to a widespread belief, photons can be localized in space with an exponential falloff of the photon energy density and of the photodetection rates. In that respect, photons are not much different from massive particles. The only constraint on the photon localizability comes from the fundamental Paley-Wiener theorem [1]. The mathematical mechanism that limits the decrease of the photon energy density is the same as in the case of the decay of quantum systems with time. The exponential falloff of the photon wave function can be of the order expf2fsrdg, where fsrd increases with r slower than linearly. I shall show that this exponential localization holds not only for the energy density and for the photodetection rates but also for the modulus of the Landau-Peierls function [2] that is viewed by some as the proper position wave function of the photon. I shall present simple analytic expressions for the photon wave functions that exhibit such an exponentially small tail. I shall also contrast the localization properties of one-photon states with those containing many photons. Adlard, Pike, and Sarkar [3] have recently constructed single photon states that exhibit arbitrarily high powerlaw falloff of the energy density and of the photodetection rates. These states were derived from the solutions of Maxwell equations that have the form of complex rational functions of coordinates. Such solutions were envisaged a long time ago by Gårding [4], Synge [5], and Trautman [6] and were used more recently by Ziolkowski [7] and Hellwarth and Nouchi [8] to describe focused electromagnetic pulses. Incidentally, the arbitrarily high power-law falloff, found in Ref. [3], could have been obtained by simply differentiating the old solutions with respect to space coordinates and by observing that a derivative of a solution of Maxwell equations is also a solution. In the present Letter I go beyond all these algebraic solutions by providing examples of much more tightly localized wave functions and by establishing the absolute limit of the photon localization. The most general one-photon (1ph) state can be described by two complex functions of the wave vector