# A Complete Solution of the Bloch Equation

## TL;DR

This paper presents a comprehensive, explicit analytical solution to the Bloch equation, addressing previous limitations and providing clear physical insights into two-level system dynamics.

## Contribution

It offers a complete, general propagator solution for the Bloch equation, explicitly handling singular parameter cases and simplifying the roots calculation.

## Key findings

- Explicit time dependence of the system is derived.
- The solution accommodates division by zero in previous models.
- Models of system dynamics as relaxation and rotation are developed.

## Abstract

The Bloch equation is the fundamental dynamical model applicable to arbitrary two-level systems. Analytical solutions to date are incomplete for a number of reasons that motivate further investigation. The solution obtained here for the propagator, which generates the time evolution of the system and embodies all the system dynamics, is compact and completely general. The parameter space that results in division by zero in previous treatments is explicitly defined and accommodated in the solution. Polynomial roots required for the solution are expressed in terms of a single real root obtained using simple functional forms. A simple graphical rendition of this root is developed that clarifies and characterizes its dependence on the physical parameters of the problem. As a result, the explicit time dependence of the system as a function of its physical parameters is immediately evident. Several intuitive models of system dynamics are also developed. In particular, the Bloch equation is separable in the proper coordinate system, written as the sum of a relaxation operator and either a null operator or a commuting rotation. The propagator thus drives either pure relaxation or relaxation followed by a rotation. The paper provides a basis for increased physical insight into the Bloch equation and its widespread applications.