Stimulated Electric Polarization and Photon Echoes

The change in the wave function of an atom or molecule which interacts with saturating incident radiation is shown to have a time evolution operator $U(t, 0)=\Sigma \mathrm{}\left|n\right)\left(n^{\prime }\right|B\left(n\alpha \right)B^{*}\left(n^{\prime }\alpha \right)\times e^{-i\left[\xi \right(n\alpha )+\frac{E\left(n\right)\mathrm{}}{\hslash }]t}$, where the sum over $n$ and $n^{\prime }$ is over both the ($2J_a+1\mathrm{}$) values of $m_a$ and ($2J_b+1\mathrm{}$) values of $m_b$, and the sum over $\alpha$ is over the $(2\mathrm{}{J}_a+1)+(2\mathrm{}{J}_b+1)$ modes indicated by the index $\alpha$. The eigenvalues $\xi \left(n\alpha \right)$ and their eigenvectors $B\left(n\alpha \right)$ depend on the intensity and polarization of the incident radiation, produce a modulation term in the electric polarization $\overrightarrow{\mathrm{P}}$ of the molecule, and give rise to anomalous polarization in the stimulated radiation. This unitary operator is used to discuss the radiation stimulated by two pulses or photon echoes. Echoes from elliptic pulses are discussed for $J<~2$, and the linear-linear sequence is compared with the theory of Gordon, Wang, Patel, Slusher, and Tomlinson. Echoes from linear-circular and circular-linear pulse sequences are discussed in detail.