Infinite-Order Excitonic Bloch Equations for Asymmetric Nanostructures

We present a new exciton-based formalism for calculating the coherent response of asymmetric semiconductor multiple quantum well structures to ultrashort optical pulses valid to infinite order in the optical field and including the self-generated intraband fields. We use these equations to calculate and explain the oscillations with time delay of peaks in the spectrally resolved degenerate four-wave mixing signals from biased semiconductor superlattices, obtaining good agreement with experiment.

Figure 1

The time-integrated intensity versus ${\tau }_{21}$ for ${\omega }_c={\omega }_0-2.27{\omega }_B$, $n_o=13$, and a density of $6.36\times {10}^9{\mathrm{c}\mathrm{m}}^{-2}$ for the three different directions, ${\mathbf{K}}_n$ indicated. For ${\mathbf{K}}_{-3}$, results for $n_0=3$ (dashed line) and 5 (dotted line) are also plotted.

Figure 2

SRFWM intensity versus frequency at ${\omega }_c={\omega }_0-2.27{\omega }_B$ and a density of $9.3\times {10}^9{\mathrm{c}\mathrm{m}}^{-2}$ for a sequence of delay times. The inset is described in the text.

Figure 3

The $\mu =-1$ SRFWM peak energy relative to $\hslash {\omega }_{-1}$ versus ${\tau }_{21}$ at ${\omega }_c={\omega }_0-2.27{\omega }_B$ for the densities given above each curve (in units of ${10}^9{\mathrm{c}\mathrm{m}}^{-2}$). The curve offsets are a real effect due to intraband renormalization of the dc field. The inset shows peak oscillation amplitude as a function of density for $({\omega }_c-{\omega }_0)/{\omega }_B=-1.24$ (circles), $-2.27$ (triangles), and $-2.83$ (squares).