If we work within the electric dipole Hamiltonian, nonlinear experiments can be expressed in terms of three or more transition matrix elements. The response of the matter in linear experiments will scale as $$|\mu_{ab}|^2$$ or $$\mu_{ab}\mu_{ab}$$, whereas in nonlinear experiments will take a form such as $$\mu_{ab}\mu_{bc}\mu_{ca}$$. Our approach to describing nonlinear spectroscopy will use the electric dipole Hamiltonian and a perturbation theory expansion of the dipole operator.