Non-Hermitian time-dependent perturbation theory: Asymmetric transitions and transitionless interactions

Abstract

The ordinary time-dependent perturbation theory of quantum mechanics, that describes the interaction of a stationary system with a time-dependent perturbation, predicts that the transition probabilities induced by the perturbation are symmetric with respect to the initial and final states. Here we extend time-dependent perturbation theory into the non-Hermitian realm and consider the transitions in a stationary Hermitian system, described by a self-adjoint Hamiltonian Hˆ0, induced by a time-dependent non-Hermitian interaction f(t)Hˆ1. In the weak interaction (perturbative) limit, the transition probabilities generally turn out to be asymmetric for exchange of initial and final states. In particular, for a temporal shape f(t) of the perturbation with one-sided Fourier spectrum, i.e. with only positive (or negative) frequency components, transitions are fully unidirectional, a result that holds even in the strong interaction regime. Interestingly, we show that non-Hermitian perturbations can be tailored to be transitionless, i.e. the perturbation leaves the system unchanged as if the interaction had not occurred at all, regardless the form of Hˆ0 and Hˆ1. As an application of our results, we provide important physical insights into the asymmetric (chiral) behavior of dynamical encircling of an exceptional point in two- and three-level non-Hermitian systems.