A non-Hermitian \mathcal{P}\mathcal{T} symmetric Bose–Hubbard model: eigenvalue rings from unfolding higher-order exceptional points

Abstract

Abstract. We study a non-Hermitian PT −symmetric generalization of an Nparticle, 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.

Information Geometry of Complex Hamiltonians and Exceptional Points

“…In the case of a complex Hamiltonian, the situation is more severe on account of the fact that the Rayleigh-Schrödinger perturbation theory breaks down altogether in the vicinities of exceptional points. Nevertheless, for a given Hamiltonian, one can expand the eigenstates and eigenvalues in the form of a Newton-Puiseux series in order to identify the metric geometry close to exceptional points (see, for example, [54,55] for effective use of the Newton-Puiseux expansion for the investigation of properties of the eigenstates of complex Hamiltonians in the vicinities of exceptional points; see, also, [56,57] for a more general discussion on related mathematical ideas). This line of investigation, therefore, leads to a new application of information geometry in the sensitivity analysis of physical systems characterised by Hermitian or more generally complex Hamiltonians (we remark that properties of exceptional points of higher order, where more than two eigenstates coalesce, can be quite intricate; see, e.g., [58,59]).…”

Bound states emerging from below the continuum in a solvablePT-symmetric discrete Schrödinger equation

“…Here we proceeded in a complementary direction of connecting these models with the simulations of the various forms of quantum phase transitions (cf., e.g., [36,37]). In the latter setting people feel most often inspired by the Bender's and Boettcher's [3] conjecture that beyond the scope of the conventional textbooks, the unitary evolution of quantum systems may still be described as generated by certain non-standard Hamiltonians H with real spectra.…”

Revisiting the Optical PT-Symmetric Dimer

“…We can follow a standard approach from non-Hermitian Bose-Hubbard dimers [58][59][60][61] used to deal with optical nonlinear P T -symmetric dimers [28], and introduce a Stokes vector, S = (S x ,S y ,S z ), with components given by,S…”