$$*$$ ∗ Nonequilibrium Green Functions Approach to Field-Matter Dynamics

Abstract

After discussing in detail the method of reduced density operators before, this chapter covers the second approach to correlated quantum many-particle systems out of equilibrium: nonequilibrium Green functions (NEGF). The starting point is the formulation of second quantization which is introduced for fermions coupled to photons within a fully relativistic framework and the definition of particle and photon Green functions, following the work of Schwinger [94, 95]. Nonequilibrium effects are introduced via the Schwinger-Keldysh time contour, and we then derive the key equations: the coupled Keldysh-Kadanoff-Baym equations (KBE) for fermions and photons. Correlation effects are included in full generality, and approximations are formulated in terms of particle and photon selfenergies. After discussing important approximations for the selfenergy, using the concept of the vertex function, we proceed to the dynamics of non-relativistic fermions, decoupling the dynamics from that of the transverse photons. This is followed by an analysis of key properties of the KBE and numerical solutions. A key result is the conservation of total energy and the relaxation towards a correlated (nonideal) equilibrium distribution. In the following more advanced numerical results are presented, including optically excited semiconductors, electrons in quantum dots and plasmas. After treating ultrafast relaxation properties we demonstrate that the nonequilibrium dynamics can also be used to compute high quality equilibrium properties such as susceptibilities, the dynamical structure factor or electronic double excitations in atoms an molecules. Finally, in Sect. 13.9 we establish the connection to the first part of the book—the method of reduced density operators. This is achieved by applying the generalized Kadanoff-Baym ansatz (GKBA) to the KBE—the same ansatz that was independently obtained already in Chaps. 7, 9 and 10. The chapter concludes with results for the ultrafast build up of dynamical screening that is obtained using selfenergies in RPA (GW), by solving for the dynamically screened Coulomb potential \(V_s(t,t')=V_C\varepsilon ^{-1}(t,t')\), as discussed before in Chap. 10. This build up of screening during a finite time resolves the problem of unphysical fast relaxation noted in Chap. 1 and has been nicely confirmed by experiments of Leitenstorfer et al. [69].