© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:2201.05140 · Journal of Physics Conference Series

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

Andreas Fring

Published Feb 1, 2023• 25 citations

Abstract

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.1

DOI: 10.1088/1742-6596/2448/1/012002

Related papers

Time-dependence in non-Hermitian quantum systems

Thomas Frith

We present a coherent and consistent framework for explicit time-dependence in non-Hermitian quantum mechanics. The area of non-Hermitian quantum mechanics has been growing rapidly over the past twenty years. This has been driven by the fact that $\mathcal{PT}$-symmetric non-Hermitian systems exhibit real energy eigenvalues and unitary time evolution. Historically, the introduction of time into the world of non-Hermitian quantum mechanics has been a conceptually difficult problem to address, as it requires the Hamiltonian to become unobservable. We solve this issue with the introduction of a new observable energy operator and explain why its instigation is a necessary and natural progression in this setting. For the first time, the introduction of time has allowed us to make sense of the parameter regime in which the $\mathcal{PT}$-symmetry is spontaneously broken. Ordinarily, in the time-independent setting, the energy eigenvalues become complex and the wave functions are asymptotically unbounded. We demonstrate that in the time-dependent setting this broken symmetry can be mended and analysis on the spontaneously broken $\mathcal{PT}$ regime is indeed possible. We provide many examples of this mending on a wide range of different systems, beginning with a $2\times2$ matrix model and extending to higher dimensional matrix models and coupled harmonic oscillator systems with infinite Hilbert space. Furthermore, we use the framework to perform analysis on time-dependent quasi-exactly solvable models. We present the "eternal life" of entropy in this thesis. Ordinarily, for entangled quantum systems coupled to the environments, the entropy decays rapidly to zero. However, in the spontaneously broken regime, we find the entropy decays asymptotically to a non-zero value. We create an elegant framework for Darboux and Darboux/Crum transformations for time-dependent non-Hermitian Hamiltonians.

arXiv:2002.01977•Jan 1, 2019•4 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

Time-dependent PT -symmetric quantum mechanics in generic non-Hermitian systems

Da-Jian Zhang, Qinghai Wang, Jiangbin Gong

A conceptual framework extending (time-independent) $\mathcal{PT}$-symmetric quantum mechanics into the time-dependent domain is presented. It is built upon a nontrivial time-dependent metric operator identified here and works for generic finite-dimensional non-Hermitian systems. All the ingredients of our framework, such as the time-dependent Hilbert space, the observable, and the measurement postulate, can be ``realized'' by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent $\mathcal{PT}$-symmetric systems. Possible applications of our framework are illustrated with well-known examples in quantum thermodynamics.

arXiv:1906.03431•Physical Review A•Dec 20, 2019•49 citations

Beyond PT-symmetry: towards a symmetry-metric relation for time-dependent non-Hermitian Hamiltonians

Luís Felipe Alves da Silva

A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

•Mar 25, 2022•1 citation

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

Eternal life of entropy in non-Hermitian quantum systems

Andreas Fring, Thomas Frith

We find a different effect for the behavior 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 $PT$-symmetric bosonic system. We show that our model is well defined even in the $PT$-broken regime with the introduction of a time-dependent metric and that it displays three distinct behaviors relating to the $PT$ 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'').

arXiv:1905.07348•Physical Review A•Jul 26, 2019•38 citations

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.1

We present a coherent and consistent framework for explicit time-dependence in non-Hermitian quantum mechanics. The area of non-Hermitian quantum mechanics has been growing rapidly over the past twenty years. This has been driven by the fact that

\mathcal{PT}

-symmetric non-Hermitian systems exhibit real energy eigenvalues and unitary time evolution. Historically, the introduction of time into the world of non-Hermitian quantum mechanics has been a conceptually difficult problem to address, as it requires the Hamiltonian to become unobservable. We solve this issue with the introduction of a new observable energy operator and explain why its instigation is a necessary and natural progression in this setting. For the first time, the introduction of time has allowed us to make sense of the parameter regime in which the

\mathcal{PT}

-symmetry is spontaneously broken. Ordinarily, in the time-independent setting, the energy eigenvalues become complex and the wave functions are asymptotically unbounded. We demonstrate that in the time-dependent setting this broken symmetry can be mended and analysis on the spontaneously broken

\mathcal{PT}

regime is indeed possible. We provide many examples of this mending on a wide range of different systems, beginning with a

2\times2

matrix model and extending to higher dimensional matrix models and coupled harmonic oscillator systems with infinite Hilbert space. Furthermore, we use the framework to perform analysis on time-dependent quasi-exactly solvable models. We present the "eternal life" of entropy in this thesis. Ordinarily, for entangled quantum systems coupled to the environments, the entropy decays rapidly to zero. However, in the spontaneously broken regime, we find the entropy decays asymptotically to a non-zero value. We create an elegant framework for Darboux and Darboux/Crum transformations for time-dependent non-Hermitian Hamiltonians.

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.

A conceptual framework extending (time-independent)

\mathcal{PT}

-symmetric quantum mechanics into the time-dependent domain is presented. It is built upon a nontrivial time-dependent metric operator identified here and works for generic finite-dimensional non-Hermitian systems. All the ingredients of our framework, such as the time-dependent Hilbert space, the observable, and the measurement postulate, can be ``realized'' by means of dilating and reinterpreting the non-Hermitian system in question as a part of a larger Hermitian system. Aided by our metric operator, we formulate the concepts of stable and unstable phases for generic non-Hermitian systems and argue that they, respectively, generalize the notions of unbroken and broken phases in time-independent

\mathcal{PT}

-symmetric systems. Possible applications of our framework are illustrated with well-known examples in quantum thermodynamics.

A new chapter in quantum mechanics has opened over the past 20 years with the fact that time-independent (TI) non-Hermitian Hamiltonians have a real spectrum and unitary time evolution when they exhibit an unbroken PT -symmetry and satisfy the pseudo-Hermiticity relation.In this Master's thesis, we first propose a method for the derivation of a general continuous antilinear time-dependent (TD) symmetry operator I(t) for non-Hermitian Hamiltonian H(t) and metric operator ρ(t) in a TD scenario.Assuming H(t) to be simultaneously ρ(t)-pseudo-Hermitian and Ξ(t)-anti-pseudo-Hermitian, we also derive the antilinear symmetry I(t) = Ξ -1 (t)ρ(t), which retrieves an important result obtained by Mostafazadeh [J.Math, Phys.43, 3944 (2002)] for the time-independent (TI) scenario.We apply our method for the derivation of the symmetry associated with TD non-Hermitian linear and quadratic Hamiltonians.In the TI scenario, we retrieve the well-known Bender-Berry-Mandilara result for the symmetry operator:].The results here derived allow us to propose a useful symmetry-metric relation for TD non-Hermitian Hamiltonians.From this relation, the TD metric is automatically derived from the TD symmetry of the problem.Our results reinforce the prospects of going beyond PT -symmetric quantum mechanics making the field of pseudo-Hermiticity even more comprehensive and promising.

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 find a different effect for the behavior 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

PT

-symmetric bosonic system. We show that our model is well defined even in the

PT

-broken regime with the introduction of a time-dependent metric and that it displays three distinct behaviors relating to the

PT

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'').