In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterpart are important for the comprehension of posed problems in quantum optics and quantum chemistry, which consists of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. We obtain the propagator for a general timedependence of the parameters and the wave-functions are obtained explicitly for constant frequency and mass and a linear time-dependence in the potential. The wave-functions are used to obtain the expectation value of the Hamiltonian. Although not PT symmetric the case of study exhibits real values of energy.In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterpart are important for the comprehension of posed problems in quantum optics and quantum chemistry, which consists of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. We obtain the propagator for a general timedependence of the parameters and the wave-functions are obtained explicitly for constant frequency and mass and a linear time-dependence in the potential. The wave-functions are used to obtain the expectation value of the Hamiltonian. Although not PT symmetric the case of study exhibits real values of energy.

Nov 7, 20030 citations

Time-dependent non-Hermitian Hamiltonians with real energies

ALVARO DE SOUZA DUTRA, Marcelo Hott, V. G. C. S. dos Santos

In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterparts are important for the comprehension of posed problems in quantum optics and quantum chemistry. They consist of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. The wave functions are used to obtain the expectation value of the Hamiltonian. Although it is neither Hermitian nor PT symmetric, the Hamiltonian under study exhibits real values of energy.In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterparts are important for the comprehension of posed problems in quantum optics and quantum chemistry. They consist of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. The wave functions are used to obtain the expectation value of the Hamiltonian. Although it is neither Hermitian nor PT symmetric, the Hamiltonian under study exhibits real values of energy.

Europhysics Letters (EPL)Jun 8, 200538 citations

A Real Expectation Value of the Time-dependent Non-Hermitian Hamiltonians*

F. Kecita, Abdelhafid Bounames, Mustapha Maamache

Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.Abstract With the aim to solve the time-dependent Schrödinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent   -symmetric one. Consequently, the solution of time-dependent Schrödinger equation becomes easily deduced and the evolution preserves the  ( t ) PT −inner product, where  ( t ) is a obtained from the charge conjugation operator  through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the  ( t ) PT normed states is guaranteed to be real. As an illustration, we present a specific quantum system given by a quantum oscillator with time-dependent mass subjected to a driving linear complex time-dependent potential.

arXiv:2112.13535Physica ScriptaNov 26, 20213 citations

Quantum theory of a non-Hermitian time-dependent forced harmonic oscillator having 𝒫𝒯 symmetry

I. A. Pedrosa, B. F. Ramos, K. Bakke +1 more

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.

International Journal of Modern Physics AOct 30, 20197 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.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, 20163 citations

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.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:2002.01977Jan 1, 20194 citations

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.

\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.