Density operator description of geometric phenomena in the ray space

Joseph Samuel

Eugene Feenberg, Srećko Kilić

Apoorva G. Wagh, Veer Chand Rakhecha

For a general quantal system, physical properties of the finite difference between a pair of density operators are derived and a complete set of generators for the associated 2-subspace is obtained. Each infinitesimal step in any general ray-space evolution takes place in the local 2-subspace and can be equated to a `spin' rotation for the equivalent spin-½ particle. Hence a completely general Hamiltonian implementing a given ray space evolution comprises Pauli operators in the local 2-subspaces, constructed using the given density operator and its differential. Dynamical phase identifies with the phase in a rotating reference frame in which the `spin' remains stationary. The transformation from this rotating frame to the laboratory frame effects a parallel transport evolution, producing geometric phase. A density operator equation is derived for geodesics. Geometric phase arises from the parallel transport of the local `spin' and equals minus half the integral of 2-subspace solid angles.

Armin Uhlmann

Vladislav Yu. Toronov, Vladimir Leonardovich Derbov

We show that such phenomena of laser dynamics as mean-phase-slope jumps and temporal phase jumps at resonance between the cavity and spectral line frequencies are intrinsically connected with the topology of attractors in the space of rays and can be interpreted as the manifestations of the geometric-phase properties of the evolution operator.

John E. Harriman