Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1703.01451

Gauge linked time-dependent non-Hermitian Hamiltonians

F. S. Luiz, M. A. de Ponte, M. H. Y. Moussa

Published Mar 4, 2017• 11 citations
arXivHTMLPDF

Abstract

In this work we address systems described by time-dependent non-Hermitian Hamiltonians under time-dependent Dyson maps. We shown that when starting from a given time-dependent non-Hermitian Hamiltonian which is not itself an observable, an infinite chain of gauge linked time-dependent non-observable non-Hermitian Hamiltonians can be derived from it. The matrix elements of the observables associated with all these non observable Hamiltonians are, however, all linked to each other, and in the particular case where global gauges exist, these matrix elements becomes all identical to each other. In this case, therefore, by approaching whatever the Hamiltonian in the chain we can get information about any other Hamiltonian. We then show that the whole chain of time-dependent non-Hermitian Hamiltonians collapses to a single time-dependent non-Hermitian Hamiltonian when, under particular choices for the time-dependent Dyson maps, the observability of the Hamiltonians is assured. This collapse thus shows that the observability character of a non-Hermitian Hamiltonian prevents the construction of the gauge-linked Hamiltonian chain and, consequently, the possibility of approaching one Hamiltonian from another.

DOI
10.48550/arxiv.1703.01451

Related papers

Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps

F. S. Luiz, M. A. de Ponte, M. H. Y. Moussa

Abstract We present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of the whole time-dependent non-Hermitian Hamiltonian or parts of it. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed, in one case, by the non-Hermitian Hamiltonian itself or, in another case, by its parts. In the former case, when the whole non-Hermitian Hamiltonian is considered, our scheme ensures the time-independence of the metric operator despite the time-dependence of the Dyson map, a necessary condition for the observability of the non-Hermitian Hamiltonian. In the later case, however, when parts of the non-Hermitian Hamiltonian is considered, our method ensures the simultaneous time-dependence of the Dyson map and the metric operator. In this latter case what is ensured is the observability of the remaining part of the non-Hermitian Hamiltonian that was not chosen for the derivation of the Dyson map. Illustrative examples, for both cases, are derived from a driven non-Hermitian Harmonic oscillator.

Physica Scripta•Mar 18, 2020•27 citations

Infinite series of time-dependent Dyson maps

Andreas Fring, Rebecca Tenney

Abstract We propose and explore a scheme that leads to an infinite series of time-dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying symmetries responsible for this feature respected by various Lewis–Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric operators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion.

arXiv:2108.06793•Journal of Physics A Mathematical and Theoretical•Oct 20, 2021•9 citations

Unitarity of the time-evolution and observability of non-Hermitian Hamiltonians for time-dependent Dyson maps

F. S. Luiz, M. A. Pontes, M. H. Y. Moussa

Here we present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of a quasi-Hermitian Hamiltonian. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed by the non-Hermitian Hamiltonian itself; despite its time-dependence our scheme ensures the time-independence of the metric operator, a necessary condition for the observability of the quasi-Hermitian Hamiltonian. As an illustrative example we consider a driven Harmonic oscillator described by a time-dependent non-Hermitian Hamiltonian. After computing the Dyson map and demonstrating the time-independence of the associated metric operator, we successfully derive an eigenvalue equation for this time-dependent Hamiltonian which enable us to analyze the $\mathcal{PT}$-symmetry breaking process.

arXiv:1611.08286•Nov 24, 2016•13 citations

Unitary quantum evolution for time-dependent quasi-Hermitian systems with nonobservable Hamiltonians

Andreas Fring, M. H. Y. Moussa

It has been argued that it is incompatible to maintain unitary time evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a nonobservable operator as it ceases to be quasi-Hermitian when the metric becomes time dependent.

arXiv:1511.08092•Physical Review A•Apr 20, 2016•120 citations

The Quantum Adiabatic Theorem for Non-Hermitian Dynamics

G. M. V. S. Aponso, B. P. W. Fernando, K. K. W. A. S. Kumara +1 more

We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.

