The Riccati equation bridges the two probability amplitudes in Landau-Zener transitions and pinpoints why the Markov approximation succeeds for one amplitude but fails for the other.
For this purpose, we first collect the corresponding ex- pressions, and then apply them to the formulae for a, b and η derived in the preceding sections of this appendix
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Hidden facts in Landau-Zener transitions revealed by the Riccati Equation
The Riccati equation bridges the two probability amplitudes in Landau-Zener transitions and pinpoints why the Markov approximation succeeds for one amplitude but fails for the other.