Metric versus observable operator representation, higher spin models

2017

Physics Letters A

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We demonstrate that non-Hermitian Hamiltonian systems with spontaneously broken PT-symmetry and partially complex eigenvalue spectrum can be made meaningful in a quantum mechanical sense when introducing some explicit time-dependence into their parameters. Exploiting the fact that explicitly time-dependent non-Hermitian Hamitonians are unobservable and not identical to the energy operators in such a scenario, we show that their corresponding non-Hermitian energy operators develop a different type of PT-symmetry from the Hamiltonians that ensures the reality of their energy spectra. For this purpose we analytically solve the fully time-dependent Dyson equation with all quantities involved being explicitly time-dependent giving rise to a time-dependent metric. The key auxiliary equation to be solved for the two level atomic system considered here is the nonlinear Ermakov-Pinney equation with time-dependent coefficients.Comment: 12 pages, 2 figure

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mending the broken PT-regime via an explicit time-dependent Dyson map

2017

Physics Letters A

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“…In this regime the system exhibits complex energy eigenvalues, becoming ill-defined and is therefore ordinarily discarded as non-physical and useless. However, it has been shown [31,32,33,34] that when a time-dependence is introduced into the central equations it is possible to make sense of the broken regime via a time-dependent metric. This allows for the definition of a Hilbert space and therefore a well-defined inner product.…”

Section: Introductionmentioning

confidence: 99%

Eternal life of entropy in non-Hermitian quantum systems

2019

Phys. Rev. A

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We find a new effect for the behaviour of Von Neumann entropy. For this we derive the framework for describing Von Neumann entropy in non-Hermitian quantum systems and then apply it to a simple interacting P T symmetric bosonic system. We show that our model is well defined even in the P T broken regime with the introduction of a time-dependent metric and that it displays three distinct behaviours relating to the P T symmetry of the original time-independent Hamiltonian. When the symmetry is unbroken, the entropy undergoes rapid decay to zero (so-called "sudden death") with a subsequent revival. At the exceptional point it decays asymptotically to zero and when the symmetry is spontaneously broken it decays asymptotically to a finite constant value ("eternal life").

“…As a concrete example we consider QES systems of E 2 -Lie algebraic type. Technically we make use of the metric picture [11,12], which is an alternative to the Schrödinger, Heisenberg and interaction picture. It will allow us to solve a Hermitian time-dependent Hamiltonian system by solving first a static non-Hermitian system as an auxiliary problem with a time-dependence in the metric operator.…”

Section: Introductionmentioning

confidence: 99%

“…PTsymmetric/quasi-Hermitian systems [13,14,15] allow for yet another equivalent variant in which the time-dependence is contained entirely in the metric operator. In order to see that we first need to solve the time-dependent Dyson relation [16,17,18,19,20,11,12,21,22,23] which in general reads…”

Section: Introductionmentioning

confidence: 99%

Quasi-exactly solvable quantum systems with explicitly time-dependent Hamiltonians

2019

Physics Letters A

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For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzedfurther and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making use of the metric picture, which is an equivalent alternative to the Schrödinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.