Solution of Non-Autonomous Bloch Equation via Multistage Differential Transformation Method

Abstract

This study explores the multistage differential transformation method (MsDTM) as an efficient approach for solving non-autonomous differential equations. The proposed method demonstrates wide-ranging applicability in fields such as computer graphics, quantum optics, biomathematics, and image processing. Specifically, the Bloch equations, which describe the interaction of a spin-1/2 system (or two-level atom) with a mono- or bichromatic laser field in the presence of an off-resonant broad-band squeezed vacuum (SV), are examined. As a system of non-autonomous ordinary differential equations, the Bloch model captures the quantum dynamics between matter and electromagnetic fields, offering insights into more complex and experimentally relevant models. The MsDTM is employed to obtain numerical solutions with high precision and computational efficiency, outperforming the classical 4th-order Runge-Kutta (RK4) method. A key advantage of the MsDTM is its adaptability; its accuracy can be further enhanced by either increasing the number of iterations or refining the time-step in the numerical scheme. Consequently, MsDTM emerges as a robust tool for computing solutions to a broad class of non-autonomous equations.