Context:
Hello, I have to present this article in class next week. I encounter some difficulty with the supplementary material at the end of section I.
There, the authors say they have a state of the form : \begin{equation}\label{eq:1} \alpha |g⟩ [|10⟩ + |01⟩] + \alpha |e⟩ [|10⟩ − |01⟩] + β|g⟩ [|10⟩ − |01⟩] + β|e⟩ [|10⟩ + |01⟩] \end{equation} Then, by passing this state through a 50:50 Beam Splitter, they obtain : $$ α|g⟩|10⟩ + α|e⟩|01⟩ + β|g⟩|01⟩ + β|e⟩|10⟩ $$
What I tried:
However, I have difficulty seeing how to pass from the first state to the final one. In effect, I use the BS action : $$ a^\dagger \rightarrow \frac{1}{\sqrt{2}}(c^\dagger + id^\dagger) $$ $$ b^\dagger \rightarrow \frac{1}{\sqrt{2}}(ic^\dagger + d^\dagger) $$
Let's say I take only the first term of the first expression I wrote above, I can rewrite it as follows : $$ \alpha |g⟩ [|10⟩ + |01⟩] \equiv \alpha |g⟩ [(a^\dagger +b^\dagger)|0⟩_a|0⟩_b|0⟩_c|0⟩_d] $$ which transforms under the action of the BS as : $$ \alpha |g⟩ [\frac{1}{\sqrt{2}}(c^\dagger + id^\dagger + ic^\dagger + d^\dagger)|0⟩_a|0⟩_b|0⟩_c|0⟩_d] $$ This is quite different from what is expected from the final expression of the authors of the article, I should rather obtain something like : $$ \alpha |g⟩ [c^\dagger|0⟩_a|0⟩_b|0⟩_c|0⟩_d] $$ but I don't see how to get rid of the imaginaries in my former expression. What am I doing wrong ?
