Landau-Zener-Stückelberg-Majorana dynamics of magnetized quarkonia

TL;DR

This work tackles the real-time nonadiabatic dynamics of charmonium states in time-dependent magnetic fields by mapping the static magnetized spectrum onto a multi-channel Landau-Zener Hamiltonian. The authors extract a compact set of LZ parameters from static spectra, then solve the time-dependent Schrödinger equation under linear ramps, Gaussian decays, and Gaussian pulses to study occupation probabilities in diabatic and adiabatic bases. They find that nonadiabatic LZ transitions and Stückelberg interference substantially redistribute populations, with dynamics strongly influenced by the number of channels, initial conditions, and field profile. The approach offers a practical framework to understand magnetized hadron dynamics and provides benchmarks and guidance for future lattice QCD and tensor-network simulations of real-time QCD phenomena.

Abstract

The mass spectrum of hadrons in magnetic fields features avoided level-crossing structures arising from the mixing of spin eigenstates. In this work, we investigate the impact of level-crossing dynamics of charmonia subjected to time-dependent magnetic fields, where we particularly focus on the occupation probabilities of two or more states as they undergo transitions at avoided crossings. Using a static spectrum of charmonia in magnetic fields, we construct a multi-channel Landau-Zener Hamiltonian. Within this framework, we analyze the time evolution under several representative magnetic-field profiles, including linear ramps and Gaussian decays corresponding to single-passage dynamics, as well as Gaussian pulses realizing double-passage dynamics, and compute the occupation probabilities over a wide range of sweep rates and initial conditions. Our results show that nonadiabatic dynamics, including Landau-Zener transitions and Stückelberg interference, strongly influences the occupation probabilities of charmonia. These findings provide new insights into the real-time dynamics of magnetized hadrons and offer useful guidance for future lattice simulation studies.

Figures (12)

• Figure 1: Comparison between the eigenvalues of the multi-channel LZ Hamiltonians and the quark-model spectrum for the two-, three-, four-, and five-channel cases. Each panel shows the fitted LZ energy levels (solid lines) together with the discrete quark-model data points (markers), demonstrating good agreement in the region of interest. The fitting parameters, energy slopes $\alpha_i$, offsets $\delta_i$, and off-diagonal couplings $\Delta_{i,i+1}$, are listed in Table \ref{['tab:LZ_parameters_full']}. The gray regions indicate data excluded from the fit. Including more channels enables the LZ model to reproduce additional avoided crossings and higher excited states.
• Figure 2: Schematic illustration of the avoided crossings in the two-channel model and the driving magnetic-field profiles considered in this work: (i) linear ramps, (ii) Gaussian decays, and (iii) Gaussian pulses. Panels (i) and (ii) correspond to single passages through the avoided crossing, while panel (iii) shows a double passage. The arrows indicate the direction of time evolution along the magnetic-field trajectories.
• Figure 3: Occupation probabilities in the two-channel model for a single passage at three sweep rates with a $\psi=(1,0)^T$ initial state: (top panels) linear ramp and (bottom panels) Gaussian decay. The gray dotted line shows the magnetic-field profiles with a maximum value of $eB_{\max}=2~\text{GeV}^2$. The red and green lines represent the occupation probabilities of the $J/\psi$- and $\eta_c(2S)$-started states, respectively, shown in the diabatic basis (solid lines) and in the adiabatic basis (dashed lines). In general, the line style and color codes for the occupation probabilities follow the notation in Table \ref{['tab:state-shorthand']}.
• Figure 4: Occupation probabilities in the three-channel model for a fast sweep, shown for three different initial diabatic states: (left) $\psi=(1,0,0)^T$ initial state, (middle) $\psi=(0,1,0)^T$ initial state, and (right) $\psi=(0,0,1)^T$ initial state. The top and bottom panels correspond to a linear ramp and a Gaussian-decay magnetic-field profile, respectively.
• Figure 5: Same as Fig. \ref{['fig:single-3level-na']}, but for a slow sweep, illustrating the adiabatic evolution.