\documentclass{revtex4}

\begin{document}


\title{Coulomb Correlations in the Quantum Kinetics of Electron--Phonon Systems}

\author{W. Sch\"{a}fer}
\affiliation{Institut f\"ur Schichten und Grenzfl\"achen,
Forschungszentrum J\"ulich, 52425 J\"ulich, Germany}

\author{P. Gartner, F. Jahnke}
\affiliation{Institute for Theoretical Physics, University of Bremen,
28334 Bremen, Germany}

\maketitle

Dynamical effects of the electron--phonon interaction on the lineshape of optical
spectra or equivalently memory effects in the time domain are a longstanding
problem, nearly as old as solid state physics. They have attracted renewed
interest in connection with recent investigations of nonlinear optical 
properties on an ultrashort time scale. The most rigorous theoretical approach is a two-time formulation
of electron-phonon quantum kinetics, based on nonequilibrium Green's function
techniques. 

Up to now, theoretical studies within the two-time formalism 
have been performed only within the random-phase
approximation for the electron-phonon self energy \cite{hartmann,gartner}. 
At this level, the Coulomb
correlation of the optically excited electron--hole pairs is taken into 
account, however, the correlations of final states, which become occupied, due
to phonon scattering, are neglected. This approximation is reasonable in the
limit of weak coupling and as long as the LO--phonon energy is large in 
comparison with the exciton-binding energy, as e.g. in III-V semiconductors.
In II-IV material systems this condition is not fulfilled, and we have to 
account for the Coulomb correlation of final states.

We present a diagrammatical analysis of the electron-phonon problem, which
takes these effects within a ladder-type approximation into account and 
derive the corresponding set of quantum kinetic equations. Various limiting 
cases can be obtained from these results by means of different approximation
schemes, which allow us to reduce two-time propagators to single time functions
- a reduction, which is applied in most versions of quantum kinetic equations. 
In particular, the application of the generalized Kadanoff--Baym ansatz
delivers well known results of density matrix theory. As we will show, such an
approximation is highly questionable in the range of intermediate electron-phonon
coupling strength, in which most recent experiments are performed (see e.g.
\cite{steinbach,betz}).
 
In our numerical studies we concentrate onto the linear case for II-IV bulk
semiconductors and quantum-well structures. These solutions are compared
with the less rigorous treatments, which are based on different approximation 
schemes. In
particular, we discuss the validity of the generalized Kadanoff--Baym ansatz
and of an alternative ansatz for two-time functions, which accounts not 
only for the polaron renormalization but also for the Coulomb correlation. 
  


\begin{thebibliography}{10}

\bibitem{hartmann}M. Hartmann, W. Sch\"afer, phys. stat. sol. (b) {\bf 173},
165 (1992)\\

\bibitem{gartner}P. Gartner, L. Banyai, H. Haug, Phys. Rev. B {\bf 60},
14234 (1999)\\

\bibitem{steinbach}D. Steinbach et al., Phys. Rev. B {\bf 60}, 12079 (1999)\\

\bibitem{betz}M. Betz et al., Phys. Rev. Lett. {\bf 86}, 4684 (2001)
\end{thebibliography}


\end{document}
