⚛️ nuclear theory

Landau-Zener-Stückelberg-Majorana dynamics of magnetized quarkonia

This paper investigates the nonadiabatic time evolution of magnetized charmonia under various time-dependent magnetic field profiles by constructing a multi-channel Landau-Zener Hamiltonian, revealing how Landau-Zener transitions and Stückelberg interference significantly influence state occupation probabilities and providing guidance for future lattice simulations.

Original authors: Ahmad Jafar Arifi, Kei Suzuki

Published 2026-07-01

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CC BY 4.0

Original authors: Ahmad Jafar Arifi, Kei Suzuki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a tiny, invisible dance floor where particles called charmonia (which are like heavy-duty "atoms" made of a charm quark and its anti-quark partner) are performing. Usually, these particles have specific energy levels, like rungs on a ladder. They can sit on one rung or jump to another, but they generally stay put unless something pushes them.

This paper explores what happens when you shake that dance floor with a magnetic field that changes over time.

The Setup: The "Avoided Crossing"

In a normal world, if two rungs on a ladder get close to each other, they might merge. But in the quantum world of these particles, something weird happens: the rungs get close, but they repel each other and never quite touch. This is called an "avoided crossing."

Think of it like two trains on parallel tracks that are about to collide. Instead of crashing, the tracks magically curve away from each other just before they meet. The paper studies what happens to the passengers (the particles) when the tracks curve.

The Problem: Static vs. Moving

Scientists already knew what happens if the magnetic field is static (stays the same forever). They could map out exactly where these "avoided crossings" happen.

However, in the real universe (like in heavy-ion collisions in particle accelerators), magnetic fields are time-dependent. They flash on and off, ramp up, or pulse. Calculating how particles move in these changing fields is incredibly hard, like trying to predict the path of a leaf blowing in a gusty, shifting wind using only a static map.

The Solution: The "Landau-Zener" Shortcut

The authors, Ahmad Jafar Arifi and Kei Suzuki, developed a clever shortcut. Instead of trying to simulate every tiny detail of the magnetic field's effect on the quarks, they built a simplified model called a Landau-Zener Hamiltonian.

Think of this model as a traffic light system for the particles:

The Map: They first looked at the "static map" (the energy levels when the field is steady) to find where the "avoided crossings" (the curves) are located.
The Rules: They created a set of rules (a mathematical formula) that says: "If the magnetic field changes at speed $X$ , and the gap between the tracks is size $Y$ , here is the probability the particle will jump tracks."
The Simulation: Using these rules, they simulated three different types of magnetic field "weather":
- Linear Ramp: The field slowly and steadily increases (like turning up a dimmer switch).
- Gaussian Decay: The field starts strong and fades away (like a flashlight battery dying).
- Gaussian Pulse: The field flashes up and then goes down (like a camera flash).

The Findings: How the Particles React

The paper reveals how the particles behave depending on how fast the magnetic field changes:

The Slow Walk (Adiabatic): If the magnetic field changes very slowly, the particle is like a careful hiker. It notices the tracks curving and smoothly follows the path of the track it started on. It doesn't jump; it just rides the curve.
The Fast Run (Non-Adiabatic): If the field changes very quickly (like a sudden gust of wind), the particle is like a startled rabbit. It doesn't have time to follow the curve. It keeps going straight, effectively "jumping" to the other track.
The Middle Ground: If the speed is just right, the particle does a mix of both, leading to a partial jump.

The "Double Take" (Interference)

The most interesting part happens with the Gaussian Pulse (the flash). Here, the magnetic field goes up and then comes back down. The particle crosses the "avoided crossing" twice.

This is like walking through a doorway, turning around, and walking back through the same doorway. The paper shows that the two trips interfere with each other, creating a complex pattern called Stückelberg interference.

