Correlation of Dirac potentials and atomic inversion in cavity quantum electrodynamics

Controlling the time evolution of the population of two states in cavity quantum electrodynamics (QED) is necessary by tunning the \textit{modified} Rabi frequency in which the $\textit{extra}$ classical effect of electromagnetic field is taken into account.The theoretical explanation underlying the perturbation of potential on spatial regime of bloch sphere is by the use of (Bagrov, Baldiotti, Gitman, and Shamshutdinova) BBGS-Darboux transformations \cite{bagrov14darboux} on the electromagnetic field potential in one-dimensional stationary Dirac model in which the Pauli matrices are the central parameters for controlling the collapse and revival of the Rabi oscillations. It is shown that by choosing $\sigma_{1}$ in the transformation generates the parabolic potential causing the total collapse of oscillations; while $\{\sigma_{2}, \sigma_{3}\}$ yield the harmonic oscillator potentials ensuring the coherence of qubits.