Model of multiphoton transitions in a current-biased Josephson junction

Abstract

We present a simple model to describe multiphoton transitions between the quasi-bound states of a current-driven 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 $\ensuremath{\omega}$ satisfies $n\ensuremath{\hbar}\ensuremath{\omega}=\ensuremath{\Delta}{E}_{10}$, where $\ensuremath{\Delta}{E}_{10}$ 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 $\ensuremath{\omega}$ and amplitude ${V}_{\text{ac}}$ 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 ${V}_{\text{ac}}$, 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.