  We investigate multiple photon-assisted Landau-Zener (LZ) transitions in a
hybrid circuit quantum electrodynamics device in which each of two interacting
transmission-line resonators is coupled to a qubit, and the qubits are driven
by periodic driving fields and also coupled to a common phonon mode. The
quantum state of the entire composite system is modeled using the multi-$\rm
D_2$ Ansatz in combination with the time-dependent Dirac-Frenkel variational
principle. Applying a sinusoidal driving field to one of the qubits, this
device is an ideal platform to study the photon-assisted LZ transitions by
comparing the dynamics of the two qubits. A series of interfering
photon-assisted LZ transitions take place if the photon frequency is much
smaller than the driving amplitude. Once the two energy scales are comparable,
independent LZ transitions arise and a transition pathway is revealed using an
energy diagram. It is found that both adiabatic and nonadiabatic transitions
are involved in the dynamics. Used to model environmental effects on the LZ
transitions, the common phonon mode coupled to the qubits allows for more
available states to facilitate the LZ transitions. An analytical formula is
obtained to estimate the short-time phonon population and produces results in
reasonable agreement with numerical calculations. Equipped with the knowledge
of the photon-assisted LZ transitions in the system, we can precisely
manipulate the qubit state and successfully generate the qubit dynamics with a
square-wave pattern by applying driving fields to both qubits, opening up new
venues to manipulate the states of qubits and photons in quantum information
devices and quantum computers

Quantum Physics;Other Condensed Matter;Atomic Physics;Chemical Physics;Computational Physics