Progress in Optics

10: Self-pulsing in the case of adiabatic elimination of both atomic variables

10.1: The general configuration

Let us set

$\begin{array}{ll}\hfill \delta f_n(\tau )& =\delta {\tilde{f}}_n(\tau )e^{-i{\alpha }_n\tau },\hfill \end{array}$

  (71a)

$\begin{array}{ll}\hfill \delta p_n(\phi ,\tau )& =\delta {\tilde{p}}_n(\phi ,\tau )e^{-i{\alpha }_n\tau },\hfill \end{array}$

  (71b)

$\begin{array}{ll}\hfill \delta d_n(\phi ,\tau )& =\delta {\tilde{d}}_n(\phi ,\tau )e^{-i{\alpha }_n\tau },\hfill \end{array}$

  (71c)

so that Eqs. (56) become

$\begin{array}{ll}\hfill \frac{d\delta {\tilde{f}}_n(\tau )}{d\tau }& =-k\left[(1+i\theta )\delta {\tilde{f}}_n(\tau )-\frac{A}{2\pi }{\int }_{-\pi }^{+\pi }d\phi {\text{e}}^{-i\phi }\delta {\tilde{p}}_n(\phi ,\tau )\right],\hfill \end{array}$

  (72a)

$\begin{array}{ll}{\gamma }_{\perp }^{-1}\frac{\partial \delta {\tilde{p}}_n(\phi ,\tau )}{\partial \tau }=& -\left(1+i\mathrm{\Delta }-i\frac{{\alpha }_n}{{\gamma }_{\perp }}\right)\delta {\tilde{p}}_n(\phi ,\tau )\end{array}$