\chapter*{Introduction}
\addcontentsline{toc}{chapter}{Introduction}

\begin{displayquote}
Why would anyone still want to study a physical phenomenon that was discovered in the 1930s, explained in the 1960s and has been the subject of numerous reviews since the 1970s?
\end{displayquote}
The introspective question~\cite{Kouwenhoven_2001} posed in the 2000s regarding the \textit{revival} of the Kondo effect in the field of condensed-matter physics is, two decades later, still remarkably contemporary. Despite the physics behind the single-impurity Kondo effect being well understood, the ubiquitousness of the effect in strongly interacting matter provides a fertile ground of \enquote{irresistible}~\cite{unknown_2014} rich physics, which still poses many challenges for theoreticians and experimentalists alike. 

The \textit{Kondo effect} first revealed itself in measurements of the resistivity of (impure) gold, which was found~\cite{de_Haas_1934} to increase at very low temperatures. Even though this perplexing observation was linked to the concentration of impurities in the material, it was at odds with the then-existing models of resistivity: the motion of conduction electrons is hampered by their scattering with impurities and lattice vibrations and it was supposed to decrease monotonically with temperature, reaching a finite value at zero temperature, related to the lattice imperfections. Jun Kondo showed~\cite{Kondo_1964} that the spin-scattering originating from an anti-ferromagnetic coupling between a magnetic impurity and the conduction electrons of the metal becomes the dominant scattering channel at low temperatures, resulting in a logarithmic increase of the resistivity as the temperature is lowered. Despite Kondo's perturbative treatment explaining the rise in resistivity, it resulted in a diverging scattering rate below a specific temperature, now referred to as the Kondo temperature $T_K$, which sets the \textit{crossover} energy scale for which the Kondo effect manifests itself. The logarithmic divergence also appeared in other physical quantities, such as susceptibility or specific heat, and efforts to eliminate it led to many advances in physics, such as the numerical or perturbative renormalisation groups. In the end, the \textit{many-body} system was found to be a \textit{local} Fermi liquid\footnote{A Fermi liquid describes an interacting system of electrons as a system of (weakly interacting) fermionic quasiparticles. A fundamental element in Fermi liquid theory is the concept of quasiparticles, long-lived and low-energy excitations of the Fermi liquid that carry the bare electron charge and spin but whose effective mass is renormalised by interactions~\cite{Coleman_2001}. These quasiparticles are generated from adiabatically turning on the Coulomb interaction between particles of a charged Fermi gas and are a suitable approximation to describe conduction electrons in ordinary metals.} at near-zero temperature, with a scattering rate proportional to the square of the temperature, and the impurity spin being fully screened and forming a spin-singlet with the conduction electrons.

Despite a well-established theoretical understanding of the \textit{single-impurity} Kondo effect, other kindred manifestations, such as when a larger density of magnetic impurities is present, are still a matter of study. In heavy-fermion compounds, neighbouring impurities can interact via the Ruderman–Kittel–Kasuya–Yosida coupling mechanism, favouring a magnetically ordered phase and directly competing with the Kondo screening of the magnetic impurities, responsible for a (lattice-coherent) \textit{heavy} Fermi-liquid phase, with an effective electron mass possibly thousands of times larger than the bare one~\cite{Schr_der_2000}. Although Doniach's picture~\cite{Doniach_1977} offers a good qualitative explanation of the competition between the two phases, questions such as how the Kondo lattice coherence forms and changes with temperature~\cite{Seiro_2018}, or the fate of the Kondo cloud~\cite{V_Borzenets_2020} or of the heavy quasiparticles~\cite{Klein_2008, Nejati_2017, Wetli_2018} at the quantum critical point, which separates both quantum phases, are still heavily debated.

