Spin and Charge Josephson effects between non-uniform superconductors with coexisting helimagnetic order

We consider the spin and charge Josephson current between two non-uniform Fulde-Ferrel-Larkin-Ovchinnikov superconductors with helimagnetic order. We demonstrate that the presence of the helimagnetic phase generates a spin Josephson effect and leads to additional contributions to both single-particle and Josephson charge current. It is shown that for such systems the AC effect differs more radically from the DC effect than in the case of a BCS superconductor with helimagnetic order considered earlier in the literature [M. L. Kuli\'c and I. M. Kuli\'c, Phys. Rev. B {\bf 63}, 104503 (2001)] where a spin Josephson current has also been found. In our system the most interesting effect occurs in the presence of an external magnetic field and in absence of voltage, where we show that the charge Josephson current can be tuned to zero while the spin Josephson current is non-vanishing. This provides a well controlled mechanism to generate a spin supercurrent in absence of charge currents.