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arXiv:2208.00481 · Scientific Reports

Majorana’s approach to nonadiabatic transitions validates the adiabatic-impulse approximation

Polina O. Kofman, O. V. Ivakhnenko, Sergey N Shevchenko, Franco Nori

Published Mar 28, 2023• 29 citations
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Abstract

The approach by Ettore Majorana for non-adiabatic transitions between two quasi-crossing levels is revisited and significantly extended. We rederive the transition probability, known as the Landau-Zener-Stückelberg-Majorana formula, and introduce Majorana's approach to modern readers. This result, typically referred as the Landau-Zener formula, was published by Majorana before Landau, Zener and Stückelberg. Moreover, we go well beyond previous results and we now obtain the full wave function, including its phase, which is important nowadays for quantum control and quantum information. The asymptotic wave function correctly describes the dynamics away from the avoided-level crossing, while it has limited accuracy in that region.

DOI
10.1038/s41598-023-31084-y

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•Jan 1, 2026•0 citations
The approach by Ettore Majorana for non-adiabatic transitions between two quasi-crossing levels is revisited and significantly extended. We rederive the transition probability, known as the Landau-Zener-Stückelberg-Majorana formula, and introduce Majorana's approach to modern readers. This result, typically referred as the Landau-Zener formula, was published by Majorana before Landau, Zener and Stückelberg. Moreover, we go well beyond previous results and we now obtain the full wave function, including its phase, which is important nowadays for quantum control and quantum information. The asymptotic wave function correctly describes the dynamics away from the avoided-level crossing, while it has limited accuracy in that region.
Since the pioneering works by Landau, Zener, Stückelberg, and Majorana (LZSM), it has been known that driving a quantum two-level system results in tunneling between its states. Even though the interference between these transitions is known to be important, it is only recently that it became both accessible, controllable, and useful for manipulating a growing number of quantum systems. Here, we systematically study various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in Ref. [504].
We study the stability of counterdiabatic drive by computing the joint dependence of the probability of nonadiabatic transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ on the adiabatic parameter $\ensuremath{\delta}$ and on the normalized amplitude of the adiabatic gauge potential (AGP) $\ensuremath{\eta}$ in the Landau-Zener-St\"uckelberg-Majorana model. We show that the Dykhne-Davis-Pechukas formula cannot be readily applied since the AGP introduces a singularity in the Hamiltonian which makes the wave function multivalued in the complex-time plane. This can be understood as the non-Abelian Aharonov-Bohm phase introduced by the AGP and leads to the counterdiabatic correction of the Landau-Zener formula. In particular, it shows that, unlike the nonperturbative suppression of transitions in the adiabatic limit $\ensuremath{\delta}\ensuremath{\rightarrow}0$, the probability is only perturbatively suppressed in the counterdiabatic limit $\ensuremath{\eta}\ensuremath{\rightarrow}1$. We then consider the extension of our results to integrable time-dependent quantum Hamiltonians. We prove that the AGP satisfies the flatness constraint which characterizes integrability in these models, which allows us to derive simple expressions for the AGP and the probability of transitions ${P}_{\ensuremath{\eta}}(\ensuremath{\delta})$ near adiabatic or counterdiabatic evolution in three- and four-state integrable examples.
Nonadiabatic transitions in multilevel systems appear in various fields of physics, but it is not easy to analyze their dynamics in general. In this paper, we propose to extend the adiabatic impulse approximation to multilevel systems. This approximation method is shown to be equivalent to a series of unitary evolutions and facilitates to evaluate the dynamics numerically. In particular, we analyze the dynamics of the Landau-Zener grid model and the multilevel Landau-Zener-St\"uckelberg-Majorana interference model, and confirm that the results are in good agreement with the exact dynamics evaluated numerically. We also derive the conditions for destructive interference to occur in the multilevel system.
We show that a density matrix formalism provides a useful description of non-adiabatic transitions in two-state quantum systems. Compared to a traditional Hamiltonian formalism, even in the absence of decoherence when there is full equivalence between the two, the density matrix formalism provides a convenient change of variables that yields a powerful general analytical solution. This solution nicely describes a transition regime between the well known Landau-Zener-Stuckelberg-Majorana (LZSM) approximation and the extremely non-adiabatic limit. Our results have very general applications, within a large variety of problems in quantum physics, neutrino physics, cosmology.
The control of nonadiabatic quantum transitions is a requirement for the development of modern quantum technologies. The paradigm to understand these dynamics is the Landau-Zener model. However, for nearly a century, the exact transition probability of this model has been derived using special functions or semiclassical approximations. These conventional methods obscure the underlying physical mechanisms and transition dynamics near the avoided crossing. This cumulative thesis, resulting from the author's research activities at the Intitute of Quantum Physics at Ulm University under the supervision of Prof. Dr. Wolfgang P. Schleich, establishes an analytical framework to study the underlying structure of two-level crossing dynamics. In contrast to the standard techniques, the framework developed in this thesis is based on elementary concepts combined with complex analysis. First, we map the Landau-Zener equations onto an integro-differential equation to decouple them and formulate the Markov approximation which reduces the complicated integrals to the evaluation of Fresnel integrals. This technique reproduces in the asymptotic limit the exact probability amplitude of the Landau-Zener model. Next, to uncover the origin behind the success of the Markov approximation, we analyze the Landau-Zener equations in terms of their underlying nonlinear dynamics. First, we introduce an amplitude-and-phase approach, which explains in a clear way the emergence of the Stueckelberg oscillations through the dynamics of the phase velocity. Second, we map the two-level dynamics onto a nonlinear Riccati differential equation and show that the Markov approximation is equivalent to a linearization of the derived Riccati equation. Moreover, our approach helps to identify the nonlinear term as the reason for the breakdown of the Markov approximation close to the avoided crossing. Finally, we derive exact asymptotic expressions for the transition amplitudes using elementary functions for both negative and positive times. Connecting these two regimes via the analytical continuation of the logarithmic phase reveals that the Landau-Zener transition is mainly governed by a logarithmic phase singularity. Based on this logarithmic phase singularity, we introduce a new time scale to map the system dynamics as a path via a logarithmic phase ansatz in the complex plane. Evaluating this path via the Cauchy’s integral formula yields the exact transition probability which demonstrates that the exact solution emerges from the underlying phase singularity rather than properties of special functions.