  We investigate the possibility to form high fidelity atomic Fock states by
gradual reduction of a quasi one dimensional trap containing spin polarized
fermions or strongly interacting bosons in the Tonk-Girardeau regime. Making
the trap shallower and simultaneously squeezing it can lead to the preparation
of an ideal atomic Fock state as one approaches either the sudden or the
adiabatic limits. Nonetheless, the fidelity of the resulting state is shown to
exhibit a non-monotonic behaviour with the time scale in which the trapping
potential is changed.
