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Photons are massless particles. However, this article states that photons can gain mass when they become entangled. How can this happen?

From the article:

Physicists create new form of light

Photons, the elementary particles that make up light, are known to be fast, weightless and to not interact with each other. But in new experiments, physicists at MIT and Harvard have now created a new form of light, demonstrating that groups of photons can be made to interact with each other, slow down and gain mass.

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    $\begingroup$ Can you cite a source that says photons gain mass when they become entangled? Seeing the context might help people decipher the intended meaning. $\endgroup$ – Dan Yand Jan 12 at 16:45
  • $\begingroup$ Related: physics.stackexchange.com/questions/273859/… Also see the linked duplicate target question. $\endgroup$ – PM 2Ring Jan 12 at 17:25
  • $\begingroup$ @PM2Ring No, I think it is unrelated... Those links deal with the Higgs mechanism operative in a superconductor, a very different setup indeed. Here, the mass is more like an attention-seeking metaphor. $\endgroup$ – Cosmas Zachos Jan 12 at 20:37
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    $\begingroup$ This one might be remotely related, as it addresses quasiparticle velocity issues. $\endgroup$ – Cosmas Zachos Jan 12 at 20:54
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I will give you my "understanding" of how this does not contradict what is known about photons at the elementary particle level.

The publication here describes a complicated system of guiding photons through a special medium

We search for a photonic trimer using an ultracold atomic gas as a quantum nonlinear medium. This medium is experimentally realized by coupling photons to highly excited atomic Rydberg states by means of electromagnetically induced transparency (EIT)

So this is a system with which photons interact and display three photon correlations which induce a mass on the photon following the mathematical model that fits the data:

enter image description here

is the effective photon mass, $a$ is the scattering length, $Ω_c$ is the control laser Rabi frequency, and $∆$ is the one-photon detuning.

So it is within a complicated mathematical model that the photons acquire mass.

To get a perspective on this I think of the $π^0$, it decays into two photons with an angle between them , and this is defined by the mass of the pion. The four vectors of the two photons added will still have the mass of the $π^0$ , the momentum and energy .If one timed the photons, it would seem that they moved with a velocity less than c, but in effect they traverse a longer path than the $π^0$ path, because of the angle between the two photons. This helps me realize that sums of four vectors are not intuitive. The four vectors involved in these photonic states are more complicated, as they are defined by interactions with a system of particles ( the cold gas). This ties up with this popularized explanation:

So why are the normally lone-ranging photons suddenly interacting with each other? The team's hypothesis is that as photons bump into the rubidium atoms, they form polaritons – quantum particles that are part-light and part-matter. Polaritons have mass, which is how they can bind to other polaritons. Once they leave the cloud, the atoms they've picked up stay behind, but the photons remain bound together.

They are not bound, they are correlated, the way the two photons of the $π^0$ are correlated.

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This newatlas.com article is a very bad article, because it says things like:

The researchers also measured the frequency of the photons' oscillation, known as their phase.

Which is completely wrong on a very basic conceptual level. Frequency is frequency and phase is phase. Sigh.

Photons can have effective mass (via Einstein's $E=\sqrt{(mc^2)^2+(pc)^2})$ if they are effectlively slowed down by being bounced around. See the answer provided by anna v for an exact explanation.

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