Beam splitter with complex parameter

Question

A non-polarizing beam splitter can usually be described by a unitary operator such as $U=e^{i\theta(a^\dagger b+b^\dagger a)}$ given a parameter $\theta\in \mathbb R$ and a pair of independent modes $a,b$.

I am interested in the more general operator $U=e^{i(\theta a^\dagger b + \theta^* b^\dagger a)}$, where this time $\theta\in \mathbb C$. What is the physical meaning of this operator? Does it always represent a beam splitter, and if so, is there any difference between this and the real case beside a change in the transmittance and reflectivity functions?

Answer 1

It means that there are phases associated with the input and output ports. If you define $\theta=R\exp(i\phi)$ where $R$ is real. Then you can get rid of the phase by absorbing it into the ladder operators. For example, you can replace $a\rightarrow a\exp(i\phi)$ and $a^{\dagger}\rightarrow a^{\dagger}\exp(-i\phi)$ and end up with the original form of the beamsplitter operator.