Influence of Resonant Frequency Shifts on Superradiant Damping

Abstract

The interaction between two electric dipoles ^→_{p 1} and ^→_{p 2} at position ^→_{r 1} and ^→_{r 2}, analyzed at frequency kc, is [1]

$$\begin{array}{l} v = - {e^{ikr}}[\left( {{{\vec p}_1} \cdot {{\vec p}_2} - \hat r \cdot {{\vec p}_1}\,\hat r \cdot {{\vec p}_2}} \right)\frac{{{k^2}}}{r}\\ + \left( {{{\vec p}_1} \cdot {{\vec p}_2} - 3\hat r \cdot {{\vec p}_1}\,\hat r \cdot {{\vec p}_2}} \right)\left( {\frac{{ik}}{{{r^2}}} - \frac{1}{{{r^3}}}} \right)]\\ = \frac{1}{{{r^3}}}\left( {{{\vec p}_1} \cdot {{\vec p}_2} - 3\hat r \cdot {{\vec p}_1}\,\hat r \cdot {{\vec p}_2}} \right) - \frac{{{k^2}}}{{2r}}\left( {{{\vec p}_1} \cdot {{\vec p}_2} - \hat r \cdot {{\vec p}_1}\,\hat r \cdot {{\vec p}_2}} \right)\\ - \frac{2}{3}\,i\,{k^3}\,{{\vec p}_1} \cdot {{\vec p}_2} + 0\left[ r \right]\\ where\,{{\vec r}_1} - {{\vec r}_2} = r\hat r. \end{array}$$

This work was supported in part by the Joint Services Electronics Program (U.S. Army, U.S. Navy and U.S. Air Force) under Contract DAAB07-69-C-0383, in part by the Atomic Energy Commission, in part by the Air Force Office of Scientific Research, OAR, under AFOSR Grant 70-1866.