Time evolution and non-Hermiticity in quantum defect theory

Quantum defect theory (QDT) is widely employed to describe Rydberg states. We show that the effective Hamiltonian employed by QDT is intrinsically non-Hermitian and examine the consequences arising in practical calculations. In particular if the eigenstates are treated in the ‘standard ’ manner, i.e. as if the Hamiltonian were Hermitian, unphysical errors may appear, such as the non-unitarity of the time evolution. Although in most cases the degree of non-Hermiticity is small, we give examples involving the autocorrelation function and the time dependence of entanglement generation where non-Hermiticity must be explicitly accounted for to avoid such unphysical errors. This is done by introducing a second basis set forming with the QDT eigenstates a biorthogonal basis. We give practical schemes to construct this basis and discuss how to express physical quantities. We illustrate the formalism by computing the time evolution of the autocorrelation function and of the linear entropy for a model Rydberg system and compare with the results obtained in the ‘standard’ manner. (Some figures in this article are in colour only in the electronic version) 1