I would like to recalculate Eq.(2.4) in PRA, 31,4,(1985), which expresses the exponential of operators as a normal ordering form. This equation reads
\begin{equation} D=e^{\alpha K_{+} - \alpha^{*} K_{-}} = e^{\xi K_{+}} e^{\gamma K_{0}} e^{- \xi^{*} K_{-}}, \end{equation}
where $\alpha =-\tau e^{-i \phi}/2$, $\gamma= \ln (1-|\xi|^2)$ and $\xi = - \tanh(\tau/2) e^{-i \phi}$. Here $K_{0}, K_{+}, K_{-}$ are the generators of a SU(1,1) Lie group which satisfy $[K_{0}, K_{\pm}] = \pm K_{\pm}$ and $[K_{-},K_{+}]=2 K_{0}$.
In my initial attempt, I used the Zassenhaus formula to derive the above relation. My calculations read
\begin{equation} e^{\alpha K_{+} - \alpha^{*} K_{-}} = e^{\alpha K_{+}} e^{-\alpha^{*} K_{-}} e^{-|\alpha|^{2} K_{0}} e^{[- \alpha^{*} |\alpha|^2 K_{+} + \alpha |\alpha|^2 K_{-}]/6} e^{-|\alpha|^4 K_{0}/2} + \ldots. \end{equation}
But I couldn't derive the expressions for $\gamma, \xi$ given in the paper so far. Could you spot the possible typo that I've made?