The Analogy: Imagine you are trying to guess a secret code. If you walk through the door once, you get a hint. If you walk through twice, the two hints combine. Depending on the exact timing (the speed of the pulse), the hints might cancel each other out or amplify each other. The authors found that by tweaking the "speed" of the pulse, they could control exactly where the particle ends up, almost like tuning a radio to get a clear signal.

Why This Matters (According to the Paper)

The authors state that this work provides a new way to understand how hadrons (particles like protons and neutrons, or in this case, charmonia) behave in real-time under extreme conditions.

They emphasize that their method is a tool for future scientists. Specifically, they suggest this model can serve as a "benchmark" or a guide for lattice QCD simulations (super-computer simulations of the strong nuclear force). Since simulating real-time changes in these systems is currently very difficult for computers, this simplified "traffic light" model offers a way to predict what should happen, helping scientists design better computer experiments for the future.

In short: The paper takes a complex quantum problem, simplifies it into a set of rules based on how the energy levels curve, and shows that by controlling the speed and shape of the magnetic field, we can predict exactly how these tiny particles will jump between energy states.

Technical Summary: Landau-Zener-Stückelberg-Majorana Dynamics of Magnetized Quarkonia

Problem Statement
The behavior of hadrons under extreme external conditions, particularly intense electromagnetic fields, offers a window into the nonperturbative regime of Quantum Chromodynamics (QCD). While the static properties of quarkonia (bound states of heavy quarks and antiquarks) in magnetic fields have been extensively studied using lattice QCD and quark models, revealing characteristic features like level repulsions and avoided crossings due to spin-state mixing, the dynamical response to time-dependent magnetic fields remains less explored. Simulating real-time dynamics in time-dependent fields is challenging for standard Monte Carlo lattice methods due to the sign problem. This work addresses the need to understand the time evolution of hadrons, specifically charmonia, as they traverse avoided level crossings induced by varying magnetic fields. The focus is on the occupation probabilities of states undergoing nonadiabatic transitions, a phenomenon relevant to heavy-ion collisions where magnetic fields change rapidly, and for designing future lattice simulations.

Methodology
The authors propose a novel approach based on a multichannel Landau-Zener (LZ) model to study the time evolution of charmonia, avoiding the need to solve the full time-dependent Schrödinger equation with a complex microscopic Hamiltonian.

Static Spectrum Construction: The study begins by utilizing a nonrelativistic constituent quark model to calculate the static charmonium spectrum in a magnetic field ( $B = B\hat{z}$ ). The Hamiltonian includes kinetic terms, a confining potential, a Coulomb term, and a spin-spin interaction. The magnetic field induces mixing between the pseudoscalar ( $\eta_c$ ) and longitudinal vector ( $J/\psi$ ) states, leading to avoided level crossings. These spectra are calculated numerically using the Cylindrical Gaussian Expansion Method (CGEM).
Effective Hamiltonian Mapping: Instead of simulating the full quark-model dynamics, the authors map the static spectrum onto an effective multichannel Landau-Zener Hamiltonian. In this effective description:
- The diagonal elements represent the energy levels of individual states, parametrized as linear functions of the magnetic field ( $H_{ii}(t) = \alpha_i eB(t) + \delta_i$ ).
- The off-diagonal elements ( $\Delta_{i,i+1}$ ) represent constant couplings between neighboring states responsible for the mixing at avoided crossings.
- The parameters ( $\alpha_i, \delta_i, \Delta_{i,i+1}$ ) are extracted by fitting the LZ Hamiltonian eigenvalues to the static quark-model spectrum for systems ranging from two to five coupled channels (e.g., $\{J/\psi, \eta'_c\}$ , $\{\eta_c, J/\psi, \eta'_c\}$ , etc.).
Time Evolution Simulation: With the effective Hamiltonian constructed, the time-dependent Schrödinger equation is solved for specific magnetic field profiles $B(t)$ $B (t)$ . The study analyzes three representative profiles:
- Linear Ramps: Constant sweep rate ( $B(t) \propto t$ ).
- Gaussian Decays: Time-dependent sweep rate, mimicking a single passage with a decaying field.
- Gaussian Pulses: Symmetric rise and fall, realizing a double passage through the avoided crossing.
Analysis of Probabilities: The authors compute occupation probabilities in both the diabatic basis (uncoupled states) and the adiabatic basis (instantaneous eigenstates) for various sweep rates and initial conditions.

