Nonorthogonality constraints in open quantum and wave systems

TL;DR

This paper investigates bounds on the overlap of energy eigenstates in open quantum systems, showing that positive semi-definiteness suffices for the bounds and exploring conditions where these bounds are violated or modified.

Contribution

It demonstrates that positive semi-definiteness, a weaker condition than positive definiteness, suffices for energy eigenstate overlap bounds and provides a geometric interpretation of nonorthogonality in complex energy space.

Findings

Positive semi-definiteness suffices for bounds on state overlap.

Quantum backflow can violate the bounds, allowing larger overlaps.

Electromagnetic systems do not require bound modifications due to dispersion relations.

Abstract

It is known that the overlap of two energy eigenstates in a decaying quantum system is bounded from above by a function of the energy detuning and the individual decay rates. This is usually traced back to the positive definiteness of an appropriately defined decay operator. Here, we show that the weaker and more realistic condition of positive semi-definiteness is sufficient. We prove also that the bound becomes an equality for the case of single-channel decay. However, we show that the condition of positive semi-definiteness can be spoiled by quantum backflow. Hence, the overlap of quasibound quantum states subjected to outgoing-wave conditions can be larger than expected from the bound. A modified and less stringent bound, however, can be introduced. For electromagnetic systems, it turns out that a modification of the bound is not necessary due to the linear free-space dispersion relation. Finally, a geometric interpretation of the nonorthogonality bound is given which reveals that in this context the complex energy space can seen as a surface of constant negative curvature.