Unitary unfoldings of a Bose-Hubbard exceptional point with and without particle number conservation

Abstract

The conventional non-Hermitian but $PT$ -symmetric three-parametric Bose-Hubbard Hamiltonian H(γ, v, c) represents a quantum system of N bosons, unitary only for parameters γ, v and c in a domain $D$ . Its boundary $\partial D$ contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v, but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H(v, v, c) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose-Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians $H\left(\lambda \right)=H(v,v,0)+\lambda V$ for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations $\lambda V$ which can be considered small.

Keywords: Bose–Hubbard Hamiltonians; dynamics near exceptional points; quantum mechanics; quasi-Hermitian observables; stability-guaranteeing perturbations.