Single-photon states with arbitrarily fast asymptotic power-law falloff of energy density and photodetection rate are explicitly constructed. This goes beyond the recently discovered tenth-power law of the Hellwarth-Nouchi photon which itself superseded the long-standing seventh-power law of the Amrein photon.
You might think that the great success of quantum electrodynamics (QED) would settle the debate on the nature of light to provide a clear view of its behavior, where the photon is regarded as the unit of excitation associated with a quantized mode of the electromagnetic (radiation) eld. However, Heisenberg’s uncertainty principle tells us that a state of denite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indenite in space and time. It suggests the diµculty of spatial localization of a photon as a particle. In fact, Newton and Wigner showed that a free photon, as a massless particle with spin 1, has no localized states on the basis of natural invariance requirements that localized states for which operators of the Lorentz group apply should be orthogonal to the undisplaced localized states, a§er a translation (Newton and Wigner 1949). According to them, one can obtain a general expression for a position operator for massive particles and for massless particles of spin 0 or 1/2, not for massless particles with nite spin, which indicates that there is no probability density for the position of the photon, and thus a position-representation wave function cannot be consistently introduced. It has also been shown that photons are not localizable, on the basis of imprimitive representations of the Euclidean group (Wightman 1962). It is now believed that photons are only weakly localizable, although single-photon states with arbitrarily fast asymptotic fallo of energy density exist, and that a lack of strict localizability is directly related to the absence of a position operator for a photon in free space and a positionrepresentation photon wave function (Hawton 1999).
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.