Interferometric Geometric Phases of $\mathcal{P}\mathcal{T}$-symmetric Quantum Mechanics

Abstract

We present a generalization of the geometric phase to pure and thermal states in $\mathcal{P}\mathcal{T}$-symmetric quantum mechanics (PTQM) based on the approach of the interferometric geometric phase (IGP). The formalism first introduces the parallel-transport conditions of quantum states and reveals two geometric phases, ${\theta }^1$ and ${\theta }^2$, for pure states in PTQM according to the states under parallel-transport. Due to the non-Hermitian Hamiltonian in PTQM, ${\theta }^1$ is complex and ${\theta }^2$ is its real part. The imaginary part of ${\theta }^1$ plays an important role when we generalize the IGP to thermal states in PTQM. The generalized IGP modifies the thermal distribution of a thermal state, thereby introducing effective temperatures. At certain critical points, the generalized IGP exhibits discrete jumps at finite temperatures, signaling a geometric phase transition. We demonstrate the finite-temperature geometric phase transition in PTQM by a two-level system and visualize its results.

arXiv comment12 pages, 3 figures

References in corpus (61)

• Topological Insulators
• Topological insulators and superconductors
• Quantum Spin Hall Effect in Graphene
• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• $Z_2$ 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
• Localization transitions in non-Hermitian quantum mechanics
• Topological phases of non-Hermitian systems
• The physics of exceptional points
• Biorthogonal Bulk-Boundary Correspondence in Non-Hermitian Systems
• Topological Band Theory for Non-Hermitian Hamiltonians
• Nonlinear waves in $\mathcal{P}\mathcal{T}$-symmetric systems
• Non-Hermitian Chern bands
• Geometric phases for mixed states in interferometry
• Topological phases in the non-Hermitian Su-Schrieffer-Heeger model
• Topological Transition in a Non-Hermitian Quantum Walk
• Kinematic approach to the mixed state geometric phase in nonunitary evolution
• Dirac Equation and Quantum Relativistic Effects in a Single Trapped Ion
• Geometric phase in open systems
• Parity-time-symmetric quantum critical phenomena
• PT-symmetry with a system of three-level atoms
• Topological invariance and global Berry phase in non-Hermitian systems
• Information Retrieval and Criticality in Parity-Time-Symmetric Systems
• Uhlmann Phase as a Topological Measure for One-Dimensional Fermion Systems
• Parity-time-symmetric topological superconductor
• An experimental observation of geometric phases for mixed states using NMR interferometry
• Continuous Phase Transition without Gap Closing in Non-Hermitian Quantum Many-Body Systems
• Geometric phases for non-degenerate and degenerate mixed states
• Topological quantum wires with balanced gain and loss
• Two-Dimensional Density-Matrix Topological Fermionic Phases: Topological Uhlmann Numbers
• Time Dependent $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• Probing the topology of density matrices
• Generalization of geometric phase to completely positive maps
• Non-hermitian quantum thermodynamics
• Measurement of geometric phase for mixed states using single photon interferometry
• Kondo effect in a parity-time-symmetric non-Hermitian Hamiltonian
• PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
• Quantum Geometric Tensor in $\mathcal{P}\mathcal{T}$-Symmetric Quantum Mechanics
• On singularities of the mixed state phase
• Spontaneous PT symmetry breaking and quantum phase transitions in dimerized spin chains
• Phases of quantum states in completely positive non-unitary evolution
• Anomalous Behaviors of Quantum Emitters in Non-Hermitian Baths
• PT-symmetry entails pseudo-Hermiticity regardless of diagonalizability
• Hermitian and Non-Hermitian Topology from Photon-Mediated Interactions
• Observation of nonadditive mixed state phases with polarized neutrons
• Reply to `Singularities of the mixed state phase'
• Ubiquity of zeros of Loschmidt amplitude for mixed states in different physical processes and their implications
• PT Symmetry as a Generalization of Hermiticity
• PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields
• Experimental Measurement of Mixed State Geometric Phase by Quantum Interferometry using NMR
• Geometric Phases for Mixed States of the Kitaev Chain
• 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
• Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states
• Kinematic approach to off-diagonal geometric phases of nondegenerate and degenerate mixed states
• Operational geometric phase for mixed quantum states
• Geometric phases of mixed quantum states: A comparative study of interferometric and Uhlmann phases
• Tunable tachyon mass in the PT-broken massive Thirring model