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American Journal of Physics

Model study of the Landau–Zener approximation

S. Geltman, Nathan D. Aragon

Published Oct 18, 2005• 3 citations
Publisher

Abstract

We reexamine the accuracy of the Landau–Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau–Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.

DOI
10.1119/1.2063047

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We reexamine the accuracy of the Landau–Zener approximation, whose analytic simplicity has made it often cited and applied over the past 70years in describing nonadiabatic transitions. We find that this approximation may be in substantial disagreement with exact numerical results for a model physical system and a wide range of coupling strengths and relative velocities. We conclude that the Landau–Zener approximation should be used with caution for quantitative estimates of nonadiabatic transition probabilities or cross sections.
This paper presents analytic formulas for various transition times in the Landau-Zener model. Considerable differences are found between the transition times in the diabatic and adiabatic bases, and between the jump time (the time for which the transition probability rises to the region of its asymptotic value) and the relaxation time (the characteristic damping time of the oscillations which appear in the transition probability after the crossing). These transition times have been calculated by using the exact values of the transition probabilities and their derivatives at the crossing point and approximations to the time evolutions of the transition probabilities in the diabatic basis, derived earlier [N. V. Vitanov and B. M. Garraway, Phys. Rev. A 53, 4288 (1996)], and similar results in the adiabatic basis, derived in the present paper.
The Landau-Zener (LZ) transition is one of the most fundamental phenomena in quantum dynamics. It describes nonadiabatic transitions between quantum states near an avoided crossing that can occur in diverse physical systems. Here we report experimental measurements and tuning of LZ transitions between the dressed eigenlevels of a Bose-Einstein condensate (BEC) that is synthetically spin-orbit (SO) coupled. We measure the transition probability as the BEC is accelerated through the SO avoided crossing and study its dependence on the coupling between the diabatic (bare) states, eigenlevel slope, and eigenstate velocity---the three parameters of the LZ model that are independently controlled in our experiments. Furthermore, we performed time-resolved measurements to demonstrate the breaking down of the spin-momentum locking of the spin-orbit-coupled BEC in the nonadiabatic regime, and we determined the diabatic switching time of the LZ transitions. Our observations show quantitative agreement with the LZ model and numerical simulations of the quantum dynamics in the quasimomentum space. The tunable LZ transition may be exploited to enable a spin-dependent atomtronic transistor.