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

## TL;DR

This paper extends quantum time-dependent perturbation theory into the non-Hermitian domain, revealing asymmetric transition probabilities and the possibility of transitionless interactions, with implications for chiral dynamics near exceptional points.

## Contribution

It introduces a non-Hermitian perturbation framework showing asymmetric and unidirectional transitions, and demonstrates how to engineer transitionless non-Hermitian interactions.

## Key findings

- Transition probabilities become asymmetric under non-Hermitian perturbations.
- Unidirectional transitions occur with one-sided Fourier spectrum of the perturbation.
- Non-Hermitian perturbations can be designed to be transitionless, leaving the system unchanged.

## 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 an 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 $\hat{H}_0$, induced by a time-dependent non-Hermitian interaction $f(t) \hat{P}$. In the weak interaction (perturbative) limit, the transition probabilities generally turn out to be {\it 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 $\hat{H}_0$ and $\hat{P}$. As an application of the results, we discuss asymmetric (chiral) behavior of dynamical encircling of an exceptional point in a two- and three-level system.