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in the study of atoms with at least two electrons , bound states and resonances are of interest . the resonances evolve into states with one free electron after a very short time . this phenomenon is called auto - ionization of the atom . with a revival of interest in the auto - ionization , fano published an appealing theoretical paper @xcite comprising an analysis of the excitation of the @xmath0 level of helium by electrons . he argued that the natural line shape contains a zero . later , the optical absorption spectra of the rare gases have been analyzed @xcite , while the paper @xcite is one of many studies dealing with the mechanism of atomic auto - ionization . a unified approach to the configuration interaction and the influence of strong lasers have been expounded in @xcite . in this framework , the studies @xcite have been realized . the quantum laser field has been taken into account in @xcite and the effect of the squeezed state has been studied in @xcite .    the fano resonances can occur also in other physical settings . the fano resonances in nanoscale structures can be mentioned @xcite . the treatment of auto - ionization and the influence of laser may be extended to a simultaneous auto - ionization , the influence of laser , and to the interaction with a neighbor two - level atom @xcite . the presence of a neighbor system may also considerably increase photo - ionization and recombination rates @xcite . in the analysis , the assumption of weak optical pumping is usually used and leads to a simpler behavior , cf . , @xcite .    in @xcite , the entanglement between an auto - ionization system and a neighbor atom is studied for a classical driving field . besides the possibility to calculate a measure of entanglement for the two atomic systems , a somewhat arbitrary , but systematic , filtering is adopted . two frequencies can be selected in the auto - ionization system and the study of entanglement reduces to the well - known two - qubit problem . in this paper , we modify this analysis by including the quantal nature of the field . in sec . ii , we describe the model . in sec . iii , we discuss photoelectron spectra and the density plots of entanglement measure . iv provides conclusion . we consider two mutually interacting atoms , @xmath1 and @xmath2 , in the presence of an electromagnetic field ( for the scheme , see fig .  1 ) . to quantize the electromagnetic field , we have to add to the usual model annihilation and creation operators of the modes which participate in the radiative interactions . indeed , although only the frequency @xmath3 of optical field is considered , an infinity of modes at this frequency can be introduced . we may suppose that the atom @xmath4 interacts with  the mode @xmath5 and the atom @xmath6 interacts with the  mode @xmath7 . we complete the levels of the atomic system by the  photon - number states , @xmath8 where @xmath9 is a photon number in the mode @xmath5 and @xmath10 is a photon number in the mode @xmath7 . in eq . ( [ 1 ] ) , @xmath11 ( @xmath12 ) is the ground state of the atom @xmath4 ( @xmath6 ) , @xmath13 is the excited state of the atom @xmath4 , @xmath14 is the auto - ionization state of the atom @xmath6 , @xmath15 @xmath16 @xmath17 is the continuum state of the atom @xmath6 , and @xmath18 is an energy difference between the ground state @xmath12 and the state @xmath15 . here we have used the photon - number states @xmath19 , @xmath20 , @xmath21 , and @xmath22 simultaneously to indicate that the hilbert space of the states can be decomposed into invariant subspaces . for @xmath23 , these subspaces have a dimension equal to 6 . each invariant subspace is a tensorial product of the subspaces corresponding to the jaynes cummings model ( dimension 2 ) and the model due to leoski and buek ( dimension 3 ) @xcite . the hamiltonian has the form @xmath24 where @xmath25 with @xmath26 and @xmath27 ( @xmath28 and @xmath29 ) being the photon annihilation ( creation ) operators . the hamiltonian @xmath30 of atom @xmath2 with auto - ionizing level in eq .  ( [ 2 ] ) is written as @xmath31 where @xmath32 means an energy difference between the ground state @xmath12 and the state @xmath14 . symbol @xmath33 gives the strength of optical excitation from the ground state @xmath34 into the auto - ionization state @xmath35 , @xmath36 is the strength of optical excitation from the ground state @xmath34 of the atom @xmath6 into the continuum state @xmath37 , and @xmath38 describes the coulomb configuration interaction between the excited states of atom @xmath6 . the hamiltonian of the neighbor two - level atom in eq . ( [ 2 ] ) reads @xmath39 where @xmath40 means an energy difference between the ground state @xmath11 and the state @xmath13 , @xmath41 is the strength of optical excitation from the ground state @xmath42 into the excited state @xmath43 .    