It is well known that a current driven through a two-dimensional electron gas with Rashba spin-orbit coupling induces a spin polarization in the perpendicular direction (Edelstein effect). This phenomenon has been extensively studied in the linear response regime, i.e., when the average drift velocity of the electrons is a small fraction of the Fermi velocity. Here we investigate the phenomenon in the nonlinear regime, meaning that the average drift velocity is comparable to, or exceeds the Fermi velocity. This regime is realized when the electric field is very large, or when electron-impurity scattering is very weak. The quantum kinetic equation for the density matrix of noninteracting electrons is exactly and analytically solvable, reducing to a problem of spin dynamics for unpaired electrons near the Fermi surface. The crucial parameter is $gamma=eEL_s/E_F$, where $E$ is the electric field, $e$ is the absolute value of the electron charge, $E_F$ is the Fermi energy, and $L_s = hbar/(2malpha)$ is the spin-precession length in the Rashba spin-orbit field with coupling strength $alpha$. If $gammall1$ the evolution of the spin is adiabatic, resulting in a spin polarization that grows monotonically in time and eventually saturates at the maximum value $n(alpha/v_F)$, where $n$ is the electron density and $v_F$ is the Fermi velocity. If $gamma gg 1$ the evolution of the spin becomes strongly non-adiabatic and the spin polarization is progressively reduced, and eventually suppressed for $gammato infty$. We also predict an inverse nonlinear Edelstein effect, in which an electric current is driven by a magnetic field that grows linearly in time. The conductivities for the direct and the inverse effect satisfy generalized Onsager reciprocity relations, which reduce to the standard ones in the linear response regime.
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