A real expectation value of the time-dependent non-Hermitian Hamiltonians

TL;DR

This paper introduces a method to compute real expectation values for time-dependent non-Hermitian Hamiltonians by transforming them into time-independent PT-symmetric forms, ensuring real physical observables.

Contribution

It presents a novel unitary transformation approach that maps time-dependent non-Hermitian Hamiltonians to time-independent PT-symmetric ones, enabling straightforward solutions and real expectation values.

Findings

The expectation value of the Hamiltonian is real in the transformed framework.

The method preserves the PT-inner product during evolution.

Application to a quantum oscillator demonstrates the approach's effectiveness.

Abstract

With the aim to solve the time-dependent Schr\"{o}dinger equation associated to a time-dependent non-Hermitian Hamiltonian, we introduce a unitary transformation that maps the Hamiltonian to a time-independent $\mathcal{PT}$-symmetric one. Consequently, the solution of time-dependent Schr\"{o}dinger equation becomes easily deduced and the evolution preserves the $\mathcal{C(}t\mathcal{)PT}$-inner product, where $\mathcal{C(}t\mathcal{)}$ is a obtained from the charge conjugation operator $\mathcal{C}$ through a time dependent unitary transformation. Moreover, the expectation value of the non-Hermitian Hamiltonian in the $\mathcal{C(}t\mathcal{)PT}$ 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.