in eq . ( [ 2 ] ) , the hamiltonian @xmath44 characterizes the dipole dipole interaction between the atoms @xmath4 and @xmath6 , @xmath45 where @xmath46 @xmath47 characterize energy transfer from the ground state @xmath12 into the state @xmath14 @xmath48 at the cost of the decay from the state @xmath13 into the state @xmath11 . we note that if @xmath49 and @xmath50 , the hamiltonian @xmath51 describes uncoupled atoms . we will treat the situation where the atoms @xmath4 and @xmath6 interact with  a single mode @xmath52 , @xmath53 . in this case , the levels written in eq . ( [ 1 ] ) simplify , @xmath54 and @xmath55 is the number of photons in the mode @xmath52 . here we have used the photon - number states @xmath56 , @xmath57 , and @xmath58 simultaneously to indicate that the hilbert space of the states can be decomposed into invariant subspaces . but in the  case of a single mode , an invariant subspace can not be investigated as a tensorial product . we can see from eq . ( [ 7 ] ) , that the atom @xmath4 at the level @xmath11 interacts with the  field in the state @xmath56 or @xmath57 in the dependence on the state of the atom @xmath6 and the atom @xmath4 at the level @xmath13 interacts with the  field in the state @xmath57 or @xmath58 in the  dependence on the state of the atom @xmath6 . the hamiltonian has the form @xmath59 where @xmath60 and @xmath61 following @xcite , we modify the schrdinger picture by considering the state vector @xmath62 in the form @xmath63 \nonumber\\ & & \mbox { } \times \bigl[c_{00}^{(n)}(t)|n\rangle_l|0,0\rangle_{ab}+ c_{10}^{(n-1)}(t)|n-1\rangle_l|1,0\rangle_{ab } \nonumber\\ & & \mbox { } + c_{01}^{(n-1)}(t)|n-1\rangle_l|0,1\rangle_{ab } \nonumber\\ & & \mbox { } + c_{11}^{(n-2)}(t)|n-2\rangle_l|1,1\rangle_{ab } \nonumber\\ & & \mbox { } + \int d_0^{(n-1)}(e_d , t)|n-1\rangle_l|0,e_{d}\rangle_{ad}\,de_d \nonumber\\ & & \mbox { } + \int d_1^{(n-2)}(e_d , t)|n-2\rangle_l|1,e_d\rangle_{ad}\,de_d\bigr ] , \label{11}\end{aligned}\ ] ] where @xmath64 . the prime indicates that for @xmath65 some of the components must be omitted . the components with @xmath66 have to be omitted for @xmath65 and those with @xmath67 have to be left out for @xmath68 . it holds that @xmath69 $ ] @xmath70 @xmath71 for @xmath72 an invariant subspace @xmath73 is the eigenspace of the operator @xmath74 related to an eigenvalue @xmath55 . we assume that @xmath75 . in this invariant subspace , the composite system is described by the equations @xmath76 where @xmath77    further @xmath78 @xmath79 we introduce the matrix @xmath80 and let @xmath81 and @xmath82 denote the eigenvalues of the matrix @xmath83 and @xmath84 , @xmath85 , be the eigenvalues of the matrix @xmath86 . let us recall the possibility of decompositions @xmath87 where @xmath88 , @xmath89 are solutions of the equations @xmath90 similarly , @xmath91 , @xmath85 , are solutions of the equations @xmath92 in eqs . ( [ 22 ] ) and ( [ 23 ] ) , @xmath93 and @xmath94 @xmath70 @xmath95 are @xmath96 and @xmath97 unit matrices , respectively .    the first vector of the components of the solution of eqs . ( [ 13 ] ) has the very simple form @xmath98    we introduce a @xmath99 matrix @xmath100 as the solution of the sylvester equation @xmath101    the solution has the form @xmath102 the dependence of the components of the amplitude spectrum on the initial state of the system with  @xmath103 is @xmath104 { \bf t}^{(n)}(e_d ) \nonumber\\ & & \mbox { } - { \bf t}^{(n)}(e_d ) \exp\left[-\frac{i}{\hbar}{\bf m}^{(n)}t \right ] \biggr\}{\bf c}^{(n)}(0 ) . \label{27}\end{aligned}\ ] ] we observe that @xmath105 where @xmath106 { \bf t}^{(n)}(e_d){\bf c}^{(n)}(0 ) . \label{29}\ ] ]    the increase of the diagonal terms by  @xmath107 means that also the eigenvalues @xmath108 , @xmath109 , @xmath110 are raised by the  same amount . in the relations like ( [ 26 ] ) , these increments mutually cancel and elsewhere they already correspond to the relation ( [ 11 ] ) . all the previous exposition beginning with ( [ 14 ] ) should be modified for @xmath111 . let us note that the initial vacuum field and the ground states of the atoms @xmath4 and @xmath6 , @xmath68 , do not lead to any transitions to the continuum states . it holds that @xmath112 @xmath70 @xmath113 and the description reduces to the equation @xmath114 @xmath70 @xmath115 . the reduction for @xmath116 is a consequence of non - existence of @xmath58 as in ( [ 11 ] ) and need not be made explicit . let us note that transition to a continuum state can occur for @xmath116 , but not simultaneously with an excitation of the atom @xmath4 . the long - time behavior is characterized by a complete ionization of the atom with an auto - ionizing level and by both the levels of the two - level atom @xmath1 being occupied . the long - time behavior is periodical due to the dynamics of the two - level atom in the cw laser field . at all times the spectra can be determined as the probability distribution of the two - level atom at its levels and of the atom with the auto - ionizing level in the continuum of levels . by the normalization , conditional spectra are defined . the difference between the conditional spectra is an effect of the atomic quantum correlation . the difference between the conditional spectra can be seen even in the case where the dipole  dipole interaction of the atoms is missing . an important case of the quantum correlation is the entanglement . we measure this entanglement using the negativity . the entanglement is conserved , even though we restrict the continuum of levels to two of them , in the most of the pairs of the selected frequencies . we calculate the negativity of the partially transposed statistical matrix for two levels of the two - level atom and selected continuum levels of the atom with an auto - ionizing level .      in the previous work , the functions @xmath117 of the parameters of the atom @xmath6 without a neighbor have been conveniently introduced . also the functions @xmath118 of the parameters of both the atoms have been defined . here we conveniently introduce a one - photon version of the usual excitation parameter @xmath119 , @xmath120 where @xmath121 by the replacements @xmath122 , @xmath123 in the function @xmath124 , the function @xmath125 originates . for @xmath126 , the parameters of the model can be expressed in the forms @xmath127 from this , @xmath128 are new parameters .    in what follows , we will assume @xmath129 and four different physically interesting cases that elucidate the behavior of the analyzed system :    \(a ) @xmath130 ; @xmath131 ; @xmath132 ; @xmath133 ; @xmath134 ,    ( a ) @xmath135 ; @xmath131 ; @xmath132 ; @xmath133 ; @xmath134 ,    \(b ) @xmath135 ; @xmath136 ; @xmath132 ; @xmath137 ; @xmath134 ,    \(c ) @xmath135 ; @xmath136 ; @xmath132 ; @xmath137 ; @xmath138 . whereas atom @xmath2 is alone in ( a ) , it feels the presence of atom @xmath1 due to the quantized optical field in ( a ) . in ( b ) , both atoms interact by the dipole - dipole interaction that includes only the continuum of states at the atom @xmath2 . finally , also the dipole - dipole interaction between the discrete levels of both atoms is taken into play in ( c ) . we note that detuning of energy levels of both atoms from the laser frequency does not qualitatively modify the behavior of the system ( for more details , see @xcite for semiclassical model ) . also , we analyze the system at time @xmath139 bellow . for the considered values of parameters , the behavior of the system at time @xmath139 already corresponds to that appropriate to the long - time limit .      in the model , atoms @xmath1 and @xmath2 are in fact mutually coupled by two types of interactions . side by side with the discussed dipole - dipole interaction , the interaction mediated by photons in the quantized field also occurs . this interaction qualitatively distinguishes the presented fully quantum model from the common semiclassical models that assume a classical predefined optical pump field @xcite . the long - time limit of the two atomic systems is described by a statistical matrix @xmath140 , with two discrete indices @xmath141 and two continuous arguments @xmath18 , @xmath142 . here @xmath143 as usual , the photoelectron spectra are identified with the distributions @xmath144 this joint description may be reduced to the marginal probability distribution of the levels of the atom @xmath4 , @xmath145 we consider also the conditional distributions or spectra @xmath146 the closed formula for @xmath147 is rather complicated , @xmath148 where @xmath149 with @xmath150 } { i\left(\xi_{k'}^{(n)}-\xi_k^{(n)}- \lambda_{{\bf m}^{(n)}j'}^*+\lambda_{{\bf m}^{(n)}j}\right ) } \nonumber\\ & & \mbox { } \times { \bf k}_k^{(n)}{\bf b}_2^{(n)}{\bf m}_j^{(n ) } { \bf c}^{(n)}(0){\bf c}^{(n)\dagger}(0 ) { \bf m}_{j'}^{(n)\dagger}{\bf b}_1^{(n)}{\bf k}_{k'}^{(n)}. \label{42}\end{aligned}\ ] ] the long - time total photoelectron spectrum is time - independent , @xmath151    in figs .  