[Paper Review] An introduction to PT-symmetric quantum mechanics -- time-dependent systems

Andreas Fring|arXiv (Cornell University)|Jan 13, 2022

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper presents a systematic framework for time-dependent ${\cal PT}$-symmetric quantum systems, demonstrating how unitary time evolution and real spectra can be preserved via time-dependent Dyson maps and metrics. It shows that in the spontaneously broken ${\cal PT}$-regime, von Neumann entropy ceases to decay completely, achieving a finite asymptotic value—offering a mechanism to suppress decoherence in quantum 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.

Motivation & Objective

• To establish a consistent quantum mechanical framework for explicitly time-dependent ${\cal PT}$-symmetric Hamiltonians.
• To resolve long-standing concerns about the viability of time-dependent non-Hermitian systems by constructing well-defined Dyson maps and metrics.
• To demonstrate that the ${\cal PT}$-symmetrically broken phase becomes physically accessible in time-dependent systems, enabling control over decoherence.
• To show that von Neumann entropy in open systems can be prolonged and stabilized at finite values, avoiding 'sudden death' in the broken regime.

Proposed method

• Construct time-dependent Dyson maps and metrics algebraically using time-dependent Darboux transformations.
• Utilize Lewis-Riesenfeld invariants and point transformations to solve the time-dependent Schrödinger equation for ${\cal PT}$-symmetric systems.
• Apply approximation methods to handle complex time-dependent Hamiltonians where exact solutions are intractable.
• Demonstrate the existence of infinite families of metric operators due to inherent ambiguities in the Dyson map construction.
• Map time-dependent double-well potentials and unstable anharmonic oscillators to equivalent ${\cal PT}$-symmetric systems via unitary transformations.
• Compute von Neumann entropy via partial traces over subsystems, analyzing its time evolution in three ${\cal PT}$-regimes.

Experimental results

Research questions

• RQ1 How can a consistent quantum mechanical framework be constructed for time-dependent ${\cal PT}$-symmetric Hamiltonians?
• RQ2 What role do time-dependent Dyson maps and metrics play in ensuring unitary time evolution and real spectra?
• RQ3 Can the ${\cal PT}$-symmetrically broken phase become physically relevant in time-dependent systems, and if so, how?
• RQ4 How does the von Neumann entropy behave in time-dependent ${\cal PT}$-symmetric systems, particularly in the broken regime?
• RQ5 To what extent can time-dependent ${\cal PT}$-symmetric systems suppress decoherence compared to standard dissipative models?

Key findings

• Time-dependent Dyson maps and metrics can be systematically constructed using time-dependent Darboux transformations and Lewis-Riesenfeld invariants, ensuring unitary evolution.
• The ${\cal PT}$-symmetrically broken regime becomes physically accessible in time-dependent systems, allowing for stable, non-decaying dynamics.
• In the broken ${\cal PT}$-regime, the von Neumann entropy does not vanish but stabilizes at a finite asymptotic value, preventing 'sudden death'.
• For a boson coupled to a bath of $N$ bosons, the entropy $S_{H,a} = -λ_-×\ln(\lambda_-) - \lambda_+×\ln(\lambda_+)$ depends on time through $\mu(t)$, with $\lambda_\pm$ defined via trigonometric functions of $\mu(t)$.
• The entropy decay is most rapid in the ${\cal PT}$-symmetric regime and is progressively delayed at the exceptional point, with complete cessation of decay in the broken regime.
• The behavior is universal across models, including non-Hermitian Jaynes–Cummings models, and aligns with observations in open quantum systems.