Pump-probe optical response and four-wave mixing in intersubband transitions of a semiconductor quantum well

We study theoretically the absorption and dispersion of a probe field in the presence of a near-resonant pump field and the four-wave mixing in intersubband transitions of a semiconductor quantum-well structure. We take into account the effects of electron-electron interactions and consider the interaction of the two-subband system with rectangular electromagnetic fields. For the description of the system dynamics we use the effective nonlinear Bloch equations, which due to the effects of electron-electron interactions contain renormalized terms for the transition energy and the applied field. We combine these equations and write the proper differential equations of the density matrix elements for the several nonlinear optical processes, under the rotating wave approximation. These equations are solved analytically, in the steady state, for a GaAs/AlGaAs quantum-well structure. We show that the probe absorption and dispersion and the four-wave mixing spectra of a strongly driven two-subband system can be significantly dependent on the frequency and the intensity of the pump field and on electron sheet density. Specifically, we have found that the absorption, dispersion and the four-wave mixing spectra are practically independent of the electron sheet density in the case that the pump field is on resonance, and in the case for a rather strong pump field independently of its detuning. In the case of moderate intensity and off-resonant pump fields the influence of the electron-electron interactions on the reported spectra is more eminent for positive detuning of the pump field.