# Is BE condensation possible for photons, phonons and magnons all with $\mu=0$ (non-conserved particle number)?

Photons have zero chemical potential and their number is not conserved. The property of zero chemical potential is also true for emergent gapless excitations such as phonons in crystals and magnons in ferromagnets. They do not obey conservation of particle number constraint. Hence, it may not be true that with the decrease in temperature, the particles occupying the excited states deplete the excited states and fall into the ground state. Particles are allowed to disappear from the system.

Is it possible for such particles to Bose condense? I mean is there a theoretical problem for photons, phonons and magnons to Bose condense?

Disclaimer: I'm no expert, but here's an answer as I understand these states.

## 1. Can massless bosonic particles such as photons, magnons, phonons, etc. with zero chemical potential Bose condense?

The short answer is no, they cannot. A BEC is characterized by an extensive population of the groundstate, and is in a confined region. Massless particles described by the Bose-Einstein distribution with $\mu=0$ will always have a vanishing occupation of the groundstate ($E\rightarrow 0$). See the answers to Why zero chemical potential does not allow the Bose-Einstein Condensation of Phonons? for more details.

## 2. Can particles such as photons, magnons, phonons, etc. Bose condense?

Yes, they can, just as the answers to the question Can a system entirely of photons be a Bose-Einsten condensate? suggest for photons. For completeness, here are some relevant experimental references:

## 3. Do we have a theoretical problem?

Points 1. and 2. are seemingly in disagreement, so perhaps we have. However, the $\mu=0$ condition is derived in equilibrium. By working away from equilibrium, the requirement is sidestepped. In particular, one can engineer a quasi-equilibrium with conserved particle number - and hence non-zero chemical potential, making Bose condensation possible. This relies on a balance of lifetimes of the particles and the rate at which new particles are pumped into the system, which is a technical challenge but demonstrably possible.