Quantum-based solution of time-dependent complex Riccati equations

TL;DR

This paper introduces a quantum-based recursive method to solve time-dependent complex Riccati equations derived from quantum systems with specific Lie algebra symmetries, enabling efficient numerical solutions and insights into system unitarity.

Contribution

It develops a recursive continued fraction approach for solving TDCREs linked to quantum systems with Lie algebra symmetries, and establishes conditions for unitarity of the time evolution operator.

Findings

Successfully applied to the Bloch-Riccati equation with hyperbolic secant pulse

Demonstrated excellent agreement with analytical solutions

Provided a framework for recognizing symmetries in quantum Hamiltonians

Abstract

Using the Wei-Norman theory we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the $\mathfrak{su}(1,1)$, $\mathfrak{su}(2)$ and $\mathfrak{so}(2,1)$ Lie algebras. Using a recently developed solution for the time evolution of these quantum systems we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra $\mathfrak{su}(2)$. As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.