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Physical Review A

Spatial dependence of weakly localized single-photon wave packets

Edward R. Pike, Sarben Sarkar

Published Jan 1, 1987• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.35.926

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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\ensuremath−7{x}^{\mathrm{\ensuremath{-}}7}x\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.
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.
We compare different measures for photon localization in terms of two-dimensional Gaussian wave packets of width Delta k. We find that all measures start to coalesce if the wave packet has evolved for times which are larger than a few (c Delta k)^{-1}. However, the Landau-Peierls wave function yields the largest initial offset <r>-ct > 0 while the Fourier transform of the helicity components of the k-space wave packet yields the smallest offset <r>-ct > 0 and converges towards <r>-ct = 0 fastest. We also discuss local detection of photons through a model detector consisting of ions in ion traps. The position-dependent detection probabilities are inferred from the scattering matrix. We find that the local detection probability for photons can be expressed in terms of three of the proposed localization measures, viz. the Landau-Peierls wave function, the energy wave function, and the Hawton density. Those three proposals also remain close throughout the time evolution of the single-photon wave packet. However, the energy wave function is locally related to electromagnetic fields and is therefore confirmed as the preferred tool for the description of local photon densities. The novelty of our observations concerns three items, viz. the possibility to express local photon absorption matrix elements through three of the proposed localization measures, the coalescence of all the proposed localization measures after sufficiently long evolution of a photon wave packet, and the striking differences of some of the proposed localization measures during early wave packet evolution.
A theory of photodetection was produced in the so-called “rotating wave approximation” which is usually an acceptable simplification for optical photons. In considering such delicate issues as the localisability of photons it may be wise to proceed initially without making such an assumption. This chapter calculates the photodetection probability without making this approximation before discussing localisation. It is necessary to have a prescription for calculating the most isotropically localised single photon. A destructive detection of one photon of the pair leaves a single photon in the field to be detected. Such an experiment provides an opportunity to study the properties of a single photon. A priori such a photon is unlikely to be as localised as one constructed mathematically from a generalised imprimitivity. The experimental dimensions are such that the correlated photons are each constructed from a certain momentum range around well-defined wavevectors.
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) œeld. However, Heisenberg’s uncertainty principle tells us that a state of deœnite momentum, energy, and polarization associated with a plane wave used as a basis function of quantization must be completely indeœnite 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 œnite 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).