Subluminal and superluminal velocities of free-space photons

Here is an explanation of the paper using simple language and creative analogies.

The Big Idea: Light Isn't Always a Straight Arrow

For a long time, we've been taught that light travels at a constant, unbreakable speed limit: $c$ (about 300,000 kilometers per second). If you shine a laser pointer, the beam zips across the room at this exact speed.

However, this paper by Konstantin Bliokh reveals a fascinating twist: When light is "squeezed" or focused into a tight beam, it actually behaves like a crowd of runners rather than a single sprinter.

In this scenario, light exhibits two different speeds simultaneously:

The "Energy" Speed (Group Velocity): This is how fast the bulk of the light packet moves. It turns out to be slower than the speed limit ( $v < c$ ).
The "Wave" Speed (Phase Velocity): This is how fast the individual ripples (the peaks and valleys) inside the light move. These ripples actually move faster than the speed limit ( $v > c$ ).

Here is the magic rule the paper proves: If you multiply the slow speed by the fast speed, you get exactly the square of the speed of light ( $v_{slow} \times v_{fast} = c^2$ ).

Analogy 1: The Marching Band (Why it gets slower)

Imagine a marching band trying to march in a perfectly straight line down a narrow hallway.

The Problem: To stay in the hallway, the musicians on the edges have to angle their steps slightly inward to avoid hitting the walls.
The Result: Even though every musician is marching at full speed, their forward progress is slower because some of their energy is being used to move sideways (to stay in the line).
The Light Connection: A "free-space" photon (like a laser beam) isn't a single, infinitely wide sheet of light. It's a focused packet. To stay focused, the light waves inside it must travel at slight angles, bouncing back and forth between the edges of the beam. Because they are zig-zagging slightly, their average forward speed (the energy speed) is slightly slower than $c$ .

Analogy 2: The Surfboard Ripples (Why the waves get faster)

Now, imagine you are surfing on a wave.

The Wave: The water is moving forward.
The Ripples: Inside that big wave, there are tiny ripples.
The Trick: If the big wave is "squeezed" (narrow), the tiny ripples inside it have to stretch out to fit the geometry of the wave. Because the ripples are traveling at an angle (like the marching band), the distance between the peaks of the ripples (the wavelength) gets longer.
The Result: If the distance between peaks gets longer, but the frequency (how often they hit you) stays the same, the ripples appear to be racing ahead faster than the wave itself.
The Light Connection: In a focused light beam, the "ripples" (the phase) spread out more than they would in a wide, flat beam. This makes the phase velocity appear superluminal (faster than light).

Crucial Note: Nothing is actually breaking the speed limit. No information or energy is traveling faster than $c$ . The "fast" speed is just a mathematical illusion of how the wave peaks line up, while the "slow" speed is the real speed of the energy.

The Three Ways the Author Proved It

The author didn't just guess; he looked at this problem through three different "lenses" to make sure the math held up.

1. The Physics Lens (Energy and Momentum)

Think of light as a fluid flowing through a pipe.

The author used the laws of conservation (energy and momentum) to show that if the light is confined (squeezed), the "center of energy" must move slower than $c$ .
He also showed that the "center of momentum" (where the push is) moves faster than $c$ .
The Takeaway: Just like a seesaw, if one side goes down (slower), the other must go up (faster) to keep the balance ( $c^2$ ).

2. The Wave Lens (The Gaussian Beam)

The author did the math for a standard "Gaussian beam" (the shape of a typical laser pointer).

He calculated exactly how much slower the beam gets based on how tight the focus is.
The Result: A tightly focused laser beam lags behind a perfect, wide beam. If you send a laser and a "perfect" beam side-by-side, the focused one will arrive a tiny fraction of a second later. This has been measured in real experiments!

3. The Quantum Lens (The "Photon" Confusion)

This is the most technical part. In quantum mechanics, we try to describe a photon as a "wavefunction" (a map of where the photon is likely to be).

