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:1610.07537 · Physical Review A

Exact analytical solutions for time-dependent Hermitian Hamiltonian systems from static unobservable non-Hermitian Hamiltonians

Andreas Fring, Thomas Frith

Published Jan 17, 2017• 54 citations
arXivHTMLPDF

Abstract

We propose a procedure to obtain exact analytical solutions to the time-dependent Schr\"odinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.

DOI
10.1103/physreva.95.010102

Related papers

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

Non-Hermitian Swanson model with a time-dependent metric

Andreas Fring, M. H. Y. Moussa

We provide further nontrivial solutions to the recently proposed time-dependent Dyson and quasi-Hermiticity relation. Here, we solve them for the generalized version of the non-Hermitian Swanson Hamiltonian with time-dependent coefficients. We construct time-dependent solutions by employing the Lewis-Riesenfeld method of invariants and discuss concrete physical applications of our results.

arXiv:1606.04807•Physical Review A•Oct 26, 2016•71 citations

Solvable two-dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime

Andreas Fring, Thomas Frith

Abstract We provide exact analytical solutions for a two-dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis–Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov–Pinney equation emerging in our and many other systems.

arXiv:1801.05828•Journal of Physics A Mathematical and Theoretical•May 17, 2018•30 citations

On the pseudo-Hermitian invariant method for the time-dependent Non-Hermitian Hamiltonians

Boubakeur Khantoul, Abdelhafid Bounames, Mustapha Maamache

We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators $H(t)$ that generate a real phase in their time-evolution. This involves the use of invariant operators $I_{PH}(t)$ that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, $H(t)$ is not generally quasi-Hermitian and does not define an observable of the system but $I_{PH}(t)$ obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.

•Oct 28, 2016•3 citations

Exactly solvable time-dependent pseudo-Hermitian su(1,1) Hamiltonian models

Roberto Grimaudo, Antonio S. M. de Castro, Marek Kuś +1 more

An exact analytical treatment of the dynamical problem for time-dependent $2\ifmmode\times\else\texttimes\fi{}2$ pseudo-Hermitian $\text{su}(1,1)$ Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter $\ensuremath{\nu}$ that allows for distinguishing two different dynamical regimes. Our results are usefully employed for exactly solving a classical propagation problem in a guided wave optics scenario. The usefulness of our procedure in a quantum context is illustrated by defining and investigating the $\text{su}(1,1)$ ``Rabi'' scenario, bringing to light analogies and differences with the standard su(2) Rabi model. Our approach, applied to the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the $\ensuremath{\nu}$-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.

arXiv:1806.02103•Physical Review A•Sep 28, 2018•27 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
We propose a procedure to obtain exact analytical solutions to the time-dependent Schr"odinger equations involving explicit time-dependent Hermitian Hamiltonians from solutions to time-independent non-Hermitian Hamiltonian systems and the time-dependent Dyson relation, together with the time-dependent quasi-Hermiticity relation. We illustrate the working of this method for a simple Hermitian Rabi-type model by relating it to a non-Hermitian time-independent system corresponding to the one-site lattice Yang-Lee model.
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.
Abstract We provide exact analytical solutions for a two-dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis–Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov–Pinney equation emerging in our and many other systems.
We propose a scheme to deal with certain time-dependent non-Hermitian Hamiltonian operators H(t)H(t)H(t) that generate a real phase in their time-evolution. This involves the use of invariant operators IPH(t)I_{PH}(t)IPH​(t) that are pseudo-Hermitian with respect to the time-dependent metric operator and which implies that the dynamics is governed by unitary time evolution. Furthermore, H(t)H(t)H(t) is not generally quasi-Hermitian and does not define an observable of the system but IPH(t)I_{PH}(t)IPH​(t) obeys a quasi-hermiticity transformation as in the completely time-independent Hamiltonian systems case. The harmonic oscillator with a time-dependent frequency under the action of a complex time-dependent linear potential is considered as an illustrative example.
An exact analytical treatment of the dynamical problem for time-dependent $2\ifmmode\times\else\texttimes\fi{}2$ pseudo-Hermitian su(1,1)\text{su}(1,1)su(1,1) Hamiltonians is reported. A class of exactly solvable and physically transparent scenarios are identified within both classical and quantum contexts. The class is spanned by a positive parameter $\ensuremath{\nu}$ that allows for distinguishing two different dynamical regimes. Our results are usefully employed for exactly solving a classical propagation problem in a guided wave optics scenario. The usefulness of our procedure in a quantum context is illustrated by defining and investigating the su(1,1)\text{su}(1,1)su(1,1) ``Rabi'' scenario, bringing to light analogies and differences with the standard su(2) Rabi model. Our approach, applied to the generalized von Neumann equation describing open quantum systems through non-Hermitian Hamiltonians, succeeds in evidencing that the $\ensuremath{\nu}$-dependent passage from a real to a complex energy spectrum is generally unrelated to the existence of the two dynamical regimes.
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.