Dressed Quantum Graphs With Optical Nonlinearities Approaching the Fundamental Limit

Abstract

We dress bare quantum graphs with finite delta function potentials and calculate optical nonlinearities that are found to match the fundamental limits set by potential optimization. We show that structures whose first hyperpolarizability is near the maximum are well described by only three states, the so-called three-level Ansatz, while structures with the largest second hyperpolarizability require four states. We analyze a very large set of configurations for graphs with quasi-quadratic energy spectra and show how they exhibit better response than bare graphs through exquisite optimization of the shape of the eigenfunctions enabled by the existence of the finite potentials. We also discover an exception to the universal scaling properties of the three-level model parameters and trace it to the observation that a greater number of levels are required to satisfy the sum rules even when the three-level Ansatz is satisfied and the first hyperpolarizability is at its maximum value, as specified by potential optimization. This exception in the universal scaling properties of nonlinear optical structures at the limit is traced to the discontinuity in the gradient of the eigenfunctions at the location of the delta potential. This is the first time that dressed quantum graphs have been devised and solved for their nonlinear response, and it is the first analytical model of a confined dynamic system with a simple potential energy that achieves the fundamental limits.

Using excited states and degeneracies to enhance the electric polarizability and first hyperpolarizability

“…The energy difference between the ground and first excited state, E 10 , sets a fundamental limit on the electric polarizability and first hyperpolarizability. These limits have been corroborated by experiment [12], potential optimization [13][14][15][16][17], and calculations on quantum graphs [18][19][20][21] though a recent Monte Carlo study utilizing filtered sampling suggests that these limits may be an overestimate by approximately 30% [22].…”

Green’s function approach for quantum graphs: An overview

“…The Hamiltonian operator on E(Γ) consists of the following unidimensional differential operators defined on each edge e s [19,161] (the dressed case) Here, V e s (x e s ) is the potential (usually assumed to be non-negative and smooth) in the interval 0 < x e s < ℓ e s . Different works have considered the above Hamiltonian for non-vanishing potentials (for instance, see [43,44,116,137,[162][163][164][165]). However, in the literature, even in papers discussing quantum chaos [37-39, 47, 166], it is usual to have for any e s that V e s = 0 (the case we assume in this review).…”

Physics of the fundamental limits of nonlinear optics: a theoretical perspective [Invited]

“…The presence of the δ functions or prongs creates a giant enhancement of the nonlinear optical response due to a effect known as phase disruption of the lowest eigenfunctions of the graph [84]. This class of graphs has the optimum topology for nonlinear optics [111][112][113]. Table II presents a compilation of results for the class of graphs depicted in Figure 5.…”

“…γ + is the value of the second hyperpolarizability's x-diagonal component when the graph is rotated to the position maximizing it. (γ − is the minimum value upon rotation [111]. were non-interacting, however, but models accounted for the Fermion properties of electrons in assembling the eigenstates.…”