  We demonstrate control of a trapped-ion quantum harmonic oscillator in a
squeezed Fock state basis, using engineered Hamiltonians analogous to the
Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for
squeezed Fock states with low $n$ the engineered Hamiltonians reproduce the
$\sqrt{n}$ scaling of the matrix elements which is typical of Jaynes-Cummings
physics, and also examine deviations due to the finite wavelength of our
control fields. Starting from a squeezed vacuum state, we apply sequences of
alternating transfer pulses which allow us to climb the squeezed Fock state
ladder, creating states up to excitations of $n = 6$ with up to 8.7 dB of
squeezing, as well as demonstrating superpositions of these states. These
techniques offer access to new sets of states of the harmonic oscillator which
may be applicable for precision metrology or quantum information science.
