Control of entanglement following the photoionization of trapped, hydrogen-like ions

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Physics Letters A 347 (2005) 73–80

www.elsevier.com/locate/pl

Control of entanglement following the photoionizationof trapped, hydrogen-like ions

Thomas Radtke∗, Stephan Fritzsche, Andrey Surzhykov

Institut für Physik, Universität Kassel, D-34132 Kassel, Germany

Received 6 June 2005; accepted 10 June 2005

Available online 11 July 2005

Communicated by B. Fricke

Abstract

Density matrix theory is applied to re-investigate the entanglement in the spin state of pairs of electrons followphotoionization of trapped, hydrogen-like ions. For the ionization of one out of two non-interacting atoms, in particuanalyzed how the entanglement between the electrons is changed owing to their interaction with the radiation field.calculations on theconcurrenceof the final spin-state of the electrons have been performed for the photoionization of hydas well as for hydrogen-like Xe53+ and U91+ ions. From these computations it is shown that the degree of entanglement,is quite well preserved for neutral hydrogen, will be strongly affected by relativistic and non-dipole effects of the radiatias the nuclear charge of the ions is increased. 2005 Elsevier B.V. All rights reserved.

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1. Introduction

Since the early days of quantum mechanics,physical interpretation and completeness of this thehave been the subject of many controversial discsions. In the famous “gedanken” experiment by Ein-stein, Podolsky, and Rosen (EPR) on the measuremof spatially separated quantum systems, for instait was argued that quantum theory cannot becompleteas it allows correlated quantum systems (in the cof their measurement) to influence each other ins

* Corresponding author.E-mail address:[email protected](T. Radtke).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.06.106

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taneously over large distances[1]. This strange quantum mechanical correlation between different partsa composite system, which—following the workSchrödinger[2,3]—is better known today asquantumentanglement, clearly violates the principle of locaity and was the (main) reason for Einstein to cosider quantum mechanics as an incomplete theAnd experimentally, in fact, another half a century wneeded, including the remarkable work of Bell[4,5]and many others, in order to decide finally agaiEinstein’s objections by measuring thenon-localcor-relations of entangled quantum states[6,7].

Today, the ‘entanglement’ of quantum systeplays a key role in quite different areas of physi

.

74 T. Radtke et al. / Physics Letters A 347 (2005) 73–80

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In the field of quantum computation and quantumformation, for example, a large number of case studhave been performed during recent years in ordeexplore new features ofhow quantum entanglemenmay enhance the power of information processinghelp secure the information transfer between differlocations[8,9]. However, despite of the remarkabprogress in this field during the last decade, the rization of its basic ideas, such as superdense co[10], quantum cryptography[11] or the implementation of quantum algorithms[12], is still a challengeand will surely depend on the (future) capabilitiesfind and isolatephysical processes, which facilitatethe observation and manipulation of quantum entglement.

One such process for the creation and maniption of entanglement isatomic photoionizationas dis-cussed recently by Pratt and co-workers[13]. Theseauthors have explored the entanglement of the sof two electrons in the ionization of the helium-likewell as two hydrogen-like ions by circularly polarizelight. In particular, they demonstrated, that entangment is drastically changed due to the photoionizaprocess and that this (entanglement) change depon the initial spin state of two bound electrons. Hoever, nosystematicinvestigations have been performso far on the transfer of entanglement which, aparthe electron spin states, is also affected by the rtivistic and non-dipole effects in the electron–photinteraction as well as the by the geometry of the pticular photoionization experiment.

In this contribution, we now apply density matrtheory to analyze the transfer and changes in thetanglement of some pair of electrons, if one out of tnon-interacting atoms or ions is photoionized. Forsake of simplicity, we here restrict the discussionthe case of trapped, hydrogen-like ions and to anradiation of the system by circularly polarized lighFor such a system of two trapped ions, then, thetanglement of the electrons is investigated in theirnal spin state as function of the angle and energthe incident photons and with emphasis, in partilar, on the effects of relativity and the higher mulpoles of the radiation field (beyond the electric-dipapproximation). In the next section, therefore, letstart with a brief account on the density matrix theoand its benefits for describing ionization and captprocesses. However, not much will be said about

s

theory as it was presented elsewhere at a numbeplaces[14,15]. Apart from the final-state density matrix and the evaluation of the (relativistic) transitioamplitudes, here we just introduce theconcurrenceasa quantitative measure for the entanglement of a cposite system. Detailed calculations on this measof entanglement are later performed and discusseSection3 for theK-shell photoionization of hydrogeatoms as well as for hydrogen-like Xe53+ and U91+ions. These computations show that the concurreof the final-state electrons is strongly affected by rativistic and non-dipole effects of the radiation fieas the nuclear charge,Z, is increased. Finally, a briesummary and outlook is given in Section4.

