In this chapter, we introduce the driven superconducting tunnel junction circuit in which we experimentally engineer the squeeze-driven Kerr oscillator (SKO) effective Hamiltonian. First, we introduce the superconducting circuit in the absence of the drive. Then, we will examine its properties when submitted to a fast-oscillating drive, thus realistically modeling our experiment. The main result of this chapter is an effective Hamiltonian that is obtained by going beyond the rotating wave approximation and which governs the interesting dynamics of our driven superconducting circuit.