Electron-phonon interaction in quantum dots: A solvable model

The relaxation of electrons in quantum dots via phonon emission is hindered by the discrete nature of the dot levels ("phonon bottleneck"). In order t o clarify the issue theoretically we consider a system of N discrete fermio nic states (dot levels) coupled to an unlimited number of bosonic modes wit h the same energy (dispersionless phonons). In analogy to the Gram-Schmidt orthogonalization procedure, we perform a unitary transformation into new b osonic modes. Since only N(N + 1)/2 of them couple to the fermions, a numer ically exact treatment is possible. The formalism is applied to a GaAs quan tum dot with only two electronic levels. If close to resonance with the pho non energy, the electronic transition shows a splitting due to quantum mech anical level repulsion. This is driven mainly by one bosonic mode, whereas the other two provide further polaronic renormalizations. The numerically e xact results for the electron spectral function compare favorably with an a nalytic solution based on degenerate perturbation theory in the basis of sh ifted oscillator states. In contrast, the widely used self-consistent first -order Born approximation proves insufficient in describing the rich spectr al features.