  We experimentally study a circuit quantum acoustodynamics system, which
consists of a superconducting artificial atom, coupled to both a
two-dimensional surface acoustic wave resonator and a one-dimensional microwave
transmission line. The strong coupling between the artificial atom and the
acoustic wave resonator is confirmed by the observation of the vacuum Rabi
splitting at the base temperature of dilution refrigerator. We show that the
propagation of microwave photons in the microwave transmission line can be
controlled by a few phonons in the acoustic wave resonator. Furthermore, we
demonstrate the temperature effect on the measurements of the Rabi splitting
and temperature induced transitions from high excited dressed states. We find
that the spectrum structure of two-peak for the Rabi splitting becomes into
those of several peaks, and gradually disappears with the increase of the
environmental temperature $T$. The quantum-to-classical transition is observed
around the crossover temperature $T_{c}$, which is determined via the thermal
fluctuation energy $k_{B}T$ and the characteristic energy level spacing of the
coupled system. Experimental results agree well with the theoretical
simulations via the master equation of the coupled system at different
effective temperatures.
