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in recent years ultrashort , high - harmonic generated vacuum and extreme ultraviolet ( vuv / xuv ) laser pulses have become available . they open the way towards time - resolved observation of electronic dynamics in plasmas , atoms and condensed matter , for an overview see e.g. @xcite . electronic motion in atoms and relaxation processes can now be studied experimentally on sub - femtosecond time scales , e.g. @xcite , for a recent overview see @xcite . this creates the need for complementary theoretical investigations . interaction of laser radiation with matter has been actively studied in recent years using a quantum kinetic theory which was derived using nonequilibrium greens functions ( negf ) methods , e.g. @xcite . in contrast , application of negf to single atoms has only recently been attempted @xcite . while the ground state properties were very well reproduced , time - dependent calculations describing the ionization dynamics of atoms during and after a laser pulse turned out to be computationally very expensive @xcite .    in this paper we give a brief overview on the application of negf to time - resolved laser - atom interaction and outline the current problems . furthermore , we discuss an alternative approach which is based on the @xmath0electron wave function  the multiconfiguration time - dependent hartree - fock method ( mctdhf ) . the latter is quite efficient in application to few - electron atoms and small molecules . this is demonstrated by numerical results for a four - electron one - dimensional model atom subject to a short laser pulse . the unperturbed hamiltonian of @xmath1 electrons in the atom is given by ( we use atomic units ) @xmath2 where the potential of the nucleus , @xmath3 , as well as the two - particle coulomb interaction @xmath4 are assumed to be spin - independent . for times @xmath5 , the atom is disturbed by a time - dependent external field , and the hamiltonian is modified : @xmath6 where the electromagnetic field is described in dipole approximation , and the field envelope is assumed to have a gaussian shape with a pulse duration @xmath7 and @xmath8 @xmath9 \exp \left[- \frac{(t - t_{\text{0}})^2}{2\tau^2}\right ] \,.\end{aligned}\ ] ]      to treat the interaction of an ensemble of atoms with the electromagnetic field a statistical theory is required . it can be based on density operators @xcite or nonequilibrium greens functions . here we discuss the latter concept . starting point are the fermionic field operators of the electrons ( the nuclei are assumed to be fixed ) @xmath10 und @xmath11 , which guarantee that the spin statistics theorem is obeyed , @xmath12 where @xmath13 and we denoted @xmath14 [ spin indices will be suppressed in the following ] . the nonequilibrium properties of the electrons are completely determined by the two - time correlation functions ( averaging is over the @xmath1-particle density operator of the system ) @xmath15 the two independent functions @xmath16 and @xmath17 contain the complete statistical information , in particular the nonequilibrium one - body density matrix , according to @xmath18 the equations of motion of @xmath16 and @xmath17 are the keldysh / kadanoff - baym equations ( kbe ) , e.g. @xcite which are the most general starting point for developing a quantum statistical theory of matter in a strong electromagnetic field . in recent years the kbe have been successfully solved numerically for spatially homogeneous plasmas , e.g. @xcite and for a weakly inhomogeneous electron plasma @xcite . yet these solutions require to propagate the correlation functions in the two - time plain which is computationally very expensive . therefore , approximate methods have been developed . one possibility consists in expressing the correlation functions by the simpler single - particle distribution function @xmath19 [ or density matrix @xmath20 , eq . ( [ rho - def ] ) ] via the so - called generalized kadanoff - baym ansatz @xcite . for plasmas interacting with a strong field this ansatz has been generalized in refs . @xcite . the emerging quantum kinetic equation for the wigner distribution which includes nonlinear field effects , inverse bremsstrahlung , collisional absorption and harmonic generation has been solved in ref . the second direction of recent research in the kinetic theory of laser - plasma interation was devoted to include neutral atoms and their excitation and ionization . starting from the kbe atomic bloch equations were derived @xcite which allow to compute the absorption spectrum of an ensemble of atoms in thermodynamic equilibrium . this approach is very useful for weak excitation near the linear response regime . however , in the case of strong field ionization of atoms we expect essential nonequilibrium dynamics effects related to ionization of several electrons and electron - electron correlations developing rapdily on femtosecond and even sub - femtosecond time scales . these phenomena which have already been observed in experiments require a fully nonequilibrium approach . starting again from the kbe we have in recent years developed a quantum kinetic approach for the electron dynamics in a single atom . the main difference to the previous applications to plasmas is that now the system of electrons under study is spatially strongly inhomogeneous  due to the confinement by the potential @xmath21 of the atomic nucleus . to solve this problem the two - time correlation functions ( [ ggtls_def ] ) were expanded into a basis of single - particle states or hartree - fock orbitals . this approach was successfully tested for `` artificial atoms '' , i.e. electrons in quantum dots in refs . more recently it was applied by the present authors to electrons in small atoms @xcite such as a one - dimensional beryllium model atom following earlier work by van leeuwen and co - workers @xcite . in these works it was found that the ground state of few - electron atoms is well reproduced with simple approximations for the selfenergy such as the second born approximation . the situation is more complex for time - dependent calculations . while lowest order calculations  using a time - dependent hartree - fock selfenergy  are well feasible @xcite with the inclusion of correlations , simulations could only be carried out for very short times . this is a particular problem of ionization dynamics where one has to simultaneously treat bound and free electron states which requires large basis sets . on the other hand , the interaction of atoms with a strong laser field can , in many cases , be assumed to be a coherent process . further , at sufficiently low density , the interaction between atoms can be neglected . also , at room temperature conditions the density matrix of the atom is dominated by the contribution of the ground state wave function , and one can instead of quantum statistics apply a pure state description .      