We probe theoretically the emergence of strong coupling in a system consisting of a topological insulator (TI) and a III-V heterostructure using a numerical approach based on the scattering matrix formalism. Specifically, we investigate the interactions between terahertz excitations in a structure composed of Bi$_{2}$Se$_{3}$ and GaAs materials. We find that the interaction between the Bi$_{2}$Se$_{3}$ layer and AlGaAs/GaAs quantum wells with intersubband transitions (ISBTs) in the terahertz frequency regime creates new hybrid modes, namely Dirac plasmon-phonon-ISBT polaritons. The formation of these hybrid modes results in anti-crossings (spectral mode splitting) whose magnitude is an indication of the strength of the coupling. By varying the structural parameters of the constituent materials, our numerical calculations reveal that the magnitude of splitting depends strongly on the doping level and the scattering rate in the AlGaAs/GaAs quantum wells, as well as on the thickness of the GaAs spacer layer that separates the quantum-well structure from the TI layer. Our results reveal the material and device parameters required to obtain experimentally-observable signatures of strong coupling. Our model includes the contribution of an extra two-dimensional hole gas (2DHG) that is predicted to arise at the Bi$_{2}$Se$_{3}$/GaAs interface, based on density functional theory (DFT) calculations that explicitly account for details of the atomic terminations at the interface. The presence of this massive 2DHG at the TI/III-V interface shifts the dispersion of the Dirac plasmon-ISBT polaritons to higher frequencies. The damping rate at this interface, in contrast, compensates the effect of the 2DHG. Finally, we observe that the phonon resonances in the TI layer are crucial to the coupling between the THz excitations in the TI and III-V materials.
