Evolution operator for time-dependent non-Hermitian Hamiltonians

TL;DR

This paper derives a method to compute the evolution operator for non-Hermitian, time-dependent quantum systems using a transformation approach based on solving a nonlinear Riccati equation, extending previous work on time-independent systems.

Contribution

It introduces a novel procedure to obtain the evolution operator for non-Hermitian, time-dependent Hamiltonians using a transformation method and Riccati equation solutions.

Findings

Provides a closed-form expression for the evolution operator in specific non-Hermitian systems.

Extends existing methods to time-dependent, non-Hermitian quantum systems.

Highlights conditions for the existence of solutions based on integrability criteria.

Abstract

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 integrabilty condition.