2 and  3 the case ( a ) of data with @xmath137 and for the initial coherent state @xmath152 is illustrated . the unconditioned and conditional photoelectron spectra have a multi - peak structure and the peak positions are about the same for both the values of the subscript @xmath153 . therefore the plot is restricted to an interval which includes a single peak of a spectrum . in fig . 2 , it is seen that the unconditioned photoelectron spectra coincide and can not be discerned in the chosen interval . in contrast , in fig .  3 it is obvious that the conditional spectra differ significantly in the selected interval . it proves the dependence of the occupation of the atom @xmath6 s level on the atom @xmath4 s level .    to reveal the role of atom @xmath1 in ionization of atom @xmath2 , we compare the long - time ionization spectra of atom @xmath2 for atom @xmath154 present and absent . we consider a greater value of single - photon rabi frequency @xmath119 to emphasize quantum features of the model ( @xmath155 ) . ionization of isolated atom @xmath2 in a quantized field leads , in general , to the occurrence of sharp peaks in the ionization spectra ( see fig .  4 ) . these peaks arise from the ionization caused by individual fock states of the optical field . this is documented in figs . 4 , 5 , in which the ionization spectra corresponding to the coherent and fock states are shown . it holds that the greater the fock number @xmath156 is , the narrower is the corresponding spectral peak and also the closer is the peak to the position of energy of the fano zero ( see fig .  4 ) . such behavior qualitatively resembles that of an ionization spectrum caused by a classical strong pump field @xcite . the presence of atom @xmath1 in the quantized pump field considerably modifies the ionization spectra of atom @xmath2 due to the mutual indirect interaction of both atoms through the quantized pump field . contrary to the spectra of isolated atom @xmath157 , the ionization spectra have contributions both below and above the pump - field frequency . moreover , the spectral peaks above and below the pump - field frequency occur in pairs which results in nearly symmetric ionization spectra ( see figs .  6 ,  7 ) . this symmetry is inherent to the fock states from which it transfers into the coherent states , as documented in figs .  6 ,  7 . it holds that the greater the fock number @xmath156 , the closer the two peaks to the pump - field frequency . if the pump - field intensity increases , the spectrum of isolated atom @xmath2 is built more and more from contributions of higher - number fock states and it moves to lower energies crossing the energy of fano zero . the more intense the pump field , the more suppressed ( smoothed ) the spectral structure of individual fock states ( see figs .  8 ,  9 ) . also the narrowing of the overall ionization spectrum in the vicinity of the energy of fano zero is observed . when atom @xmath1 is present , the ionization spectra also gradually lose their peaked structure with the increasing pump - field intensities ( see figs .  10 ,  11 ) . for sufficiently high pump - field intensities , the spectrum approaches that of the isolated atom @xmath1 .    when the interaction mediated by the quantized field is weaker , the behavior of ionization spectra with the increasing pump - field intensities is qualitatively similar to the usual one discussed in the fano model . the spectra move towards lower energies with the increasing pump - field intensities and cross at certain intensity the energy of fano zero , as documented in figs . 12 ,  13 . whereas the isolated atom @xmath2 has a one - peak spectrum , the spectrum of atom @xmath2 influenced by atom @xmath1 consists of two peaks that form a spectral dublet at greater pump - field intensities clearly visible in fig .  13 . the dipole - dipole interaction between atoms @xmath1 and @xmath2 , in general , splits the peaks in the ionization spectra of isolated atom @xmath2 into two parts ( see figs . 