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arXiv:0811.1388 · Physical Review A

Integral definition of transition time in the Landau-Zener model

Yue Yan, Biao Wu

Published Feb 26, 2010• 10 citations
arXivHTMLPDF

Abstract

We give a general definition for the transition time in the Landau-Zener model. This definition allows us to compute numerically the Landau-Zener transition time at any sweeping rate without ambiguity in both diabatic and adiabatic bases. With this new definition, analytical results are obtained in both the adiabatic limit and the sudden limit.

DOI
10.1103/physreva.81.022126

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We give a general definition for the transition time in the Landau-Zener model. This definition allows us to compute numerically the Landau-Zener transition time at any sweeping rate without ambiguity in both diabatic and adiabatic bases. With this new definition, analytical results are obtained in both the adiabatic limit and the sudden limit.
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 time of Zener tunneling, ${\ensuremath{\tau}}_{z}$, is calculated for a general two-level quantum-mechanical system by two methods. In the first we determine the width of the transition profile in time. In the second we apply an oscillating perturbation and examine how the final transition probability depends upon the perturbation frequency and phase. Both methods show, given that the coupling energy between the diabatic levels is \ensuremath{\Delta} and the time rate of change of the energy is \ensuremath{\alpha}, that in the adiabatic limit ${\ensuremath{\tau}}_{z}$ scales as \ensuremath{\Delta}/\ensuremath{\alpha} and in the sudden limit ${\ensuremath{\tau}}_{z}$ scales as (\ensuremath{\Elzxh}/\ensuremath{\alpha})1/2{)}^{1/2})1/2. We calculate the approximate transition probability at finite times for the two limits. The results can be applied to specific mesoscopic electronic systems.
The quantum dynamics of the Landau-Zener transition in a dissipative environment is studied and analytical results for the transition probability are given in terms of temperature, coupling strength, and the Landau-Zener time. For short Landau-Zener time there is no effect of dissipation. In the oppotise limit we distinguish various temperature regimes: The adiabatic limit is shown to be restricted to low temperatures and no effect of dissipation is present at zero temperature. At intermediate temperatures thermal transitions dominate for weak coupling and high temperatures correspond to the strong-coupling limit.
We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.
An innovative microscopic model with a minimal number of parameters: tunneling splitting gap, external field sweeping velocity, and decoherence rate is used to describe dynamics of the dissipative Landau-Zener transition in the presence of the decoherence. In limiting cases, the derived equation of motion gives rise to the well-known Landau-Zener and Kayanuma formula. In a general case, the description demonstrates a non-monotonic flipping probability with respect to the sweeping velocity, which is also found in some other models. This non-monotony can be explained by considering the competition and timescale of the quantum tunneling, crossing period, and decoherence process. The simplicity and robustness of the theory offer a practical and novel description of the Landau-Zener transition. In addition, it promises an alternative method to the electron paramagnetic resonance in measuring the effective decoherence rate of relevant quantum systems.