Maxwell–Bloch equations, C Neumann system and Kaluza–Klein theory

Abstract

The Maxwell–Bloch equations are represented as the equation of motion for a continuous chain of coupled C Neumann oscillators on the three-dimensional sphere. This description enables us to find new Hamiltonian and Lagrangian structures of the Maxwell–Bloch equations. The symplectic structure contains a topologically non-trivial magnetic term which is responsible for the coupling. The coupling forces are geometrized by means of an analogue of Kaluza–Klein theory. The conjugate momentum of the additional degree of freedom is precisely the speed of light in the medium. It can also be thought of as the strength of the coupling. The Lagrangian description has a structure similar to that of the Wess–Zumino–Witten–Novikov action. We describe two families of solutions of the Maxwell–Bloch equations which are expressed in terms of the C Neumann system. One family describes travelling non-linear waves whose constituent oscillators are the C Neumann oscillators in the same way as the harmonic oscillators are the constituent oscillators of the harmonic waves. The 2π-pulse soliton is a member of this family.