The Momentum Paradox, Revisited

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A research team calculates that a light pulse passing through a nondispersive medium gives rise to a mass density wave that propagates with the light. The picture offers a potential new approach for resolving a long-standing paradox in physics. [Image: Jyrki Hokkanen / CSC - IT Center for Science]

For more 100 years, physicists have argued about a seemingly straightforward quantity: the momentum of a light pulse propagating through a dielectric medium. The controversy rests on two simple yet apparently irreconcilable equations for that quantity, proposed early in the 20th century. And while a possible resolution to the paradox seven years ago gained considerable attention, debate has continued in the physics community since then.

Now, a group of scientists from Finland and Denmark have put another possible solution on the table. The difference between the two momentum expressions, they argue, stems from a “photon mass drag” effect, in which a light pulse transfers a mass component to the surrounding medium that’s driven forward in a sort of optoelestic shock wave (Phys. Rev. A, doi: 10.1103/PhysRevA.95.063850). Thus far the researchers have explored these ideas only computationally. But they also propose an experimental test, based on a titanium-sapphire pulsed laser and a waveguide—and suggest that, if the theory passes that test, it could represent “a fundamental change in our understanding of light propagation in a medium.”

A century of debate

At issue is the so-called Abraham-Minkowski dilemma. In the early 20th century, in the wake of Max Planck’s initial articulation of quantum theory, two possible candidates for the momentum of a photon propagating through a medium emerged. One, proposed by Hermann Minkowski in 1908, held that the photon’s momentum was given by pM = nω/c (where n is the material’s refractive index, is Planck’s constant, ω is the photon’s angular frequency and c the speed of light in vacuum). A year later, Max Abraham came up with a different expression for the photon’s momentum: pA = ω/nc.

Needless to say, for any value of n other than 1, both expressions can’t be correct. Yet over the years physicists have found evidence, both from first-principles arguments and from experiment, for each expression in different contexts. And some have proposed ways that the apparent paradox might be resolved.

A particularly interesting and much-discussed recent example came from Stephen Barnett, now at the University of Glasgow, U.K. (Phys. Rev. Lett., doi: 10.1103/PhysRevLett.104.070401). Barnett argued that both the Abraham and Minkowski formulations are correct—but that they express two different kinds of momentum. The Abraham form expresses the pure kinetic momentum of the photon impinging on the material, and the Minkowski form expresses the so-called canonical momentum, tied to dipole interactions in the medium.

Coupled dynamics

Now, researchers led by Jukka Tulkki of Aalto University have proposed an alternative take on this century-old issue. The Tulkki-led team holds that previous attempts to crack the problem, including Barnett’s, have not completely accounted for the dynamics of the photon-matter interaction, and the resulting kinetic and elastic energies, as the photon propagates through the medium. Doing so, the researchers suggest, offers a different kind of resolution of the Abraham-Minkowski paradox.

The reserchers tested out these ideas through two different computer simulations. In one, they looked at the light-matter system quantum mechanically, tracking the pulse of coupled light and matter passing through a nondispersive, transparent medium (specifically diamond) as a quasiparticle, or “mass polariton” (MP), and modeling the MP’s behavior using conservation laws and special relativity. In the other, they took a semiclassical, continuum-dynamics approach, calculating the effect of the passing photon on the material’s atoms using Newton’s equations of motion, and taking into account elastic forces and strain and recoil energy as atoms are displaced by the photon’s energy.

An optoelastic “shock wave”

The simulations, according to the team, suggest a model in which a light pulse, passing through a nondispersive medium, transfers a component of mass to the medium. That mass component is carried along in the photon’s wake in the form of a mass density wave (MDW)—a sort of optoelastic shock wave that moves through the medium at the speed of light. Under this view, the Minkowski momentum is the total momentum of the system, the Abraham momentum is the portion of the momentum carried by the light field, and the difference between the two is carried by the MDW.

The possibility of this atomic MDW in photonic materials, according to the researchers, raises the prospect of building “novel photonic components” based on the wave action. And, while the experiments have thus far taken place only in a computer, the MDW effect gives rise to displacements that, in the context of a confined zone such as a waveguide, could be used to verify the simulations and add weight to the MDW theory. (Interestingly, the theory also predicts another potentially measurable parameter—a sound wave that would be generated after the photon passes, as the optoelastic wave relaxes at acoustic-wave speeds.)

Cosmological implications?

In an e-mail communication with OPN, Tulkki and lead author Mikko Partanen noted that, while the team’s numerical experiments considered light passing through a hypothetical diamond block, the theory is “very general” and could apply to virtually any nondispersive material. A press release accompanying the paper suggested that the photon mass drag effect might even have relevance in the gases that make up the dilute interstellar medium—and that it gives cosmological-redshift predictions qualitatively similar to those of Hubble’s Law.

That law, of course, is usually cited as foundational evidence that the universe is expanding—which in turn suggests that the findings of Tulkki’s team, providing an alternative possible explanation for the redshift, conceivably might have cosmic implications. But don’t throw out the expanding-universe concept quite yet. In an e-mail to OPN, the research team stressed that “detailed computer simulations are needed to find out if the theory can really reproduce the experimentally observed redshift” underpinning Hubble’s law. And, they add, there are other lines of evidence besides Hubble’s Law that also point to an expanding universe.

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