  We propose an ideal scheme for preparing vibrational $\mathrm{SU(1,1)}
\otimes \mathrm{SU(1,1)}$ states in a two-dimensional ion trap using red and
blue second sideband resolved driving of two orthogonal vibrational modes.
Symmetric and asymmetric driving provide two regimes to realize quantum state
engineering of the vibrational modes. In one regime, we show that time
evolution synthesizes so-called $\mathrm{SU}(1,1)$ Perelomov coherent states,
that is separable squeezed states and their superposition too. The other regime
allows engineering of lossless 50/50 $\mathrm{SU}(2)$ beam splitter states that
are entangled states. These ideal dynamics are reversible, thus, the
non-classical and entangled states produced by our schemes might be used as
resources for interferometry.

Quantum Physics