Application of machine learning to find exceptional points

Abstract

In open quantum systems, resonances can occur. These are quasi-bound states which can decay. By introducing a complex scaling, e.g. according to Reinhardt, and thus non-Hermitian operators, the complex energy eigenvalues of the resonances can be calculated. Here, the real part represents their energy, while the imaginary part unveils their lifetime. Resonances can degenerate, where a special case is the so-called exceptional point (EP) at which not only the eigenvalues but also the eigenvectors degenerate. Thus, the two resonances coalesce at the EP. An isolated EP can be described by a two-dimensional matrix model. A property of such an EP is that the two associated eigenvalues exchange their positions after one adiabatic orbit in parameter space around the EP. In 2007 the existence of these EPs was proven for the hydrogen atom in electric and magnetic fields by Cartarius. Due to limitations especially in magnetic field strengths, EPs in the hydrogen atom are not experimentally accessible. In 2014, a remarkable discovery by Kazimierczuk et al. revealed a mesmerizing hydrogen-like spectrum within cuprous oxide. This revelation stemmed from the resemblance between an exciton, a quasi-particle in a semiconductor consisting of electron and hole, and their atomic counterpart, the hydrogen atom. However, the fact that the excitons are environed by cuprous oxide necessitated consideration of the band structure to precisely describe the observed spectrum. This discovery kindled excitement as it provided a rare opportunity to bridge the realms of experimental and theoretical physics, inviting an enthralling dialogue between theory and experiment. For cuprous oxide the field strengths to observe EPs of resonances with small quantum numbers are much lower compared to the field strengths for the hydrogen atom, which is why it is favorable to find EPs in this system. This was already done for a hydrogen-like model, but to obtain experimentally comparable results the above mentioned band structure terms need to be considered. However, this increases the computational cost drastically for each diagonalization of the Hamiltonian due to its complexity. The existing methods to find EPs are based on a Taylor expansion around the EP. Due to the computational expensive diagonalizations of the Hamiltonian, these methods are inefficient or even not applicable. Hence, a new method is required to accurately and efficiently identify EPs in cuprous oxide. Inspired by the remarkable advances in machine learning, especially within the realm of physics, a novel method on the foundation of Gaussian process regression (GPR) is developed. As a prominent member of the supervised machine learning family, GPR serves as a powerful and innovative approach to predict the positions of EPs in cuprous oxide. The used data to train a GPR model is obtained by simulations. Hence, the error is only due to numerical inaccuracies, which can be neglected. Unlike neural networks, GPR offers the advantage of precisely passing through the provided training points, which is a key motivation for its utilization. Yet, the optimization of the searching process goes beyond the new method. An efficient algorithm is devised to enhance the search for EPs in cuprous oxide, which contributes to the discovery of promising EPs and thus enables a possible experimental verification of these data.