Selection rules for the excitation of quantum dots by spatially structured light beams: Application to the reconstruction of higher excited exciton wave functions

Spatially structured light fields applied to semiconductor quantum dots yield fundamentally different absorption spectra than homogeneous beams. In this paper, we provide a detailed theoretical discussion of the resulting spectra for different light beams using a cylindrical multipole expansion. For the description of the quantum dots we employ a model based on the envelope function approximation including Coulomb interaction and valence band mixing. The combination of a single spatially structured light beam and state mixing allows all exciton states in the quantum dot to become optically addressable. Furthermore, we demonstrate that the beams can be tailored such that single states are selectively excited, without the need of spectral separation. Using this selectivity, we propose a method to measure the exciton wave function of the quantum dot eigenstate. The measurement goes beyond electron density measurements by revealing the spatial phase information of the exciton wave function. Such an extraction of phase information is known from polarization-sensitive measurements; however, here the infinitely large spatial degree of freedom can be accessed by the beam profile in addition to the two-dimensional polarization degree of freedom.