The physical meaning of the action of the following operator I've been thinking about different realisations of the two-mode bosonic Lie algebras and faced the following misunderstanding.
It is possible to consider different unitary operators, such as the squeeze operator
$$
S(z) = \exp(z^* ab - z a^\dagger b^\dagger),
$$
which action on the Fock or on the coherent state is clear to me.
I'm interested in the result of the action of the following operator
$$
?(z) = \exp(z^* ab^\dagger \pm z a^\dagger b ),
$$
on the Fock state $|n,k\rangle$ or on the coherent state $|\alpha, \beta\rangle = e^{-|\alpha|^2/2 - |\beta|^2/2}\sum_{n,k} \frac{\alpha^n \beta^k}{\sqrt{n! k!}} | n,k\rangle$?
P.S. I'm looking for something like this (journals.aps.org/pra/abstract/10.1103/PhysRevA.49.4957) or (journals.aps.org/pra/abstract/10.1103/PhysRevA.43.3854), but for the mentioned operator $?()$.
 A: Well that one's easy.
Using
\begin{align}
L_+=\hat a^\dagger \hat b\, \quad 
L_-=\hat a \hat b^\dagger\, ,\quad 
L_z=\frac{1}{2}[L_+,L_-]
\end{align}
you operator $S(z)$ can in fact be written as
$$
\exp(i \alpha L_z)\exp(-i\beta L_y)\exp(-i \alpha L_z) \tag{1}
$$
where $\alpha$ and $\beta$ are related to $z$ and $z^*$.
First use
$z=\vert z\vert (\cos\theta+i\sin\theta)$ and then rewrite $z^*L_-+zL_+$ as a combination of $L_x$ and $L_y$, then use
$$
\exp(i \alpha L_z) L_y \exp(-i \alpha L_z)=
L_y\cos\alpha + L_x\sin\alpha
$$
to reach (1).
Next, using the Schwinger realization, the 2-mode state $\vert n_1,n_1\rangle$ maps to the angular momentum state $\vert j m\rangle$ where $j=\frac{1}{2}(n_1+n_2), m=\frac{1}{2}(n_1-n_2)$.
so it's then a simple matter to go from
\begin{align}
\exp(z^* ab^\dagger + z a^\dagger b)\vert n_1,n_2\rangle&\to 
R_z(\alpha)R_y(\beta)R_z(\alpha)\vert j m\rangle_{a,b}\, ,\\
&=\sum_{m'}\vert jm'\rangle D^{j}_{m'm}(\alpha,\beta,-\alpha)\\
&\to \sum_{a'b'}\vert a'b'\rangle D^{j}_{m'm}(\alpha,\beta,-\alpha)\, .
\end{align}
Thus, your operator unitarily scatters a state with a fixed number of excitations to other states with the same total number of excitations.  For applications see for instance the seminal work of

Yurke B, McCall SL, Klauder JR. SU (2) and SU (1, 1) interferometers. Physical Review A. 1986 Jun 1;33(6):4033.

