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

We calculate both the dc and the ac Josephson current through a one-dimensional 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 zero temperature, the critical current is found to decay algebraically with increasing distance d between the junctions. The decay is characterized by an exponent which depends on the strength of the interaction. At finite temperatures T, lower than the superconducting transition temperature ${\mathit{T}}_{\mathit{c}}$, there is a crossover from algebraic to exponential decay of the critical current as a function of d, at a distance of the order of ħ${\mathit{v}}_{\mathit{F}}$/${\mathit{k}}_{\mathit{BT}}$. Moreover, the dependence of critical current on temperature shows nonmonotonic behavior. If the Luttinger liquid is confined to a ring of circumference 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 π junction. These features are robust against thermal fluctuations up to temperatures on the order of ħ${\mathit{v}}_{\mathit{F}}$/${\mathit{k}}_{\mathit{BL}}$. 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 spin-charge separation. © 1996 The American Physical Society.