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We investigate multiple photon-assisted Landau-Zener (LZ) transitions in ahybrid circuit quantum electrodynamics device in which each of two interactingtransmission-line resonators is coupled to a qubit, and the qubits are drivenby periodic driving fields and also coupled to a common phonon mode. Thequantum state of the entire composite system is modeled using the multi-$\rmD_2$ Ansatz in combination with the time-dependent Dirac-Frenkel variationalprinciple. Applying a sinusoidal driving field to one of the qubits, thisdevice is an ideal platform to study the photon-assisted LZ transitions bycomparing the dynamics of the two qubits. A series of interferingphoton-assisted LZ transitions take place if the photon frequency is muchsmaller than the driving amplitude. Once the two energy scales are comparable,independent LZ transitions arise and a transition pathway is revealed using anenergy diagram. It is found that both adiabatic and nonadiabatic transitionsare involved in the dynamics. Used to model environmental effects on the LZtransitions, the common phonon mode coupled to the qubits allows for moreavailable states to facilitate the LZ transitions. An analytical formula isobtained to estimate the short-time phonon population and produces results inreasonable agreement with numerical calculations. Equipped with the knowledgeof the photon-assisted LZ transitions in the system, we can preciselymanipulate the qubit state and successfully generate the qubit dynamics with asquare-wave pattern by applying driving fields to both qubits, opening up newvenues to manipulate the states of qubits and photons in quantum informationdevices and quantum computers

Photon-assisted Landau-Zener transitions in a periodically driven Rabi dimer coupled to a dissipative mode