Journal of Physics B Atomic and Molecular Physics

On the probability of non-adiabatic transitions in multiple level crossings

Yōsuke Kayanuma, Susumu Fukuchi

Published Oct 28, 1985• 45 citations

Abstract

The authors calculate the probability of nonadiabatic transitions in a multiple-crossing Landau-Zener model, in which a diabatic state mod 0) crosses a set of diabatic states mod k) (k=1, 2, . . ., N) with a constant velocity. It is shown that the probability P that the system stays in mod 0) after the multiple crossings under the condition it starts from mod 0) is exactly given by P=exp(-2 pi Sigma k=1 N J k 2 /h(cross) mod u mod ) irrespective of the configuration of mod k), where J k is the off-diagonal coupling constant between mod 0) and mod k( and u is the velocity of the change of the energy differences.

DOI: 10.1088/0022-3700/18/20/012

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Journal of Physics B Atomic and Molecular Physics

On the probability of non-adiabatic transitions in multiple level crossings

Yōsuke Kayanuma, Susumu Fukuchi

Published Oct 28, 1985• 45 citations

Publisher

Abstract

DOI: 10.1088/0022-3700/18/20/012

Related papers

Nonadiabatic Transitions in Level Crossing with Energy Fluctuation. I. Analytical Investigations

Yōsuke Kayanuma

A simple stochastic model is proposed describing the nonadiabatic transitions in level crossing with energy fluctuation. The model is an extension of Zener's model having a diagonal energy term fluctuating as a Markoffian Gaussian process. The transition rate P ∞ defined in terms of the diabatic basis is calculated exactly by the formal perturbation expansion with respect to the off-diagonal coupling in the two limiting cases: In the slow fluctuation limit, P ∞ coincides with the Landau-Zener formula, $P_{\infty}=P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)$ , where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference. In the strong damping limit, which is a limiting case of large fluctuation amplitude in the rapid fluctuation limit, P ∞ is given by the formula, $P_{\infty}=P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2$ .A simple stochastic model is proposed describing the nonadiabatic transitions in level crossing with energy fluctuation. The model is an extension of Zener's model having a diagonal energy term fluctuating as a Markoffian Gaussian process. The transition rate P ∞ defined in terms of the diabatic basis is calculated exactly by the formal perturbation expansion with respect to the off-diagonal coupling in the two limiting cases: In the slow fluctuation limit, P ∞ coincides with the Landau-Zener formula, $P_{\infty}=P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)$, where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference. In the strong damping limit, which is a limiting case of large fluctuation amplitude in the rapid fluctuation limit, P ∞ is given by the formula, $P_{\infty}=P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2$.

Journal of the Physical Society of JapanJan 1, 1984121 citations

Nonadiabatic Transitions in Level Crossing with Energy Fluctuation. II. Numerical Investigations

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The transition rate P ∞ in the level crossing with energy fluctuation is investigated numerically by the Wiener-Hermite expansion technique for the stochastic variable. It is shown that the fluctuation tends to modify the transition rate as if the nonadiabaticity is effectively increased and that the values of the Landau-Zener formula $P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)$ and the strong damping formula $P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2$ give the upper and the lower bounds of P ∞ , respectively, for given values of J 2 /| v |, where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference.The transition rate P ∞ in the level crossing with energy fluctuation is investigated numerically by the Wiener-Hermite expansion technique for the stochastic variable. It is shown that the fluctuation tends to modify the transition rate as if the nonadiabaticity is effectively increased and that the values of the Landau-Zener formula $P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)$ and the strong damping formula $P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2$ give the upper and the lower bounds of P ∞ , respectively, for given values of J 2 /| v |, where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference.

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M. B. Kenmoe, S. E. Mkam Tchouobiap, C. Kenfack-Sadem +2 more

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.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.

Journal of Physics A Mathematical and TheoreticalFeb 10, 201512 citations

Landau-Zener transitions in a multilevel system: An exact result

A. V. Shytov

We study the $S$ matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].We study the $S$ matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].

arXiv:cond-mat/0312011Physical Review ANov 17, 2004115 citations

P_{\infty}=P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)

, where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference. In the strong damping limit, which is a limiting case of large fluctuation amplitude in the rapid fluctuation limit, P ∞ is given by the formula,

P_{\infty}=P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2

P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)

and the strong damping formula

P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2

give the upper and the lower bounds of P ∞ , respectively, for given values of J 2 /| v |, where J is the off-diagonal coupling constant and v is the velocity of the change of the mean energy difference.

P{=}1-\{1+(4{\pi}J^{2}/\hbar|v|)\}^{-1/2}

, where J is the averaged amplitude of the off-diagonal term and v is the velocity of the change of the energy difference between the crossing levels. In the rapid fluctuation limit, P is given by

P{=}\{1-{\exp}(-4{\pi}J^{2}/\hbar|v|)\}/2

. The intermediate case is studied numerically by the Wiener-Hermite expansion method. Generally, P approaches not 1 but 1/2 in the limit of slow passage, namely, v →0.

We study the

S

matrix for the transitions at an avoided crossing of several energy levels, which is a multilevel generalization of the Landau-Zener problem. We demonstrate that, by extending the Schr\"odinger evolution to complex time, one can obtain an exact answer for some of the transition amplitudes. Similar to the Landau-Zener case, our result covers both the adiabatic (slow evolution) and the diabatic (fast evolution) regimes. The form of the exact transition amplitude coincides with that obtained in a sequential pairwise level crossing approximation, in accord with the conjecture of Brundobler and Elser [J. Phys. A 26, 1211 (1993)].