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August  2020, 13(8): 2285-2301. doi: 10.3934/dcdss.2020189

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

Katherine A. Newhall 1,, , Gregor Kovačič 2, and  Ildar Gabitov 3,

1. 

University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, USA
2. 

Rensselaer Polytechnic Institute, Troy, NY 12180, USA
3. 

University of Arizona, Tucson, AZ 85721-0089, USA

* Corresponding author: Katherine A. Newhall

Received  July 2018 Revised  December 2018 Published  August 2020 Early access  November 2019

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

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.

Keywords: Maxwell-Bloch equations, dressing method, solitons, polarization switching.
Mathematics Subject Classification: 78A40, 78A60, 78M35.
Citation: Katherine A. Newhall, Gregor Kovačič, Ildar Gabitov. Polarization dynamics in a resonant optical medium with initial coherence between degenerate states. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2285-2301. doi: 10.3934/dcdss.2020189
References:
[1]

M. J. Ablowitz, D. J. Kaup and A. C. Newell, Coherent pulse propagation, a dispersive, irreversible phenomenon, J. Math. Phys., 15 (1974), 1852-1858.  doi: 10.1063/1.1666551.

[2]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover Publications, New York, 1987.

[3]

E. P. Atkins, P. R. Kramer, G. Kovačič and I. R. Gabitov, Stochastic pulse switching in a degenerate resonant optical medium, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.043834.

[4]

A. M. Basharov and A. I. Maimistov, Self-induced transparency when the resonance energy levels are degenerate, Sov. Phys. Jetp, 60 (1984), 913-919. 

[5]

L. A. Bolshov, V. V. Likhanskii and M. I. Persiantsev, Contribution to the theory of coherent interaction of light pulses with resonant multilevel media, Sov. Phys. Jetp, 57 (1983), 524-528. 

[6] R. W. Boyd, Nonlinear Optics, Elsevier/Academic Press, Amsterdam, 2008.  doi: 10.1016/B978-0-12-121682-5.X5000-7.
[7] P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge Studies in Modern Optics, 9, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139167994.
[8]

J. A. Byrne, I. R. Gabitov and G. Kovačič, Polarization switching of light interacting with a degenerate two-level optical medium, Phys. D, 186 (2003), 69-92.  doi: 10.1016/S0167-2789(03)00245-8.

[9]

S. Chakravarty, B. Prinari and M. J. Ablowitz, Inverse scattering transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening, Phys. D, 278-279 (2014), 58-78.  doi: 10.1016/j.physd.2014.04.003.

[10]

V. Chernyak and V. Rupasov, Polarization effects in self-induced transparency theory, Phys. Lett. A, 108 (1985), 434-436.  doi: 10.1016/0375-9601(85)90032-5.

[11]

B. D. Clader and J. H. Eberly, Two-pulse propagation in a partially phase-coherent medium, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.033803.

[12]

I. Gabitov, V. Zakharov and A. Mikhailov, The Maxwell-Bloch equation and the method of the inverse scattering problem, Teoret. Mat. Fiz., 63 (1985), 11-31. 

[13]

M. J. Konopnicki and J. H. Eberly, Simultaneous propagation of short different-wavelength optical pulses,, Phys. Rev. A, 24 (1981), 2567-2583.  doi: 10.1103/PhysRevA.24.2567.

[14]

M. J. Konopnicki, P. D. Drummond and J. H. Eberly, Theory of lossless propagation of simultaneous different-wavelength optical pulses, Opt. Commun., 36 (1981), 313-316.  doi: 10.1016/0030-4018(81)90382-5.

[15]

G. Lamb, Coherent-optical-pulse propagation as an inverse problem, Phys. Rev. A, 9 (1974), 422-430.  doi: 10.1103/PhysRevA.9.422.

[16]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure and Appl. Math, 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503.

[17]

S. Li, G. Biondini, G. Kovačič and I. Gabitov, Inverse-scattering transform for the $\Lambda$-configuration Maxwell-Bloch system with nonzero background, work in progress.

[18]

A. Ma$\breve{\rm i}$mistov and A. M. Basharov, Present state of self-induced transparency theory, Phys. Reports - Rev. Sect. Phys. Lett., 191 (1990), 1-108. 

[19]

A. I. Ma$\breve{\rm i}$mistov and Y. M. Sklyarov, The coherent interaction of light-pulses with a 3-level medium, Optika I Spektroskopiya, 59 (1985), 760-763. 

[20]

A. I. Ma$\breve{\rm i}$mistov, Rigorous theory of self-induced transparency in the case of a double resonance in a three-level medium, Sov. J. Quantum Electron., 14 (1984), 385-389.  doi: 10.1070/QE1984v014n03ABEH004908.

[21]

A. C. Newell and J. V. Moloney, Nonlinear Optics, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company, Redwood City, CA, 1992.

[22]

K. A. Newhall, E. P. Atkins, P. R. Kramer, G. Kovačič and I. R. Gabitov, Random polarization dynamics in a resonant optical medium, Optics Lett., 38 (2013), 893-895.  doi: 10.1364/OL.38.000893.

[23]

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984.

[24]

A. V. Rybin, Application of the Darboux transformation to inverse problems with variable spectral parameters, J. Phys. A, 24 (1991), 5235-5243.  doi: 10.1088/0305-4470/24/22/007.

[25]

K. Shimoda, Introduction to Laser Physics, Springer Series in Optical Sciences, 44, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-662-13548-8.

