Non-standard quantum algebras and finite dimensional PT -symmetric systems

Abstract

In this work,PT-symmetric Hamiltonians defined on quantum sl(2, R) algebras are presented. We study the spectrum of a family of non-Hermitian Hamiltonians written in terms of the generators of the non-standard Uz(sl(2,R)) Hopf algebra deformation of sl(2, R). By making use of a particular boson representation of the generators of Uz(sl(2, R)), both the co-product and the commutation relations of the quantum algebra are shown to be invariant under the PT-transformation. In terms of these operators, we construct several finite dimensional PT-symmetry Hamiltonians, whose spectrum is analytically obtained for any arbitrary dimension. In particular, we show the appearance of Exceptional Points in the space of model parameters and we discuss the behaviour of the spectrum both in the exact PT-symmetry and the broken PT-symmetry dynamical phases. As an application, we show that this non-standard quantum algebra can be used to define an effective model Hamiltonian describing accurately the experimental spectra of three-electron hybrid qubits based on asymmetric double quantum dots. Remarkably enough, in this effective model, the deformation parameter "z" has to be identified with the detuning parameter of the system.

Features, Paradoxes and Amendments of Perturbative Non-Hermitian Quantum Mechanics

“…Temporarily let us simplify the mathematics and assume that dim K < ∞ [59]. Then, for the reasons explained in diagram (A13) of Appendix C, we must distinguish between the equations at λ = 0 (the unperturbed limit) and at λ ̸ = 0 (the perturbed regime).…”