Nonabelian Sine-gordon Theory and Its Application to Nonlinear Optics

Using a field theory generalization of the spinning top motion, we construct nonabelian generalizations of the sine-Gordon theory according to each symmetric spaces. A La-grangian formulation of these generalized sine-Gordon theories is given in terms of a deformed gauged Wess-Zumino-Witten action which also accounts for integrably perturbed coset conformal field theories. As for physical applications, we show that they become precisely the effective field theories of self-induced transparency in nonlinear optics. This provides a dictionary between field theory and nonlinear optics. Among many integrable equations, the sine-Gordon equation is one of the most well-known equation which finds countless applications in a wide range of physics due to its ubiquitous nature. In many cases, however, it is desirable to have a " generalized " sine-Gordon equation in order to ac-comodate more realistic physical systems. In this talk, I will show that such generalizations are indeed possible. They are constructed according to each coset G/H and shown to provide a Lagrangian formulation of in-tegrably perturbed coset conformal field theories. Surprisingly, when G/H is restricted to a Hermitian symmetric space, the generalized sine-Gordon equation finds unexpected applications in nonlinear optics. In particular, when G/H = SU (2)/U (1), it describes the two-level self-induced transparency (SIT), a phenomenon of anomalously low energy loss in coherent optical pulse propagation[1]. In order to gain physical insight as well as a handle for the generalization, we first introduce the SIT equation which in the sharp line limit is given by the Maxwell equation under the " slowly varying envelope approximation " ¯ ∂E + 2βP = 0 (1) and the optical Bloch equation ∂D − E * P − EP * = 0 ∂P + 2iξP + 2ED = 0 (2) D represent the electric field, the polarization and the population inversion respectively. Here, we will not concern about the details of nonlinear optics and for the sake of this talk, it suffices to say that Eqs.(1) and (2) are coupled nonlinear partial differential equations in 1+1-dimensional spacetime 1