  We investigate the multi-photon resonance and multi-ion excitation in a
single-mode cavity with identical vibrating ion-qubits, which enables the
tripartite interaction among the internal states of ions, the cavity mode and
the ions' vibrational motion. Under particular resonant conditions, we derive
effective Hamiltonians for the three-photon and the three-excitation cases,
respectively, and find that the magnitude of the effective coupling energy can
be tuned through the vibration mode, allowing for manipulations of ion-photon
coupling in experiments. Furthermore, we analyze the system dynamics of our
proposed setups and demonstrate the Rabi oscillation behaviors in these systems
with dissipation effects. We propose our system as a versatile platform for the
exploration of entangled multi-qubit physics.
