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Journal of the Physical Society of Japan

Stochastic Theory for Nonadiabatic Level Crossing with Fluctuating Off-Diagonal Coupling

Yōsuke Kayanuma

Published May 15, 1985• 62 citations
Publisher

Abstract

A simple stochastic model is presented describing the nonadiabatic transitions in the level crossing with fluctuating off-diagonal coupling, in which the fluctuation is assumed to obey the Markoffian Gaussian process. The probability P that the transition occurs from one diabatic state to another is calculated exactly in the two limiting situations: In the slow fluctuation limit, P is given by \(P{=}1-\{1+(4{\pi}J^{2}/\hbar|v|)\}^{-1/2}\), 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.

DOI
10.1143/jpsj.54.2037

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A simple stochastic model is presented describing the nonadiabatic transitions in the level crossing with fluctuating off-diagonal coupling, in which the fluctuation is assumed to obey the Markoffian Gaussian process. The probability P that the transition occurs from one diabatic state to another is calculated exactly in the two limiting situations: In the slow fluctuation limit, P is given by (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.
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∞=PLZ≡1−exp⁡(−2πJ2/ℏ∣v∣)P_{\infty}=P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)P∞​=PLZ​≡1−exp(−2πJ2/ℏ∣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∞=PSD≡{1−exp⁡(−4πJ2/ℏ∣v∣)}/2P_{\infty}=P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2P∞​=PSD​≡{1−exp(−4πJ2/ℏ∣v∣)}/2.
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 PLZ≡1−exp⁡(−2πJ2/ℏ∣v∣)P_{\text{LZ}} \equiv 1 - \exp (-2 \pi J^{2}/\hbar |v|)PLZ​≡1−exp(−2πJ2/ℏ∣v∣) and the strong damping formula PSD≡{1−exp⁡(−4πJ2/ℏ∣v∣)}/2P_{\text{SD}} \equiv \{1 - \exp (-4 \pi J^{2}/\hbar |v|)\}/2PSD​≡{1−exp(−4πJ2/ℏ∣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.
A comprehensive investigation of the transition dynamics of a level-crossing system with quantum dissipation is carried out for the Landau-Zener model coupled with a system of phonons. Analytical study by the formal perturbation expansion series with respect to the off-diagonal matrix element shows that the transition dynamics is characterized by the competition between the energy fluctuation and the energy dissipation. Closed expressions of the transition rate are derived for various limiting situations in a unified way. In the case of strong decoherence, a formula of the transition rate is obtained, which covers the high-temperature limit and the low-temperature, strong-coupling limit. Numerical calculation by utilizing the damping hyperoperator technique is performed, which clarifies the overall features of the time evolution of the system.
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.
The effect of a slow noise in nondiagonal matrix element J(t)J(t)J(t) that describes the diabatic level coupling on the probability of the Landau-Zener transition is studied. For slow noise, the correlation time ${\ensuremath{\tau}}_{c}$ of J(t)J(t)J(t) is much longer than the characteristic time of the transition. Existing theory for this case suggests that the average transition probability is the result of averaging of the conventional Landau-Zener probability, calculated for a given constant JJJ, over the distribution of JJJ. We calculate a finite-${\ensuremath{\tau}}_{c}$ correction for this classical result. Our main finding is that this correction is dominated by sparse realizations of noise for which J(t)J(t)J(t) passes through zero within a narrow time interval near the level crossing. Two models of noise, random telegraph noise and Gaussian noise, are considered. Naturally, in both models the average probability of transition decreases upon decreasing ${\ensuremath{\tau}}_{c}$. For Gaussian noise we identify two domains of this falloff with specific dependencies of average transition probability on ${\ensuremath{\tau}}_{c}$.
Dissipative effects on the nonadiabatic transition for the two- and three-level systems are studied. When the system is affected by a strong dissipation through the diabatic states, the exact transition probability is enumerated making use of the effective master equation. In the two-level system, we consider the case where the external field is swept from not only a negative large value but also from the resonant field, and the exact transition probabilities in these cases are derived. The transition probabilities are derived for the three-level system where the three diabatic states form only one avoided-level crossing point. These probabilities are compared with that in the pure quantum case obtained by Carroll and Hioe.