We consider a time-dependent model that describes a qubit time-dependently interacting
with a cavity containing finite entangled pair coherent parametric converter fields. The
dynamics of some quantum phenomena, as phase space information, quantum entanglement, and squeezing, are explored by atomic Husimi function, atomic Wehrl entropy, variance, and entropy squeezing. The influences of the unitary qubit-cavity interaction, the difference between the two-mode photon numbers, the initial atomic coherence, and the time-dependent qubit location are investigated. It is found that the regularity, amplitudes, and frequency of the quantum phenomena can be controlled by the physical parameters. For the initial atomic pure state, the qubit-cavity entanglement, the qubit phase space information, and atomic squeezing can be generated strongly compared to those of the initial atomic mixed state. The time-dependent location parameters enhance the generated quantum phenomena, and their effect can be enhanced by the parameters of the two-mode photon numbers and the initial atomic coherence.