The Classical Bloch Equations

TL;DR

This paper shows that key quantum phenomena in two-level systems like Rabi oscillations and echoes can be explained using a classical harmonic oscillator model, clarifying their fundamental nature.

Contribution

It introduces a classical harmonic oscillator framework that reproduces quantum coherent phenomena, highlighting the differences between classical and quantum descriptions.

Findings

Classical model reproduces Rabi oscillations, Ramsey fringes, and Hahn echoes.

Longitudinal and transverse relaxation times are equal in the classical analogy.

The analysis provides an intuitive understanding of quantum coherent control.

Abstract

Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled harmonic oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times ($T_1$ and $T_2$) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories.