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

  • * Corresponding author: Katherine A. Newhall

    * Corresponding author: Katherine A. Newhall 

The second author is supported by National Science Foundation (NSF) grant number DMS1615859

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  • We investigate the polarization switching phenomenon for a one-soliton solution to the $ \Lambda $-configuration Maxwell-Bloch equations in the case of initial coherence in the material, corresponding to the forbidden transition between the two lower energy levels. We find two polarization states that are stationary. They are not the purely right- and left-circularly polarized solitons, as in the case of zero initial coherence, but rather two mixed, elliptically polarized, states. These polarization states, of which only one is asymptotically stable, depend on both the initial population levels of the lower states and the coherence value. We also find the existence of superluminal soliton propagation, but through numerical simulations show this solution to be unstable, and therefore likely not realizable physically.

    Mathematics Subject Classification: 78A40, 78A60, 78M35.

    Citation:

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  • 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)

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