Coherent control for a two-level system coupled to phonons

The interband polarization induced by two phase-locked pulses in a semiconductor shows strong interference effects depending on the time, ${\tau }_1,$ separating the pulses. The four-wave mixing signal diffracted from a third pulse delayed by $\tau$ is coherently controlled by tuning ${\tau }_1.$ The four-wave mixing response is evaluated exactly for a two-level system coupled to a single LO phonon. In the weak coupling regime it shows oscillations with the phonon frequency which turn into sharp peaks at multiples of the phonon period for a larger coupling strength. Destructive interferences between the two phase-locked pulses produce a splitting of the phonon peaks into a doublet. For fixed $\tau$ but varying ${\tau }_1,$ the signal shows rapid oscillations at the interband transition frequency, whose amplitude exhibits bursts at multiples of the phonon period.