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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 andblue second sideband resolved driving of two orthogonal vibrational modes.Symmetric and asymmetric driving provide two regimes to realize quantum stateengineering of the vibrational modes. In one regime, we show that timeevolution synthesizes so-called $\mathrm{SU}(1,1)$ Perelomov coherent states,that is separable squeezed states and their superposition too. The other regimeallows engineering of lossless 50/50 $\mathrm{SU}(2)$ beam splitter states thatare entangled states. These ideal dynamics are reversible, thus, thenon-classical and entangled states produced by our schemes might be used asresources for interferometry.

Engineering $\mathrm{SU}(1,1) \otimes \mathrm{SU}(1,1)$ vibrational states