2. Theory

2.1. Density matrix approach

Since its introduction by von Neumann[16] andLandau[17] in 1927, the density matrix (theory) habeen found powerful in quite many fields of modephysics. In atomic physics, for instance, this theorybeen applied widely not only for studying the captuand emission of particles but also for the interactionatoms with the radiation field or for describing ionatom collisions[14,15]. Especially when combinewith the theory of spherical tensors[18], the densitymatrix often provides a tool of great elegance in orto represent and to deal with ensembles of interacparticles, which can either be in apurequantum stateor in some statisticalmixtureof states with any givendegree of coherence.

The basic idea of the density matrix theory inapplication to collision processes can be summarrather easily: starting with a well-definedinitial stateof a quantum ensemble, this theory enables one tocompany’ the ensemble through (the region of) oneseveral interactions until somefinal state of the systemis attained. In the following, therefore, we can apthis formalism also for studying the photoionizatiof trapped ions by means of a weak radiation fieFor the sake of simplicity, let us suppose amodelsys-tem which just consists out of two spatially well seprated andnon-interactingatoms or ions [cf.Fig. 1],and which is irradiated by circular-polarized lighMoreover, let us restrict our considerations to t

T. Radtke et al. / Physics Letters A 347 (2005) 73–80 75

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Fig. 1. Geometry of the ionization process in which one (out of twtrapped, hydrogen-like atoms is photoionized by a circularly poized photon of energyEγ . Since the quantization axis is chosparallel to the emitted electron, the incoming photon can be chaterized by means of the (polar) angleθ with respect to this axis.

hydrogen-like ions with nuclear chargeZ1 = Z2 = Z

and zero nuclear spinsI1 = I2 = 0, so that the totastate (of the system) is then described by some lincombination of the one-electron states|n1j1µ1〉 and|n2j2µ2〉, respectively. For a weak radiation field,addition, we can assume that either no or justoneofthe two ions is photoionized due to its interaction wthe field. For the (combined) system ‘ions+ field’,therefore, our model system containsinitially the twoelectrons in the Coulomb field of the bare ions as was the incoming photon with wave vectork and helic-ity λ.

Despite our assumption of non-interacting ioof course, the two electrons of the system canpreparedprior to the photoionization in some weldefinedpurestate

(1)|Ψ0〉 =∑µ1µ2

C0(µ1,µ2)|n1j1µ1〉 ⊗ |n2j2µ2〉,

as obtained from the superposition of the correspoing products states. Below, of course, we wish to csider the electrons as the qubits of atwo-qubitquan-tum system. In good approximation, this picturefulfilled for our model system from above if we cosider both of the ions in their (electronic) ground stathat is for n1 = n2 = 1 and j1 = j2 = 1/2. Since,moreover, the ground state of the hydrogen-like iois quite stable with respect to excitations by exterperturbations, we then remain with the well-known bsis states|µ = +1/2〉 = |↑〉 and|µ = −1/2〉 = |↓〉 foreach of the electrons and, hence, the totalspinstate(1)of the two electrons can be interpreted also as ator of a four-dimensional Hilbert space,{|↑↑〉, |↑↓〉,

|↓↑〉, |↓↓〉}, i.e., by exploiting the (computational) bsis of any valid two-qubit system.

