  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
$L_s = \hbar/(2m\alpha)$ is the spin-precession length in the Rashba spin-orbit
field with coupling strength $\alpha$. If $\gamma\ll1$ 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 $\gamma\to \infty$. 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.
