Analytic solutions of the Bloch equations * Two-level system * Time-harmonic field * Torrey solution * Analytic solutions of the Bloch equations

Supporting Information for my manuscript

``On unorthodox solutions of the Bloch equations"


A systematic, rigorous, and complete investigation of the Bloch equations in time-harmonic driving classical field was performed. Our treatment is unique in that it takes full advantage of the partial fraction decomposition over real number field [1], which makes it possible to find and classify all analytic solutions. Torrey's analytic solution in the form of exponentially damped harmonic oscillations [2] 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 <= (\gamma-\gamma_t)^2/27 and small field strengths \Omega^2<= 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.


  1. A rather complete information on the partial fraction decomposition can be found on wikipedia.
  2. H. C. Torrey, Transient Nutations in Nuclear Magnetic Resonance, Phys. Rev. 76, 1059-1068 (1949).
  3. A. Moroz, On unorthodox solutions of the Bloch equations, submitted for publication. [arXiv:1208.5736]

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Alexander Moroz, last updated September 10, 2012

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