Nonlinear susceptibility

Definition

Consider a medium of nonlinear optics with an instantaneous and local response. So the polarization density $\mathbf{P}$ at a point is a smooth function of the electric field $\mathbf{E}$ at that point, with $\mathbf{P}(\mathbf{0}) = \mathbf{0}$. Let $\varepsilon_0$ be the vacuum permittivity, and expand $\mathbf{P}$ in a power series: $$P_i = \varepsilon_0 \sum_{n \ge 1} \sum_{j_1, \dots, j_n} \chi^{(n)}_{i j_1 \cdots j_n} E_{j_1} \cdots E_{j_n}.$$ Here each tensor $\chi^{(n)}$ has $3^{n+1}$ components, and it can be chosen symmetric in its last $n$ indices, since the product $E_{j_1} \cdots E_{j_n}$ is. For $n \ge 2$, $\chi^{(n)}$ is the nonlinear susceptibility of order $n$. With this choice, $\varepsilon_0 \chi^{(n)}$ is $1/n!$ times the $n$-th derivative of $\mathbf{P}$ at $\mathbf{E} = \mathbf{0}$.