An introduction to PT-symmetric quantum mechanics -- time-dependent systems

Abstract

I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT$-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.

37 pages, Lecture notes, Contribution to the proceedings of the Quantum Fest 2021, International Conference on Quantum Phenomena, Quantum Control and Quantum Optics, virtually held at Cinvestav Mexico, 25/10-29/10 2021 (eds. Marco Enríquez, Sara Cruz y Cruz, Francisco Delgado, Víctor Manuel Velázquez, Journal of Physics: Conference Series (IoP Publishing))

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Cited by in corpus (9)

• Multicriticality in Yang-Lee edge singularity
• Emergent parallel transport and curvature in Hermitian and non-Hermitian quantum mechanics
• The Floquet Baxterisation
• Real energies and Berry phases in all PT-regimes in time-dependent non-Hermitian theories
• Non-Hermitian Quantum Quenches in Holography
• Discrete-coordinate crypto-Hermitian quantum system controlled by time-dependent Robin boundary conditions
• Non-Hermitian Quantum Fermi Accelerator
• Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions
• Emergence and localization of exceptional points in an exactly solvable toy model