Geometric phases in two-level atoms excited by pulses propagating without loss.

Abstract

We consider the geometrical phases arising in the state vector of two-level atoms due to their interaction with a self-consistently generated classical electrical field propagating without loss through the atomic medium. Three conservation laws are shown to exist generally and are used to solve for the individual quantum amplitudes, phases, and the electric field. We calculate the geometrical phases in two situations: (a) where the atoms are initially in the ground state and (b) where the initial state is a coherent superposition of the ground and excited states. In both cases the geometrical phase is the Aharonov-Anandan phase resulting from the atomic state vector tracing out a closed curve on the projective Hilbert space--here the Bloch sphere. We show that geometric quantities associated with the curve on the Bloch sphere are directly related to physical observables. The solid angle subtended by the closed curve (shown to equal twice the geometric phase) is a measure of the maximum atomic inversion, while the speed with which the curve is traced is related to the energy uncertainty in the state. An experimental method to observe the total phase change in a two-level subsystem is outlined, using photon echoes in a three-level medium.