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Microscopic theory of the current-voltage characteristics of Josephson tunnel junctions

Deep theoretical understanding of the electrical response of Josephson junctions is indispensable regarding both recent discoveries of new kinds of superconductivity and technological advances such as superconducting quantum computers. Here, we study the microscopic theory of the DC current-biased $I$-$V$ characteristics of Josephson tunnel junctions. We derive an analytical formula of the $I$-$V$ characteristics of generic junctions. We identify subharmonics of the $I$-$V$ characteristics and their underlying mechanism as the feedback effect of intrinsic AC currents generated by voltage pulses in the past. We apply our theory to analytically solve the Werthamer equation and describe various DC current-biased $I$-$V$ characteristics as a function of softening of the superconducting gap. Strikingly, we identify voltage staircases of the $I$-$V$ characteristics in a genuine Josephson junction without AC current bias or qubit dynamics. Our general analytical formalism opens new avenues for a microscopic understanding of $I$-$V$ characteristics of Josephson junctions that have been limited to phenomenological models so far.