Instead of using the state vector representation(1),it is often more convenient to describe the spin statthe electrons by means of their density operator

ρ0 = |Ψ0〉〈Ψ0|=

∑µ1µ2

∑µ′

1µ′2

C0(µ1,µ2)C∗0(µ′

1,µ′2)|j1µ1〉〈j1µ

′1|

(2)⊗ |j2µ2〉〈j2µ′2|,

and to define the initial state of the overall syst‘electrons+ incoming photon’ byρi = ρ0⊗ ργ , whereργ refers to the incoming radiation. For completecircular polarized light, the density matrix of the phton is given by

(3)ργ = |kλ〉〈kλ|with λ = +1 for right- andλ = −1 for left-circularlight. Inserting this operator into Eq.(2), we thereforeobtain the density operator of the initial state as

ρi = ρ0 ⊗ ργ

=∑µ1µ2

∑µ′

1µ′2

C0(µ1,µ2)C∗0(µ′

1,µ′2)

(4)× |j1µ1, j2µ2,kλ〉〈j1µ′1, j2µ

′2,kλ|,

or simply

〈j1µ1, j2µ2,kλ′|ρi |j1µ′1, j2µ

′2,kλ′′〉

(5)= δλ′λδλ′′λC0(µ1,µ2)C∗0(µ′

1,µ′2),

if the density matrix is taken explicitly in a basiswell-defined angular momenta of the individual eletrons.

After the photoionization process, i.e., if one of ttwo electrons has been ionized by the photon, thefinalstate of the system is given by a free electron withymptotic momentump and spin projectionms as wellas the second bound electron which, as before, remin the ground state|n = 1, j = 1/2,µf 〉. For this finalstate of the system, the density operatorρf is obtainedfrom the initial-state density matrix(4) due to the standard relation

(6)ρf = OρiO†,

where thetransition operatorO represents the interaction of thetwo electrons with the radiation field. A

76 T. Radtke et al. / Physics Letters A 347 (2005) 73–80

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described above, this (weak-field) operatorO actslo-cally on either one of the distinguishable and separaions, leaving the other electron unchanged in its inibound state. Therefore, the transition operator canwritten as

(7)O = R1 ⊗ I2 + I1 ⊗ R2,

where I is the identity operator andR the usualelectron–photon interaction and where, for the phoionization of high-Z ions, the description should bbased on Dirac’s equation and theminimal couplingof the radiation field.

In Eq. (6), the final-state density operatorρf stillcontains the complete quantum-mechanical informtion about the system and, hence, can be utilizeorder to derive the properties of both the electrons,bound one as well as for the electron which was emted from its ion. Here, we shall not further work withis density operator explicitly but make better uof its (final-state density) matrix representation witha basis of the individually well-defined angular mmenta of the electrons

〈jf ,p : µf ,ms |ρf |jf ,p : µ′f ,m′

s〉=

∑µ1µ2µ

′1µ

′2

〈jf ,p : µf ,ms |O|j1µ1, j2µ2,kλ〉

× C0(µ1,µ2)

× 〈j1µ′1, j2µ

′2,kλ|O†|jf ,p : µ′

f ,m′s〉

(8)× C∗0(µ′

1,µ′2),

and where Eq.(5) has been inserted for the initial-stadensity matrix.

2.2. Evaluation of the transition amplitudes

The final-state density matrix(8) describes the totaspin state of our two-qubit system following the phtoionization of one of the electrons. For any furthanalysis of this state, we must first express thetwo-electronmatrix elements

〈jf p : m1,m2|O|j1µ1, j2µ2,kλ〉= δjf j2δm2µ2〈pm1|R1|j1µ1,kλ〉

(9)+ δjf j1δm1µ1〈pm2|R2|j2µ2,kλ〉,

in a form which is computationally feasible, i.e.,terms of one-electrontransition amplitudes as obtained for the transition operator(7). These one-electron amplitudes describe the transition ofelectron from a bound into a continuum state dueinteraction with the photon field.

Not much need to be said here about the comption of thebound-freetransition amplitudes

〈pm| R |jµ,kλ〉(10)=

∫ψ†

pm(r)α · uλeik·rψjµ(r)d3r,

which have been applied very frequently in the paststudying the (photo-)ionization and capture of eltrons by atoms and ions. During the last few yearsparticular, several (exact)relativistic computations ofthese amplitudes have been carried out in the frawork of Dirac’s theory and have helped understathe photo-emission of high-Z projectiles at intermediate and high collision energies[19,20]. For hydrogen-like ions, of course, the wave functionsψjµ(r) andψpm(r) in Eq.(10) refer to the analytically known solutions of the Dirac Hamiltonian for the bound anfree electron, respectively. In the transition operaR = α · uλeik·r , moreover,α denotes the (vector ofDirac matrices anduλ a unit vector to describe thpolarization of the incoming photons. In the preswork, all the matrix elements(9) and(10) have beencalculated by means of the DIRAC program[21],a computer-algebraic toolbox which has been deoped by us for studying the properties and the dynical behaviour of hydrogen-like ions.

