Two Different Fock State Inputs into a Beam Splitter

[BeyondBelief96]

Hello everyone, I am an undergraduate doing research in quantum optics, and my topic involves 50:50 beam splitters and studying entanglement for different input states. I came across a paper which I am using as a guide for now, but I wanted to derive a result they had and have been working on it but have been stuck for quite some time. The paper I am referring to is this: https://arxiv.org/pdf/quant-ph/0106136.pdf

The section of which I am referring to in the paper is II: Fock State Input where they express the output as a superposition of two-mode fock states. What I am trying to determine is that coefficient in that expression B_(n_1,n_2)^(N_1,N_2). If you look at this link to my work so far, I have been able to make some decent progress and some of my terms match thiers but I can't seem to get what they have. If anyone has the time to look through my work and compare and maybe try to offer some insight that would be great. Also in using the Baker-Campbell-Hausdorff Formula, from what I've read, you can only decompose e^{A +B} = e^A*e^B*e^([A,B])/2 if [A,B] is a scalar? If that is correct, then I may have decomposed the beam splitter operator wrong. Or my commutation relation is incorrect. Anyways, any help would be great, here is the link to my work thus far:

LINK TO MY WORK:

 

[PeterDonis]

If you look at this link to my work

We can't accept links to work posted elsewhere. We need to be able to quote parts of your work in response. That means you need to use the PF LaTeX feature to post your work, or at least the part of it that you are asking questions about, including equations, here directly.