The Trap: If you use the wrong mathematical map for a photon, you get confusing results that seem to break physics.
The Fix: The author clarified that we must distinguish between the "energy map" (where the light's power is) and the "probability map" (where the photon might be).
The Conclusion: When you use the correct definitions, the quantum math agrees perfectly with the classical wave math. The "slow" and "fast" speeds are real features of how light behaves, not a glitch in the math.

Why Does This Matter?

You might ask, "So what? It's just a tiny difference."

It's Fundamental: It proves that the speed of light ( $c$ ) is a limit for information, but the geometry of light beams creates natural variations in how the energy and the waves move.
It Explains Experiments: Scientists have been measuring "slow light" in free space for years and were puzzled. This paper explains why it happens without needing special materials or weird quantum tricks. It's just the nature of a focused beam.
Higher Order Modes: The paper also shows that if you twist the light (like a corkscrew beam, called a Laguerre-Gaussian beam), the effect gets even stronger. The more "twisted" or complex the beam, the slower the energy moves.

The Bottom Line

Light in a focused beam is like a marching band in a narrow hallway.

The band's forward progress (Energy) is slower than the speed limit because they are zig-zagging to stay in line.
The individual steps (Phase) look like they are faster than the speed limit because of how they are spaced out.
The two speeds balance each other out perfectly to keep the universe's speed limit ( $c$ ) intact.

This paper confirms that these "subluminal" (slower) and "superluminal" (faster) speeds are not paradoxes, but natural, predictable consequences of how waves behave when they are confined in space.

Here is a detailed technical summary of the paper "Subluminal and superluminal velocities of free-space photons" by Konstantin Y. Bliokh.

1. Problem Statement

The paper addresses the long-standing question of how electromagnetic wavepackets (and single photons) propagate in free space. While the speed of light in a vacuum ( $c$ ) is a fundamental constant, various definitions of "velocity" exist (phase, group, signal, energy-transfer).

The Core Paradox: It is often assumed that light in a vacuum travels strictly at $c$ . However, spatially localized wavepackets (such as Gaussian beams) inherently possess a transverse momentum spread due to diffraction.
The Question: How do the group velocity ( $v_g$ , associated with energy transport) and phase velocity ( $v_{ph}$ , associated with wavefronts) behave for these localized structures? Specifically, does the group velocity remain $c$ , or does it deviate? If it deviates, what is the relationship between $v_g$ and $v_{ph}$ ?

2. Methodology

The author employs three complementary theoretical frameworks to analyze rectilinear free-space propagation of electromagnetic wavepackets without engineered internal structures:

Electromagnetic Field Theory (Relativistic):
- Utilizes conservation laws derived from Noether's theorem (energy, momentum, and boost momentum).
- Analyzes the motion of the energy centroid ( $R_E$ ) and the momentum centroid ( $R_P$ ).
- Derives velocities based on the Poynting vector and energy density.
Classical Wavepacket Analysis (Scalar Approximation):
- Treats the wavepacket as a superposition of plane waves with a dispersion relation $\omega(k) = ck$ .
- Calculates the average group velocity as the weighted average of local group velocities in momentum space.
- Calculates the average phase velocity as the ratio of mean frequency to mean wavevector.
- Performs explicit calculations for Gaussian beams and Laguerre-Gaussian (LG) beams, utilizing paraxial approximations and exact solutions to the wave equation.
Quantum-Mechanical Formalism:
- Uses the Riemann–Silberstein (RS) vector $\mathbf{F} = (\mathbf{E} + i\mathbf{H})/\sqrt{2}$ as a "photon wavefunction" satisfying a Dirac-like equation for a massless particle.
- Investigates the Heisenberg equation of motion for the velocity operator.
- Critically examines the distinction between the energy centroid (derived from the RS vector intensity) and the probability centroid (derived from a proper photon wavefunction $\tilde{\psi} = \tilde{F}/\sqrt{\omega}$ ).

