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arXiv:2602.17576

Subluminal and superluminal velocities of free-space photons

Konstantin Y. Bliokh

quant-phphysics.opticsSubmitted Feb 19, 2026• Announced Feb 20, 2026• Revised May 9, 2026 (v3)
v1 · Feb 19, 2026v2 · Mar 5, 2026v3 · May 9, 2026
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Abstract

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 $c^2$. 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'.

Journal reference
Low Temp. Phys. 52, 1048 (2026)
DOI
10.1063/10.0044812
Comments
7 pages, 3 figures, to appear in Low Temp. Phys

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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'.
It is shown that there are no dispersing media with «subluminal only» or «superluminal only» real group velocity – if in some medium at some frequencies the real group velocity is subluminal, then in the same medium there are necessarily frequencies at which the real group velocity is superluminal. Conversely, if a medium has frequencies with a superluminal group velocity, then it necessarily has frequencies with a subluminal group velocity. In other words, there is a necessary connection between subluminal and superluminal group velocity – if any medium is capable of generating subluminal group velocity, then the same medium is capable of generating superluminal group velocity. Similarly, there are no media with «only positive» or «only negative» imaginary part of the complex group velocity – if at some frequencies the imaginary part of the group velocity is positive, then at other frequencies in the same medium it is negative and vice versa. This means that if there is a carrier frequency in the medium at which some linearly frequency-modulated signal propagates at a speed lower than the group speed, then in the same medium there is necessarily a carrier frequency at which the same signal will propagate at a speed higher than the group speed and vice versa.
We discuss the propagation of electromagnetic waves in a space-time modulated medium, for which neither energy nor momentum remains conserved. We have found a conservation law when modulation moves at a constant speed vvv in a traveling-wave fashion. We use this law to study how light is scattered inside such a medium. We consider both the subluminal (v<cv<cv<c) and superluminal (v>cv>cv>c) cases and find that they differ considerably in terms of what remains conserved. We show that the total number of photons is conserved in the subluminal case. In contrast, it is the difference between the forward- and backward-moving photons that remains conserved in the superluminal case. We also study the reciprocity issue for a space-time modulated medium and find that reciprocity holds in all situations. We develop a quantum formulation of our scattering problem and show that the subluminal case is similar to the action of a beam splitter. In the superluminal case, photon pairs can be generated from vacuum in a fashion analogous to the two-mode frequency down-conversion process.
For Klein?Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability Psuper, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. Psuper?depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein?Gordon waves and 1?1/ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for Psuper.
In this paper, the quantum dynamics was obtained in the framework of the general theory of relativity, where a quantum particle is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spatial coordinates and of the momentum, called wave functions. For a free particle, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time-dependent phases proportional to the relativistic Lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the particle wave functions, Lorentz’s force and the Maxwell equations were obtained. From the wave/group equation in the momentum space describing the Lorentz force, the expressions of the electric and magnetic fields as functions of the electric potential conjugated to time and of the vector potential conjugated to the coordinates in the particle-field Lagrangian were obtained. With these expressions, the electric and magnetic fields that satisfy the Faraday-Maxwell law of electromagnetic induction and the two Gauss-Maxwell laws of these fields were obtained. The Ampère-Maxwell law is obtained only by taking into account the physical consistency of the matter-field interaction of the equality of the propagation field velocity with the maximum relativistic velocity c. For a quantum particle in the electromagnetic field, dynamic equations in the coordinate and momentum spaces and the particle and antiparticle wave functions were obtained. It was shown that the electromagnetic potentials as functions of the coordinates describing the matter distribution of the quantum particle do not alter this distribution – under the action of an electromagnetic a quantum particle moves as a whole. The scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion, were obtained. This description of a quantum particle as a distribution of matter with a density amplitude/wavefunction of the form of a wave packet, with the time-dependent phase proportional to the relativistic Lagrangian as a function of the metric tensor including also the gravitational field, enables the application of this theory in quantum gravity and quantum field theory in agreement with general relativity.
We show that the group velocity of a weak pulse can be manipulated by controlling the phases of two weak optical fields applied to a V-shaped three-level system. Such control can even cause the probe propagation to change from subluminal to superluminal. We consider two schemes: in the first, the excited states are coupled by decay-induced coherence, which is an inherent property of the medium, and in the second, quantum coherence is created by coupling the excited states to each other by a strong microwave field. We also discuss the group velocity reduction experienced by a single weak propagating probe due to decay-induced coherence.