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We consider superconducting nanowire proximity coupled to superconductor /ferromagnet / superconductor junction, where the magnetization penetrates intosuperconducting segment in nanowire decaying as $\sim\exp[-\frac{\mid n\mid}{\xi}]$ with site index $n$ and the decay length $\xi$. We tune chemicalpotential and spin-orbit coupling so that topological superconducting regimehosting Majorana fermion is realized for long $\xi$. We find that when $\xi$becomes shorter, zero energy state at the interface between superconductor andferromagnet splits into two away from zero energy. Accordingly, the behavior ofJosephson current is drastically changed due to this "zero mode-non-zero modecrossover". By tuning the model parameters, we find an almost second-harmoniccurrent-phase relation, $\sin2\varphi$, with phase difference $\varphi$. Basedon the analysis of Andreev bound state (ABS), we clarify that current-phaserelation is determined by coupling of the states within the energy gap. We findthat the emergence of crossing points of ABS is a key ingredient to generate$\sin2\varphi$ dependence in current-phase relation. We further study both theenergy and $\varphi$ dependence of pair amplitudes in the ferromagnetic region.For long $\xi$, odd-frequency spin-triplet $s$-wave component is dominant. Themagnitude of the odd-frequency pair amplitude is enhanced at the energy levelof ABS.

Josephson current in a normal-metal nanowire coupled to superconductor/ferromagnet/superconductor junction