quant-pharXiv:2607.19737•Jul 23, 2026•0 citations

Metric versus observable operator representation, higher spin models

Andreas Fring, Thomas Frith

We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

arXiv:1612.06122•The European Physical Journal Plus•Feb 1, 2018•14 citations
In this work we address systems described by time-dependent non-Hermitian Hamiltonians under time-dependent Dyson maps. We shown that when starting from a given time-dependent non-Hermitian Hamiltonian which is not itself an observable, an infinite chain of gauge linked time-dependent non-observable non-Hermitian Hamiltonians can be derived from it. The matrix elements of the observables associated with all these non observable Hamiltonians are, however, all linked to each other, and in the particular case where global gauges exist, these matrix elements becomes all identical to each other. In this case, therefore, by approaching whatever the Hamiltonian in the chain we can get information about any other Hamiltonian. We then show that the whole chain of time-dependent non-Hermitian Hamiltonians collapses to a single time-dependent non-Hermitian Hamiltonian when, under particular choices for the time-dependent Dyson maps, the observability of the Hamiltonians is assured. This collapse thus shows that the observability character of a non-Hermitian Hamiltonian prevents the construction of the gauge-linked Hamiltonian chain and, consequently, the possibility of approaching one Hamiltonian from another.
Abstract We present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of the whole time-dependent non-Hermitian Hamiltonian or parts of it. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed, in one case, by the non-Hermitian Hamiltonian itself or, in another case, by its parts. In the former case, when the whole non-Hermitian Hamiltonian is considered, our scheme ensures the time-independence of the metric operator despite the time-dependence of the Dyson map, a necessary condition for the observability of the non-Hermitian Hamiltonian. In the later case, however, when parts of the non-Hermitian Hamiltonian is considered, our method ensures the simultaneous time-dependence of the Dyson map and the metric operator. In this latter case what is ensured is the observability of the remaining part of the non-Hermitian Hamiltonian that was not chosen for the derivation of the Dyson map. Illustrative examples, for both cases, are derived from a driven non-Hermitian Harmonic oscillator.
Abstract We propose and explore a scheme that leads to an infinite series of time-dependent Dyson maps which associate different Hermitian Hamiltonians to a uniquely specified time-dependent non-Hermitian Hamiltonian. We identify the underlying symmetries responsible for this feature respected by various Lewis–Riesenfeld invariants. The latter are used to facilitate the explicit construction of the Dyson maps and metric operators. As a concrete example for which the scheme is worked out in detail we present a two-dimensional system of oscillators that are coupled to each other in a non-Hermitian PT -symmetrical fashion.
Here we present an strategy for the derivation of a time-dependent Dyson map which ensures simultaneously the unitarity of the time evolution and the observability of a quasi-Hermitian Hamiltonian. The time-dependent Dyson map is derived through a constructed Schrödinger-like equation governed by the non-Hermitian Hamiltonian itself; despite its time-dependence our scheme ensures the time-independence of the metric operator, a necessary condition for the observability of the quasi-Hermitian Hamiltonian. As an illustrative example we consider a driven Harmonic oscillator described by a time-dependent non-Hermitian Hamiltonian. After computing the Dyson map and demonstrating the time-independence of the associated metric operator, we successfully derive an eigenvalue equation for this time-dependent Hamiltonian which enable us to analyze the PT\mathcal{PT}PT-symmetry breaking process.
It has been argued that it is incompatible to maintain unitary time evolution for time-dependent non-Hermitian Hamiltonians when the metric operator is explicitly time dependent. We demonstrate here that the time-dependent Dyson equation and the time-dependent quasi-Hermiticity relation can be solved consistently in such a scenario for a time-dependent Dyson map and time-dependent metric operator, respectively. These solutions are obtained at the cost of rendering the non-Hermitian Hamiltonian to be a nonobservable operator as it ceases to be quasi-Hermitian when the metric becomes time dependent.
We establish a quantitative adiabatic estimate for a class of finite-dimensional non-Hermitian Schrödinger dynamics. The original non-Hermitian Hamiltonian is assumed to be diagonalizable with real spectrum and non-crossing eigenvalues. We then construct a dynamically compatible time-dependent metric operator, and its positive square root defines a Dyson map. The associated Dyson-transformed Hamiltonian is Hermitian. When this Dyson-transformed Hamiltonian satisfies the Hermitian adiabatic assumption, the standard resolvent-projection method gives an explicit adiabatic estimate in the Hermitian representation. Pulling this estimate back through the Dyson map gives an approximation in the original non-Hermitian representation. The resulting Dyson-pulled-back projections are then compared with the spectral projections of the original non-Hermitian Hamiltonian by using a contour-resolvent estimate. The final bound contains two contributions: the pulled-back Hermitian adiabatic error and the projection-comparison error. A two-level non-Hermitian model is presented to illustrate the hypotheses of the theorem and the two contributions appearing in the final estimate.
We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.