Key Contributions

Spectrum-Driven LZ Framework: The paper establishes a method to construct an effective time-dependent LZ Hamiltonian directly from static spectral data (either from quark models or potentially lattice QCD), bypassing the need for the full microscopic time-dependent Hamiltonian.
Multichannel Extension: The study extends the standard two-channel LZ analysis to multichannel systems (up to five channels), capturing the complexity of multiple avoided crossings and level repulsions involving excited states like $\eta'_c$ and $\psi'$ .
Systematic Profile Comparison: The work systematically compares single-passage dynamics (linear ramps vs. Gaussian decays) with double-passage dynamics (Gaussian pulses), highlighting the emergence of Stückelberg interference in the latter.

Results

Sweep Rate Dependence: The occupation probabilities are strongly controlled by the magnetic field sweep rate ( $\nu_B$ $ν_{B}$ ).
- Fast Sweeps: Lead to strongly nonadiabatic evolution where the system remains in its initial diabatic state, and adiabatic probabilities undergo sharp redistribution.
- Slow Sweeps: Result in predominantly adiabatic evolution, where the system follows the instantaneous eigenstates, leading to near-complete transfer between diabatic channels.
- Intermediate Sweeps: Exhibit partial nonadiabatic transitions.
Profile Effects:
- Linear Ramps: Serve as the standard benchmark.
- Gaussian Decays: Due to the time-dependent sweep rate, the avoided crossing occurs earlier, enhancing nonadiabatic effects compared to linear ramps with the same maximum field.
- Gaussian Pulses (Double Passage): The system traverses the avoided crossing twice, leading to Landau-Zener-Stückelberg (LZS) interference. The final occupation probability is highly sensitive to the pulse width and sweep rate, allowing for control over the final state population. Small variations in the pulse width can cause the final probability to oscillate sharply (e.g., from 0.4 to 1.0).
Multichannel Dynamics:
- In multichannel systems, the presence of additional states (like $\eta_c$ and $\psi'$ ) introduces new transition pathways and interference patterns.
- Initial State Dependence: The choice of initial state (diabatic vs. adiabatic) significantly alters the dynamics. Diabatic initial states in slow sweeps exhibit large-amplitude oscillations due to coherent interference between components, whereas adiabatic initial states evolve more smoothly.
- Channel Roles: In fast sweeps, lower channels have minor influence while transitions to higher channels dominate. In slow sweeps, nearby channels play a significant role in shaping the evolution through level repulsion in weak-field regions.

Significance and Claims
The authors claim that their work provides new insights into the real-time dynamics of magnetized hadrons. By demonstrating that nonadiabatic dynamics, including LZ transitions and Stückelberg interference, strongly influence charmonium occupation probabilities, the study offers a theoretical framework for understanding hadron behavior in time-dependent environments.

Crucially, the paper positions this LZ-based approach as a useful guide for future lattice QCD simulations. Since direct real-time simulation of hadrons in time-dependent fields is currently hindered by the sign problem, the authors suggest that their results can serve as benchmarks for future sign-problem-free approaches (such as tensor-network methods) and for simulations in higher spatial dimensions. The study emphasizes that the LZ framework is versatile, as it relies on spectral input rather than a specific microscopic model, making it applicable to spectra derived from lattice QCD or effective field theories. The authors explicitly distinguish these coherent nonadiabatic transitions from irreversible decays, noting that their analysis focuses on probability redistribution within the bound-state subsystem.