Impact of the phonon environment on the nonlinear quantum-dot cavity QED. II. Analytical approach

TL;DR

This paper presents an analytical method to account for phonon effects on the nonlinear optical response of quantum dot-cavity systems in the strong coupling regime, enabling analysis at higher energy levels and across various temperatures.

Contribution

The authors develop a generalized analytical approach that accurately describes phonon effects in nonlinear quantum dot-cavity systems, extending the analysis to higher Jaynes-Cummings ladder rungs.

Findings

Good agreement with exact solutions across temperature ranges

Effective in analyzing higher rungs of the Jaynes-Cummings ladder

Useful for studying coherent dynamics in nonlinear optical systems

Abstract

The effect of phonons on a nonlinear optical response of a quantum dot-cavity system in quantum strong coupling regime can be accounted for by a fully analytical treatment, provided that the exciton-phonon dynamics is much faster than the exciton-cavity dynamics. Modern experiments involving semiconductor quantum dots embedded in optical microcavities typically meet this criterion. We find that, for a relatively small exciton-cavity coupling, the effect of phonons is concentrated mainly in the polaron shift of the exciton frequency and reduction of exciton-cavity coupling by the Huang-Rhys factor. We have generalized this result to an arbitrary optical nonlinearity and demonstrated a good agreement with the exact solution in a wide range of temperatures. This generalization provides access to higher rungs of the Jaynes-Cummings ladder, where exact numerical approaches are impractical or even impossible. At larger coupling strengths and low temperatures, our approximation is also in good agreement with the exact solution, which makes it a very useful tool for addressing the phonon contribution to the coherent dynamics of a nonlinear optical system. We demonstrate our results for third-order optical polarization with varying observation time and delay time between excitation pulses in the form of two-dimensional spectra. These spectra provide useful information about the coherent coupling between the exciton and the cavity modes. The presented analytical approach is also compared with another useful approximation, having the form of a matrix product, which is a special case of the asymptotically exact solution in the limit of a short phonon memory time.