  We show that the motion of a cold trapped ion can be squeezed by modulating
the intensity of a phase-stable optical lattice placed inside the trap. As this
method is reversible and state selective it effectively implements a
controlled-squeeze gate. We show how to use this resource, that can be useful
for quantum information processing with continuous variables, in order to
prepare coherent superpositions of states which are squeezed along
complementary quadratures. We show that these states, which we denote as
"${\mathcal X}$-states", exhibit high sensitivity to small displacements along
two complementary quadratures which make them useful for quantum metrology.