14 ,  15 ,  16 ) . as a consequence , there occur two major peaks in the ionization spectra for greater pump - field intensities . these spectral peaks are broken into many sub - peaks for low pump - field intensities as a consequence of quantum character of the pump field ( see fig . individual sub - peaks can be connected with the appropriate fock states , similarly as in the previous section . two major peaks approach each other with the increasing pump - field intensity and form a spectral dublet at certain moment ( see fig . if only the dipole - dipole interaction between the discrete level of atom @xmath1 and the continuum of states of atom @xmath2 ( @xmath158 ) is considered , the fano zero of isolated atom @xmath2 is inevitably lost . however , when also the dipole - dipole interaction between the discrete levels of atoms @xmath1 and @xmath2 occurs ( @xmath159 ) , the fano zero can be preserved under certain conditions found in @xcite . the two mentioned dipole - dipole interactions compete in ionizing atom @xmath2 in certain sense . if the strengths of two interactions equal for the energy of fano zero formed at atom @xmath2 , the fano zero is preserved . the appropriate condition was derived in @xcite for the semi - classical model in the form @xmath160 numerical computations have revealed that the condition in eq . ( [ 44 ] ) is valid also in the analyzed quantum model ( see fig . 17 ) . here , we would like to note that the original fano zero of isolated atom @xmath2 is usually replaced by a broad deep minimum in the ionization spectra provided that the condition in eq . ( [ 44 ] ) is not fulfilled ( see fig . such behavior originates in the weakness of dipole - dipole interactions compared to the coulomb and optical dipole interactions that form the fano zero of isolated atom @xmath2 . we have assessed the entanglement by the computable measure of entanglement , i.  e. , the negativity @xcite . it is recommended as such in the case of two parties ( components ) each possessing a finite number of levels . we mark a difference , because in our analysis one of the two parties has an infinite number of levels . the straightforward approach was successful on the assumption of a classical light field @xcite , because the two components are in a joint pure quantum state . to our knowledge , such an approach can not be based on simple formulas on inclusion of the quantum nature of the field which leads to a mixed quantum state describing the involved parties . numerical calculation would be a challenging task . recently , a selection of the frequencies has been realized in a somewhat arbitrary , but systematic , way @xcite . two states with these frequencies are just the levels needed for producing a qubit . in such a way , we return to the well - known two - qubit problem . for @xmath161 @xmath162 @xmath163 $ ] , @xmath164,[-5,10 ] , [ -1.5,1.5]$ ] , we generate a density plot of the negativity at @xmath165 that is @xmath166 where @xmath167 are eigenvalues of the partially transposed statistical matrix @xmath168 here @xmath169 } \nonumber\\ & & \mbox { } \times\left(\begin{array}{cc } \rho_{jk}^{\rm out}(e_d , e_d , t)&\rho_{jk}^{\rm out}(e_d , e'_d , t)\\ \rho_{jk}^{\rm out}(e'_d , e_d , t)&\rho_{jk}^{\rm out}(e'_d , e'_d , t)\end{array } \right ) , \label{47}\end{aligned}\ ] ] with @xmath170 both the dipole - dipole interaction and the interaction mediated by the quantized pump field create the entanglement between the bound electron at atom @xmath1 and the ionized electron at atom @xmath2 . suitable conditions for creating highly entangled states have been revealed in @xcite concerning a classical pump field . it holds that the stronger the dipole - dipole interaction , the more entangled state is reached . however , also a weaker dipole - dipole interaction can provide highly entangled states provided that the ionization process is sufficiently slow . this can be reached when the strengths of the direct ionization path ( connecting the ground state of atom @xmath2 with the continuum ) and the indirect ionization path ( that ionizes an electron from the ground state of atom @xmath2 through the auto - ionizing discrete state of atom @xmath2 ) are balanced . similarly as in the semiclassical model analyzed in @xcite , the overall negativity can roughly be composed of negativities of qubit - qubit systems obtained from the qubit of atom @xmath1 and all possible qubits found in the continuum of atom @xmath2 . such densities of negativity give us information about the spectral distribution of entanglement . the density of negativity for the ionization spectrum shown in fig .  6 and appropriate for the interaction mediated by the quantized field is plotted in fig . we can see in fig . 18 that the negativity is distributed in the whole area of energies present in the ionization spectrum . it is remarkable that the values of density of negativity are very low for the degenerate energies of qubits inside the continuum of atom @xmath2 ( @xmath171 ) . this behavior can be explained by the long - time energy conservation that does not allow to entangle such qubits in the continuum with the qubit of atom @xmath172 . the densities of negativity appropriate to the coherent and fock states completely differ , as demonstrated in figs .  