  Understanding how and to what magnitude solid-state qubits couple to metallic
wires is crucial to the design of quantum systems such as quantum computers.
Here, we investigate the coupling between a multi-level system, or qudit, and a
superconducting (SC) resonator's electromagnetic field, focusing on the
interaction involving both the transition and diagonal dipole moments of the
qudit. Specifically, we explore the effective dynamical (time-dependent)
longitudinal coupling that arises when a solid-state qudit is adiabatically
modulated at small gate frequencies and amplitudes, in addition to a static
dispersive interaction with the SC resonator. For the first time, we derive
Hamiltonians describing the longitudinal multi-level interactions in a general
dispersive regime, encompassing both dynamical longitudinal and dispersive
interactions. These Hamiltonians smoothly transition between their adiabatic
values, where the couplings of the n-th level are proportional to the level's
energy curvature concerning a qudit gate voltage, and the substantially larger
dispersive values, which occur due to a resonant form factor. We provide
several examples illustrating the transition from adiabatic to dispersive
coupling in different qubit systems, including the charge (1e DQD) qubit, the
transmon, the double quantum dot singlet-triplet qubit, and the triple quantum
dot exchange-only qubit. In some of these qubits, higher energy levels play a
critical role, particularly when their qubit's dipole moment is minimal or
zero. For an experimentally relevant scenario involving a spin-charge qubit
with magnetic field gradient coupled capacitively to a SC resonator, we
showcase the potential of these interactions. They enable
close-to-quantum-limited quantum non-demolition (QND) measurements and remote
geometric phase gates, demonstrating their practical utility in quantum
information processing.
