Josephson effect in two-band superconductors

We study theoretically the Josephson effect between two two-band superconductors respecting time-reversal symmetry, where we assume a spin-singlet $s$-wave pair potential in each conduction band. The superconducting phase at the first band ${\ensuremath{\varphi}}₁$ and that at the second band ${\ensuremath{\varphi}}₂$ characterize a two-band superconducting state. We consider a Josephson junction where an insulating barrier separates two such two-band superconductors. By applying the tunnel Hamiltonian description, the Josephson current is calculated in terms of the anomalous Green's function on either side of the junction. We find that the Josephson current consists of three components which depend on three types of phase differences across the junction: the phase difference at the first band $\ensuremath{\delta}{\ensuremath{\varphi}}₁$, the phase difference at the second band $\ensuremath{\delta}{\ensuremath{\varphi}}₂$, and the difference at the center-of-mass phase $(\ensuremath{\delta}{\ensuremath{\varphi}}₁+\ensuremath{\delta}{\ensuremath{\varphi}}₂)/2$. A Cooper pair generated by the band hybridization carries the last current component. We discuss the relation between the Josephson current calculated in theories and that observed in experiments.