On unorthodox solutions of the Bloch equations

TL;DR

This paper provides a comprehensive analysis of the Bloch equations under time-harmonic driving, classifying all analytic solutions and highlighting the dominance of Torrey's damped harmonic oscillation solution in most parameter regimes.

Contribution

It introduces a systematic method using partial fraction decomposition to find and classify all analytic solutions of the Bloch equations, including unorthodox solutions in specific parameter ranges.

Findings

Torrey's solution dominates most parameter space.

Unorthodox solutions occur at small detunings and field strengths.

Unorthodox solutions have featureless time dependence.

Abstract

A systematic, rigorous, and complete investigation of the Bloch equations in time-harmonic driving classical field is performed. Our treatment is unique in that it takes full advantage of the partial fraction decomposition over real number field, which makes it possible to find and classify all analytic solutions. Torrey's analytic solution in the form of exponentially damped harmonic oscillations [Phys. Rev. {\bf 76}, 1059 (1949)] is found to dominate the parameter space, which justifies its use at numerous occasions in magnetic resonance and in quantum optics of atoms, molecules, and quantum dots. The unorthodox solutions of the Bloch equations, which do not have the form of exponentially damped harmonic oscillations, are confined to rather small detunings $\delta^2\lesssim (\gamma-\gamma_t)^2/27$ and small field strengths $\Omega^2\lesssim 8 (\gamma-\gamma_t)^2/27$, where $\gamma$ and $\gamma_t$ describe decay rates of the excited state (the total population relaxation rate) and of the coherence, respectively. The unorthodox solutions being readily accessible experimentally are characterized by rather featureless time dependence.