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 $ u$ and $z$, scales as $n sim tau^{-alpha u d/ (alpha z u+1)}$ [$n sim (alpha g^{(alpha-1)/alpha}/tau)^{ u d/(z u+1)}$], if the quench takes the system across the critical point at time $t=0$ [$t=t_0 e 0$], 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.
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