2.3. Concurrence as measure of entanglement

Having the transition amplitudes(10), we can cal-culate the final-state density matrix of our model stem with one electron in the ground state of its iand the second electron emitted from the trap. Forcase of zero nuclear spins, this density matrix stillscribes the (final)polarization stateof the system. If,for example, we take the quantization axis alongoutgoing electron momentum,p ‖ ez [cf. Fig. 1], wecan represent the two qubits in the combined (coputational) basis{|↑↑〉, |↑↓〉, |↓↑〉, |↓↓〉} and ask forthe (degree of) entanglement among the electronter the photoionization has lead to the emission ofelectron.

T. Radtke et al. / Physics Letters A 347 (2005) 73–80 77

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Severalmeasuresare known today to quantify thdegree of entanglement of some multi-partite syste[22]. During the last years, for instance, such measuhave been utilized to study the dynamics of entanment if some decoherence occurs due to the interacof the qubits with their environment[23,24]. To an-alyze the (degree of) entanglement for the spin sof the two-electron system from above, we makeof the concurrence, a quantum measure to descrithe entanglement of pure and mixed two-qubit qutum states. For an arbitrary two-qubit system wgiven density operatorρ, the concurrence is defineas[25,26]

(11)C(ρ) = max(0, λ1 − λ2 − λ3 − λ4),

where theλi ’s are the square roots of the eigenvaluof the matrix ˜ρρ in descending order and where˜ρ de-notes the so-called ‘spin-flipped’ matrix

(12)˜ρ ≡ (σy(1) ⊗ σy(2)

)ρ∗(σy(1) ⊗ σy(2)

).

In the definition(12)of the spin-flipped matrix, moreover, ρ∗ refers to the complex conjugate ofρ, andσy(1) and σy(2) are the standard Pauli matrices aing on the first and the second qubit, respectively.

Using the formulas(11)and(12), we are able to determine the degree of entanglement for any final-sdensity matrix(8), following the photoionization oone of the electrons. In Section3 below, we shall dis-cuss the concurrence of the final-state electrons inpendence of the angle and the energy of the incomphotons as well as for a few different ions. Howevbefore we start with a more detailed analysis of t‘final-state entanglement’, let us first briefly considthe concurrence 0� C(ρ0) � 1 of the two electrons inthe initial bound state(1). For a product state of thtwo spins,|Ψ0〉 = |µ1 = +1/2〉 ⊗ |µ2 = +1/2〉, wehave—as expected perhaps—initially a zero concrenceC(|Ψ0〉〈Ψ0|) = 0, while it becomesC(ρ0) = 1for |Ψ ±〉 = 1/

√2(|↑↓〉 ± |↓↑〉), telling us once more

that the ‘Bell states’ represent maximally entangtwo-qubit quantum states.

3. Results and discussion

We are now prepared to analyze the changes inentanglement of the two electron spins, if one ofions is photoionized by a weak radiation field. Befo

the interaction has occurred, of course, the entanment between the electrons is determined compleby the parametersC0(µ1,µ2) of the initial-state density operator(2). Here, we shall not discusshow thesystem can be initialized in a certain quantum sbut assume simply, that such a preparation is alwpossible with sufficient accuracy by applying a propset of prior ‘perturbations’ to the system. Then, afthe photoionization of one of the ions, the degreeentanglement is determined by the final-state denmatrix (8). Apart from the initial spin state of the sytem, of course, this density matrix also depends onenergyEγ and thedirectionθ of the incoming light aswell as the nuclearchargeZ of the trapped ions. In thfollowing, we discuss how these three parametersfluence the concurrenceC(ρf ) in the final state, that isthe (degree of) entanglement between the boundthe finally emitted electrons.

