Quantum calculations of the dispersion of the fundamental limits of nonlinear susceptibilities

Using sum rules and a new dipole-free sum-over-states expression, we introduce a method for calculating the fundamental limits of the dispersion of the real and imaginary parts of all electronic nonlinear-optical susceptibilities. As such, these general results can be used to study any nonlinear optical phenomena at any wavelength, making it possible to push both applications and our understanding of such processes to the limits. These results reveal the ultimate constraints imposed by nature on our ability to control and use light.