[Paper Review] On the pseudo-Hermitian invariant method for the time-dependent Non-Hermitian Hamiltonians

Boubakeur Khantoul, A. Bounames+1 more|arXiv (Cornell University)|Oct 28, 2016

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper introduces a pseudo-Hermitian invariant method for time-dependent non-Hermitian Hamiltonians that generate real phases in time evolution. By constructing time-dependent pseudo-Hermitian invariants $I_{PH}(t)$ with respect to a dynamic metric, the framework ensures unitary time evolution even when the Hamiltonian $H(t)$ is not quasi-Hermitian or observable, illustrated via a time-dependent complex linear potential harmonic oscillator.

ABSTRACT

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.

Motivation & Objective

• To address the challenge of time-evolution unitarity in time-dependent non-Hermitian Hamiltonians that do not generally qualify as quasi-Hermitian or observable.
• To develop a framework where the dynamics remains unitary despite the non-Hermitian nature of $H(t)$, by introducing a time-dependent pseudo-Hermitian invariant $I_{PH}(t)$.
• To generalize the quasi-Hermitian transformation structure from time-independent to time-dependent systems, preserving key dynamical features.
• To provide a systematic method for analyzing non-Hermitian systems with time-dependent parameters, particularly those with complex time-dependent potentials.

Proposed method

• The method introduces a time-dependent metric operator that defines pseudo-Hermiticity for the invariant operator $I_{PH}(t)$.
• The invariant $I_{PH}(t)$ is constructed to be pseudo-Hermitian with respect to the time-dependent metric, ensuring unitary time evolution.
• The framework leverages a quasi-Hermiticity condition for $I_{PH}(t)$, analogous to the time-independent case, even though $H(t)$ itself is not quasi-Hermitian.
• The time evolution is governed by a unitary operator derived from the pseudo-Hermitian invariant, ensuring real energy spectra and consistent dynamics.
• The method applies to systems where $H(t)$ generates a real phase in time evolution, even if $H(t)$ is not observable or quasi-Hermitian.

Experimental results

Research questions

• RQ1 Can a time-dependent non-Hermitian Hamiltonian with a real phase in time evolution be described via a unitary dynamics framework?
• RQ2 How can a pseudo-Hermitian invariant be constructed for time-dependent non-Hermitian systems when the Hamiltonian is not quasi-Hermitian?
• RQ3 Does the invariant $I_{PH}(t)$ preserve the quasi-Hermiticity structure seen in time-independent systems under time evolution?
• RQ4 What role does the time-dependent metric operator play in ensuring unitarity for such systems?

Key findings

• The time-dependent invariant $I_{PH}(t)$ is pseudo-Hermitian with respect to the time-dependent metric, ensuring unitary time evolution despite $H(t)$ not being quasi-Hermitian.
• The dynamics governed by $H(t)$ remains unitary due to the pseudo-Hermitian structure of $I_{PH}(t)$, even when $H(t)$ does not define an observable.
• The invariant $I_{PH}(t)$ satisfies a quasi-Hermiticity transformation law similar to that in time-independent systems, generalizing the known formalism.
• The framework successfully describes the harmonic oscillator with time-dependent frequency under a complex time-dependent linear potential as a concrete example.