Motivated by recent experiments [I. J. Vera-Marun, V. Ranjan, and B. J. van Wees, Nat. Phys. 8, 313 (2012)], we formulate a nonlinear theory of spin transport in quantum coherent conductors. We show how a mesoscopic constriction with energy-dependent transmission can convert a spin current injected by a spin accumulation into an electric signal, relying neither on magnetic nor exchange fields. When the transmission through the constriction is spin independent, the spin-charge coupling is nonlinear, with an electric signal that is quadratic in the accumulation. We estimate that gated mesoscopic constrictions have a sensitivity that allows to detect accumulations much smaller than a percent of the Fermi energy.