The Josephson junction as a quantum engine

We treat the Cooper pairs in the superconducting electrodes of a Josephson junction (JJ) as an open system, coupled via Andreev scattering to external baths of electrons. The disequilibrium between the baths generates the direct-current bias applied to the JJ. In the weak-coupling limit we obtain a Markovian master equation that provides a simple dynamical description consistent with the main features of the JJ, including the form of the current–voltage characteristic, its hysteresis, and the appearance under periodic voltage driving of discrete Shapiro steps. For small dissipation, our model also exhibits a self-oscillation of the JJ’s electrical dipole with frequency Ω = 2 e V / ℏ around mean voltage V . This self-oscillation, associated with ‘hidden attractors’ of the nonlinear equations of motion, explains the observed production of monochromatic radiation with frequency Ω and its harmonics. We argue that this picture of the JJ as a quantum engine resolves open questions about the Josephson effect as an irreversible process and could open new perspectives in quantum thermodynamics and in the theory of dynamical systems.