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arXiv:2306.00490 · Journal of Physics A Mathematical and Theoretical

Photon centroids and their subluminal propagation

Konstantin Yu Bliokh

Published Aug 11, 2023• 3 citations
arXivHTMLPDF

Abstract

Abstract We examine properties of the energy-density and photon-probability centroids of electromagnetic wavepackets in free space. In the second-order paraxial approximation, both of these centroids propagate with the same subluminal velocity because of the transverse confinement of the wavepacket and its diffraction. The tiny difference between the energy and probability centroid velocities appears only in the fourth order. We consider three types of wavepackets: Gaussian, Bessel, and non-diffracting Bessel. In all these cases, the subluminal propagation is clearly visible in the intensity distributions and can be measured experimentally in both classical-light and single-photon regimes. For Gaussian wavepackets, the half-wavelength delay is accumulated after propagation over about 12 Rayleigh lengths.

DOI
10.1088/1751-8121/acef7f

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Abstract We examine properties of the energy-density and photon-probability centroids of electromagnetic wavepackets in free space. In the second-order paraxial approximation, both of these centroids propagate with the same subluminal velocity because of the transverse confinement of the wavepacket and its diffraction. The tiny difference between the energy and probability centroid velocities appears only in the fourth order. We consider three types of wavepackets: Gaussian, Bessel, and non-diffracting Bessel. In all these cases, the subluminal propagation is clearly visible in the intensity distributions and can be measured experimentally in both classical-light and single-photon regimes. For Gaussian wavepackets, the half-wavelength delay is accumulated after propagation over about 12 Rayleigh lengths.
We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals c2c^2c2. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian and higher-order beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.
That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wave vectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam's transverse spatial structure. Using time-correlated photon pairs, we show a reduction in the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several micrometers over a propagation distance of ~1 meter. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves.
The speed of light is nominally given by c/n, where n is the refractive index of the medium in which the light is traveling. Since the refractive index of free space is 1, it is natural to expect that light travels at c in free space. We show that this is not the case if one considers real beams that have finite transverse extent. The modification of the wavevectors comprising the beam result in a change in the phase and group velocities. We study the group velocity of single photons, whose arrival times are unambiguously established by a click in a single-photon detector.We generate time-correlated photon pairs via spontaneous parametric down-conversion. One photon is sent through a fiber delay of a fixed length; the arrival time of this photon is the reference time to which we compare the arrival time of the other photon, which we can choose to be either unstructured or structured. The structuring is done by spatial light modulators that can be switched on and off, hence allowing photons to travel through exactly the same optical path, much like in a race. We consider photons in a Bessel mode and a focused Gaussian mode, and show that in both cases, the reduction in group velocity results in a delay of several micrometers over a propagation distance of 1 m or ~30 femtoseconds in terms of arrival time. The effect is well-understood from a geometric argument and from a rigorous calculation of the harmonic average of the group velocity, and hence, is not anomalous at all. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves.
In this work, the paraxial approximation of the free Dirac equation is examined. The results are first obtained by constructing superpositions of exact solutions with suitable profiles, which are borrowed from paraxial optics. In this manner, the paraxial Dirac beams are obtained in four cases: as Gaussian, Bessel-Gaussian, modified Bessel-Gaussian and elegant Laguerre-Gaussian beams. In the second part of the work, the paraxial Dirac equation is derived, and then its solutions in the aforementioned cases are directly obtained. All the resulting wave functions conform to those derived formerly by virtue of superpositions, except for terms that are negligible upon the assumption that the paraxial functions along the propagation axis vary only slightly over a distance equal to the de Broglie wavelength, which is the standard paraxial requirement.
The spreading of a one-dimensional wavepacket solution of Schrödinger’s equation is related to the diffraction of light, as can be verified by considering the three-dimensional spreading of a wavepacket for an arbitrary dispersion relation. This investigation uncovers a special property of Schrödinger’s equation for a free particle: A wavepacket with initial spherical symmetry will preserve this symmetry in all Galilean reference frames. This property leads to a derivation of de Broglie’s postulate that wave number is proportional to momentum (or velocity). The application to non-Gaussian wavepackets and to Fraunhoffer diffraction also is discussed.