Relation between phenomenological few-level models and microscopic theories of the nonlinear optical response of semiconductor quantum wells

We analyze a generic atomic few-level model for the third-order $\left({\chi }^{\left(3\right)}\right)$ optical response of semiconductor quantum wells. The purpose is to seek a good understanding of the physical nature of the model’s parameters in terms of the material’s microscopic constituents and their motions. The strategy is to bring the algebraic expression of the ${\chi }^{\left(3\right)}$ interband polarization in the few-level model to a form similar to that derived in microscopic theories. Most importantly, in the coherent limit, we make the “time evolution structure” of the interband polarization in the few-level and microscopic formalisms coincide, whereby the atomic model’s parameters can be interpreted microscopically through a comparison of terms of similar structure in the two formalisms. We also discuss how the conclusion of this comparison can be changed by the introduction of phenomenological dephasing and decay into both theories.

Figure 1

Sketch of the measurement configurations. The wave vectors are shown with their in-plane components. The angles of the pump and probe propagation relative to normal incidence are exaggerated for clarity.

Figure 2

Level schematic of the few-level model with ground state, one-exciton states, and two-exciton states. The symbols are defined in the text.

Figure 3

Fourier transforms of retarded two-exciton correlation functions calculated in microscopic theory vs frequency $\Omega$ (after Ref. 71). In the $+-$ channel, the contributions from the bound biexciton (dash-dotted line) and the two-exciton continuum (dotted line) are shown separately; the total is shown as solid line.

Figure 4

Fourier transforms of the phenomenological two-exciton correlation functions evaluated for the seven-level system vs frequency. The parameter values are given in the text.