[Paper Review] Nonadiabatic Landau-Zener-Stückelberg-Majorana transitions, dynamics, and interference

O. V. Ivakhnenko, S. N. Shevchenko+1 more|arXiv (Cornell University)|Mar 30, 2022

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper provides a comprehensive theoretical review of nonadiabatic Landau-Zener-Stückelberg-Majorana (LZSM) transitions in driven quantum two-level systems, systematically analyzing transition dynamics, interference effects, and control mechanisms across diverse physical platforms. It derives and compares key analytical methods—including the adiabatic-impulse model, rotating-wave approximation, and Floquet theory—and establishes quantitative connections between interference patterns and measurable transition probabilities, especially in periodic driving and dissipative regimes.

ABSTRACT

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 engineering quantum systems. Here, we study systematically various aspects of LZSM physics and review the relevant literature, significantly expanding the review article in [Shevchenko, S. N., S. Ashhab, and F. Nori (2010), "Landau-Zener-Stückelberg interferometry," Phys. Rep. 492, 1].

Motivation & Objective

• To systematically analyze nonadiabatic LZSM transitions in driven quantum systems, extending beyond the original Landau-Zener model to include interference and dynamical effects.
• To unify and compare multiple theoretical approaches—such as the adiabatic-impulse model, rotating-wave approximation, and Floquet theory—for describing periodic driving and multi-passage dynamics.
• To clarify the role of interference in transition probabilities, particularly in the double-passage regime, and to connect analytical solutions with experimental observables like occupation probabilities and Rabi oscillations.
• To examine the impact of decoherence, dissipation, and temperature on LZSM dynamics, especially in the context of quantum control and quantum information processing.
• To provide a detailed analytical framework linking the Bessel function and Airy function representations of transition probabilities, enabling quantitative predictions for superconducting qubits, quantum dots, and atomic systems.

Proposed method

• Derives the LZSM transition probability using multiple analytical techniques: near-adiabatic limit (Landau), parabolic cylinder functions (Zener), contour integrals (Majorana), and WKB approximation (Stückelberg).
• Applies the adiabatic-impulse model (AIM) to describe double- and multiple-passage transitions, particularly in periodic driving fields, and derives the upper-level occupation probability in the fast-passage limit.
• Uses the rotating-wave approximation (RWA) to analyze multi-photon Rabi oscillations and maps the transition dynamics to a Bloch equation framework.
• Employs Floquet theory to describe the effective Hamiltonian of periodically driven systems and to analyze the stability and periodicity of solutions.
• Transforms Bessel function expressions into Airy function representations via asymptotic approximations, enabling analytical treatment of interference patterns in the presence of detuning and damping.
• Derives a rate equation approach with white noise and connects it to the transition rate via the Bessel-Airy transformation, particularly in the double-passage regime with finite decoherence.

Experimental results

Research questions

• RQ1 How do interference effects manifest in nonadiabatic transitions of a periodically driven two-level system, and what determines the periodicity of the resulting oscillations?
• RQ2 What is the quantitative relationship between the Airy function approximation and the Bessel function representation of the transition probability in the double-passage regime?
• RQ3 How does decoherence (via relaxation and dephasing rates) affect the visibility and phase of LZSM interference fringes in driven qubits?
• RQ4 In what conditions do the adiabatic-impulse model and the rotating-wave approximation yield equivalent predictions for multi-photon transitions?
• RQ5 How can the transition probability be analytically expressed in terms of the driving amplitude, frequency, and detuning, especially in the limit of strong driving and finite relaxation?

Key findings

• The double-passage transition probability is accurately described by $ P_{+}^{ ext{double}} acksimeq rac{2 au}{ au_{ ext{Rabi}}} rac{ ilde{ u}^2}{ au_{ ext{Rabi}}} ext{ with a phase } ilde{ u} = rac{2 au}{ au_{ ext{Rabi}}} $, showing sinusoidal oscillations dependent on detuning and driving frequency.
• The Airy function approximation of the Bessel function yields $ W acksimeq rac{ au}{ au_{ ext{Rabi}}} ext{Ai}^2ig( ext{argument} ig) $, providing a smooth, analytically tractable form for interference patterns.
• In the double-passage regime with $ au_1 acksimeq 2T_d $, the transition rate $ W $ matches the adiabatic-impulse model prediction: $ W acksimeq rac{ au}{ au_{ ext{Rabi}}} ext{ with phase } rac{2 au}{ au_{ ext{Rabi}}} $, confirming consistency between methods.
• The phase of the interference fringes is determined by $ rac{2 au}{ au_{ ext{Rabi}}} $, and the oscillation frequency is proportional to $ rac{A}{ au_{ ext{Rabi}}} $, with a correction factor $ rac{2 au}{ au_{ ext{Rabi}}} $.
• The analytical expression $ P_{+}^{ ext{double}} acksimeq rac{2 au}{ au_{ ext{Rabi}}} ext{ with phase } rac{2 au}{ au_{ ext{Rabi}}} $ is shown to be consistent with the Airy function-based derivation, validating the approach.
• The transition probability in the double-passage regime is quantitatively linked to the relaxation time $ T_1 $, with $ P_+ acksimeq T_1 W $, and the oscillation depth depends on $ rac{ au}{ au_{ ext{Rabi}}} $, confirming the role of decoherence in visibility.