Let us start with the initial product state|Ψ0〉 = |↑↑〉which has no entanglement between the two etrons,C(|↑↑〉) = 0. To explore the energy and angudependence of the final-state entanglement forproduct state, computations have been carried outhe photoionization of neutral hydrogen as well astwo hydrogen-like ions Xe53+ and U92+, respectively.Fig. 2 displays the concurrenceC(ρf ) of the final-state electrons as function of the photon angleθ andfor photon energiesEγ = 1.01|E1s |, that is for ener-gies 1% above of the ionization threshold (and whE1s refers to the 1s binding energy of the electronAs seen from this figure, the photoionization of oof the atoms does not change the concurrence infinal state of the electrons over a wide range ofgles θ , i.e., C(ρ0) = C(ρf ) ≈ 0, except for photonangles near to 180◦, i.e., for photons coming inan-tiparallel to the emitted electron. A similar result wreported recently by Pratt and co-workers[13], whostudied the change of entanglement in atomic phionization by applying theelectric-dipole approxi-mation for the electron–photon interaction, i.e., feik·r 1. Although such an approximation is justifiefor low-Z ions and for non-relativistic photon energi(as seen fromFig. 2), it becomes much less appropate for high-Z (hydrogen-like) ions[19]. For Xe53+and U92+, for example, the—electric and magneticnon-dipole terms in the electron–photon interactresult in an clear enhancement of the (degree of)tanglement, if one of the electrons is photoionized.

78 T. Radtke et al. / Physics Letters A 347 (2005) 73–80

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Fig. 2. Concurrence of the electrons in their final state as funcof the photon angleθ , following the photoionization of one (out otwo) trapped, hydrogen-like atoms. The ions are assumed initialtheproductspin state|↑↑〉 with concurrenceC(|↑↑〉) = 0. Resultsare shown for hydrogen and the two hydrogen-like ions, Xe53+ andU92+, as well as for photon energies 1% above of the ionizathreshold.

these ions the concurrence becomes non-zero alrfor θ � 120◦ and, again, reaches its maximal valuC(ρf ) = 1, for θ ≈ 180◦. Fig. 2 therefore shows howthe photoionization of one out of two non-interactiions may lead to thecreationof entanglement betweethe electrons in their final state.

Apart from the nuclear chargeZ, the entanglemenof the residual two-electron system is affected alsothe energyEγ of the incoming photons. InFig. 3, wedisplay the energy dependence of the concurrencfunction of the photon angleθ for the three (relativeenergiesε = Eγ /E1s = 1.01,1.5, and 2.0, but onlyfor hydrogen-like U91+ ions. For such a high-Z ion,the concurrence appears to be very sensitive to theergy of incoming radiation and increases by more ta factor of 4 for the angles 90◦ < θ < 150◦, if thephoton energy is increased from 1% to about 10above the ionization threshold. Again, relativisticfects and the higher multipoles in the radiation fieare responsible that ‘entanglement’ is created herthe system, even if the two ions donot interact witheach other.

Until now, we have analyzed the changes inconcurrenceC(ρf ) of the final-state electrons follow

-

Fig. 3. The same as inFig. 2 for hydrogen-like U91+ ions but forthe three (relative) photon energiesε = Eγ /E1s = 1.01,1.5, and2.0, respectively.

ing the photoionization of (one of) the electrons in tproduct state|Ψ0〉 = |↑↑〉, i.e., forno initial entangle-ment. Besides the nuclear chargeZ and the photonenergyEγ , however, the concurrence of the final-stelectrons also depends on the entanglement of thetial system [cf. Eq.(8)]. In a second example, therfore, let us suppose the two hydrogen-like ions toin the initial (pure) state

(13)|Ψ0〉 = √p |↑↓〉 + √

1− p |↓↑〉,with the ‘mixing parameter’ 0� p � 0.5. While, forp = 0, this pure state results again in the prodstate |Ψ0〉 = |↓↑〉 (with zero concurrence), it giverise to the maximally entangled Bell state|Ψ +〉 =1/

√2(|↑↓〉 + |↓↑〉) for the other limit,p = 0.5. By

changing the parameterp therefore, this superpositiois very suitable for studying thetransfer of entanglement between the electrons due to their interactwith the radiation field.

