We compare numerically the carrier-LO-phonon quantum kinetics as provided by the twotime Green's function formalism on the one hand and its one-time approximation given by the generalized Kadanoff-Baym ansatz on the other hand. Self-consistent RPA self-energies are used in both cases. The aim of the analysis is to check the correct thermalization behaviour of the solutions. It is shown that both formalisms give rise to steady states in the long time limit. It is shown that these states correspond to the correct thermal equilibrium for the two-time solutions, while the one-time approach fails to thermalize the carriers in the interemediate-coupling regime. The thermalization criterion used is the KMS condition.
We compare numerically the carrier-LO-phonon quantum kinetics as provided by the twotime Green's function formalism on the one hand and its one-time approximation given by the generalized Kadanoff-Baym ansatz on the other hand. Self-consistent RPA self-energies are used in both cases. The aim of the analysis is to check the correct thermalization behaviour of the solutions. It is shown that both formalisms give rise to steady states in the long time limit. It is shown that these states correspond to the correct thermal equilibrium for the two-time solutions, while the one-time approach fails to thermalize the carriers in the interemediate-coupling regime. The thermalization criterion used is the KMS condition.
In a previous publication [H. S. K\"ohler, Phys. Rev. C 51, 3232 (1995)] equilibration rates were studied for an extended system of nonequilibrium nuclear matter. The two-time Green's functions in the Kadanoff-Baym (KB) formalism were solved numerically. These quantum-mechanical calculations were compared with the Markovian (semi)classical Uehling-Uhlenbeck collision term often used in the analysis of collisions between heavy ions. The equilibration was found to be substantially slowed down by quantum effects. The theoretical as well as experimental study of nonequilibrium systems is of course of great interest in many areas of physics, e.g., within the realm of quantum transport in solid state devices. On the theoretical side various methods and approximations are the subject of intensive study. By the KB ansatz, for example, the equations can be simplified to a one-time kinetic equation while preserving the non-Markovian character. The generalized KB (GKB) ansatz of Lipavsky, Spicka, and Velicky [Phys. Rev. B 34, 6933 (1986)] is often preferred. In this paper approximations based upon the GKB ansatz are discussed in relation to the exact KB formalism. Numerical comparisons are made for both the strongly interacting nuclear medium as well as an electron plasma. A non-Markovian collision integral, including memory effects in a quasiparticle approximation, is often discussed in the literature. It is found that this approximation appreciably overestimates the collision rates even more than the classical (Boltzmann) approximation. The inclusion of a width in the spectral function, i.e., going beyond the quasiparticle approximation, gives considerable improvement. The competition between the memory and correlation effects is found to lead to an anomalous saturation of the relaxtion rate. \textcopyright{} 1996 The American Physical Society.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.
We provide a systematic approach to compute different kinds of non-equilibrium Green's functions for open quantum systems which are essentially two-point correlation functions in time. We reveal that the definition of Green's functions based on the Born approximation does not provide the correct results in the leading order of the system-bath coupling. We next provide a systematic correction term in Green's functions by going beyond the Born approximation and incorporating a finite correlation between the system and the bath. We primarily focus on two paradigmatic models of open quantum systems, namely, the dissipative Caldeira-Leggett model and the dissipative spin-boson model. We show that the inclusion of such a correction correctly reproduces the Kadanoff-Baym type equation for the so-called lesser or greater components of the Green's functions and provides the correct long-time result up to the first non-zero order of the system-bath coupling, satisfying the detailed balance condition in thermal equilibrium. We further extend our study to a system coupled to multiple reservoirs simultaneously that are maintained at different temperatures and obtain expressions for non-equilibrium steady-state energy current, once again correct up to the first non-zero order of the system-bath coupling.
ABSTRACT We benchmark nonequilibrium Green's function (NEGF) approaches for interaction quenches in the half‐filled Fermi–Hubbard model in one and two dimensions. We compare fully self‐consistent two‐time Kadanoff–Baym equations (KBE), the generalized Kadanoff–Baym ansatz (GKBA), and the recently developed NEGF‐based quantum fluctuations approach (NEGF‐QF) with second‐order Born (2B) and GW self‐energies. As references, we use DMRG in 1D and time‐dependent perturbation theory at weak coupling. While all methods reproduce the early‐time build‐up of correlations (prethermalization), different deficiencies emerge during the slower relaxation in the long‐time limit: KBE shows overdamped oscillations, GKBA exhibits a negative overshoot of the Fermi‐surface jump, and NEGF‐QF develops a prethermalization plateau with delayed relaxation. In 1D, we find that GKBA + 2B agrees best with DMRG across the explored parameter range, while QF + GW deviates the most, consistent with its reduced self‐consistency. In 2D, all approaches correctly capture the momentum‐dependence of the relaxation dynamics.
