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Laser Physics

How small a packet of photons can be made?

Peeter Saari

Published Apr 1, 2006• 4 citations
Publisher

Abstract

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.

DOI
10.1134/s1054660x06040049

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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.
We review and sharpen the concept of a photon wave function based on the quantum theory of light. We argue that a point-like atom serves as the archetype for both the creation and detection of photons. Spontaneous emission from atoms provides a spatially localized source of photon states that serves as a natural wave packet basis for quantum states of light. Photodetection theory allows us to give operational meaning to the photon wave function which, for single photons, is analogous to the electric field in classical wave optics. Entanglement between photons, and the uniquely quantum phenomena that result from it, are exemplified by two-photon wave functions.
A review is given of the space-time wave mechanics of single photons, a subject with an almost century long history. The Landau-Peierls photon wave function, which is related nonlocally to the electromagnetic field is first described, and thereafter the so-called energy wave function, based on the positive-frequency Riemann-Silberstein vectors, is discussed. Recent attempts to understand the birth process of a photon emerging from a single atom are summarized. The polychromatic photon concept is introduced, and it is indicated how the wave mechanics of polychromatic photons can be upgraded to wave train quantum electrodynamics. A brief description of particle (photon) position operators is given, and it is shown that photons usually are only algebraically confined in an emission process. Finally, it is demonstrated that the profile of the birth domain of a radio-frequency photon emitted in a hyperfine spin transition shows an exponential falloff with the distance from the atomic nucleus. The length parameter in this falloff is the Bohr radius.
The classical free-space solutions of Maxwell’s equations for light propagation in one dimension include wave packets of any shape that travel at the speed of light. This includes highly-localised wave packets that remain localised at all times. Motivated by this observation, this paper builds on recent work by Southall et al. [J. Mod. Opt. 68, 647 (2021)] and shows that a local description of the quantised electromagnetic field, which supports such solutions and which must overcome several no-go theorems, is indeed possible. Starting from the assumption that the basic building blocks of photonic wave packets are so-called bosons localised in position (blips), we identify the relevant Schrödinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of states.