dc and ac Josephson effect in a superconductor-Luttinger-liquid-superconductor system

We calculate both the dc and the ac Josephson current through a 1-D system of interacting electrons, connected to two superconductors by tunnel junctions. We treat the (repulsive) Coulomb interaction in the framework of the one-channel, spin-1/2 Luttinger model. The Josephson current is obtained for two geometries of experimental relevance: a quantum wire and a ring. At T=0, the critical current is found to decay algebraically with increasing distance {ital d} between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures {ital T}, lower than the superconducting transition temperature {ital T}{sub {ital c}}, there is a crossover from algebraic to exponential decay of the critical current as a function of {ital d}, at a distance of the order of {h_bar}{ital v}{sub {ital F}}/{ital k}{sub {ital BT}}. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circumference {ital L}, coupled capacitively to a gate voltage and threaded by a magnetic flux, the Josephson current shows remarkable parity effects under the variation of these parameters. For some values of the gate voltage and applied flux, the ring acts as a {pi} junction. These features are robust against thermal fluctuations up to temperatures on the order of {h_bar}{ital v}{sub {ital F}}/{ital k}{sub {ital BL}}. For the wire geometry, we have also studied the ac-Josephson effect. The amplitude and the phase of the time-dependent Josephson current are affected by electron-electron interactions. Specifically, the amplitude shows pronounced oscillations as a function of the bias voltage due to the difference between the velocities of spin and charge excitations in the Luttinger liquid. Therefore, the ac-Josephson effect can be used as a tool for the observation of {ital spin}-{ital charge} separation. {copyright} {ital 1996 The American Physical Society.}