  We report our direct study of the compressibility on ultrahigh mobility
two-dimensional electron system ($\mu_{e} \sim 1 \times 10^{7}$ cm$^{2}$/Vs) in
GaAs/AlGaAs quantum wells under microwave (MW) irradiation. The field
penetration current results show that the quantum capacitance oscillates with
microwave induced resistance oscillations (MIRO), however, the trend is
opposite with respect to the compressibility for usual equilibrium states in
previous theoretical explanations. The anomalous phenomena provide a platform
for study on the non-equilibrium system under microwave, and point to the
current domains and inhomogeneity induced by radiation. Moreover, the quantum
capacitance indication for multi-photon process around $j = 1/2$ is detected
under intensive microwave below 30 GHz.
