# Unitary Transformation of an Interfering Beam Splitter

I was reading this research paper Quantum interference enables constant time quantum information processing and was confused by one particular expression involving the Hamiltonian of a beam splitter.

We consider two interfering modes $$a$$ and $$b$$ on a beam splitter device. The hamiltonian of the beam splitter is then

$$\hat{H}_{B S}=\frac{i \hbar}{2}\left(\hat{a}^{\dagger} \hat{b} e^{-i \varphi}-\hat{a} \hat{b}^{\dagger} e^{i \varphi}\right)$$

where $$\hat{a}$$, and $$\hat b$$ are respective annihilation operators of states $$a$$ and $$b$$.

Then the unitary transformation generated by the beam splitter is given by $$U_{bs}=e^{-i \theta H_{B S} / \hbar}$$ where $$\sin\left(\frac{\theta}{2}\right)=\sqrt{r}$$ (the reflection coefficient).

My question given this how is it possible to derive the expression given on page 32 for the annihilation operatorrs for output states $$a_{r}$$ and $$a_{t}$$

$$\begin{array}{l}{a_{r}=U_{B S}^{\dagger} a U_{B S}=a \cos \frac{\theta}{2}+b e^{-i \varphi} \sin \frac{\theta}{2}} \\ {a_{t}=U_{B S}^{\dagger} b U_{B S}=-a e^{i \varphi} \sin \frac{\theta}{2}+b \cos \frac{\theta}{2}}\end{array}$$

How can we derive this result using only the information given above? I tried evaluating the matrix exponential in Mathematica for some low dimensional cases and got several contradictions (mostly sign errors) which just made me more confused.

1. Write the evolution equation in the Heisenberg picture for $$a$$ (and the same for $$b$$): $$\dot a = i[H,a]\ .$$
2. Simplify the expression for the given Hamiltonian (omitting $$\hbar$$): \begin{aligned} \dot a &= -\tfrac12[a^\dagger b e^{-i\phi},a]+\tfrac12[ab^\dagger e^{i\phi},a] \\ & = -\tfrac12[a^\dagger,a] b e^{-i\phi} \\ & = \tfrac12 b e^{-i\phi}\ . \end{aligned}
3. Do the same for $$b$$: $$\dot b = -\tfrac12 a e^{i\phi}\ .$$
4. Integrate the two coupled differential equations $$\begin{pmatrix}\dot a\\ \dot b\end{pmatrix} = \frac12\begin{pmatrix} & e^{-i\phi}\\-e^{i\phi} \end{pmatrix} \begin{pmatrix} a\\ b\end{pmatrix}\ .$$