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We study new types of Josephson junctions composed of helical $p$-wavesuperconductors with $k_{x}\hat{x}\pm k_{y}\hat{y}$ and $k_{y}\hat{x}\pmk_{x}\hat{y}$-pairing symmetries using quasiclassical Green's functions withthe generalized Riccati parametrization. The junctions can host rich groundstates: $\pi$ phase, $0+\pi$ phase, $\varphi_{0}$ phase and $\varphi$ phase.The phase transition can be tuned by rotating the magnetization in theferromagnetic interface. We present the phase diagrams in the parameter spaceformed by the orientation of the magnetization or by the magnitude of theinterfacial potentials. The selection rules for the lowest order current whichare responsible for the formation of the rich phases are summarized from thecurrent-phase relations based on the numerical calculation. We construct aGinzburg-Landau type of free energy for the junctions with ${\bf{d}}$-vectorsand the magnetization, which not only reveals the interaction forms ofspin-triplet superconductivity and ferromagnetism but also can directly leadsto the selection rules. In addition, the energies of the Andreev bound statesand the novel symmetries in the current-phase relations are also investigated.Our results are helpful both in the prediction of the novel Josephson phasesand in the design of the quantum circuits.

Tunable ground states in helical $p$-wave Josephson junctions