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Middle English quenchen, "to extinguish, put out" (heat, light, fire, also of desire, hunger, thirst), also figurative, "to bring to naught, eliminate, render ineffectual" (c. 1200), Old English acwencan "to quench" (of fire, light), from Proto-Germanic *kwenkjanan, probably a causative form from the source of Old English cwincan "to go out, be extinguished," Old Frisian kwinka. No certain cognates outside Germanic; perhaps a substratum word. Especially "to cool or extinguish by means of cold water," hence "to drench in water" (late 15c.). Related: Quenched; quenching.
Quarks cannot be observed as free particles in nature because they are confined into baryons and mesons, as a result of the fact that the strong interaction between them increases with their separation. However, it is less known that this phenomenon also occurs in condensed matter and statistical physics as experimentally proved in several quasi-1D compounds1,2. Most of the theoretical and experimental studies so far concentrated on understanding the consequences of confinement for the equilibrium physics of both high-energy and condensed matter systems. Here, instead we show that confinement has dramatic consequences for the non-equilibrium dynamics following a quantum quench and that these effects could be exploited as a quantitative probe of confinement.
How can confinement change qualitatively the spreading of correlations? Exactly as in the standard scenario, the initial state emits pairs of quasiparticles starting off in opposite directions. However, these are eventually turned back by the confining potential, leading to an oscillatory behaviour as shown in Fig. 1. In analogy to strong interaction, the bound states formed by this process are called mesons. Mesons produced after a quench typically move much slower than the elementary quasiparticles due to their larger mass.
where the coupling J sets the energy scale and σjα are the Pauli matrices. This model has been engineered with cold atoms15 and quench protocols have also been implemented9.
For hx = 0, the Hamiltonian (1) is diagonalized by a mapping to free fermions (see Supplementary Information). For hz hx induces a linear potential between pairs of domain walls confining them into mesons16. In this Letter, we report numerical results for several observables showing that the non-equilibrium quench dynamics is drastically altered by confinement.
Similarly, the connected correlation function of the transverse component of the spin also reflects the change of the light cone due to the modified velocity of the mesons (see Supplementary Information). Furthermore, for quenches from and to several other values of the magnetic fields the overall picture is unchanged, so our conclusions are very general and not limited to the reported cases.
Entanglement entropy provides another smoking gun for the quasiparticle propagation and light-cone effects14. For zero hx, we observe in Fig. 4 a linear growth in time of the half-chain entanglement entropy in agreement with the known exact results19. By turning on the interaction hx, the growth of the entanglement entropy for quenches within the ferromagnetic phase is strongly slowed down and, after a transient, it appears to oscillate around a saturation value. The frequencies of these oscillations are in rough agreement with the meson masses and their differences as reported in Supplementary Information. This is a consequence of the fact that mesons are predominantly produced at rest and it is consistent with the strong suppression of the light cone of the two-point function. Actually, the small fraction of mesons with non-negligible velocities should produce a very slow increase of the entanglement that, however, is too small to be observed numerically. Conversely, for a quench across the critical point, the growth of the entanglement entropy is not negligible, but it is considerably reduced due to the production of many mesons with non-vanishing velocities. Overall, the data for the entanglement are compatible with the confinement scenario drawn for the correlations.
To conclude, we speculate on a few other consequences and applications of our work. It was noticed some time ago20 that in some quenches within the Hamiltonian (1) the system does not approach asymptotically a thermal stationary state as expected from the non-integrability of the model. A possible explanation is that due to confinement there are rare states in the spectrum that prevent the eigenstate thermalization hypothesis21 from being applied. Along the same line of thought, even if these confined systems eventually thermalize, the standard prethermalization scenario22,23 for weak integrability breaking cannot be applied, since a small perturbation changes not only the long-time asymptotic expectation values, but completely alters the dynamics even at short timescales.
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