Effective discrete-level density matrix model for unipolar quantum optoelectronic devices

Increasingly, unipolar quantum optoelectronic devices such as quantum cascade lasers are employed for the targeted generation of dynamic waveforms in the mid-infrared and terahertz regime. These include for example short-pulse trains, frequency combs and solitons. For the theoretical investigation and targeted development of these devices, suitable semiclassical models such as Maxwell-Bloch type equations have been developed, which employ a two- or multilevel density matrix description for the electron dynamics and a classical propagation equation for the optical resonator field. Unipolar devices typically utilize quantized conduction band states as optical levels. For quantum well and wire structures, the electron states are additionally characterized by a wavevector associated with free motion in the non-confined directions. This degree of freedom can give rise to nonparabolicity effects as well as Bloch gain, both leading to gain asymmetry and linewidth enhancement. However, fully accounting for the wavevector greatly increases the computational cost of the density matrix approach. Here, we introduce an effective discrete-level density matrix model, which includes these effects via correction factors obtained by suitable wavevector averaging. These parameters can be extracted from carrier transport simulations along with other required input data, yielding a self-consistent model. Coupling the effective density matrix description to optical propagation equations results in an effective Maxwell-density matrix approach, which is well-suited for dynamic simulations of quantum optoelectronic devices.