Dephasing in Quantum Dots: Quadratic Coupling to Acoustic Phonons

A microscopic theory of optical transitions in quantum dots with a carrier-phonon interaction is developed. Virtual transitions into higher confined states with acoustic phonon assistance add a quadratic phonon coupling to the standard linear one, thus extending the independent boson model. Summing infinitely many diagrams in the cumulant, a numerically exact solution for the interband polarization is found. Its full time dependence and the absorption line shape of the quantum dot are calculated. It is the quadratic interaction which gives rise to a temperature-dependent broadening of the zero-phonon line, calculated here for the first time in a consistent scheme.

Figure 1

Polarization amplitude calculated for an InAs QD with $l=3.3\mathrm{n}\mathrm{m}$ at $T=100\mathrm{K}$ with linear and quadratic coupling to acoustic phonons. For the dashed curve, a finite decay $1/\Gamma =12.3\mathrm{p}\mathrm{s}$ has been added by hand to the linear result. The inset shows the (dominant) imaginary part of the eigenvalues ${\Lambda }_j$ at ${\omega }_j$ for $t=10\mathrm{p}\mathrm{s}$ (circles) and $20\mathrm{p}\mathrm{s}$ (diamonds), superimposed to the Fourier transform ${\tilde{D}}_Q(\omega )$ (solid curve).

Figure 2

Absorption spectra of an InAs QD with carrier confinement length of $l=3.3\mathrm{n}\mathrm{m}$ at different temperatures. The ZPL transition energy is taken as zero of energy. Inset: Calculated broadening of the ZPL compared with the experimental results by Borri et al. [2] (circles). A radiative rate of $0.85\mu \mathrm{e}\mathrm{V}$ is added to the calculated phonon-mediated decay rate.