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.
Matrix exponential solution of the Landau-Zener problem
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abstract
We present a derivation of the Landau-Zener solution through the explicit evaluation of the time ordered propagator. The result is exact and does not involve the solution of the differential equation for the spin amplitudes.
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The Landau-Zener amplitude a_LZ equals exp[-π/(2ε)] because of a logarithmic phase singularity from a contour integral encircling a simple pole.
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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.
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A logarithmic phase singularity at the heart of Landau-Zener transitions
The Landau-Zener amplitude a_LZ equals exp[-π/(2ε)] because of a logarithmic phase singularity from a contour integral encircling a simple pole.