A non-Hermitian $PT-$symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

TL;DR

This paper investigates a non-Hermitian $PT$-symmetric Bose-Hubbard model, revealing how exceptional points of high order unfold into complex eigenvalue structures influenced by particle interactions and perturbations.

Contribution

It provides an analytical study of the unfolding of high-order exceptional points into eigenvalue rings in a non-Hermitian Bose-Hubbard system, highlighting the effects of interactions and perturbations.

Findings

High-order EPs of order N+1 exist at zero interaction.

EPs unfold into eigenvalue pairs, triplets, or rings depending on perturbation.

Analytical perturbation techniques describe eigenvalue ring formation.

Abstract

We study a non-Hermitian $PT-$symmetric generalization of an $N$-particle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order $N+1$ which under perturbation unfold either into $[(N+1)/2]$ eigenvalue pairs (and in case of $N+1$ odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and $(N+1)\mod 3$ single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to $(N+1)$th order are indicated.