Helical vortex phase in the non-centrosymmetric CePt$_3$Si

The recently discovered heavy fermion superconductor CePt$_3$Si [{\it E. Bauer et al., Phys. Rev. Lett. {\bf 92}, 027003 (2004)}] has triggered many experimental and theoretical studies. There are two features which have caused this attention: the absence of inversion symmetry; and the comparatively high upper critical magnetic field ($H_{c2}$). We have studied the role of magnetic fields on the broken inversion superconductor CePt$_3$Si and found that the upper critical field for a field along the c-axis exhibits a strong suppression of the paramagnetic effect, while a field applied perpendicular to the c-axis still exhibits a strong paramagnetic effect. This in-plane paramagnetic effect is reduced by the appearance of helical structure in the order parameter. We have proposed a Josephson junction experiment that can be used to detect this helical order. In particular, we have predicted that Josephson current will exhibit a magnetic interference pattern for a magnetic field applied {\it perpendicular} to the junction normal.