\chapter{Afterword}

The interaction of light with matter is not a new problem. In fact, it is pretty old, going back all the way to the foundations of quantum mechanics, with the theory of blackbody radiation by Max Planck and the photoelectric effect by Albert Einstein. Yet, more than a hundred years after these pioneering works, the problem has transformed itself into countless new ones, re-inventing the world on the way. Motivated by recent experiments~\cite{Wetli_2018, Pal2019}, one of those problems lies at the intersection of quantum optics and many-body physics -- both emerging from the exchange of light quanta with matter -- and is the subject matter of this thesis. It is on how Kondo lattice coherence, a delicate many-body effect, is disrupted out-of-equilibrium by a travelling quantum pulse of light. Even though Kondo, non-equilibrium and quantum radiation physics are well-known problems, the theoretical journey was on \textit{how} to properly unite them from first principles. Then, too, from its onset, two technical challenges that ought to be overcome were already known: low temperatures and long timescales had to be resolved for Kondo physics to emerge.

Despite an abundance of different methods in the theoretical condensed-matter toolbox, only a few can take on a Kondo problem -- i.e., a strongly-interacting continuum system -- driven out-equilibrium. Two-point (Green) functions, generated from non-equilibrium field theory, offer a good trade-off between accuracy -- they are perturbative, hence approximate, and system size -- the formalism is semi-analytic, which allows transformations and summing over continuous degrees of freedom. This thesis's inaugural application of non-equilibrium field theory techniques is with auxiliary particles, introduced to embed strongly-interacting models, typically non-perturbative, with a perturbative expansion. There were already extensions of auxiliary particles to non-equilibrium field theory at the level of Green functions~\cite{Langreth_1991, Wingreen_1994, Eckstein_2010}. However, in~\cref{sec:aux-particles}, this is reformulated in a modern yet comprehensible manner, congruous with their equilibrium formulation. This is particularly important for non-equilibrium problems which start from an equilibrium configuration. First, the main components of the auxiliary particle's non-equilibrium Green function are identified by their \enquote{$\zeta$-scaling}. Secondly, a connection is established between the Matsubara Green function and the $\zeta$-scaled non-equilibrium Green function components. Finally, a formulation of the components of the non-equilibrium Green function related to thermal observables as a function of real frequencies is obtained, which precludes the use of (numerical) analytic continuation methods.

A significant difficulty in calculating non-equilibrium Green functions -- which measure correlations between two points in time -- is that general non-equilibrium regimes contain no time symmetries, and the two-point functions are hence dense matrices, of elevated computational cost, in time. In~\cref{sec:vide}, adaptivity is extended to the two-time integrodifferential solvers of non-equilibrium Green functions. This critical improvement can dynamically adjust the step size to the non-equilibrium regime and dramatically reduce the time-steps required in long-time integrations, allowing Kondo coherence timescales to be reached with a laptop.

In~\cref{sec:thz}, the centrepiece model of this thesis -- which revolves around formalisms and approximations to solve it -- is finally brought to light. First, an Anderson lattice model is expressed with recourse to auxiliary particles due to a non-perturbative interaction term related to the repulsion between neighbouring electrons. Then, the Anderson lattice model is coupled to the electromagnetic field, namely to a time-dependent external quantum pulse of radiation. Furthermore, dissipative channels are introduced to counteract spurious heating from the interaction of the external pulse of light with the system.

The model is solved by two different techniques. First, in~\cref{sec:thz-mf}, it is solved by non-equilibrium saddle-point theory. The use of the Konstantinov-Perel' contour results in two different sets of equations, one describing an initial thermal state and another describing the dynamical evolution of the auxiliary saddle-point fields, which are known to capture the Kondo coherence in the limit of vanishing temperature. Despite not being discussed in the thesis, the set of equations in thermal equilibrium is obtained via automatic differentiation, which may prove to be a valuable resource for aiding theoreticians in generating saddle-point equations for large systems. The theory, however, ended up being quite limited, as it requires a series of approximations that make it independent of the frequency content of the external pulse. Still, this is not a fundamental limitation of the formalism, and, together with decoherence processes such as those found in open Markovian systems, could be added in further studies. However, the fact the solutions are found in strong-perturbation regimes is unphysical, and some sort of criterion that can gauge whether the saddle-point approximation remains valid throughout the time evolution is yet required to validate the theory. Finally, in~\cref{sec:thz-nca}, the driven-dissipative Anderson lattice model is solved in its full glory. The generalised formalism of non-equilibrium auxiliary-particle Green functions, introduced in medias res, is employed to treat dynamical mean-field theory and the non-crossing approximation properly. The resulting formulation can reach the low temperatures required for Kondo-coherent regimes, and observables of general physical particles can be computed. Aspects regarding the non-equilibrium mechanisms that collapse and later allow the revival of Kondo coherence, namely enhanced hybridisation and decoherence, are discussed. Moreover, the intensity of the renormalised incident light pulse after interacting with the system is studied as a function of several system parameters. It is shown that the system can emit a non-superradiant echo pulse at low temperatures. The Kondo-related origin of the echo pulses observed in experiments~\cite{Wetli_2018, Pal2019} is confirmed, legitimising the theoretical formalism developed for studying future experiments of low-energy pulses of light interacting with many-body systems.