Whereas the field of strongly-interacting electrons -- encompassing, e.g., Kondo physics, spin liquids, superconductivity --  has been at the forefront of modern theoretical and experimental research, the last decade has seen a steady increase in the study of \textit{non-equilibrium} phenomena, namely concerning the dynamics of interacting electrons. Arguably, this shift was made possible theoretically through a renewed interest in -- and irreverence towards -- problems once deemed too \enquote{violent} to be treated with quantum field theory, and experimentally by advances in the techniques and instrumentation of ultrafast measurement. In particular, recent experiments with time-resolved spectroscopy~\cite{Wetli_2018, Pal2019, Yang2020, Yang_2022} have cast new light onto the yet unsettled questions about heavy-fermion compounds. The intricate non-equilibrium dynamics observed, related to the collapse and subsequent \textit{revival} of the Kondo effect, following irradiation by an electromagnetic pulse, warrants a detailed theoretical study. However, due to the difficulty of resolving the low temperatures and long timescales characteristic of Kondo systems, non-equilibrium theoretical studies were mainly confined to quasi-equilibrium or quantum-dot configurations~\cite{Nordlander_1999, Kroha_2002, Rosch_2005, Souto_2018}. Such difficulties stem from the strongly interacting character of Kondo physics and were addressed by combining and extending varied field-theoretical and numerical\footnote{Despite the solemn words of the \enquote{pope}~\cite{Zangwill_2021} Philip Anderson, reprehending the then -- and now -- status of the field of condensed-matter, where he remarks~\cite{Anderson_1999} a \enquote{prejudice in favour of heavy computer use and the existence of the oxymoron \enquote{computational physics}}, bridging the world of many-body physics and transient dynamics without recourse to some sort of numerics appears to be a task worthy only of his apostolic successor.} methods in order to uncover the physics intrinsic to heavy-fermion systems.

\section*{\ref*{part:non-eq}\quad\nameref*{part:non-eq}}

Virtually all systems found in nature are in a state of non-equilibrium -- an indispensable prerequisite of life -- and will likely remain so until the heat-death of the universe. The distribution of a non-equilibrium state, described by a density matrix $\hat \rho$, is best understood by what it is \textit{not}: a Gibbs ensemble
\begin{equation*}
    \hat \rho \ne \frac{e^{-\beta \hat H}}{\tr e^{-\beta \hat H}}~,
\end{equation*}
or associated generalisations~\cite{Thompson_2015}. These maximise entropy under dynamical constraints, depending only on the values of a few conserved charges, such as energy~\cite{Berges_2004_2}, with the former arising for a system described by a time-independent Hamiltonian $\hat H$ in thermal equilibrium with an infinitely large heat reservoir at inverse temperature $\beta$. At the macroscopic scale, however, many systems are \textit{near} thermal equilibrium -- keeping no memory about their past and being described by Gibbs ensembles. 
This is perhaps expected for macroscopic classical systems due to the large phase space available for fast particle collisions that nonlinearly scramble the system's information through its constituents, resulting in a memoryless ensemble. More surprisingly, despite the linearity and reversibility of time evolution in Quantum Mechanics, these also appear to be the dynamical fixed point, or asymptotic state, for most quantum many-body systems. This is behind the success of thermal quantum field theory -- paired with statistical physics and a profound knowledge of many-body and physical processes -- explaining the physics of emergent phenomena and properties of commonly found states of matter. A non-equilibrium setting brings further challenges, such as incorporating non-trivial initial conditions and time-dependent Hamiltonians, identifying different non-equilibrium regimes and their timescales, or understanding the influence of external, time-dependent fields. Moreover, systems in non-equilibrium lack underlying solid physical principles found in thermal equilibrium, which facilitate the development of theoretical techniques, such as time translational invariance, conservation of energy and/or particle number, or the principle of minimum energy, where the total energy of a system with fixed entropy is minimised.

An attempt to condense all of the relevant works of non-equilibrium quantum field theory appropriately into a handful of pages would be nothing less than a Sisyphean task, and in view of the extensiveness and comprehensiveness of the field, \cref{sec:neqft} aims to be a modest, self-contained introduction. Nevertheless, it lays the foundation on which the thesis rests and is driven by one of the foundational questions of the field: how to calculate -- from first principles -- experimentally-accessible observables of quantum systems in \textit{non-equilibrium}. Non-equilibrium quantum field theory is intimately related with \textit{time evolution}, arguably the most fundamental transformation in quantum physics. Barring grand questions, such as how the irreversibility of time in the macroscopic world arises from a time-reversible microscopic quantum world, its utilitarian goal is to describe the time evolution of a system from some initial, arbitrary state via its transient dynamics to its long-time, asymptotic state.

