The paradox of the many-state catastrophe of fundamental limits and the three-state conjecture

TL;DR

This paper explores the limitations of the three-level ansatz in quantum nonlinear susceptibilities, revealing a many-state catastrophe that suggests the true fundamental limits are smaller than previously thought.

Contribution

It demonstrates that relaxing the three-level ansatz leads to divergence in hyperpolarizability, challenging existing fundamental limit calculations and proposing the ansatz may be unprovable.

Findings

Divergence occurs when more than three states contribute to hyperpolarizability.

The three-level ansatz acts as a physical constraint excluding nonphysical Hamiltonians.

The true fundamental limit of nonlinear response may be smaller than current estimates.

Abstract

The calculation of the fundamental limits of nonlinear susceptibilities posits that when a quantum system has a nonlinear response at the fundamental limit, only three energy eigenstates contribute to the first and second hyperpolarizability. This is called the three-level ansatz and is the only unproven assumption in the theory of fundamental limits. All calculations that are based on direct solution of the Schrodinger equation yield intrinsic hyperpolarizabilities less than 0.709 and intrinsic second hyperpolarizabilities less than 0.6. In this work, we show that relaxing the three-level ansatz and allowing an arbitrary number of states to contribute leads to divergence of the optimized intrinsic hyperpolarizability in the limit of an infinite number of states - what we call the many-state catastrophe. This is not surprising given that the divergent systems are most likely not derivable from the Schrodinger equation, yet obey the sum rules. The sums rules are the second ingredient in limit theory, and apply also to systems with more general Hamiltonians. These exotic Hamiltonians may not model any real systems found in nature. Indeed, a class of transition moments and energies that come form the sum rules do not have a corresponding Hamiltonian that is expressible in differential form. In this work, we show that the three-level ansatz acts as a constraint that excludes many of the nonphysical Hamiltonians and prevents the intrinsic hyperpolarizability from diverging. We argue that this implies that the true fundamental limit is smaller than previously calculated. Since the three-level ansatz does not lead to the largest possible nonlinear response, contrary to its assertion, we propose the intriguing possibility that the three-level ansatz is true for any system that obeys the Schrodinger equation, yet this assertion may be unprovable.