  We explore and develop the mathematics of the two multi-level ions. In
particular, we describe some new features of quantum entanglement in two
three-level trapped ions confined in a one-dimensional harmonic potential,
allowing the instantaneous position of the center-of-mass motion of the ions to
be explicitly time-dependent. By solving the exact dynamics of the system, we
show how survivability of the quantum entanglement is determined by a specific
choice of the initial state settings.
