Polarization dynamics in a resonant optical medium with initial coherence between degenerate states

Figure 1.  The solution to the Maxwell-Bloch equations appearing in (10), plotted in the lab frame of reference for an initially circularly-polarized soliton pulse ($ a = 1 $, $ b = 0 $, $ \beta = 1/3 $). With the virtual polarization initially $ \mu_0 = 0.05 $, and the initial population offset $ \alpha = \tfrac12 $, the soliton switches its polarization to a mixed-polarization state

Figure 2.  The solution to the Maxwell-Bloch equations appearing in (8) plotted in the lab frame of reference for an initially linearly-polarized soliton pulse ($ a = 1/2 $, $ b = 1/2 $, $ \beta = 1/3 $). The color bar is the same for each row. The other parameters are $ \alpha = -0.25 $, and $ \mu_0 = 0.4 $. The corresponding value of $ \sin(2\eta) $ for this soliton as it propagates is shown in Fig. 3(b)

Figure 3.  (a) The limiting value of $ \sin(2\eta) $ for the stable invariant polarization state given by the expression in (19), as a function of $ \alpha $ for the indicated values of $ \mu_0 $ starting at 0 and increasing by $ 0.1 $ until $ \mu_0 = 0.7 $. (b) Dynamics of $ \sin(2\eta) $ approaching its limiting value in (19) for the soliton depicted in Fig. 2, $ \alpha = -0.25 $, and $ \mu_0 = 0.4 $

Figure 4.  The dynamics of the angles $ \eta $ and $ \psi $ for the case of a soliton with the eigenvalue $ \lambda_0 = \tfrac13 + \tfrac13i $. Two cases are shown, one for $ \mu_0 = 0 $ and one for $ \mu_0 = 0.02 $ showing the difference between the behavior of the angle $ \psi $. The other parameters are $ a = 1 $, $ b = \tfrac{1}{10} $, and $ \alpha = \tfrac12 $

Figure 5.  An unphysical solution to the Maxwell-Bloch equations appearing in (8) plotted in the lab frame of reference for an initially linearly polarized soliton pulse ($ a = 1/2 $, $ b = 1/2 $, $ \beta = 1/3 $). The color bar is the same for each row. The other parameters are $ \alpha = 0.75 $, and $ \mu_0 = 0.7 $. The dashed lines in the top row are drawn at the speed of light ($ c = 1 $) highlighting the fact that the soliton is propagating faster than the speed of light

Figure 6.  The electric-field envelopes computed by numerically solving the Maxwell-Bloch equations in (1) using the method described in Appendix C with $ \Delta x = \Delta t = 0.05 $. The faster-than-light soliton solution shown in Fig. 5 ($ a = 1/2 $, $ b = 1/2 $, $ \beta = 1/3 $, $ \alpha = 0.75 $, and $ \mu_0 = 0.7 $) is unstable and sheds radiation, slowing down to propagate at the speed of light (the dashed line)