Geometric Phases of the Uhlmann and Sjoqvist et al Types for O(3)-Orbits of n-Level Gibbsian Density Matrices
Abstract
Description
We accept the implicit challenge of A. Uhlmann in his 1994 paper, "Parallel Lifts and Holonomy along Density Operators: Computable Examples Using O(3)-Orbits," by, in fact, computing the holonomy invariants for rotations of certain n-level Gibbsian density matrices (n = 2,...,11). From these, we derive, by the tracing operation, the associated geometric phases and visibilities, which we analyze and display. We then proceed analogously, implementing the alternative methodology presented by E. Sjoqvist et al in their letter, "Geometric Phases for Mixed States in Interferometry" (Phys. Rev. Lett. 85, 2845 [2000]). For the Uhlmann case, we are able to also compute several higher-order invariants. We compare the various geometric phases and visibilities for different values of n, and also directly compare the two forms of analysis. By setting one parameter (a) in the Uhlmann analysis to zero, we find that the so-reduced form of the first-order holonomy invariant is simply equal to (-1)^{n+1} times the holonomy invariant in the Sjöqvist et al method. Additional phenomena of interest are reported too.
61 pages, LaTeX, 105 (compressed) encapsulated postscript MATHEMATICA figures. This is a greatly expanded and enhanced version, now incorporating analyses of the Sjoqvist et al geometric phases for mixed states to accompany those of the Uhlmann geometric phases, as well as analyses of higher-order Uhlmann holonomy invariants et al
61 pages, LaTeX, 105 (compressed) encapsulated postscript MATHEMATICA figures. This is a greatly expanded and enhanced version, now incorporating analyses of the Sjoqvist et al geometric phases for mixed states to accompany those of the Uhlmann geometric phases, as well as analyses of higher-order Uhlmann holonomy invariants et al