Horseshoe-shaped maps in chaotic dynamics of atom-field interaction.

Abstract

We show that in addition to the length of the Bloch vector there exists an energylike integral of motion for the Maxwell-Bloch equations. This allows us to recast the semiclassical problem of atom-field interaction into a classical Hamiltonian problem of two coupled oscillators which contain both periodic and homoclinic orbits. Melnikov's method is then used to demonstrate the existence of Smale horseshoe-shaped maps in its dynamics which preclude the existence of any further analytic integral of motion. The treatment yields a systematic procedure for understanding the mechanism of instability in this fundamental model of quantum optics.