A path-integral formulation of non-equilibrium quantum field theory is developed for a closed system by specifying a density matrix, encoding its initial condition, and the system's Hamiltonian, which entirely generates its unitary dynamics. Different initial density matrices are encoded by different time contours, namely the Schwinger, Keldysh or Konstantinov-Perel' contour, which arise naturally when calculating observables perturbatively, a necessity since interacting quantum field theories generally do not have closed-form solutions. The $n$-particle irreducible effective action formalism is introduced to go beyond bare perturbative expansions, which display spurious error growth with time in non-equilibrium regimes~\cite{Berges_2004} and fail in being controllable approximations or ensuring conservation laws. Namely, stationarity conditions of the two-particle irreducible effective action yield self-consistent equations for the one-point (mean-fields) and two-point functions (Green functions), resulting in a conserving theory, valid for all orders of the perturbative expansion and fully general regarding arbitrarily fast or slow modulation of the system parameters or external fields. As such, non-equilibrium dynamics are studied through these two-point functions, which encode spectral and statistical information and can be used to understand coherence and decoherence, and calculate any physical observable related to single-particle excitations, such as densities and currents.

Interacting theories typically do not have closed-form solutions and require perturbative expansions when calculating $n$-point functions. This thesis focuses on the study of strongly-interacting matter, where it is common to come across conditions where the Coulomb interaction is the largest energy scale in the system, and perturbative expansions fail -- whether due to non-convergence of the perturbative series or failure of Wick's theorem due to the Hamiltonian operators not satisfying \textit{canonical} commutation relations. In~\cref{sec:aux-particles}, the method of auxiliary particles is introduced, which, compared to alternative ways~\cite{Bickers_1987} of tackling these issues, directly allows the construction of a path integral and perturbative expansions, and hence access to the whole machinery of quantum field theory. The addition of auxiliary particles and subsequent truncation of the enlarged Hilbert space results in distinctive properties of auxiliary-particle two-point functions. These encompass unusual equations of motion in non-equilibrium regimes and non-trivial two-point function components related to thermal initial conditions.

The equations of motion of two-point functions in non-equilibrium are 2-time integrodifferential equations, which are highly nonlinear and typically scale at least cubically with time due to their 2-time and integral nature. Reducing the number of numerical steps -- a significant bottleneck for studying long-time dynamics -- required for integrating the equations of motion can dramatically increase the accessible integration time. In~\cref{sec:vide}, the technique of adaptivity, common in the solution of standard ordinary differential equations, is extended to integrodifferential equations, and some other techniques to reduce computational complexity are presented.

\section*{\ref*{part:thz}\quad\nameref*{part:thz}}

%Whereas the standard picture of such measurements involves the "collapse of the wave function" following Von Neumann's projection postulate, such strongly measuring probes are rarely implemented in the laboratory. More typically, a continuous probe interacts with the system which is then detected as a macroscopic signal.~\cite{Silberfarb_2003}
A measurement always requires some sort of interaction with matter, unavoidably driving the system out of equilibrium, however varying exactly \textit{how much}. In weak perturbation regimes, where the system's response is proportional to the perturbation, linear-response theory~\cite{Bruus_2004} can successfully describe the underlying physics, as the interactions of excited states with the rest of the system are neglected. However, much deeper insight into the behaviour of matter can be obtained with stronger perturbations, driving the system out of equilibrium and generating nonlinear responses, requiring the more refined quantum field theory. An example is through optical or terahertz (THz) spectroscopy, which, beyond the technological prospects of manipulating conduction and optical properties through the interplay of matter and radiation~\cite{Kaindl_2007, Sal_n_2019}, is playing an ever-increasing role in providing an understanding of the microscopic processes and quasiparticle dynamics that determine the physics of complex materials such as strongly-interacting materials.

