Model of multiphoton transitions in a current-biased Josephson junction

We present a simple model to describe multiphoton transitions between the quasi-bound states of a currentdriven Josephson junction. The transitions are induced by applying an ac voltage with controllable frequency and amplitude across the junction. The voltage induces transitions across the junction when the frequency satisfies n = E10, where E10 is the splitting between the ground and first excited quasi-bound state of the junction. We calculate the matrix elements of the transitions as a function of the dc bias current I, and the frequency and amplitude Vac of the microwave voltage, for representative junction parameters. We also calculate the frequency-dependent absorption coefficient by solving the relevant Bloch equations when the ac voltage is sufficiently weak. In this regime, the absorption coefficient is a sum of Lorentzian lines centered at the n-photon absorption frequency, of strength proportional to the squared matrix elements. For fixed Vac, the transition rate for an n-photon process usually decreases with increasing n. We also find a characteristic even-odd effect: The absorption coefficient typically increases with I for n even but decreases for n odd. Our results agree qualitatively with recent experiments.