We investigate resonant time-resolved four-wave mixing on GaAs quantum wells at T=300K and at elevated carrier densities using excitation with 11 fs optical pulses. The experiments are compared with non-Markovian quantum-kinetic calculations including carrier-carrier and carrier-LO-phonon scattering. A self-consistently determined two-time-dependent screened random-phase approximation Coulomb potential is used for the carrier-carrier scattering. In experiment and theory we find an unusual real-time response, which is far from an ideal photon echo. Surprisingly, we also find that the dependence of the decay time \ensuremath{\tau} of the four-wave-mixing signal versus the optically excited nonequilibrium electron-hole pair density neh is quantitatively similar to the corresponding behavior of bulk GaAs. Thus both data sets can be fitted by the same form ${\ensuremath{\tau}}^{\ensuremath{-}1}={\ensuremath{\tau}}_{0}^{\ensuremath{-}1}{+cn}_{\mathrm{eh}}^{1/3}.$ This is not consistent with previous experimental work in a more limited range of carrier densities, which determined different behavior for quasi-two-dimensional (2D) and 3D, i.e., exponents of 0.3 and $0.55\ifmmode\pm\else\textpm\fi{}0.04,$ respectively.
The method of nonequilibrium Green functions (NEGFs) constitutes a solid framework to describe closed and open quantum systems. It captures situations far from equilibrium including arbitrary perturbations and correlations - effects higher than first order in the binary interactions. On the contrary, in the absence of external fields, it reduces to the imaginary-time Matsubara formalism. The abilities (and successes) of the method originate from its self-consistency and the conserving character of many-body approximations that are introduced by diagram technique. On this account, NEGFs have, now for a long time, brought insight into the dynamics of nuclear matter, non-ideal quantum gases and (semiconductor) plasmas. However, applications to localized, finite and strongly interacting systems have emerged only within the recent decade. The dynamical properties follow from quantum kinetic equations - the two-time Kadanoff-Baym equations for the one-particle NEGF - the numerical solution of which is the central topic of this thesis. The non-Markovian structure of these equations thereby inhibits an unlimited, gradual propagation of the nonequilibrium Green function: Any self-consistent, future dynamics of the system is tightly linked to the (complete) past by non-local self-energies. Further, the success in obtaining solutions is being diminished if the system under investigation is not isotropic or inhomogeneous and, in addition, the spatial topology (e.g., the charge carrier or spin density) needs to be resolved in time. On top of this, small isolated systems are generally not candidates showing fast decoherence which offers little hope for smooth and well-damped integration kernels. Despite these difficulties, the thesis at hand is devoted to the computation of the two-time NEGF for finite, closed systems of which ensembles of electrons bound by nuclei forming atoms or molecules are the most natural representatives. Here, in the spirit of the well-studied one-dimensional helium atom, the electron dynamics in model atoms and molecules is addressed in ab initio fashion. To render calculations possible in the first place, (i) a novel hybrid representation of the NEGF is introduced based on the conception of finite elements (FE) and the discrete variable representation (DVR), and (ii) a parallel algorithm is developed that allows for an efficient, well scalable distributed-memory computation of the full two-time Green function. The first point attributes plenty of flexibility and makes the method readily adjustable to different Hamiltonians avoiding the need to, by hand or in intricate numerical manner, calculate the whole bunch of matrix elements of one- and two-particle operators. Also, in consequence, it drastically simplifies the evaluation of self-energies. The second point makes the NEGF approach ready for large-scale, high-performance computing facilities, the resources (of computer power: mainly memory!) of which are indispensable for (some of) the calculations presented. The applicability of the developed methods is demonstrated inter alia on the basis of helium, beryllium, hydrogen and lithium hydride modeled in one spatial dimension whereby electron-electron correlations are treated in the second(-order) Born approximation. After a thorough discussion on the self-consistent ground states, the thesis focuses on the electron dynamics in the linear response regime as well as in the presence of strong external fields. Overall, to assess the the performance of the second Born approximation, the results are compared to the Hartree-Fock approximation, on the one hand, and to the exact solution of the time-(in)dependent Schrödinger equation, on the other hand. In addition, aiming at the possibility of extended time-dependent calculations, the behavior of the generalized Kadanoff-Baym ansatz is discussed.