Hillery-type squeezing in fan-states

We study the Hillery-type, i.e. $N$-th power, amplitude squeezing in the fan-state $| \xi ;2k,f>_{F}$ characterized by $\xi \in \mathcal{C}$, $k=1,2,3,...$ and $f$ a nonlinear operator-valued function. We show that for a given $k$ there exists a critical $\xi_c$ such that for $0<|\xi|\leq|\xi_c|$ squeezing occurs simultaneously in $2k$ directions for the powers $N$ which are a multiple of $2k$. This result does not depend on the concrete form of $f$, i.e. it holds true for both $f\equiv 1$ and $f\neq 1$. However, for $f\neq 1$, which is realized here in the ion trap context, the squeezing directions as well as the magnitude of $\xi$ can be controlled by adjusting the system driving parameters.