Inner-shell ionization of rotating linear molecules in the presence of ...

Eur. Phys. J. Special Topics 169, 117–121 (2009) c EDP Sciences, Springer-Verlag 2009
DOI: 10.1140/epjst/e2009-00980-1

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Inner-shell ionization of rotating linear molecules in the presence of spin-dependent interactions: Entanglement between a photoelectron and an auger electron R. Ghosh, N. Chandraa , and S. Parida Department of Physics & Meteorology, Indian Institute of Technology, Kharagpur 721302, India

Abstract. This paper reports results of a theoretical study of angle- and spinresolved photo-Auger electron coincident spectroscopy in the form of entanglement between these two particles emitted from a linear molecule. First, we develop an expression for a density matrix needed for studying spin-entanglement between a photoelectron and an Auger electron. In order to properly represent the molecular symmetries, nuclear rotation, and the spin-dependent interactions (SDIs), we have used symmetry adapted wavefunctions in Hund’s coupling scheme (a) for all the species participating in this two-step process. This expression shows that spin-entanglement in a photo-Auger electron pair in the presence of SDIs very strongly depends upon, among other things, polarization of the ionizing radiation, directions of motion and of spin polarization of two ejected electrons, and the dynamics of photoionization and of Auger decay. We have applied this expression, as an example, to a generic linear molecule in its J0 , M0 = 0 state. This model calculation clearly brings out the salient features of the spin-entanglement of a photo-Auger electron pair in the presence of the SDIs.

1 Introduction We have investigated the well known and powerful technique of inner-shell photoionization, followed by the non-radiative decay of the consequent vacancy, for producing an EinsteinPodolsky-Rosen-Bohm [1,2] (EPRB) pair of electrons from a linear molecule with its both spindependent interactions (SDIs) and rotation of nuclei taken into account. This molecule belongs to either of the C∞v or D∞h point group. Two particles forming an EPRB pair are said [3] to be in an entangled state possessing the non-classical phenomena of quantum entanglement (QE). The QE plays important roles in many diverse areas of science. For example: the EPRB pairs of particles are a valuable and indispensable resource [4] for studies in the newly emerging field [3] of quantum information and computation; Sch¨ offer et al. [5] have shown that localization or delocalization of a core hole created in N2 in photoionization depends upon the way the entangled state formed by the photoelectron with the subsequently emitted Auger electron is detected in a measurement; QE has experimentally been shown [6] to be capable of affecting the macroscopic properties of systems in their solid phase and to lead to further insight in areas of physics like statistical mechanics and quantum field theory [7,8], etc. A recent investigation [9] has shown that, in the absence of SDIs, the entanglement in a photo-Auger electron pair can be generated only by the Coulombic forces experienced by these two escaping particles in side the molecule. A knowledge merely of the spin multiplicities of the a

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electronic states of a linear molecule, say AB, its excited photoion ∗AB + , and of the dication AB ++ formed in the subsequent non-radiative decay of the AB + is sufficient for complete characterization of this Coulombic quantum entanglement (CQE). On the other hand, the SDIs are known to greatly influence not only the properties of, but also the physical and chemical processes taking place in, any atomic or molecular system. For a linear molecule, spin-orbit and spin-rotation are the two most important SDIs [10]. Both of these interactions are properly taken into account in any theoretical framework if one is formulating a problem in Hund’s coupling scheme (a) [10]. In order to properly learn about the influences of these important interactions on the CQE [9] generated between a photoelectron and an Auger electron in their (i.e., SDIs’) absence, we have studied the QE properties of a photo-Auger electron pair emitted from a linear molecule by including both the rotational degrees of freedom and the SDIs in the Hund’s coupling schemes (a) [10]. The effects of various symmetries present in a rotating C∞v /D∞h molecule are also included in our study by representing each of the participating ∗ species (i.e., AB, AB + , AB ++ ) by its symmetry adapted wavefunction [10]. The entanglement properties of a photo-Auger electron pair are now shown to be totally different from those found earlier [9] when no SDIs were taken into account.

