  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.
