I am interested in explicitly doing the math for what is decribed in this answer.
Specifically, I am interested in the simplest model for the creation of a coherent state from stimulated emission in a laser.
As @pyramids describes, a photon can either cause stimulated emission in a gain medium or not (the "boring case"). Therefore a photon will be in a superposition of the interesting case and the boring case, and as the photon goes through the medium (or loops in a cavity), this process is repeated over and over, creating a superposition of (near) infinite possible photon numbers.
I could swear that I worked this out by hand a few years ago, but I can't find my notes anywhere and I've been having some trouble reproducing the math.
My attempt:
my photon has a chance of creating a second photon with chance p, and has the "boring case" of no interacting with probability 1-p
$|1\rangle \rightarrow (1-p)|1\rangle + p|2\rangle$
which for now we will say p is small enough that we can replace (1-p) with 1.
$|1\rangle \rightarrow |1\rangle + p|2\rangle$
and using the second quantization notation:
$a^{\dagger^2}|0\rangle = \sqrt{2}|2\rangle \implies |2\rangle = \frac{1}{\sqrt{2}}a^{\dagger^2}|0\rangle $
we get that this:
$a^\dagger|0\rangle \rightarrow a^\dagger|0\rangle + p\frac{1}{\sqrt{2}} a^{\dagger^2}|0\rangle $
which I'm going to interpret as a rule that:
$a^\dagger \rightarrow a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2} $
which means that in the second step, this interaction is going to happen again:
$\begin{align}a^\dagger &\xrightarrow{1} a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2} \\ &\xrightarrow{2} \Big(a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2}\Big) + p\frac{1}{\sqrt{2}} \Big( a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2} \Big) \Big( a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2} \Big) \\ &= \Big(a^\dagger + p\frac{1}{\sqrt{2}} a^{\dagger^2}\Big) + p\frac{1}{\sqrt{2}} \Big( a^{\dagger^2} + 2p\frac{1}{\sqrt{2}} a^{\dagger^3} + 4p^2\frac{1}{2} a^{\dagger^4} \Big) \\ &= a^\dagger + 2 p\frac{1}{\sqrt{2}} a^{\dagger^2} + p^2 a^{\dagger^3} + 4p^3\frac{1}{2\sqrt{2}} a^{\dagger^4} \end{align} $
so I can keep repeating this step over-and-over and write an infinite series, but it doesn't seem like it is approaching the coefficients of a coherent state.
