Dynamic symmetries of the vacuum field Jaynes-Cummings model

Abstract

We use the standard Jaynes-Cummings model with the field initially in the vacuum state and the atom in an arbitrary superposition state to illustrate the concept of dynamic symmetry in quantum optics. By dynamic symmetry we mean a situation in which a Hamiltonian is expressible as a linear combination of operators obeying a Lie algebra. This circumstance permits a psuedospin (or coherence vector) treatment to be given of the dynamics of the system. We give two such interpretations of the Jaynes-Cummings model. In the first we regard the atom and the field as a pair of interacting two-level systems and formulate a SU(2)-SU(2) dynamic symmetry. In the second we interpret the composite atom-field system is a three-level system and formulate a SU(3) dynamic symmetry. The former model illustrates the ideas of Fano1 and the latter those of Hioe and Eberly.2 We give a complete pseudospin description from both points of view and also obtain all the constants of motion.