**It can**, it's just hard as you have to engineer it. <br>
See, for instance, [here][1].

A Bose-Einstein Condensate is more than just a state described by a macroscopic wavefunction. It's the result of a *phase transition*. Indeed, it's the phase transition that defines the BEC phase, as the breaking of the $U(1)$ symmetry is what guarantees the choice of a specific phase and all that. In this context, Bose-Einstein condensation is different from "Bose-Einstein proliferation", i.e. just a lot of bosons in the ground state.

So you have to show that your system can go from a thermal state to a BEC as you decrease the temperature (or increase particle number). 

### Crash course in Bose-Einstein condensation

A BEC is essentially a **saturation** effect; the occupancy of bosons $$f(E) = \frac{1}{\mathrm{e}^\frac{E-\mu}{k_{\mathrm{B}}T}-1}$$
*has* to be positive, which means that $E-\mu \geqslant 0 \quad \forall E$. So you if fix $\mu$ and choose your ground energy to be $E_0 = 0$ (hence $\mu \leqslant 0$), ***any*** energy level is **capped** at $\mathrm{max}[f(E)] = (\mathrm{e}^\frac{E}{k_{\mathrm{B}}T}-1)^{-1}$.

Because the occupancy is capped, energy levels may run out of places to accommodate particles. Decreasing the temperature $T$ helps bringing this cap sooner. So as you decrease the temperature $T$, the $n^{\mathrm{th}}$ energy level doesn't have any free spots any longer. Etc. At some point (critical temperature $T_{\mathrm{c}}$), all excited states ($E>0$) are *full*. <br> **If particle number is conserved**, then particles cannot disappear. They have to go somewhere. Indeed, they go to the only state with infinite acceptance, i.e. the ground state with $E_0 = 0$ and hence $f(E_0) \rightarrow \infty$. It is this saturation that triggers the macroscopic occupation of the ground state. 

### Ok so what about the chemical potential?

Essentially, the chemical potential $\mu$ is the change in Helmholtz free energy $F = U-TS$ when a particle is added to the system. Adding a particle at a particular temperature increases the internal energy $U$, but this extra particle results in many more possible arrangements of the particles in the system, which in turn increases the entropy $S$. In the thermal phase, the entropy change is larger than the energy term, hence the chemical potential is negative $\mu < 0$. This agrees with what found above from the mathematical requirement of $f(E) > 0$.

When you hit condensation, then new particles can *only* be allocated in the ground state. Which has energy zero, so $U=0$. The certainty of the state where it ends up in, also, means that entropy does not increase$^\dagger$. Hence $\mu = 0$ only for $T \leqslant T_{\mathrm{c}}$.

### Photons

For (**free**) photons, $\mu = 0$ *always*. It is not a function of temperature. It does not entail any interesting dynamics. 

The Planck's distribution, indeed, tends to zero for $T \rightarrow 0$. Which is the same thing, really: photons just "vanish". Objects do not radiate as much when cold.

So how do you **make** a gas photons undergo Bose-Einstein condensation? <br> You *force* them to adpot $\mu \neq 0$. For example by placing them in a cavity, where different modes interact via a dye -- as done in the reference in the first line.

### Addenda:

1) Is a laser a BEC? <br>
No. While both a laser and a BEC are coherent states, the latter is an *equilibrium* state of matter while the former is a "steady-state" -- meaning pumping and stimulated emission are balanced but both need to be non-zero. And pumping is external.

2) In your eq. (1) you forgot an essential part; the density of states $g(E)$. Its functional dependence with dimensionality $d$ is what determines which trap geometries can have a BEC.

----------

$^\dagger$: Indeed, the BEC is a coherent state with zero entropy. Experimentally reaching a BEC therefore not only requires loss of energy via cooling, but also, more importantly, the removal of entropy. This dictates which cooling mechanisms are useful and which are not (e.g. adiabatic relaxations).

  [1]: https://www.nature.com/articles/nature09567