Electron-Phonon Interacation in Quantum Dots: A Solvable Model

TL;DR

This paper introduces a solvable model for electron-phonon interactions in quantum dots, revealing complex spectral features and limitations of common approximation methods through an exact numerical approach.

Contribution

The paper develops a numerically exact method for analyzing electron-phonon interactions in quantum dots with multiple levels, providing insights beyond traditional approximations.

Findings

Quantum mechanical level splitting occurs near resonance.

One bosonic mode primarily drives the level repulsion.

Standard Born approximation fails to capture spectral complexities.

Abstract

The relaxation of electrons in quantum dots via phonon emission is hindered by the discrete nature of the dot levels (phonon bottleneck). In order to clarify the issue theoretically we consider a system of $N$ discrete fermionic states (dot levels) coupled to an unlimited number of bosonic modes with the same energy (dispersionless phonons). In analogy to the Gram-Schmidt orthogonalization procedure, we perform a unitary transformation into new bosonic modes. Since only $N(N+1)/2$ of them couple to the fermions, a numerically exact treatment is possible. The formalism is applied to a GaAs quantum dot with only two electronic levels. If close to resonance with the phonon energy, the electronic transition shows a splitting due to quantum mechanical level repulsion. This is driven mainly by one bosonic mode, whereas the other two provide further polaronic renormalizations. The numerically exact results for the electron spectral function compare favourably with an analytic solution based on degenerate perturbation theory in the basis of shifted oscillator states. In contrast, the widely used selfconsistent first-order Born approximation proves insufficient in describing the rich spectral features.