3. Key Contributions and Results

A. Fundamental Velocity Relationship

The paper demonstrates that for any spatially localized free-space wavepacket:

The group velocity (energy velocity) is subluminal ( $v_g < c$ ).
The phase velocity (momentum velocity) is superluminal ( $v_{ph} > c$ ).
Their product is strictly equal to the square of the speed of light:
$v_g v_{ph} = c^2$
This relationship holds regardless of the specific beam shape, provided it is spatially confined.

B. Physical Origin: Transverse Confinement

The deviation from $c$ is caused by the transverse confinement of the beam.

A localized beam requires a spread in transverse momentum ( $k_\perp$ ).
Consequently, the longitudinal momentum component ( $k_z = \sqrt{k^2 - k_\perp^2}$ ) is slightly less than the total wavenumber $k$ .
Group Velocity: The energy centroid moves at the average group velocity $\langle v_g \rangle \approx c(1 - \langle k_\perp^2 \rangle / 2k^2)$ .
Phase Velocity: The phase fronts move at $\langle v_{ph} \rangle \approx c(1 + \langle k_\perp^2 \rangle / 2k^2)$ .
The paper explicitly links this retardation to the Gouy phase shift and the phase curvature of the beam.

C. Quantitative Analysis for Specific Beams

Gaussian Beams: The paper derives that the retardation of the group velocity is proportional to $1/(k z_R) $, where$ z_R $is the Rayleigh range. A Gaussian beam accumulates a half-wavelength retardation after propagating approximately$ 6$ Rayleigh ranges.
Higher-Order Modes (Laguerre-Gaussian): For beams with orbital angular momentum (indices $\ell$ and radial index $p$ ), the effect is enhanced by a factor of $(N+1)$ , where $N = |\ell| + 2p$ .
$v_g \approx c \left( 1 - \frac{N+1}{2kz_R} \right)$
This confirms that higher-order modes propagate significantly slower than fundamental Gaussian modes.

D. Resolution of Quantum Mechanical Ambiguities

The paper clarifies a subtle issue in the quantum description of photons:

The Riemann–Silberstein vector $\mathbf{F}$ is often treated as a wavefunction, but its intensity $|\mathbf{F}|^2$ represents energy density, not probability density.
Using $\mathbf{F}$ directly yields the energy-centroid velocity (subluminal).
To obtain the probability centroid (true photon position), one must define the photon wavefunction in momentum space as $\tilde{\psi} = \tilde{F}/\sqrt{\omega}$ .
The author shows that while the definitions differ formally, for paraxial Gaussian wavepackets, the difference between the energy centroid and probability centroid is negligible (appearing only at higher orders), justifying the use of classical field theory for calculating group velocities in most practical scenarios.

4. Significance and Conclusion

Resolution of Paradox: The paper establishes that subluminal group velocity and superluminal phase velocity in free space are not paradoxical but are fundamental, internally consistent features of wave propagation. They arise naturally from the geometry of localized wavepackets.
Unification of Concepts: It successfully unifies concepts from relativistic field theory (boost momentum conservation), classical optics (diffraction and Gouy phase), and quantum mechanics (photon wavefunctions).
Experimental Relevance: The results provide a theoretical foundation for experimental observations of "slow light" in vacuum (e.g., structured photons traveling slower than $c$ ) and clarify the interpretation of "momentum velocity" introduced by Milton and Schwinger.
Practical Implication: The findings are crucial for precision metrology, ultrafast optics, and the interpretation of photon trajectories in quantum optics, particularly when dealing with tightly focused or high-order structured light.

In summary, Bliokh proves that the speed of a photon in free space is not a single fixed value $c$ but depends on the definition of velocity and the spatial structure of the wavepacket. The energy transport is always slower than $c$ due to diffraction, while the phase fronts move faster than $c$ to compensate, maintaining the invariant product $c^2$ .