## Nonlinear optics near the fundamental limits

##### Abstract

The nonlinear optical response of materials on the fastest time scale is ultimately determined by the quantum mechanical properties of the material's electronic configuration. This response is limited by the amount of oscillator strength available per electron, but this limit is an order of magnitude higher than has been achieved in real systems. Here we develop analytic and numerical techniques necessary to investigate a collection of novel quantum systems to bridge the gap between observed nonlinear optical responses and the fundamental limits.
The connection between energy spectrum and nonlinear optical response is explored through the space of power law potentials in one dimension. The deficiency in the truncated sum-over-state expressions of perturbation theory to accurately calculate the class of singular power-law systems is discussed and a method of determining a proxy state constrained by quantum mechanical sum rules is developed.
Without the sum-over-states method of calculating nonlinear optical susceptibilities, other analytic methods of perturbation theory are necessary. We describe and specialize the Dalgarno-Lewis method of perturbation theory for calculating nonlinear optical susceptibilities, including frequency dependence, where only the unperturbed ground state of the system is required.
As an application of the fundamental limits on nonlinear susceptibilities, we describe a device figure of merit for a polymer based electro-optic modulator. This serves as a prescription for how one would use the fundamental limits and a three-state model to consider device figures of merit in the context of their quantum limitations.
The outstanding foundational question in the theory of fundamental limits lies in the inability of Hamiltonian systems to reach the limit. We extend the quantum Monte Carlo algorithm to include off-diagonal sum rules in a self-consistent filtering process to show that the fundamental limits determined in the three-level model are not reachable and the Hamiltonian limits are indeed the true limits.
Taking these lessons, we describe ongoing efforts to determine and characterize novel paradigms for future development in nonlinear optical materials. Quantum graphs as a toy model for branching nanostructure systems and a hybrid molecular-nanostructure system are investigated.