In conclusion, the work presented provides, most importantly, a complete theoretical description of how to tackle non-equilibrium Kondo physics at the level of 2-point functions. It required consolidating several existing ideas and techniques that had to work in unison to resolve the low temperature and long timescales of Kondo physics in non-equilibrium. Theoretical condensed-matter physicists would most likely go extinct if it were possible to solve a strongly-interacting, 3-dimensional, non-equilibrium problem exactly in finite time. In present times, however, things could not be more different. The physics of Kondo coherence melting is a breadbasket of profound and fascinating transient many-body effects which this thesis could only scratch the surface of -- hopefully serving as a starting point for future pilgrimages through the beautiful world of non-equilibrium Kondo physics.

\section*{Outlook}
Prosperous times lay ahead for theoreticians and experimentalists, with the matrimony of the old and vast fields of non-equilibrium and many-body physics. For one, time-resolved terahertz (THz) spectroscopy experiments such as~\cite{Wetli_2018, Pal2019, Yang2020, Yang_2022} will continue delivering access to previously inaccessible physical regimes and unveil the secrets of many-body systems. Second, there is yet a world of fertile and unexplored physics, with exotic steady or transient non-equilibrium quantum phases harbouring new physics and technological potential. 

The most evident and pressing case where a similar analysis to this thesis could be applied, both at the saddle-point~\cite{Secchi2018} or fully-interacting~\cite{Sentef2016} level, is in the study of transients in high-Tc superconductors. Possible research avenues in these systems perfectly capture the possibilities borne by the \textit{pair}ing of non-equilibrium and many-body physics. First, it is possible to probe these systems with electromagnetic radiation to gain insight into the nature of their ground state. Namely, recovering superconductivity dynamics following photo-excitation could give a definite answer to the open question of what is the pairing mechanism, or \enquote{glue}, which binds electron pairs in high-Tc superconductors. Second, it has also been reported~\cite{Demsar_2020} to be possible to either enhance or induce superconductivity with low-frequency pulses at temperatures far above the superconducting critical temperature, which could be technologically significant.

From a numerical standpoint, there are yet two crucial improvements to the adaptive Kadanoff-Baym solver, which could radically slash computation time and memory requirements by orders of magnitude, especially for very long integration times. The first is a direct critique and revision of the adaptive scheme presented in~\cref{sec:vide}. The stepping scheme is formulated with a \textit{global} time-step in the vertical and centre-of-mass time directions. However, the natural direction for controlling adaptivity is the relative-time direction since Green functions and self-energies typically decay in this direction. For most systems, the Green functions decay in any direction away from the time-diagonal -- a signal of loss of correlation over time -- however, the relative-time direction is the most natural direction to inspect decay. Therefore, a more refined adaptive algorithm could have a \textit{local} time-step for each relative-time direction. This would significantly increase efficiency by allowing larger time-steps or even stopping the integration in a particular direction if correlations have completely died out. Secondly, a compression scheme similar to~\cite{Kaye_2021} could be extended to non-equidistant time grids. The compression scheme was introduced as an alternative representation of the dense two-time grid due to the decay property along the relative-time direction. Extension of the technique to a non-equidistant two-time grid would allow it to be used directly with the developed time-stepper, and implementing a compression similar to the hierarchical mesh of~\cite{Doelz_2021} would properly leverage the decay along the relative-time direction.

From a theoretical standpoint, there is yet some work to be resolved. For example, ultrafast THz pulses are somewhat incompatible with the mono-chromatic/narrow-band approximation, i.e., the pulse bandwidth being smaller than the carrier frequency. This approximation greatly simplified the treatment of the external pulse in~\cref{sec:thz} as a \textit{single} mode / quantum field but may not yield accurate quantitative results. It would be interesting to see how the results change when many pulse modes are coupled to matter -- or an alternative treatment that can consider fast-varying amplitude envelopes. In a similar vein, a DMFT-like formulation of the THz pulse could be formulated for intense pulses, where there could be transient but coherent effects that require temporal correlations to be considered, identical to quantum theories of superfluorescence~\cite{Benediktovitch_2019}. Finally, a comparison between the intensity of the output pulse coming from the auxiliary-particle projection and from an ensemble average of jump operators when the input pulse is treated within a Markovian master equation~\cite{Chan_2018, Kiilerich_2020} would be of interest, especially regarding the validation of the non-crossing approximation to treat such systems.

Finally, the \enquote{holy grail}~\cite{Nejati_phd_2017} of Kondo physics would be a conserving approximation that includes the non-local Ruderman–Kittel–Kasuya–Yosida interaction. Given the challenges associated with resolving Kondo physics and adding non-local extensions to DMFT, capturing criticality-induced breakdown of Kondo regimes is a pipedream. However, renormalisation-group theory~\cite{Nejati_2017} hints at a feasible introduction of a magnetic-instability channel within a conserving approximation. Such a theory could provide equilibrium and non-equilibrium descriptions of the breakdown of the Kondo effect in microscopic detail and bring definitive answers to -- or, suspecting to be victims of some sort of Poincare recurrence, \textit{revive}~\cite{Kouwenhoven_2001} once again -- the field of \enquote{irresistible}~\cite{unknown_2014} physics. % Moreover- an anti-ferromagnetic Bethe lattice could be described by two nested ferromagnetic lattices.