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Physical Review A

Quantum transitions at a level crossing of decaying states

Curt A. Moyer

Published Aug 2, 2001• 14 citations
Publisher

Abstract

Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.

DOI
10.1103/physreva.64.033406

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Using a comparison equation method rooted in the Landau-Zener model, we report results for the probability of transition and survival in two-state models for a crossing of levels having some width, to account for dissipation. The theory allows arbitrary forms for the state energies, their rate of decay, and coupling, subject only to the limitation inherent in the Landau-Zener class (coupling with infinite ``pulse area''). We show how the strong decay result of adiabatic elimination can be recovered, but with important restrictions as to its usability. We also confirm the validity of the well-known Dykhne formula for crossings of real energy levels, and use the new theory to extend the Dykhne result to a class of dissipative crossings we call pseudoreal. Finally, we examine the overall probability for transition at two successive crossings, as occurs in atomic collisions. Here, interference of the crossing states leads to oscillations in the transition probability with collision time. For slow collisions we find that even a modest level of dissipation heavily damps these oscillations, but at the same time enhances the average probability of transition.
An analytical study of the dissipative Landau-Zener model is presented. The model where two energy levels at constant speed are brought to cross is a standard model used to describe a large variety of phenomena. In many cases of interest the presence of coupling of the two-state system to an environment is of importance, the accounting for which shall be done here from first principles. Analytical results for the excitation transition from the ground state of the two-state system at large negative times to the excited state at large positive times (as well as the opposite, decay, transition) are obtained in terms of the speed by which the two levels approach each other, the energy gap between the adiabatic energies, the coupling strength to the environment, and its temperature. For the excitation transition we find the following results: In the slow-passage limit of small sweeping speed it is shown that adiabaticity is limited to low temperatures and the quantitative adiabatic criterion is established. Particularly, at zero temperature there is no influence of the environment on the transition probability as a consequence of a compensation property shown to be peculiar to the linear-sweep model. The transition is at low temperatures due to quantum tunneling and, with an increase in temperature, an intermediate region appears where the transition is dominated by thermally assisted transitions across the energy gap before finally at high temperatures a saturated regime is reached with equal population of the levels. In contrast to the dependence on the temperature the dependence of the transition probability as a function of coupling strength is nonmonotonic with maximum influence at intermediate strength. In the fast-passage limit with rapid sweep speed there is no influence of the environment on the transition probability. For the decay transition the adiabatic limit does not exist for the linear-sweep model for physically relevant spectra of the environment and the decay transition is dominated by spontaneous emission, except in the fast-passage limit and the high-temperature limit where the decay transition probability equals the excitation probability.
The Landau-Zener formula provides an analytical expression for the final excitation of a quantum system after passage of an avoided crossing of two energy levels. If the two levels correspond to a ground state and to an excited state which is subject to radiative decay, the probability of exciting the system by adiabatic passage of the level crossing is reduced. In this article we use a stochastic master equation to study the level crossing dynamics when the system is subject to continuous probing of the emitted radiation. The measurement back action on the system associated with the fluctuating homodyne detection record alters the level crossing dynamics, leading to significant excitation in spite of decay and imperfect transfer.
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 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 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.
For a Landau-Zener transition in a two-level system, the probability for a particle, initially in the first level, to survive the transition and to remain in the first level, does not depend on whether or not the second level is broadened [V. M. Akulin and W. P. Schleich, Phys. Rev. A 46, 4110 (1992)]. In other words, the seminal Landau-Zener result applies regardless of the broadening of the second level. The same question for the multistate Landau-Zener transition is addressed in the present paper. While for integrable multistate models, where the transition does not involve interference of the virtual paths, it can be argued that the independence of the broadening persists, we focus on nonintegrable models involving interference. For a simplest four-state model, which allows an analytical treatment, we demonstrate that the decay of the excited states affects the survival probability provided that the widths of the final states are different.