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Understanding how and to what magnitude solid-state qubits couple to metallicwires 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 asuperconducting (SC) resonator's electromagnetic field, focusing on theinteraction involving both the transition and diagonal dipole moments of thequdit. Specifically, we explore the effective dynamical (time-dependent)longitudinal coupling that arises when a solid-state qudit is adiabaticallymodulated at small gate frequencies and amplitudes, in addition to a staticdispersive interaction with the SC resonator. For the first time, we deriveHamiltonians describing the longitudinal multi-level interactions in a generaldispersive regime, encompassing both dynamical longitudinal and dispersiveinteractions. These Hamiltonians smoothly transition between their adiabaticvalues, where the couplings of the n-th level are proportional to the level'senergy curvature concerning a qudit gate voltage, and the substantially largerdispersive values, which occur due to a resonant form factor. We provideseveral examples illustrating the transition from adiabatic to dispersivecoupling in different qubit systems, including the charge (1e DQD) qubit, thetransmon, the double quantum dot singlet-triplet qubit, and the triple quantumdot exchange-only qubit. In some of these qubits, higher energy levels play acritical role, particularly when their qubit's dipole moment is minimal orzero. For an experimentally relevant scenario involving a spin-charge qubitwith magnetic field gradient coupled capacitively to a SC resonator, weshowcase the potential of these interactions. They enableclose-to-quantum-limited quantum non-demolition (QND) measurements and remotegeometric phase gates, demonstrating their practical utility in quantuminformation processing.

Longitudinal (curvature) couplings of an $N$-level qudit to a superconducting resonator at the adiabatic limit and beyond