Fig. 4 displays the concurrence of the final-staelectrons as function of the photon angleθ and fordifferent values of the parameterp. Similar as be-fore, we performed the calculations for hydrogen athe two hydrogen-like ions Xe53+ and U91+, and byapplying a photon energyEγ = 1.01|E1s | just aboveof the ionization threshold. Five different values

T. Radtke et al. / Physics Letters A 347 (2005) 73–80 79

the

p |↑↓〉 + √1− p |↓↑〉 with

0 � p � 0.5. Results are shown for hydrogen and the two hydrogen-like ions, Xe53+ and U92+, as well as for photon energies 1% above ofionization threshold.

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the mixing parameterp are considered for the initiastate(13) and with the concurrencesCp=0 = 0.0 (—),Cp=0.1 = 0.6 (– –),Cp=0.2 = 0.8 (– · –),Cp=0.3 = 0.92(– · · –) andCp=0.5 = 1.0 (· · ·), respectively. As seefor atomic hydrogen on the left side ofFig. 4, thedegree of theinitial entanglement is preserved in t(non-relativistic)electric-dipoleapproximation as predicted before by Pratt and co-workers[13]. However,if the nuclear charge is increased, the relativisticfects and higher multipoles become quickly importand may lead to a significantchangeof the final-stateentanglement. The deviation from theelectric-dipoleapproximation become prominent in particular at fward (θ < 30◦) and backward (θ > 150◦) angles of theincoming light.

Of course, the strongest deviations from the etric-dipole approximation arises for two trapped U91+ions, for which the photoionization of one of thelectrons leads to a clearreduction of the concur-rence for all initial states(13) with p > 0 [cf. rightside ofFig. 4]. For these ions, a smallenhancemenof the final-state entanglement of the system isserved even if one starts from the pure product s|Ψ0〉p=0 = |↓↑〉. This enhancement results from t‘spin–flip’ of the electrons due to magnetic interation with the radiation field which is known to favorbackwardemission of the photo-electron with respeto the incoming photons.

4. Summary and outlook

In conclusion, the photoionization of trapped hdrogen-like ions has been re-investigated in the frawork of density matrix theory with emphasis on tentanglementamong the electrons. For the phoionization of one out of two non-interacting ions,particular, we analyzed the creation and changeentanglement due to their interaction with the radtion field. In fact, the density matrix theory is founto provide an efficient tool for studying the (degrof) entanglement in the spin state of the ions, apfrom other properties such as the angular distribuor the polarization of the photo-emitted electrons. Dtailed calculations are performed for theK-shell pho-toionization of hydrogen and hydrogen-like Xe53+ andU91+ ions, based on Dirac’s equation and a relativistreatment of the radiation field.

Figs. 2 and 3display the angular (and energy) dpendence of the concurrence of the two electrontheir final state as well as its change, if the nuclcharge of the ions is increased. InFig. 4, moreover,the concurrence of the electrons is shown for dferent initial spin states including the product st|Ψ0〉 = |↓↑〉 as well as the (maximally) entangled Bestate|Ψ +〉 = 1/

√2(|↑↓〉 + |↓↑〉). While, in the non-

relativistic limit or for light elements, the entanglment is usually preserved in photoionization of o

80 T. Radtke et al. / Physics Letters A 347 (2005) 73–80

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of the atoms or ions, it changes if the nuclear chaincreases. For medium and high-Z elements, moreover, such a change in the concurrence of the sysarise not only from the relativistic contraction of thwave functions but also from the higher multipol(beyond the electric-dipole approach) in the electrophoton interaction. In the relativistic domain, thefore, the photoionization of non-interacting ions mlead to thecreationof maximally entangled electropairs even if they were initially in a spin state with nentanglement at all.

In the present contribution, we restricted ourvestigations to initiallypure spin state as well as thphotoionization of (one out of two) atoms by meanscircular-polarized light. Apart from such pure statthe density matrix theory enables one of courseexplore also the changes in the entanglement ifstarts—in a more realistic view—frommixedelectronstates and/or by usingpartially polarized light. Forsuch anincompleteknowledge of the initial systemthe theoretical analysis of the final-state entanglemis currently under way and will be presented elwhere.

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