Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states

Abstract

While a pure quantum state may accumulate both the Berry phase and dynamic phase as it undergoes a cyclic path in the parameter space, the situation is more complicated when mixed quantum states are considered. From the Ulhmann bundle, a mixed quantum state can accumulate the Ulhmann phase if the parallel-transport condition is satisfied. However, we show that the Ulhmann process is in general not compatible with the evolution equation of the density matrix governed by the Hamiltonian. Thus, a mixed quantum state usually accumulates a dynamic phase during its time evolution. We present the expression of the dynamic phase for mixed quantum states. In examples of quasi-static one-dimensional two-band models and simple harmonic oscillator, the dynamic phase can take multiple discrete values at infinitely high temperature due to the resonant points. However, the behavior differs if the energy spectrum is continuous without a band gap. Moreover, there is no natural analog of the dynamic phase in classical systems.

17 pages, 3 figures

References in corpus (33)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• $Z_2$ Topological Order and the Quantum Spin Hall Effect
• Topological Insulators in Three Dimensions
• Classification of topological quantum matter with symmetries
• Topological invariants of time-reversal-invariant band structures
• Quantum Spin Hall Effect
• Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond
• Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems
• Conductivity of disordered quantum lattice models at infinite temperature: Many-body localization
• On quantumness in multi-parameter quantum estimation
• Geometric phases for non-degenerate and degenerate mixed states
• Topology of density matrices
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Probing the topology of density matrices
• Topological indices for open and thermal systems via Uhlmann's phase
• Observation of topological Uhlmann phases with superconducting qubits
• Comparison and unification of non-Hermitian and Lindblad approaches with applications to open quantum optical systems
• Geometric phase distributions for open quantum systems
• On singularities of the mixed state phase
• Symmetry-protected Topological Phases at Finite Temperature
• The Uhlmann connection in fermionic systems undergoing phase transitions
• Quantum correlations at infinite temperature: the dynamical Nagaoka effect
• Reply to `Singularities of the mixed state phase'
• Uhlmann number in translational invariant systems
• Fidelity and Uhlmann connection analysis of topological phase transitions in two dimensions
• Thermal Uhlmann Chern number from the Uhlmann connection for extracting topological properties of mixed states
• Kinematic approach to off-diagonal geometric phases of nondegenerate and degenerate mixed states
• Uhlmann fidelities from tensor networks
• Topological order, mixed states and open systems
• Momentum maps for mixed states in quantum and classical mechanics
• Nonequilibrium quantum order at infinite temperature: spatiotemporal correlations and their generating functions

Cited by in corpus (14)

• Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications
• Finite-temperature topological phase transitions of spin-$j$ systems in Uhlmann processes: General formalism and experimental protocols
• Local geometry and quantum geometric tensor of mixed states
• Uhlmann phase of coherent states and the Uhlmann-Berry correspondence
• Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases
• Uhlmann quench and geometric dynamic quantum phase transition of mixed states
• Uhlmann holonomy against Lindblad dynamics of topological systems at finite temperatures
• Generalizations of Berry phase and differentiation of purified state and thermal vacuum of mixed states
• Density operator approach to turbulent flows in plasma and atmospheric fluids
• The Uhlmann Phase Winding in Bose-Einstein Condensates at Finite Temperature
• 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
• Comparison of finite-temperature topological indicators based on Uhlmann connection