Josephsonic diagnostic of competing orders in quantum critical multiband superconductors

Motivated by the recent experimentally observed manifestations of the quantum critical point fluctuations in the thermodynamic properties of multiband superconductors, we derive a general expression for the Josephson current of various junctions between two superconductors in the phase of superconductivity coexistence with the spin-density-wave. We demonstrate that the critical current peaks at the quantum critical point that separates pure and mixed superconducting phases. We argue that our results are generic and, in particular, can be adopted to explain the recent observations of a nonmonotonic dependence of the supercurrent on the external pressure in the heavy fermion superconductor CeRhIn$_5$, and on the chemical doping in iron-based superconductors such as Ba(Fe$_{1-x}$Co$_x$)$_2$As$_2$.