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International Journal of Modern Physics B

Analytical solutions and expressions of the propagator for Bloch equations

Heung‐Ryoul Noh

Published Jul 1, 2020• 1 citation
Publisher

Abstract

In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.

DOI
10.1142/s0217979220501581

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In this paper, we present analytical solutions to the Bloch equations. After solving the secular equation for the eigenvalues, derived from the Bloch equations, analytical solutions for the temporal evolution of the magnetization vector are obtained at arbitrary initial conditions. Subsequently, explicit analytical expressions of the propagator for the Bloch equations and optical Bloch equations are obtained. Compared to the results of existing analytical studies, the present results are more succinct and rigorous, and they can predict the behavior of the propagator in different regions of parameter spaces. The analytical solutions to the propagator can be directly used in composite laser-pulse spectroscopy.
An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.
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.
The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.
We discuss a new way of solving the optical Maxwell–Bloch equations governing the interaction of two-level inhomogeneously broadened atoms with a laser by expanding the components of the Bloch vector in power series of the detuning. This method leads to a set of infinitely many coupled ordinary differential equations in which the detuning no longer appears explicitly. These equations are solved numerically, reproducing some analytic solutions reported previously and obtaining a variety of other shape-preserving steady-state solutions. These new solutions, however, are neither factorable nor simply periodic. In this connection we discuss the implications of the factorization assumption, which appears to be the only condition leading to closed-form analytic solutions.
We present a comprehensive analytical study of the optical Bloch equations for a closed two-level atom. We obtain a map in phase space of the Rabi frequency and the laser frequency detuning, discriminating the characteristics of solutions to the optical Bloch equations separated by a boundary in phase space. Outside the boundary, oscillatory behavior is shown, while simple decaying behavior is shown inside the boundary. The behavior of the solutions is also found to be dependent on the relative magnitude of the decay rates of the optical coherence and the excited state population.