  We derive a general Hamiltonian that governs the interaction between an
$N$-ion chain and an externally controlled laser field, where the ion motion is
quantized and the laser field is considered beyond the plane-wave
approximation. This general form not only explicitly includes terms that are
used to drive ion-ion entanglement, but also a series of unwanted terms that
can lead to quantum gate infidelity. We demonstrate the power of our
expressivity of the general Hamiltonian by singling out the effect of axial
mode heating and confirm this experimentally. We discuss pathways forward in
furthering the trapped-ion quantum computational quality, guiding hardware
design decisions.
