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Journal of the Physical Society of Japan

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

Yōsuke Kayanuma

Published Jan 1, 1984• 27 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.53.118

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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.
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.
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+(4πJ2/ℏ∣v∣)}−1/2P{=}1-\{1+(4{\pi}J^{2}/\hbar|v|)\}^{-1/2}P=1−{1+(4πJ2/ℏ∣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⁡(−4πJ2/ℏ∣v∣)}/2P{=}\{1-{\exp}(-4{\pi}J^{2}/\hbar|v|)\}/2P={1−exp(−4πJ2/ℏ∣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 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.
We examine the effect of nonlinearity at a level crossing on the probability for nonadiabatic transitions P. By using the Dykhne-Davis-Pechukas formula, we derive simple analytic estimates for P for two types of nonlinear crossings. In the first type, the nonlinearity in the detuning appears as a perturbative correction to the dominant linear time dependence. Then appreciable deviations from the Landau-Zener probability PLZ{P}_{\mathrm{LZ}}PLZ​ are found to appear for large couplings only, when P is small; this explains why the Landau-Zener model is often seen to provide more accurate results than expected. In the second type of nonlinearity, called essential nonlinearity, the detuning is proportional to an odd power of time. Then the nonadiabatic probability P is qualitatively and quantitatively different from PLZ{P}_{\mathrm{LZ}}PLZ​ because, on the one hand, it vanishes in an oscillatory manner as the coupling increases, and, on the other, it is much larger than PLZ.{P}_{\mathrm{LZ}}.PLZ​. We suggest experimental situations where the predicted deviations can be observed.
Some aspects of the Landau-Zener-type transition of a periodically driven two-level system with dephasing are clarified. Under the condition that the transition is well localized around the level-crossing region, the transition dynamics for the typical cases is described by simple formulas; namely, by the transfer-matrix method for the long-range coherence, by the application of the incoherent Landau-Zener formula for the short-range coherence, and by the strong-dephasing formula in the limit of complete decoherence.