Intro; Preface; Contents; Contributors; 1 Introduction to the Volume; 1.1 Introduction; References; 2 Towards the Understanding of Superconductors and Correlated Materials out of Equilibrium: Mean Field Approaches; 2.1 Motivation and Introduction; 2.2 The Gutzwiller Variational Method; 2.2.1 The Time-Independent Gutzwiller Approximation; 2.2.2 An Explicit Example: The Single Band Hubbard Model; 2.2.3 Time-Dependent Gutzwiller Approximation; 2.2.4 Quantum Quench of the Interaction in the Hubbard Model; 2.3 Dynamical Mean-Field Theory; 2.3.1 The Hubbard Model

2.3.2 Antiferromagnetism in the Hubbard Model2.3.3 The Attractive Hubbard Model; 2.4 Quantum Quenches for the Hubbard Model; 2.4.1 Quench Dynamics in the Gutzwiller Approximation; 2.4.2 Dynamical Phase Transition Within DMFT; 2.4.3 From Adiabatic Switching to Quantum Quenches: Ramping up the Interaction; 2.4.4 Interaction Quench in the Antiferromagnetic Hubbard Model; 2.5 Non-equilibrium Superconductors: Mean-Field Theory; 2.5.1 Anderson Pseudospin Representation; 2.5.2 Equations of Motions for the Pairing Amplitudes; 2.5.3 Dynamics After a Quantum Quench

2.5.4 The Phase Diagram of s-wave Superductors After a Quantum Quench2.5.5 Comparison Between s-wave and d-wave Superconductors; 2.5.6 Dynamics After a Small Perturbation; References; 3 From the Keldysh Formalism to Non-equilibrium Dynamical Mean-Field Theory; 3.1 Introduction; 3.2 Green's Functions and Many-Body Systems Out of Equilibrium; 3.2.1 The Thermal Equilibrium State; 3.2.2 Green's Functions and Electronic Structure; 3.2.3 Probabilistic Interpretation of Real-Time Green's Functions; 3.3 The Keldysh Formalism; 3.3.1 The Time-Evolution Operator

3.3.2 Time-Dependent Expectation Values and the Keldysh Contour3.3.3 Contour-Ordered Green's Functions; 3.3.4 Noninteracting Green's Function; 3.3.5 The Two-Time Self-energy; 3.3.6 Self-energy of the Open Quantum System; 3.3.7 Diagrammatic Perturbation Theory; 3.4 The Dyson Equation; 3.4.1 Langreth Rules; 3.4.2 Kadanoff-Baym Equations; 3.4.3 Steady-State Formalism; 3.5 Nonequilibrium Dynamical-Mean-Field Theory; 3.5.1 The Dynamical Mean-Field Formalism; 3.5.2 Bethe Lattice; 3.5.3 Numerical Implementation and Impurity Solvers; 3.6 Photo-Doped Mott Insulators; 3.6.1 The Mott Transition in DMFT

3.6.2 Paramagnetic Phase
Dynamics of Photo-Excited Doublons3.6.3 Antiferromagnetic Case; 3.6.4 The Buildup of the Fermi Liquid; 3.7 Concluding Remarks; References; 4 Master Equations Versus Keldysh Green's Functions for Correlated Quantum Systems Out of Equilibrium; 4.1 Introduction; 4.2 Master Equations; 4.3 Density Matrix; 4.3.1 Time Dependence; 4.3.2 Reduced Density Matrix; 4.4 Lindblad Equation; 4.4.1 Heuristic Derivation; 4.4.2 Solution of the Lindblad Equation by Exact Diagonalization; 4.4.3 Fermionic Model Described by the Lindblad Equation