Superconductivity exhibits elegant macroscopic quantum coherence in such a way that the many-body physics can be understood in a one-body way, described by the superconducting phase, and its quantum conjugate variable, the charge. When two superconductors are connected to each other through a tunnel junction, the charge tunneling can be controlled by the phase difference φ, leading to many interesting phenomena. For decades, because of the robustness of phase coherence in large junctions, a simple classical approach by modeling the system by a damped pendulum to the problem is successful and overwhelmed.(Tinkham 1996) However, as the sub-micro fabrication techniques had emerged in the 1990’s, ultra small junctions were found exhibiting stronger quantum fluctuations in phase due to charging effect, which cannot be overlooked. For exploring the novel phenomena in the opposite limit, people have made devices with robust charging effect. These phenomena can be well understood by treating the charge tunneling as a noncoherent perturbation to the quantum states described by charge. However, in the case when the Josephson energy and charging energy are competing, neither approach gives a satisfactory description. One of the attempts is to include the coherent nature in the charge tunneling processes by introducing a phase correlation function in time, which quantifies the robustness of the phase coherence.(Ingold & Nazarov 1991) The correlation function, which has been studied in many other fields, has a universal relation to the dissipation of the system, called fluctuation-dissipation theorem. Taken in this sense, dissipation is an important controlling parameter in the phase coherence robustness. If the environment impedance of the junction is much smaller than the quantum resistance (for Cooper pairs) RQ=h/4e2, the phase fluctuation is strongly damped, leading to pronounced phase coherence.(Devoret, Esteve et al. 1990) Indeed, why the classical model is so successful? The pioneer work done by Cadeira and Leggett(Caldeira & Leggett 1983) has pointed out that the environment impedance plays an important role. Their work stimulated many studies in dissipation-driven phase transitions in various systems including Josephson junctions.(Leggett, Chakravarty et al. 1987; Schon & Zaikin 1990) In this article, we will focus on the responses of Josephson junctions under microwave irradiation. Theoretically the phenomenon can be explained by the phase dynamics under an ac driving. In section 2, we will start from a classical picture by considering a Langevin
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