Time evolution operator for a $\left\{ h(1) \oplus h(1) \right\} \uplus u(2)$ time-dependent quantum Hamiltonian; a self-consistent resolution method based on Feynman's disentangling rules

TL;DR

This paper develops a self-consistent method using Feynman's disentangling rules to explicitly compute the time evolution operator for a complex two-oscillator quantum system, reducing it to solving a Riccati differential equation.

Contribution

It introduces a novel approach to derive explicit disentangled expressions for the evolution operator of a complex quantum system within a specific algebraic framework.

Findings

Derived explicit disentangling expressions for the evolution operator.

Reduced the problem to solving a Riccati-type differential equation.

Analyzed the time evolution of coherent states in the system.

Abstract

In this article the time evolution operator of two interacting quantum oscillators, whose Hamiltonian is an element of the complex $\left\{ h(1) \oplus h(1) \right\} \uplus u(2)$ algebra, is analyzed using the Feynman time ordering operator techniques. This method is consistently used to generate the conditions and to formally find explicit disentangled expressions for such operator. In this way, it is shown that all the problem reduces to solve a complex Riccati-type differential equation. Some closed solutions to this differential equation are found and then concrete disentangling expressions for the time-ordered evolution operator are given. Finally, the time evolution of the coherent states linked to the isotropic 2D quantum oscillator are analyzed under alternative time-independent an time-dependent Hamiltonian systems.