18 ,  19 . we note that a pump field in the fock state with one photon can not create entanglement due to the energy conservation . however , higher - number fock states are already suitable for the entanglement creation .    the densities of negativities formed by the dipole - dipole interaction behave similarly as those created by the interaction mediated by the quantized field . it holds also here that appreciable values of the density of negativity are found for energies appreciably present in the ionization spectra . also very low values of the density of negativity occur around the degenerate energies @xmath171 ( see figs .  20 ,  21 ) . thus , the spectral concentration of negativity is observed as the pump - field intensity increases ( compare figs . 20 , 21 ) . when the spectrum forms a spectral dublet , the entanglement is encoded between the two peaks of the dublet .    as follows from the above results , effects stemming from quantum features of the pump optical field are clearly visible both in ionization spectra and entanglement provided that the one - photon excitation parameter @xmath119 is greater or comparable to 0.1 and the mean number of photons is smaller or comparable to 10 . both coherent laser fields and highly - nonclassical fock - state fields are suitable for the observation of quantum signatures of auto - ionization process . as for the fock - state fields , they can be generated , e.g. , in heralded single - photon sources @xcite or their generalizations @xcite and in qed cavities @xcite . greater values of the one - photon excitation parameter @xmath119 represent experimental challenge as the values reached in current ionization experiments are much smaller . however , modern photonic band - gap structures @xcite give a hope here . they allow to dramatically increase electric - field amplitudes inside due to constructive interference on one side . on the other side , they form photonic bands with continuum of states which are similar to those participating in ionization . we have studied quantum correlations of two atoms . we have assumed that one atomic system contains an auto - ionizing level whereas the other atom does not comprise any auto - ionizing level . both the atoms interact with the same mode of the quantized field . we have concentrated ourselves to the long - time behavior of the atomic systems . the long - time behavior exhibits quantum correlations of the two atoms even in the case where the atoms do not interact directly . we have illustrated quantum correlations comparing the one - peak spectrum appropriate for the neighbor atom without optical excitation with the two - peak spectrum occurring for the optically - excited neighbor atom . in the classical limit of strong field the differences vanish . * we have identified conditions for the observation of quantum features in long - time electron ionization spectra . * also the fano zero has been found in these spectra for the quantized optical field considering the same conditions as for the classical optical field . the authors acknowledge the financial support by the project operational program research and development for innovations - european social fund ( project cz.1.05/2.1.00/03.0058 ) of the ministry of education , youth and sports of the czech republic and the project iga no . prf-014 - 05 . a. luk , v. peinov , j. peina , jr . , j. kepelka , and w. leoski , in _ wave and quantum aspects of contemporary optics _ , edited by j. mllerov , d. senderkov , and s. jureka , proceedings of spie vol . 7746 ( bellingham : spie ) p 77460w ( 2010 ) . g. brida , i. p. degiovanni , m. genovese , f. piacentini , p. traina , a. della  frera , a. tosi , a. bahgat  shehata , c. scarcella , a. gulinatti , m. ghioni , s. v. polyakov , a. migdall , and a. giudice , appl . * 101 * , 221112 ( 2012 ) .
quantum correlations between two neighbor atoms are studied . 
 it is assumed that one atomic system comprises a single auto - ionizing level and the other atom does not contain any auto - ionizing level . 
 the excitation of both atoms is achieved by the interaction with the same mode of the quantized field . 
 it is shown that the long - time behavior of two atoms exhibits quantum correlations even when the atoms do not interact directly . 
 this can be shown using the optical excitation of the neighbor atom . 
 also a measure of entanglement of two atoms can be applied after reduction of the continuum to two levels .