[26]

L. J. Wang, A. Kuzmich and A. Dogariu, Gain-assisted superluminal light propagation, Nature, 406 (2000), 277-279.  doi: 10.1038/35018520.

show all references

References:
[1]

M. J. Ablowitz, D. J. Kaup and A. C. Newell, Coherent pulse propagation, a dispersive, irreversible phenomenon, J. Math. Phys., 15 (1974), 1852-1858.  doi: 10.1063/1.1666551.

[2]

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms, Dover Publications, New York, 1987.

[3]

E. P. Atkins, P. R. Kramer, G. Kovačič and I. R. Gabitov, Stochastic pulse switching in a degenerate resonant optical medium, Phys. Rev. A, 85 (2012). doi: 10.1103/PhysRevA.85.043834.

[4]

A. M. Basharov and A. I. Maimistov, Self-induced transparency when the resonance energy levels are degenerate, Sov. Phys. Jetp, 60 (1984), 913-919. 

[5]

L. A. Bolshov, V. V. Likhanskii and M. I. Persiantsev, Contribution to the theory of coherent interaction of light pulses with resonant multilevel media, Sov. Phys. Jetp, 57 (1983), 524-528. 

[6] R. W. Boyd, Nonlinear Optics, Elsevier/Academic Press, Amsterdam, 2008.  doi: 10.1016/B978-0-12-121682-5.X5000-7.
[7] P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics, Cambridge Studies in Modern Optics, 9, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9781139167994.
[8]

J. A. Byrne, I. R. Gabitov and G. Kovačič, Polarization switching of light interacting with a degenerate two-level optical medium, Phys. D, 186 (2003), 69-92.  doi: 10.1016/S0167-2789(03)00245-8.

[9]

S. Chakravarty, B. Prinari and M. J. Ablowitz, Inverse scattering transform for 3-level coupled Maxwell-Bloch equations with inhomogeneous broadening, Phys. D, 278-279 (2014), 58-78.  doi: 10.1016/j.physd.2014.04.003.

[10]

V. Chernyak and V. Rupasov, Polarization effects in self-induced transparency theory, Phys. Lett. A, 108 (1985), 434-436.  doi: 10.1016/0375-9601(85)90032-5.

[11]

B. D. Clader and J. H. Eberly, Two-pulse propagation in a partially phase-coherent medium, Phys. Rev. A, 78 (2008). doi: 10.1103/PhysRevA.78.033803.

[12]

I. Gabitov, V. Zakharov and A. Mikhailov, The Maxwell-Bloch equation and the method of the inverse scattering problem, Teoret. Mat. Fiz., 63 (1985), 11-31. 

[13]

M. J. Konopnicki and J. H. Eberly, Simultaneous propagation of short different-wavelength optical pulses,, Phys. Rev. A, 24 (1981), 2567-2583.  doi: 10.1103/PhysRevA.24.2567.

[14]

M. J. Konopnicki, P. D. Drummond and J. H. Eberly, Theory of lossless propagation of simultaneous different-wavelength optical pulses, Opt. Commun., 36 (1981), 313-316.  doi: 10.1016/0030-4018(81)90382-5.

[15]

G. Lamb, Coherent-optical-pulse propagation as an inverse problem, Phys. Rev. A, 9 (1974), 422-430.  doi: 10.1103/PhysRevA.9.422.

[16]

P. D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Commun. Pure and Appl. Math, 21 (1968), 467-490.  doi: 10.1002/cpa.3160210503.

[17]

S. Li, G. Biondini, G. Kovačič and I. Gabitov, Inverse-scattering transform for the $\Lambda$-configuration Maxwell-Bloch system with nonzero background, work in progress.

[18]

A. Ma$\breve{\rm i}$mistov and A. M. Basharov, Present state of self-induced transparency theory, Phys. Reports - Rev. Sect. Phys. Lett., 191 (1990), 1-108. 

[19]

A. I. Ma$\breve{\rm i}$mistov and Y. M. Sklyarov, The coherent interaction of light-pulses with a 3-level medium, Optika I Spektroskopiya, 59 (1985), 760-763. 

[20]

A. I. Ma$\breve{\rm i}$mistov, Rigorous theory of self-induced transparency in the case of a double resonance in a three-level medium, Sov. J. Quantum Electron., 14 (1984), 385-389.  doi: 10.1070/QE1984v014n03ABEH004908.

[21]

A. C. Newell and J. V. Moloney, Nonlinear Optics, Advanced Topics in the Interdisciplinary Mathematical Sciences, Addison-Wesley Publishing Company, Redwood City, CA, 1992.

[22]

K. A. Newhall, E. P. Atkins, P. R. Kramer, G. Kovačič and I. R. Gabitov, Random polarization dynamics in a resonant optical medium, Optics Lett., 38 (2013), 893-895.  doi: 10.1364/OL.38.000893.

[23]

S. P. Novikov, S. V. Manakov, L. P. Pitaevskii and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method, Contemporary Soviet Mathematics, Consultants Bureau [Plenum], New York, 1984.

[24]

A. V. Rybin, Application of the Darboux transformation to inverse problems with variable spectral parameters, J. Phys. A, 24 (1991), 5235-5243.  doi: 10.1088/0305-4470/24/22/007.

[25]

K. Shimoda, Introduction to Laser Physics, Springer Series in Optical Sciences, 44, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-662-13548-8.

[26]

L. J. Wang, A. Kuzmich and A. Dogariu, Gain-assisted superluminal light propagation, Nature, 406 (2000), 277-279.  doi: 10.1038/35018520.

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 $, the soliton switches its polarization to a mixed-polarization state
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Figure 2.  The solution to the Maxwell-Bloch equations appearing in (8) plotted in the lab frame of reference for an initially linearly-polarized soliton pulse ($ 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)
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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 $
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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 $
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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
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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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