Topological superconducting phases and Josephson effect in curved superconductors with time reversal invariance

We consider a Rashba spin-orbit coupled nanowire with anisotropic spin-singlet superconducting pairing and with time reversal invariance. We explore the evolution of the topological superconducting phases of this system due to geometric deformations for the representative case of a wire bent in a semielliptical shape. We find that when the system is in its topological superconducting phase, strong inhomogeneities in the profile curvature can produce a pair of localized eigenmodes, which can be attributed to a nonuniform topological phase. The curved geometric profile also allows us to tune the spin correlations of the superconducting state via the induced inhomogeneity of the spin-orbit coupling (SOC). The geometric control of the superconducting pair correlations leads to the manipulation of the critical current in Josephson junctions made up of two time reversal invariant topological superconductors separated by a spin-orbit coupled normal metal. In particular, we find that the curvature inhomogeneity can be exploited for amplifying the current intensity but also to generate a $0\text{\ensuremath{-}}\ensuremath{\pi}$ transition, and a second harmonic contribution, which generates, for some specific geometric configurations, a $\ensuremath{\varphi}$-junction behavior.