The prototypical strongly-interacting materials are heavy-fermion compounds, a class of metals formed between actinide/rare-earth elements and transition/noble metals. The interplay of localisation, due to strong Coulomb interaction within valence electrons of the former and itineracy of the conduction electrons, leads to exotic quantum behaviour, such as Kondo, anti-ferromagnetic and superconductor physics -- all characteristic of heavy-fermion compounds. These are also particularly rich in low-energy excitations such as collective lattice (phonon), electronic (plasmon) and many-body (e.g., Kondo) excitations, typically in the THz\footnote{According to Planck's law, every physical body in thermal equilibrium spontaneously emits electromagnetic radiation.
Chances are that this thesis's medium mainly radiates THz ($\sim 10^{12}$ Hz) photons, given an ambient temperature of roughly $300$ K on the Earth's surface.} regime. In the past, experiments probed these compounds with near-infrared or optical light pulses. However, such photons carried too much energy and tended to over-excite the system by either overheating or destroying any low-energy excitation -- a problem which has been overcome by THz spectroscopy, which uses photons with up to a hundred times less energy and hence can target only low-energy excitations.

In particular, the heavy-fermion compound CeCu$_6$ responded~\cite{Wetli_2018} to an incident ultrafast THz pulse by the emission of a time-delayed reflex pulse. It was understood that such an \enquote{echo} response arose from the reconstruction of the Kondo regime following its destruction by the incident pulse. This intriguing observation was the \textit{driving force} behind this thesis, which aims at a complete microscopic description of the transient collapse and subsequent revival of Kondo lattice coherence by a single-pulse of quantum radiation, far away from the quantum critical point and deep in the heavy Fermi liquid phase.
At first glance, a heavy-fermion lattice driven by a single pulse of radiation may look like an innocuous theoretical problem. However, it is well-known that heavy-fermion problems are already some of the most demanding in thermal equilibrium. Moreover, in this non-equilibrium setting, several more obstacles are present, namely modelling a travelling pulse of quantum radiation and the predestined necessity of coupling the system to reservoirs. Moreover, there is also the technical problem of resolving previously inaccessible low temperatures and long timescales required for the emergence of Kondo physics.
In~\cref{sec:thz}, this problem is set out in greater detail, with a closer inspection of the time-resolved spectroscopy experiment~\cite{Wetli_2018} and the microscopic model and phenomenology that describes the low-energy physics of CeCu$_{6}$. The difficulty of solving the microscopic models of such systems is rooted in the failure of many-body perturbative expansions due to the Coulomb interaction strength vastly exceeding the other energy scales -- an ideal playground for \textit{auxiliary particles}. This system is to be perturbed by a single Gaussian pulse of THz radiation, which arrives, destroys the heavy quasiparticles and then flies away, carrying information about the interaction. Without dissipative dynamics, the interaction of the external pulse of radiation with matter could deposit excess energy into the system, which would heat up due to being closed. Dissipative dynamics also appear to be at odds with the Hamiltonian formulation of quantum mechanics, which is unitary. A naive quantisation of classical dissipative mechanics can even lead to the decay of the fundamental Heisenberg uncertainty relation~\cite{Carmichael_1999}. However, a \enquote{system plus reservoir} approach can describe dissipative dynamics by introducing an environment, a reservoir of infinitely many modes the system can couple with, but cannot renormalise due to its vastness. This introduces quantum scattering channels through which information/energy of the system is irreversibly lost, allowing the excess energy to be dissipated into the environment and bringing the formulation closer to experimental settings. 

