Solvable dilation model of $\mathcal{P}\mathcal{T}$-symmetric systems

Abstract

The dilation method is a practical way to experimentally simulate non-Hermitian, especially $\mathcal{P}\mathcal{T}$-symmetric quantum systems. However, the time-dependent dilation problem cannot be explicitly solved in general. In this paper, we present a simple yet non-trivial exactly solvable dilation problem with two dimensional time-dependent $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian. Our system is initially set in the unbroken $\mathcal{P}\mathcal{T}$-symmetric phase and later goes across the so-called exceptional point and enters the broken $\mathcal{P}\mathcal{T}$-symmetric phase. For this system, the dilated Hamiltonian and the evolution of $\mathcal{P}\mathcal{T}$-symmetric system are analytically worked out. Our result clearly showed that the exceptional points do not have much physical relevance in a \textit{time-dependent} system.

arXiv comment9 pages, 4 figures, close to the published version

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• Lewis-Riesenfeld invariants for PT-symmetrically coupled oscillators from two dimensional point transformations and Lie algebraic expansions
• Topological state permutations in time-modulated non-Hermitian multiqubit systems with suppressed non-adiabatic transitions