  The center of mass motion of trapped ions and neutral atoms is suitable for
approximation by a time-dependent driven quantum harmonic oscillator whose
frequency and driving strength may be controlled with high precision. We show
the time evolution for these systems with continuous differentiable
time-dependent parameters in terms of the three basic operations provided by
its underlying symmetry, rotation, displacement, and squeezing, using a Lie
algebraic approach. Our factorization of the dynamics allows for the intuitive
construction of protocols for state engineering, for example, creating and
removing displacement and squeezing, as well as their combinations, optimizing
squeezing, or more complex protocols that work for slow and fast rates of
change in the oscillator parameters.
