Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases

Abstract

Two geometric phases of mixed quantum states, known as the interferometric phase and Uhlmann phase, are generalizations of the Berry phase of pure states. After reviewing the two geometric phases and examining their parallel-transport conditions, we specify a class of cyclic processes that are compatible with both conditions and therefore accumulate both phases through their definitions, respectively. Those processes then facilitate a fair comparison between the two phases. We present exact solutions of two-level and three-level systems to contrast the two phases. While the interferometric phase exhibits finite-temperature transitions only in the three-level system but not the two-level system, the Uhlmann phase shows finite-temperature transitions in both cases. Thus, using the two geometric phases as finite-temperature topological indicators demonstrates the rich physics of topology of mixed states.

12 pages, 2 figures. Eqs. (44), (45), (46), (53), and (54) have been updated

References in corpus (13)

• Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond
• An experimental observation of geometric phases for mixed states using NMR interferometry
• Topology of density matrices
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Measurement of geometric phase for mixed states using single photon interferometry
• Symmetry-protected Topological Phases at Finite Temperature
• Macroscopic Distinguishability Between Quantum States Defining Different Phases of Matter: Fidelity and the Uhlmann Geometric Phase
• Observation of nonadditive mixed state phases with polarized neutrons
• Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications
• Topological Uhlmann phase transitions for a spin-j particle in a magnetic field
• Finite-temperature topological phase transitions of spin-$j$ systems in Uhlmann processes: General formalism and experimental protocols
• Uhlmann phase of coherent states and the Uhlmann-Berry correspondence
• Generalizations of Berry phase and differentiation of purified state and thermal vacuum of mixed states

Cited by in corpus (4)

• Thermal Uhlmann phase in a locally driven two-spin system
• Mixed-state geometric phases of coherent and squeezed spin states
• Uhlmann and scalar Wilczek-Zee phases of degenerate quantum systems
• Interferometric Geometric Phases of $\mathcal{P}\mathcal{T}$-symmetric Quantum Mechanics