at the conditions described above the initial state @xmath22 of an atom is given by the ground state wave function or a coherent superposition of atomic eigenfunctions . then the dynamics following an external exciation is described by the time - dependent @xmath1-electron schrdinger equation ( tdse ) with the hamiltonian ( [ h ] ) @xmath23 electrons , depending on the dimensionality . therefore , approximate methods have been developed . a particularly successful approach is the _ multiconfiguration time - dependent hartree - fock _ ( mctdhf ) method due to cederbaum @xcite , see also @xcite and references therein . we have , therefore , implemented the mctdhf method for photoionzation of atoms in a strong field and briefly outline the main idea . starting point is the following ansatz for the many - electron wave function as linear combination of time - dependent slater determinants which are denoted in terms of the occupation numbers @xmath24 of the single - particle orbitals @xmath25 the sum runs over an integer number @xmath26 of determinants . in the limit @xmath27 the method yields the exact wave function , whereas for @xmath28 one recovers the time - dependent hartree - fock approximation ( tdhf ) . thus , by increasing @xmath29 one can systematically improve the treatment of electronic correlations . the equations of motion for the coefficients @xmath30 and the single - particle orbitals @xmath31 are obtained by a variational principle and read , with a matrix notation ( with respect to slater determinants ) @xmath32\ , ,   \end{aligned}\ ] ] where @xmath33 and @xmath34 denote the single - particle and two - particle density matrix , and @xmath35 is the single - particle hamiltonian . further , @xmath36 is the operator of the mean - field potential which , in coordinate representation , is given by @xmath37 . eqs . ( [ eq_wavefunction],[eq_orbitals ] ) show that the expansion coefficients @xmath38 obey a time - dependent schrdinger equation , whereas the single - particle orbitals @xmath39 are the solutions of a more complicated non - linear integro - differential equation .    [ be_ion0 ] we now present numerical results for photoionization of a four - electron one - dimensional model atom ( beryllium , @xmath40 ) for which we presented nonequilibrium greens functions results in ref . @xcite . for the @xmath41 simulations we use regularized coulomb potentials , @xmath42 , and @xmath43^{-1/2 } \ $ ] where the screening parameters are introduced to avoid divergencies . this model has been used successfully in many studies of atom - laser interaction where also the influence of @xmath44 has been investigated . following refs . @xcite we use @xmath45 . a sketch of the atomic core potential and the ( square of the ) occupied hf basis functions ( shifted by the orbital energies ) is shown in fig . [ be_ion0 ] together with the total electron density . there are two bound states at the energies @xmath46 and @xmath47 , each occupied by two electrons . the first task is to obtain accurate results for the ground state . the results of our mctdhf calculations for the ground state energy are presented in the table . convergence with respect to the correlation energy is reached for @xmath48 . in comparison , the ground state energy for the same system from negf calculations in second born approximation is found to be equal to @xmath49 , i.e. on the level of mctdhf with @xmath50 . .ground state energy of the 1d beryllium atom with @xmath45 for different numbers of determinants @xmath29 . we used a basis with 50 eigenfunctions of the single - particle hamiltonian which assured convergence with respect to the basis size . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ]     as a next step , we use the ground state wave function as the initial state and expose the atom to a time - dependent laser field of the form ( [ eq : gauss ] ) with a fixed number of cycles choosing the pulse width according to @xmath51 and an amplitude of @xmath52 . we consider two frequencies , @xmath53 and @xmath54 which corresponds to a keldysh - parameter @xmath55 , [ @xmath56 is the ionization potential , and @xmath57 the ponderomotive potential , @xmath58 of @xmath59 and @xmath60 . in the first case we expect that direct photoionization is not possible and the dominant process will be tunnel ionization . in contrast , in the second case multiphoton ionization should be expected . the time - dependent numerical results of our mctdhf calculations are shown in fig . [ be_ion ] . in the lower row we show the ionization yield [ time - dependent expectation value of the number of ionized electrons @xmath61 . interestingly , @xmath62 is much larger for the smaller photon energy ( case 1 ) which confirms our previous results @xcite . this is due to the much larger electron excursion amplitude in the low frequency laser field [ recall that , for a free electron , the amplitude is @xmath63 . for the larger photon frequency @xmath64 , @xmath65 is strongly reduced . nevertheless , the two electrons in the upper state with energy @xmath66 are driven substantially away from the nucleus during one half - cycle of the field but they are driven back during the following half - cycle . as a consequence , the ionization probability is substantially reduced . in this second case , mctdhf calculations with different numbers of determinants @xmath29 are very close to each other , and even a tdhf calculation yields the qualitatively correct result . more interesting is the behavior for the smaller frequency . here two electrons are driven far away from the nucleus and they occupy continuum states which leads to strong correlation effects . as a result , tdhf calculations  while overall giving the correct picture  become inaccurate already after about one laser cycle ( @xmath67 ) . [ be_ion ]     summarizing , we have presented mctdhf calculations for a four electron model atom exposed to strong laser pulses with two different frequencies . these calculations are rather efficient and allow to follow the time - resolved ionization dynamics and the history of electronic correlations in single atoms . they show that , overall , tdhf gives the correct tendencies . however , it fails to accurately describe the dynamics of electrons at longer times , in particular in cases of strong ionization . in future work we will concentrate on correlation dynamics and on the inneratomic relaxation as well as on further comparisons with nonequilibrium greens functions results . this work is supported by the deutsche forschungsgemeinschaft via sfb - tr 24 and by the u.s . department of energy award de - fg02 - 07er54946 .
theoretical approaches to the photoionization of few - electron atoms are discussed . 
 these include nonequilibrium greens functions and wave function based approaches . in particular , the multiconfiguration time - dependent hartree - fock method is discussed and applied to a model one - dimensional atom with four electrons . 
 we compute ground state energies and the time - dependent photoionization by the field a strong laser pulse with two different frequencies in the ultraviolet .