  The high fidelity generation of strongly entangled states of many particles,
such as cat states, is a particularly demanding challenge. One approach is to
drive the system, within a certain final time, as adiabatically as possible, in
order to avoid the generation of unwanted excitations. However, excitations can
be generated also by the presence of dissipative effects such as dephasing.
Here we compare the effectiveness of Local Adiabatic and the FAst QUasi
ADiabatic protocols in achieving a high fidelity for a target superposition
state both with and without dephasing. In particular we consider trapped ions
set-ups in which each spin interacts with all the others with the uniform
coupling strength or with a power-law coupling. In order to mitigate the
effects of dephasing, we complement the adiabatic protocols with dynamical
decoupling and we test its effectiveness. The protocols we study could be
readily implemented with state-of-the-art techniques.
