The evolution operator U(t) for a time-independent parity-time-symmetric systems is well studied in the literature.However, for the non-Hermitian time-dependent systems, a closed form expression for the evolution operator is not available.In this paper, we make use of a procedure, originally developed by A.R.P. Rau [Phys.Rev.Lett, 81, 4785-4789 (1998)], in the context of deriving the solution of Liuville-Bloch equations in the product form of exponential operators when time-dependent external fields are present, for the evaluation of U(t) in the interaction picture wherein the corresponding Hamiltonian is time-dependent and in general non-Hermitian.This amounts to a transformation of the whole scheme in terms of addressing a nonlinear Riccati equation the existence of whose solutions depends on the fulfillment of a certain accompanying integrability condition.The evolution operator U(t) for a time-independent parity-time-symmetric systems is well studied in the literature.However, for the non-Hermitian time-dependent systems, a closed form expression for the evolution operator is not available.In this paper, we make use of a procedure, originally developed by A.R.P. Rau [Phys.Rev.Lett, 81, 4785-4789 (1998)], in the context of deriving the solution of Liuville-Bloch equations in the product form of exponential operators when time-dependent external fields are present, for the evaluation of U(t) in the interaction picture wherein the corresponding Hamiltonian is time-dependent and in general non-Hermitian.This amounts to a transformation of the whole scheme in terms of addressing a nonlinear Riccati equation the existence of whose solutions depends on the fulfillment of a certain accompanying integrability condition.

Letters in High Energy PhysicsNov 13, 20186 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

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

Quantum-based solution of time-dependent complex Riccati equations

D. Martínez-Tibaduiza, Carlos González-Arciniegas, Carlos Farina +2 more

Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathrm{su}(1,1)$ , $\mathrm{su}(2)$ , and $\mathrm{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathrm{su}(2)$ . As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathrm{su}(1,1)$, $\mathrm{su}(2)$, and $\mathrm{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathrm{su}(2)$. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.

arXiv:2209.03504Physical Review AApr 14, 20233 citations

Generalized adiabatic product expansion: A nonperturbative method of solving the time-dependent Schrödinger equation

Ali Mostafazadeh

We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.We outline a method based on successive canonical transformations which yields a product expansion for the evolution operator of a general (possibly non-Hermitian) Hamiltonian. For a class of such Hamiltonians this expansion involves a finite number of terms, and our method gives the exact solution of the corresponding time-dependent Schrödinger equation. We apply this method to study the dynamics of a general nondegenerate two-level quantum system, a time-dependent classical harmonic oscillator, and a degenerate system consisting of a spin 1 particle interacting with a time-dependent electric field E⃗(t) through the Stark Hamiltonian H=λ(J⃗⋅E⃗)2.

arXiv:quant-ph/9904026Journal of Mathematical PhysicsJul 1, 199912 citations

On the η pseudo P T symmetry theory for non-Hermitian Hamiltonians: time-dependent systems

Mustapha Maamache

Abstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi SmailAbstract In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo P T symmetry, i.e. the non-Hermitian Hamiltonian H is related to its adjoint H † via the relation, H † = P T H P T . We propose a derivation of pseudo P T symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new symmetry operator η ˜ ( t ) = P T η ( t ) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1, 1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new symmetry operator and discuss a concrete physical applications of our results. Dedicated to the memory of my mother Djabou Zoulikha and to Djabi Smail

Physica ScriptaDec 19, 20230 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.

\mathrm{su}(1,1)

\mathrm{su}(2)

, and

\mathrm{so}(2,1)

Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra

\mathrm{su}(2)

. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.