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the prospect of a semiconductor laser in the infrared and thz region has been one of the key reasons for the development and study of semiconductor heterostructure elements , since the first proposal of semiconductor superlattices in 1970 @xcite . a possible gain mechanism may be based on two different ideas : ( i ) at certain electrical fields resonant tunneling between different subbands can lead to population inversion associated with gain at the transition energy @xcite . this idea was realized in the quantum cascade laser @xcite , which has become an important device in the infrared region . lasing in the thz region has been demonstrated very recently as well @xcite . for further details , see the review @xcite . ( ii ) the occurrence of negative differential conductivity in superlattices gives raise to gain in the low frequency range extending up to frequencies of the order of the bloch frequency @xcite . despite a strong effort by several groups all over the world , this idea is still not realized . the difficulty in its realization is attributed to the instability of the operating state leading to domain formation @xcite , which may be circumvented by different strategies @xcite . for further references and a detailed discussion see ref .  . these concepts for gain are commonly described in different ways : ( i ) an external electromagnetic field causes transitions between different levels , where the transition rate @xmath0 is evaluated by fermi s golden rule . if stimulated emission dominates stimulated absorption ( i.e. , for population inversion ) , gain occurs at the transition frequency . for a pair of states @xmath1 with energies @xmath2 , occupation probabilities @xmath3 , and a dipole matrix element @xmath4 , the contribution to the material gain ( i.e. , increase of light intensity per length ) is given by @xmath5 where @xmath6 is the normalization volume , @xmath7 is the electron charge , @xmath8 is the vacuum speed of light , and @xmath9 the background dielectric constant . \(ii ) transport theory in the presence of alternating electric fields provides the complex dynamical conductance @xmath10 . standard electrodynamics ( see , e.g. , section 7.5 of ref . @xcite ) gives the material gain ( the negative absorption coefficient @xmath11 , which is assumed to be small compared to the wave vector here ) @xmath12 thus , gain is equivalent to a negative differential conductivity in the respective frequency range . this concept has been frequently applied to superlattice transport . ( for superlattices eq . ( [ eqgainsimple ] ) gives zero gain in the basis of wannier stark states , as the translational symmetry gives identical @xmath13 for all wannier - stark levels . ) here the current is either evaluated from a semiclassical miniband transport model @xcite or by sequential tunneling between different layers @xcite . while these approaches use a macroscopic current density in eq . ( [ eqgainsigma ] ) , the consideration of polarization currents allows for a derivation of eq . ( [ eqgainsimple ] ) , see , e.g. , ref .  .    for complicated semiconductor heterostructures , with a variety of different tunneling and optical transitions , gain may result both from optical transitions in the spirit of ( i ) and macroscopic currents in the spirit of ( ii ) . the aim of this paper is to show the feasibility of a general approach based on a quantum transport theory which treats both mechanisms on a equal footing . the method of nonequilibrium green functions @xcite is used here , which has been frequently applied to light emission in semiconductors ( see , e.g. , refs .   and references given therein ) . while these works consider transitions between the conduction and valence band , the focus in this paper is on transitions within the conduction band in current - driven heterostructure devices . the calculations are performed in two steps : first the transport problem is solved by evaluating a self - consistent stationary solution of the quantum kinetic equations for a given applied bias . this provides us with the green functions describing the electronic state far from equilibrium . in a second step , an additional weak radiation field is taken into account . the time - dependent kinetic equations are linearized around the stationary nonequilibrium state in oder to study the linear response of the current - driven system . the formulation used here employs the full two - time structure of the theory without any sort of ( generalized ) kadanoff - baym ansatz @xcite . the capability of the approach is demonstrated by calculations of the gain spectra in a quantum cascade laser and a superlattice . we describe a general system by a set of orthonormal states labeled @xmath11 with energies @xmath14 and write the hamiltonian in the form : @xmath15 where @xmath16 and @xmath17 are electron annihilation and creation operators in the state @xmath11 . @xmath18 represents the matrix elements of the kinetic energy and the potential part of the hamiltonian @xmath19 . in the following , @xmath20 will be split into a constant part @xmath21 and a small time - dependent perturbation @xmath22 describing the interaction with a radiation field . finally , @xmath23 contains the part of the hamiltonian which will be solved perturbatively ( such as impurity , phonon , or electron - electron scattering matrix elements ) .    within the theory of nonequilibrium green functions @xcite the key quantities are the correlation function ( or lesser green function ) @xmath24 and the retarded / advanced green functions @xmath25\\ & \times \big\langle   \hat{a}_{\alpha_1}(t_1)\hat{a}^{\dag}_{\alpha_2}(t_2 ) + \hat{a}^{\dag}_{\alpha_2}(t_2)\hat{a}_{\alpha_1}(t_1 ) \big\rangle\ , , \end{split}\ ] ] respectively , where the heisenberg picture is used . first consider a stationary state with a time independent matrix @xmath26 , neglecting the radiation field . in this case all functions depend only on the time difference @xmath27 [ these stationary state functions are labeled by a tilde , e.g. , @xmath28 , in the following ] , and it is convenient to work in fourier space defined by @xmath29 then eqs . ( [ eqgret - time - tilde],[eqkeldysh - time - tilde ] ) of the appendix become the matrix equations @xmath30 these have to be solved self - consistently together with the equations for the self - energies which are functionals of the green functions @xmath31 this standard approach allows for a self - consistent evaluation of the green functions @xmath32 in a nonequilibrium situation caused by an applied bias ( contained in @xmath21 ) . details , such as the specific form of the functionals in the self - consistent born approximation , can be found in , e.g. , @xcite .    the current density ( in the @xmath33-direction ) can be evaluated from the expectation value of the momentum operator divided by the mass : @xmath34\rangle\\ = & \frac{e}{\hbar v}\sum_{\alpha\beta}w_{\alpha\beta } g^<_{\beta\alpha}(t , t)+\mbox{scattering currents } \label{eqcurrent } \end{split}\ ] ] where @xmath35 . the scattering currents result from the scattering part of the hamiltonian and can be expressed in terms of self - energies . details will be given elsewhere . now we consider the influence of an additional time - dependent potential @xmath36 in the hamiltonian . ( bold capital symbols denote matrices in the state indices @xmath1 ) . we treat the change of the system in linear response , and set @xmath37 the chance in the self - energies is described within the linearization of the functional ( here in the time domain ) @xmath38 such a decomposition has been used in ref .   for hartree - fock self - energies , e.g .. in this case , the time - dependence of the self - energies allows for a reduction to density matrix equations by setting @xmath39 . in contrast , in our case , where scattering effects are considered , @xmath40 exhibits an explicit time dependence in both arguments . this two time structure is fully taken into account here . therefore we apply the fourier decomposition in both times via @xmath41 the same decomposition is used for @xmath42 . it is shown in appendix [ appder ] that @xmath40 is determined within linear response by the following equations :    @xmath43 \tilde{\bf g}^{\text{ret / adv}}(e ) \label{eqgretlinresp}\\ \delta { \bf g}^{<}(\omega , e)&= & \tilde{\bf g}^{\text{ret}}(e+\hbar \omega)\delta { \bf u}(\omega ) \tilde{\bf g}^{<}(e ) + \tilde{\bf g}^{<}(e+\hbar \omega)\delta{\bf u}(\omega ) \tilde{\bf g}^{\text{adv}}(e)\nonumber \\ & + & \tilde{\bf