Common use of Thesis Outline Clause in Contracts

Thesis Outline. This thesis studies the non-equilibrium dynamics of composite quantum systems following the quench described in the introduction at early times specifically. The time scales of interest are well before the validity of ETH and even hydrodynam- ics, so the analysis is based on the evolution of energy, the von Neumann and relative entropies, as well as the information spread between different parts of the system. This was initially motivated by an observation in numerical simula- tions of a peculiar early-time energy increase in the hotter of two quench-coupled systems. This thesis not only provides a detailed explanation of this quantum phenomenon but also explores how it can be used for experimentally measuring quantum correlations, both as a feature for detecting a lab realization of an SYK system as well as the implications it has on the formation of black holes. Those results have been published in three different papers [71, 72, 73] plus one yet un- published manuscript. Before elaborating on them we provided a brief overview of each of the four chapters. In the first Chapter 2, we introduce this paradoxical early time rise in energy in the hotter of the two baths and the quantum thermodynamic approach to studying post-quench dynamics. To understand the essence of this phenomenon, without the interference of model-dependent peculiarities, we use the 1D free fermion model as a case study. Conveniently, in the time regime of interest, we obtain analytical expressions for the energy and for the von Neumann entropies, which is an additional benefit of this model. In addition to the theoretical study, we have suggested an experimentally realizable quench protocol that can measure entanglement between two subsystems using the energy increase and its relation to von Neumann entropy. In this quench, one starts with two originally independent systems of free fermions A and B initially prepared in quantum thermal states at temperatures TA and TB. At low temperatures, when quenched, the increasing entanglement contribution to the von Neumann entropy is dominant over the decreasing thermal one. As a consequence the von Neumann entropy of each subsystem increases for a certain period after the subsystems are coupled. If in this period one decouples the subsystems there is an energy transfer to the system in the amount set by the von Neumann entropy accumulated during the joint evolution of A and B. This energy transfer appears as work produced by the quench to decouple the reservoirs. Once A and B are disconnected, the information about their mutual correlations – von Neumann entropy – is stored in the energy increment of each reservoir which allows a direct readout of quantum correlations by measuring the energy of the subsystems. While this thesis doesn’t cover the details on the feasibility of the experimental realization and subtleties when the temperature of either subsystem approaches T 0, interested readers are advised to consult [71]. Next, in Chapter 3 we study the post-quench quantum dynamics of both strongly correlated SYK systems and weakly correlated mixed field Ising chains. As previ- ously mentioned, the quantum thermodynamic relations require exact knowledge of the time evolution of energy and the ▇▇▇ ▇▇▇▇▇▇▇ and relative entropies. For those systems these cannot be obtained analytically, therefore, we resort to exact diagonalization. While this approach is limited to finite sized systems, it allows us to study their evolution at an arbitrary time-scale allowing us to distinguish two qualitatively different behaviours. Namely, the early time polynomial increase of energy in both sub-systems is followed by a conventional classical-like evaporation with the energy of the hotter/colder system exponentially decreasing/increasing. At the transition time tm the energy of the hotter system peaks, a feature known as the energy bump. We show that even in the quantum regime the origin of this energy bump is not due to thermal flux from the hot to the cold, contrary to what has been reported before [13]. Instead the early-time energy increase of the hotter subsystem is not related to a temperature increase but results from the potential energy gained by coupling the two systems. The size of the energy increase is set by the entropy gain and lasts until correlations between the subsystems saturate. When the systems of interests are SYK dots we have numerically found that the energy bump appears regardless of the initial temperature difference, which isn’t surprising, given the ultra quantum nature of this system.8 To answer the ques- tion “Why we haven’t seen this phenomenon in everyday life?”, we use the MFI model which has classical, integrable, chaotic and critical regimes. At the critical point, same as with SYK, the energy bump always appears, but moving even slightly away from criticality there is a distinct temperature Tc such that, when the hotter system is initiated above TA > Tc, its energy decreases from the very beginning in full agreement with our (classical) intuition. If MFI is tuned to the classical regime this distinct temperature vanishes Tc = 0 and classical dynamics is recovered at any initial configuration. In order to better understand the numerical results for the SYK and MFI models we analyze in Chapter 4 their properties with a perturbative time expansion of each subsystem’s energy. We derive the coefficients up to the third order and focus on situations where both subsystems are initialized in a thermal state. In this case the first and third coefficients vanish and the second coefficient captures the relation between the appearance of the early time energy increase in the hotter subsystem and before-quench thermodynamic states. More precisely, the bump appears whenever the second coefficient is larger than zero. Using this approach we have shown that, for two quench-coupled SYKs, the existence of the quantum regime isn’t conditioned on the temperature difference. since the second coefficient of each subsystem depends only on its own parameters and they’re always positive. Consequently, the early time energy increase will always appear, as we’ve suspected from the numerical results. However, if the systems under consideration are MFI models, the second coefficient has two competing contributions. One is always positive, similar to the SYK case, and depends only on the analyzed subsystem, whereas the other is negative and depends on the initial parameters of both, hence, the bump persists as long as the former term is dominant over the former. Both terms have the same magnitude at the critical temperature Tc, after which the second term dominates resulting in the bump disappearance, for any temperature T ≥ Tc. Here, using the perturbative 8This is intrinsically linked to the ANEC inequality when modeling evaporative black hole formation with SYK dots [14, 15]. analytical expressions, we were able to compute the critical temperature as a function of quantities evaluated in the initial thermal state and prove it matches with results obtained from numerical time evolution.