Evidence for a finite-momentum Cooper pair in tricolor $d$-wave superconducting superlattices

Abstract: Fermionic superfluidity with a nontrivial Cooper-pairing, beyond the conventional Bardeen-Cooper-Schrieffer state, is a captivating field of study in quantum many-body systems. In particular, the search for superconducting states with finite-momentum pairs has long been a challenge, but establishing its existence has long suffered from the lack of an appropriate probe to reveal its momentum. Recently, it has been proposed that the nonreciprocal {\cred electron} transport is the most {\cred powerful} probe for the finite-momentum pairs, {\cred because it directly couples} to the supercurrents. Here we reveal such a pairing state by the non-reciprocal transport on tricolor superlattices with strong spin-orbit coupling combined with broken inversion-symmetry consisting of atomically thin $d$-wave superconductor CeCoIn$_5$. We find that while the second-harmonic resistance exhibits a distinct dip anomaly at the low-temperature ($T$)/high-magnetic field ($H$) corner in the $HT$-plane for ${\bm H}$ applied to the antinodal direction of the $d$-wave gap, such an anomaly is absent for ${\bm H}$ along the nodal direction. By meticulously isolating extrinsic effects due to vortex dynamics, we reveal the presence of a non-reciprocal response originating from intrinsic superconducting properties characterized by finite-momentum pairs. We attribute the high-field state to the helical superconducting state, wherein the phase of the order parameter is spontaneously spatially modulated.