The quantization scheme of the mesoscopic mutual inductance circuit with superconducting Josephson junctions is given. By means of squeezing unitary transformation,the energy level and the ground state vector of the system are obtained. Then,the quantum fluctuations of excess charges and phase difference between the sides of Josephson junction in the ground state are investigated. It is found that the ground state of the system is a rotated two single-mode squeezed vacuum state.