Complex sine-Gordon Theory for Coherent Optical Pulse Propagation

It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency. 1 E-mail address; [email protected] 2 E-mail address; [email protected] Self-induced transparency(SIT), a phenomenon of anomalously low energy loss in coherent optical pulse propagation, was first discovered by McCall and Hahn[1] and the integrability of the SIT equation was demonstrated by employing the inverse scattering method[2]. When phase variation is ignored in the case for a symmetric frequency distribution g(∆w) of inhomogeneous broadening, McCall and Hahn have proved an area theorem for pulse propagation. In the sharp line limit where the frequency distribution is sharply peaked at the carrier frequency w0 such that g(∆w) = δ(w − w0), the SIT equation reduces to the wellknown sine-Gordon equation and the 2π area pulse becomes a 1-soliton of the sine-Gordon theory. However, when phase variation is included, the area theorem no longer holds and the structure of SIT in general has not been well understood except for the construction of explicit solutions by the inverse scattering method[2][3]. In particular, despite its integrability, the SIT theory in the sharp line limit has not been identified with a known 1+1 dimensional integrable field theory, which made a systematic understanding of SIT in terms of a lagrangian field theory impossible. The purpose of this Letter is to show that the SIT theory with phase variation can be identified with the complex sine-Gordon theory in the sharp line limit. The complex sine-Gordon theory, a generalization of the sine-Gordon theory with a phase degree of freedom, can be reformulated in terms of a nonlinear sigma model which is known as the integrably deformed gauged Wess-Zumino-Witten(WZW) model associated with the coset SU(2)/U(1)[4][5]. This allows us to address various new aspects of SIT in terms of characteristics of the complex sine-Gordon theory; e.g. topological v.s. non-topological solitons, local gauge symmetry, the U(1)-charge conservation, the chiral symmetry and the Krammers-Wannier duality for dark v.s. bright solitons. We also explain the off-resonance effect and inhomogeneous broadening of SIT in the context of the local gauge symmetry of the present formulation. The SIT equation is given by ∂̄E + 2β < P > = 0 ∂D − E∗P − EP ∗ = 0 ∂P + 2i∆wP + 2ED = 0 (1) where ∆w = w−w0 , ∂ ≡ ∂/∂z , ∂̄ ≡ ∂/∂z̄ , z = t−x/c, z̄ = x/c. E, P and D represent the electric field, the polarization and the population inversion respectively. The bracket denotes an averaging over the distribution function of inhomogeneous broadening. Since Eq.(1) is invariant under the interchange (β,E, P,D) ↔ (−β,E,−P,−D), we assume the coupling constant β to be positive which we set to one by rescaling E, P and D. In a simpler case where phase variation is ignored to make E real and the frequency distribution is sharply peaked at the carrier frequency(∆w = 0), we may parametrize E, P and D by E = E∗ = ∂φ , < P >= P = − sin 2φ , D = cos 2φ . (2) Then the SIT equation reduces to the well-known sine-Gordon equation ∂̄∂φ− 2β sin 2φ = 0. (3)