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Physical Review A

Landau-Zener transitions in a level-crossing system with periodic modulations of the diagonal energy

Yōsuke Kayanuma, Y. Mizumoto

Published Oct 30, 2000• 35 citations
Publisher

Abstract

We clarify some aspects of population dynamics of a level-crossing system, in which a periodic modulation of the relative energy is superimposed on the energy change with a constant velocity. For moderate values of the off-diagonal coupling, the temporal evolution of the system is well described by a transfer-matrix formalism. It is shown that the existence probability on one level shows a series of steplike changes both in the high-frequecy limit and the low-frequency limit of the periodic modulation, reflecting the quasiquantization of the oscillating field in the former case, while reflecting the multiple real crossings in the latter case.

DOI
10.1103/physreva.62.061401

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We clarify some aspects of population dynamics of a level-crossing system, in which a periodic modulation of the relative energy is superimposed on the energy change with a constant velocity. For moderate values of the off-diagonal coupling, the temporal evolution of the system is well described by a transfer-matrix formalism. It is shown that the existence probability on one level shows a series of steplike changes both in the high-frequecy limit and the low-frequency limit of the periodic modulation, reflecting the quasiquantization of the oscillating field in the former case, while reflecting the multiple real crossings in the latter case.
We present an analytic study of the population dynamics of a two-state system interacting with an external field and subjected to periodic level crossings. We apply an evolution matrix approach to calculate the excited-state population at the crossings (the nodes) and at the antinodes. The results are expressed in terms of only two parameters: the transition probability p for a quarter period from a crossing to an antinode, and the transition probability P for a half period between two successive crossings. We find that the values of the excited-state population at the antinodes can form global (gross) structures. We show that these structures and the population dynamics as a whole are very sensitive to the initial phase \ensuremath{\varphi} of the frequency-modulated field, particularly in the limits \ensuremath{\varphi}=0 (cosine modulation) and \ensuremath{\varphi}=\ensuremath{\pi}/2 (sine modulation). We calculate the parameters p and P by using two analytic approaches: one based on the original Landau-Zener model, and the other based on the finite Landau-Zener model. Both approaches unexpectedly lead to the same results. The notion of the global structures and the relevant parametrization in terms of p and P allow us to find various distinctive cases of population dynamics, such as population swapping, completely periodic evolution, superpositional trapping, and stepwise evolution.
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.
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.
We study a two-level system described by a Hamiltonian which exhibits a time-periodic variation of its diagonal matrix elements in the small coupling regime. We investigate the transition probability, the Bloch–Siegert shift, and the Rabi frequency of the multiquantum transition as functions of the amplitude and frequency of the oscillations. These quantities all exhibit a very distinct behavior depending upon whether or not the amplitude of the oscillations allows for the crossing of the levels. This behavior is monotonic for small amplitudes while it presents oscillations as a function of the frequency for larger amplitudes. These oscillations are unambiguously attributed to the Landau–Zener–Stueckelberg interference effects between the crossing point and the turning point. A discussion of the relevance of these oscillations to possible experimental observations, notably in electron-transfer reactions, is presented.
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 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.