g}^{\text{ret}}(e+\hbar\omega ) \delta\bm{\sigma}^{\text{ret}}(\omega , e)\tilde{\bf g}^{<}(e ) + \tilde{\bf g}^{\text{ret}}(e+\hbar\omega ) \delta\bm{\sigma}^{<}(\omega , e)\tilde{\bf g}^{\text{adv}}(e)\nonumber \\ & + & \tilde{\bf g}^{<}(e+\hbar \omega ) \delta\bm{\sigma}^{\text{adv}}(\omega , e)\tilde{\bf g}^{\text{adv}}(e ) \label{eqglesslinresp}\end{aligned}\ ] ]    the changes in self - energy @xmath42 are functionals of @xmath40 , which have to be evaluated self - consistently with @xmath40 . the derivation of the functionals from the linearization of eq . ( [ eqfuncsigmatime ] ) is straightforward and can be performed for arbitrary self - energies . the situation is particularly simple for the self - consistent born approximation , where the functional @xmath44 is linear in @xmath45 , and one finds : @xmath46 with the same functional as eq . ( [ eqfuncsigmae ] ) . ( note , that @xmath47^{\dag}$ ] does _ not hold _ for the time - dependent quantities . thus , the advanced quantities should be calculated explicitely )    the terms related to @xmath48 correspond to the ladder corrections in the evaluation of diagrams for the kubo formula in equilibrium . it is a general advantage of linear response within nonequilibrium green functions that these terms are obtained directly , see also @xcite . now we consider the linear response to an external radiation field where the electric field points in the @xmath33 direction . this gives additional terms in the hamiltonian @xmath49 , where @xmath50 are the electromagnetic potentials . neglecting quadratic terms in the radiation field and terms containing @xmath51 ( i.e. , assuming that the wavelength is large compared to the size of the active region ) , the following perturbation potentials are obtained ( see appendix [ appgauge ] ) : in the _ coulomb gauge _ @xmath52 neglecting the scattering part of @xmath19 ; in the _ lorentz gauge _ @xmath53    finally , the change in current density is given by the change in eq . ( [ eqcurrent ] ) @xmath54\\ & \cdot\tilde{\bf g}^<(e)+{\bf w}\cdot\delta { \bf g}^<(\omega , e ) \big\}\ , . \end{split}\ ] ] the scattering current in eq . ( [ eqcurrent ] ) has not been included here . with these ingredients , one obtains the complex conductivity @xmath55 , as well as the gain coefficient via eq . ( [ eqgainsigma ] ) . in the following we consider two different structures : ( i ) the quantum - cascade - laser structure of ref .  . ( ii ) the superlattice used in ref .   with 2 nm al@xmath56ga@xmath57as barriers and 8 nm gaas wells and a doping of @xmath58 . in both cases , we use a basis set consisting of products of wannier functions ( in the growth direction @xmath33 ) and plane waves with wave vector @xmath59 ( in the @xmath60-plane ) , and obtain @xmath61 from nominal sample parameters @xcite . the self - consistent solution for the electrostatic potential is included as well . for the scattering hamiltonian we include interface - roughness scattering and phonon scattering ( including both optical phonons and a second phonon with low energy to mimic acoustic phonons ) . the self - energies are evaluated in the self - consistent born approximation . furthermore , we use momentum - independent scattering matrix elements ( evaluated for a typical momentum transfer ) and diagonal self - energies . in this approximation the self - energies are not @xmath59-dependent , which significantly reduces the numerical effort . results and further details for superlattices @xcite and quantum cascade lasers @xcite have been given previously . the current - field characteristics are displayed in fig . [ figkenn](a , b ) for both structures , which agree reasonably well with the respective experimental data . note that domain formation in the region of negative differential conductivity modifies the behavior of the experimental curve for the superlattice . such effects are beyond the scope of the general approach discussed here , but can be easily studied using simpler transport models @xcite . similar results for the current - field relation of this quantum cascade laser have been reported in ref .  . for the superlattice structure , the simple 2-time miniband transport model @xcite gives @xmath62 for a non - degenerate electron gas . here @xmath63 is the electrical field , @xmath64 the miniband width and @xmath65 the period of the superlattice . @xmath66 ps and @xmath67 ps represent fitted , phenomenological energy and momentum scattering times . @xmath68 are the modified bessel functions . in fig . [ figkenn](b ) this model shows reasonable agreement with the full quantum transport calculation for @xmath69 mev , the field range where miniband transport holds true @xcite . ( for lower temperatures the shape of the current - field relation becomes more complicated both in the semiclassical boltzmann and the quantum transport model @xcite so that the simple 2-time model can not fit the data . )    ) . ]    in the following the response to an external radiation field is studied : first let us use the coulomb gauge and neglect the terms with @xmath42 in eq . ( [ eqglesslinresp ] ) . the gain spectrum for the quantum cascade laser of ref . is displayed in fig . [ figgainqcl ] . gain sets in for current densities above 0.8 ka/@xmath70 and increases with current . the peak gain coefficient is 57 @xmath71 at 6.5 ka/@xmath70 and a photon energy of 130 mev , in excellent agreement with the findings of ref . ( estimated losses divided by confinement factor @xmath72  @xmath71 and threshold current @xmath73=7.2 ka/@xmath70 , lasing at @xmath74 mev ) . the width of the gain spectrum agrees well with the findings of ref .  . results for the superlattice are given in fig . [ figsigmasl ] . the real part of the conductivity essentially follows the result from the simple 2-time miniband model @xcite @xmath75 with @xmath76 . thus , this simple model gives , at least for wide minibands and high temperatures , reliable results . mv and 20 mv , respectively . dotted and dash - dotted line : corresponding results from the 2-time miniband model @xcite . ]    now we want to study the relevance of the terms @xmath42 in eq . ( [ eqglesslinresp ] ) , which have been neglected so far . [ figgaincomp ] shows that their inclusion barely changes the result in the coulomb gauge ( the full and dashed lines are almost indistinguishable ) . therefore these self - energy corrections may be neglected ( at least for diagonal self - energies assumed here ) allowing for a significant reduction of the numerical effort . in contrast , in the case of the lorentz gauge , the terms evaluated with self - energy corrections ( dash - dotted line ) differ substantially from the simpler version without these terms ( dotted line ) . the difference is particularly strong in the case of the superlattice , where the diagonal terms in the perturbation potential ( [ eqdeltau - lorentz ] ) are essential for the transition . furthermore , the results of coulomb gauge and lorentz gauge are almost identical if the self - energy corrections are included . this demonstrates that the inclusion of these terms is essential to obtain a consistent theory , which must be independent of the choice of the gauge .     for ( a ) the superlattice at @xmath77 and ( b ) the quantum cascade laser at @xmath78 mv . full line : coulomb gauge neglecting @xmath79 ; dotted line : lorentz gauge neglecting @xmath79 ; dashed line : coulomb gauge with @xmath79 ( falls together with the full line ) ; dash - dotted line : lorentz gauge with @xmath79 . ] a formulation of the theory of nonequilibrium green functions has been given which allows for a quantitative description of current - induced gain in semiconductor heterostructures . quantitative agreement with the experiment for the gain spectrum of a quantum cascade laser was obtained . the standard 2-time miniband model for the superlattice could be verified for a large miniband width and under room temperature operation . the numerical results are not sensitive to the choice of the gauge if the variations of self - energy terms are taken into account . these terms are of particular importance for the lorentz gauge , while their contribution is small in the coulomb gauge in the calculations presented here . helpful discussions with a.  knorr , a .- jauho , s .- c . lee , m.  pereira and e.  schll as well as financial support by dfg within for394 are gratefully acknowledged . the green functions are determined by the dyson equation ( see , e.g. , chapter 5 of ref .  ) @xmath80 and the keldysh relation , see also appendix [ appremark ] . @xmath81    now we consider the stationary state , where @xmath82 . ( [ eqgret - time],[eqkeldysh - time ] ) become : @xmath83 and their fourier - transformation ( [ eqfourier - tilde ] ) provides us with eqs . ( [ eqdysone - tilde],[eqkeldyshe - tilde ] ) . eq .  ( [ eqkeldysh - time ] ) is a particular solution of the inhomogeneous differential equations ( 5.3 ) of ref .  , which are given more explicitly in eqs . ( 106 - 107 ) of ref .   in the notation used here . the keldysh relation ( 5.11 ) derived in ref .   contains an additional term : @xmath84\\ \times{\bf g}_0^<(t_b , t_c ) \left[\hbar\delta(t_c - t_2){\bf 1 } + { \bf u}(t_c){\bf g}^{\mathrm{adv}}(t_c , t_2 ) + \int \frac{{{\rm d}}t_d}{\hbar}\bm{\sigma}^{\mathrm{ret}}(t_c , t_d ) { \bf g}^{\mathrm{adv}}(t_d , t_2)\right]\end{gathered}\ ] ] this term constitutes a homogeneous solution of the inhomogeneous differential equations ( for fixed @xmath85 and @xmath86 ) which guarantees that @xmath87 for @xmath88 . this satisfies a general assumption underlying the perturbation expansion of the @xmath89-matrix . in the following we show that @xmath90 vanishes for finite times @xmath91 .    using eqs . ( 108,109 ) of ref . one obtains @xmath92_{t_a=-\infty}^{t_a=\infty}\ , , \end{split}\ ] ] where @xmath93 means that the derivative operates to the left . in the second line , partial integration has been used . as @xmath94 , the first term vanishes . furthermore , at the upper bound @xmath95 holds for finite times @xmath96 . applying the same manipulations for @xmath97 , we find @xmath98 for a real system the retarded and advanced green functions decay for large time differences and we find @xmath99 for finite @xmath91 . therefore this additional term is spurious .    a second , more physical argument relates to the idea , that the stationary state of a real physical system should not depend on the initial conditions used in a formal theory . except for specific situations exhibiting a hysteresis , scattering processes will install a stationary state which is independent of initial conditions . therefore all additional homogeneous solutions should reflect transient effects , which vanish if the initial state is chosen at @xmath100 and @xmath101 . setting @xmath102 , @xmath103 and @xmath104 in eqs . ( [ eqgret - time],[eqkeldysh - time ] ) gives in linear order @xmath105 : @xmath106 this constitutes an inhomogeneous differential equation , where the homogeneous part ( left - hand side ) is identical with the defining equation ( [ eqgret - time - tilde ] ) for @xmath107 . thus @xmath107 is the corresponding green function and the solution is given by : @xmath108         \tilde{\bf g}^{\text{ret}}(t'-t_2)\ ] ] and fourier transformation using eq . ( [ eqfourier ] ) gives eq . ( [ eqgretlinresp ] ) after some algebra . @xmath109 is evaluated in the same way with ret replaced by adv . the linearization of the keldysh relation ( [ eqkeldysh - time ] ) provides us with @xmath110\end{gathered}\ ] ] fourier transformation gives eq . ( [ eqglesslinresp ] ) after some algebra . we study the linear response to an external radiation field propagating in @xmath111-direction @xmath112 thus we have two reasonable choices of the electromagnetic potentials @xcite : @xmath113 in the spirit of a dipole approximation ( i.e. , assuming that the wavelength is large compared to the size of the active region ) the terms containing @xmath51 are neglected , providing the expressions ( [ eqdeltau - coulomb],[eqdeltau - lorentz ] ) in the perturbation hamiltonian , where quadratic terms @xmath114 are neglected .
gain in current - driven semiconductor heterostructure devices is calculated within the theory of nonequilibrium green functions . in order to treat the nonequilibrium distribution self - consistently the full two - time structure of the theory 
 is employed without relying on any sort of kadanoff - baym ansatz . 
 the results are independent of the choice of the electromagnetic field if the variation of the self - energy is taken into account . 
 excellent quantitative agreement is obtained with the experimental gain spectrum of a quantum cascade laser . 
 calculations for semiconductor superlattices show that the simple 2-time miniband transport model gives reliable results for large miniband widths at room temperature .