# Level shift and decay dynamics of a quantum emitter around plasmonic nanostructure

## TL;DR

This paper introduces a new, efficient method for calculating quantum energy level shifts of emitters near nanostructures, avoiding complex integrals and requiring less frequency data, with demonstrated applications to nanospheres and nanocavities.

## Contribution

A novel approach for computing quantum energy level shifts that simplifies calculations by eliminating principal value integrals and reducing frequency range requirements.

## Key findings

- The new method avoids principal value integrals and imaginary frequency calculations.
- It requires a narrower frequency range for accurate results.
- Reversible quantum dynamics are observed in the studied systems.

## Abstract

We put forward a general approach for calculating the quantum energy level shift for emitter in arbitrary nanostructures, in which the energy level shift is expressed by the sum of the real part of the scattering photon Green function (GF) and a simple integral about the imaginary part of the photon GF in the real frequency range without principle value. Compared with the method of direct principal value integral over the positive frequency axis and the method by transferring into the imaginary axis, this method avoids the principle value integral and the calculation of the scattering GF with imaginary frequency. In addition, a much narrower frequency range about the scattering photon GF in enough to get a convergent result. It is numerically demonstrated in the case for a quantum emitter (QE) located around a nanosphere and in a gap plasmonic nanocavity. Quantum dynamics of the emitter is calculated by the time domain method through solving Schr\"{o}dinger equation in the form of Volterra integral of the second kind and by the frequency domain method based on the Green's function expression for the evolution operator. It is found that the frequency domain method needs information of the scattering GF over a much narrower frequency range. In addition, reversible dynamics is observed. These findings are instructive in the fields of coherent light-matter interactions.