  We show that the defect density $n$, for a slow non-linear power-law quench
with a rate $\tau^{-1}$ and an exponent $\alpha>0$, which takes the system
through a critical point characterized by correlation length and dynamical
critical exponents $\nu$ and $z$, scales as $n \sim \tau^{-\alpha \nu d/
(\alpha z\nu+1)}$ [$n \sim (\alpha g^{(\alpha-1)/\alpha}/\tau)^{\nu
d/(z\nu+1)}$], if the quench takes the system across the critical point at time
$t=0$ [$t=t_0 \ne 0$], where $g$ is a non-universal constant and $d$ is the
system dimension. These scaling laws constitute the first theoretical results
for defect production in non-linear quenches across quantum critical points and
reproduce their well-known counterpart for linear quench ($\alpha=1$) as a
special case. We supplement our results with numerical studies of well-known
models and suggest experiments to test our theory.