Due to the THz pulse matching the timescales associated with Kondo physics, no separation of timescales can alleviate the non-equilibrium formulation, which must be kept fully general. Despite the field-theoretical framework introduced being adequate to describe such non-separable regimes, the low-energy auxiliary-particle Hamiltonian that captures a general heavy-fermion system with a \textit{drive by} quantum radiation is still far too complicated to be solved. It is plagued by the same mathematical intractability of interacting quantum systems encountered by Paul Dirac almost a century ago, 
\begin{displayquote}
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry
are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.~\cite{Dirac_1929}
\end{displayquote}
Theoreticians are thus forced into simplifying approximations that reduce the complexity of the problem, coming to a workable set of equations. A common tactic is first to try to describe the physics in a classical limit, and in~\cref{sec:thz-mf}, the problem is solved at the saddle point. This approximation shackles quantum fluctuations and dramatically simplifies the problem, at the cost of being only an educated guess of the underlying Kondo physics. Then, as perhaps foreseen, one must go beyond the classical limit, leading to~\cref{sec:thz-nca} and a concoction of all the previous chapters. However, the unruliness of quantum fluctuations complicates the theory exponentially, which must yet somehow be tamed: in the complex world of strongly-interacting matter, no model is \textit{correct}, and the art is in constructing a theory that is \textit{not} wrong. An indispensable approximation to lattice systems is provided by dynamical mean-field theory, where the problem is mapped to an effective interacting single-site. However, a solution of strong interactions in a single site also requires some approximation or truncation of its perturbative series, as it does not have a closed form. For that, the non-crossing approximation, known to reproduce Kondo physics qualitatively, is employed. Finally, the dynamics of the driven-dissipative heavy-fermion lattice system can be resolved, and long-time temporal coherence aspects of Kondo physics are investigated.

\newpage
\section*{Summary of manuscripts and publications}
  
\noindent The contents of \cref{sec:aux-particles} appear in
\begin{leftbar}
\begin{quote}%
    \fcite[false]{Meirinhos_2022_prep1}
\end{quote}
\end{leftbar}
\vspace{5mm}

\noindent The contents of \cref{sec:vide} appear in
\begin{leftbar}
\begin{quote}%
    \cite{Meirinhos_2022}~\fcite[true]{Meirinhos_2022}
\end{quote}
\end{leftbar}
\vspace{5mm}

% \noindent The contents of \cref{sec:thz-nca} appear in
% \fcite{Meirinhos_2022_prep2}
% \vspace{5mm}

\noindent \paragraph{Non-thesis research:}
I have also contributed to the following publications. These are excluded from the remainder of this thesis.
\begin{leftbar}
\begin{quote}%  
  \begin{refsection}
  \fcite[true]{Gorjao2019} \\[5mm]
  % \fcite[true]{Meirinhos_2022_prep3} \\[5mm]
  \fcite[false]{Bode_2022_prep}
  \end{refsection}
\end{quote}
\end{leftbar}

% \newpage
\section*{Summary of open source contributions}

The advice that \enquote{software is just another kind of experimental apparatus and should be built, checked, and used as carefully as any physical apparatus}~\cite{Greg_2014} was regarded throughout the code development, and as much intensity and \textit{commit}ment were invested in these implementations as in the theory presented in this thesis. Only then would it be possible to ascertain that the obtained results are \textit{true} and not artefacts of either bugs or unidentified numerical limitations. As such, the entirety of the code and experiments used to generate the results presented is open-source and adequately tagged for full reproducibility:

\begin{itemize}
\item \url{https://doi.org/10.60507/FK2/Q2VE6X}\\
A Julia implementation of an adaptive two-time Volterra integrodifferential equation solver presented in \cref{sec:vide}, as well as the Wigner-Ville transform presented in \cref{sec:neqft}.
\item \url{https://doi.org/10.60507/FK2/C0YZ1B}\\ 
A Julia implementation of the Kondo problems presented in \cref{sec:thz-mf} and \cref{sec:thz-nca}.
\end{itemize}


% \vspace{5mm}
% \noindent
% I have also created/worked on the following open-source projects during my PhD:

% \begin{itemize}
% \item \url{https://github.com/LRydin/KramersMoyal}\\
% A Python implementation for the calculation of Kramers-Moyal coefficients for stochastic data of any dimension.
% % \item \url{https://github.com/fmeirinhos/nnfrg}\\
% % A Python implementation of a neural-network functional differential equation solver.
% \item \url{https://github.com/fmeirinhos/pytorch-hessianfree}\\
% A PyTorch implementation of Hessian-free optimisation for artificial neural-networks.
% \end{itemize}

\vfill
\noindent
\begin{center}
The source code for this thesis document is available at:

\url{https://doi.org/10.60507/FK2/S27IOB}
\end{center}
