General considerations

A response function is a measure of how a property of a system changes in the presence of one or more perturbations. With our notation (see e.g. Ref. [48]), $\langle\!\langle A;B\rangle\!\rangle_{\omega_b}$, $\langle\!\langle A;B,C\rangle\!\rangle_{\omega_b,\omega_c}$, and $\langle\!\langle A;B,C,D\rangle\!\rangle_{\omega_b,\omega_c,\omega_d}$ denote linear, quadratic and cubic response functions, respectively, which provide the first, second, and third-order corrections to the expectation-value of $A$, due to the perturbations $B$, $C$, and $D$, each of which is associated with a frequency $\omega_b$, $\omega_c$, and $\omega_d$. Often the perturbations are considered to be external monochromatic fields, or static (e.g. relativistic) perturbations, in which case the frequency is zero. In general, the perturbations $B$, $C$, and $D$ represent Fourier components of an arbitrary time-dependent perturbation.