Using tensor network simulations, we compute the time evolution of the bottom of the entanglement spectrum and the return rate function after global quantum quenching in the Ising model. We consider ground-state quenching towards the meson parameter range with confined fermion pairs as non-perturbative bound states in the semiclassical regime and relativistic E$_8$ theory. In both cases, we find that only the dominant eigenvalues ​​of the modular Hamiltonian fully encode the mesonic content of the quantum many-body or quantum field theory, giving rise to nearly identical entanglement oscillations in the entanglement entropy. When the initial state is prepared in the paramagnetic phase, the return rate density exhibits regular cusps at unequally spaced positions, indicating the emergence of a dynamic quantum phase transition in which the entanglement spectrum remains gapped. Our analysis provides a deeper understanding of the role of quantum information content on the dynamics of emergent phenomena reminiscent of systems in high-energy physics.

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