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

On a class of time-dependent non-hermitian hamiltonians

F. Kecita, Boubakeur Khantoul, Abdelhafid Bounames

arXiv:2510.02494Reports on Mathematical PhysicsOct 1, 20260 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.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:1511.08092Physical Review AApr 20, 2016120 citations

Non-Hermitian Hamiltonians with unitary and antiunitary symmetries

Francisco Marcelo Fernández, Javier Garcia

arXiv:1309.0808Annals of PhysicsJan 10, 201420 citations

Non-Hermitian Degeneracies in unitary quantum dynamics interrupted by repetitive quantum measurements

S. Kar, Supervisor Dr. Sourin Das

The effect of repeated measurements on unitary evolution of a quantum system at regular intervals has been studied picking a specific example of a quantum, particle moving on a lattice of three sites. Dividing the system into two subsystems, we obtain the survival probability of the particle (i.e. particle not being detected) by time evolution followed by projection onto one subsystem. Using this evolution operator we obtain the expression for effective non-Hermitian Hamiltonian whose dynamics could be mapped to a general firs-order perturbation approach. We obtained a good argument between three approaches followed for calculating the survival probability, namely, the first order perturbative approach, the exact non-hermitian evolution and repeated the non-hermitian measurements interrupting the unitary evolution. Further we obtained the non-Hermitian degeneracies called Exceptional Points using the effective Hamiltonian. Finally, the presence of exceptional point is verified by decomposing the initial state into eigenstates of effective Hamiltonian and observing the behavior of the coefficients of decomposition while traversing through each parameter on which the exceptional point depends.The effect of repeated measurements on unitary evolution of a quantum system at regular intervals has been studied picking a specific example of a quantum, particle moving on a lattice of three sites. Dividing the system into two subsystems, we obtain the survival probability of the particle (i.e. particle not being detected) by time evolution followed by projection onto one subsystem. Using this evolution operator we obtain the expression for effective non-Hermitian Hamiltonian whose dynamics could be mapped to a general firs-order perturbation approach. We obtained a good argument between three approaches followed for calculating the survival probability, namely, the first order perturbative approach, the exact non-hermitian evolution and repeated the non-hermitian measurements interrupting the unitary evolution. Further we obtained the non-Hermitian degeneracies called Exceptional Points using the effective Hamiltonian. Finally, the presence of exceptional point is verified by decomposing the initial state into eigenstates of effective Hamiltonian and observing the behavior of the coefficients of decomposition while traversing through each parameter on which the exceptional point depends.

Apr 1, 20180 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.