Illuminating sequential freeze-in dark matter with dark photon signal at the CERN SHiP experiment

Single-field freeze-in dark matter barely leaves observable footprints in dark matter direct detection, collider or fixed-target experiments, which can be altered in the two-field context. In this work, we consider sequential freeze-in dark matter through signals of dark photon mediator with a mass range of $m_{A'}\sim 10^{-2}-10$ GeV covered by the proposed SHiP experiment. We show that the dark charge is fixed to be $e'\sim 1.3\times 10^{-12}$ and the mixing parameter is restricted to $10^{-11}\leq \epsilon< 10^{-8}-10^{-7.5}$, as a result of the out-of-equilibrium condition of dark photon and the observed relic abundance of dark matter. Within this $\epsilon$ region, the 5(15)-year data of proton bremsstrahlung process for the dark photon, assuming vector meson (dipole) dominance, excludes $\epsilon\geq 10^{-8.5} (10^{-7.9})$ at 90\% confidence level, implying only a narrow region of $\epsilon$ close to $\sim 10^{-11}$ left for alternative tests.