Can you create an actual coherent state? If I understood correctly, a coherent state $\lvert\alpha\rangle $ is an eigenstate of the creation and annihilation operator, meaning that adding or removing a particles does not change it. Mathematically, this is constructed by using a superposition of all possible numbers of particles.
$$\lvert\alpha\rangle = \sum\limits_{n=0}^\infty c_n \lvert n\rangle $$
Is it possible to actually create a real system in such a state? If it were, would you be able to add take out any number of particles to from that system without ever changing it? (That sounds like an infinite energy source.)
Or is this just an abstract mathematical concept?
 A: In a laser lab setting, it's very difficult to get individual photons, and laser light is typically coherent state light, unless you want to prepare it.
An answer to a similar question outlines how to a coherent state can be created by stimulated emission. Basically, we know from introductory physics that a laser works by having photons create more photons in a cascading reaction. But each time the photon has a probability of creating stimulated emission, it also has a probability of not stimulating (and then this superposition traverses the system again in a loop). The quantum superposition of all the possibilities continue to add up and form an infinite series. 
A: There is a general principle by which coherent states can be manufactured: All types of coherent states can be produced by coupling of the system under consideration to a classical (C-number) source. 
For the electromagnetic radiation, this principle was already explained by Glauber in his original work  on coherent states (equations 9.16-9.21). 
A more transparent derivation is given by Zhang (section $3$): When a free electromagnetic field originally at the vacuum state $|0\rangle$(no photons) is adiabatically coupled to a classical current $j^{\mu}$:
$$\mathcal{L} = - \frac{1}{4} F_{\mu\nu}F^{\mu\nu} - A_{\mu}j^{\mu},$$
its state after the current application becomes:
$$|0\rangle_{\mathrm{out}} = e^{-\frac{1}{2}\int d^3k\sum_{\lambda}|z_k^{\lambda}|^2}
e^{\int d^3k\sum_{\lambda}z_k^{\lambda} a_k^{\lambda \dagger}}|0\rangle,$$
where: $z_k^{\lambda} = \epsilon^{\lambda}_{\mu}(k) j^{\mu}$ and $\epsilon^{\lambda}_{\mu}$ is the polarization vector.
Another example is for spin coherent states which can be manufactured by adiabatically applying a magnetic field to a spin originally at the highest weight state $|0\rangle = |j, j\rangle$.
The equations of motion
$$\frac{dS_i}{dt} = \frac{i}{\hbar} \mu \epsilon_{ijk}B_j(t) S_k$$
($\mu$ is the magnetic moment). In this case, the system will reach a state $|0\rangle_{\mathrm{out}} $ given by:
$$|0\rangle_{\mathrm{out}}  = T e^{\frac{i}{\hbar}  \mu \int dt \mathbf{B}(t) \cdot \mathbf{S} }|0\rangle $$
($T$ denotes time ordering). This is a spin coherent state, because it is obtained by the action of a group element on the vacuum state.