2 Theory and application 2.1 Theory The density operator

ρf = KFa Fp (ρr ⊗ ρ0 )(Fa Fp )†

(1)

simultaneously describes both the inner-shell photoionization ∗

γr (|1 mr ) + AB (|0) −→ AB + (|e) + e1 (|µ1 u ˆ 1 k1 )

(2a)

of a molecule AB and the non-radiative decay ∗

ˆ2 k2 ) , AB + (|e) −→ AB ++ (|f ) + e2 (|µ2 u

(2b)

∗

of its excited photoion AB + , leading to the emission of the photoelectron e1 and Auger electron e2 , respectively. In (1), K depends upon some fundamental constants, on the energies γr of the ionizing radiation in (2a) and that of the Auger electron e2 , but is totally independent of the dynamics of the processes (2). The operators Fp and Fa in (1) are for photoionization (2a) in the electric dipole (E1) approximation and for the spontaneous Auger emission (2b). Further in (1), ρr = |1 mr 1 mr | is the density operator of the ionizing radiation whose polarization is represented by mr [= 0 for plane polarization, +1 (−1) for positive (negative) helicities]; ρ0 ≡ |00| is the density operator of the molecule AB. Both, in ρ0 and in Eqs. (2), |0, |e, and ∗ |f  represent states of the AB, AB + , and of the dication AB ++ , respectively. Further in (2), ki (θi , φi ), with i = 1 and 2, is the propagation vector of the electron ei with its spin component µi along the direction u ˆi (ϑi , ϕi ). In a linear molecule, the SDIs include both spin-orbit plus spin-rotation interactions. In ∗ order to properly describe each of (AB, AB + , AB ++ ) in (2) by properly including the SDIs and the rotation of its nuclei, one needs to work with Hund’s coupling scheme (a) [10]. A symmetry adapted state of AB in this case is given by [10] 1 |0 = √ [|n0 Λ0 |J0 Ω0 M0 |S0 Σ0  + (−1)p0 |n0 − Λ0 |J0 − Ω0 M0 |S0 − Σ0 ] . 2

(3)

Here, J0 = N0 (= L0 +R0 )+S0 is the total angular momentum of the molecule AB, with L0 and S0 the electronic total orbital and total spin angular momenta, respectively, and R0 the nuclear ˆ Λ0 = L0 · R, ˆ Σ0 = S0 · R, ˆ rotational angular momentum. Further in the state (3), Ω0 = J0 · R, ˆ where R ˆ and R ˆ are, respectively, the direction of the line joining all the nuclei and M0 = J0 · R,

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in the linear molecule AB and the polar axis of a space fixed frame of reference with respect ˆ is defined. [R ˆ is taken to be along the electric field to which each of the vectors ki , u ˆi , and R vector of the linearly (mr = 0) or direction of propagation of circularly polarized (mr = ±1) or unpolarized radiation used in (2a) for ionization.] Finally [10] in (3), p0 is the parity of the state |0, and |J0 Ω0 M0  as well as |J0 − Ω0 M0  are the rotational states of the molecule AB in the Hund’s case (a). ∗ Expressions similar to (3) are applicable also to the states |e of the photoion AB + and ++ |f  of the dication AB , participating in the two-step process (2) described in the Hund’s coupling scheme (a), after replacing the subscript “0” with “e” and “f ”, respectively. Using these wavefunctions, we [11] obtain the following matrix for the density operator (1) 1 1 
S1   µ1 +µ2 2 2 f ; µ1 u ˆ1 k1 ; µ2 u ˆ2 k2 |ρf |f ; µ1 u ˆ1 k1 ; µ2 u ˆ2 k2  = (−1) µ1 −µ1 MS1 S1 MS1 NS1 S2 MS2 NS2  1 1 ∗  ∗ S2 S1 S2 2 2 (ω ) (ω ) ASN1SS2NS (mr ; k1 k2 ). (4a) D D 1 2 MS1 NS1 MS2 NS2 1 2 µ2 −µ2 MS2 Here, D’s are the rotational harmonics [12] with their arguments given by the Euler angles ωi = (ϕi , ϑi , 0) and K ASN1SS2NS (mr ; k1 ; k2 ) = (−1)1+mr +J0 +2Je +Jf (2S1 + 1)(2S2 + 1)(2Je + 1)(2Jf +1) 1 2 4π(2 − δ0 Λ0 )
   (−1)1 +2 −j2+jt (2J1 + 1)(2J2 + 1)(2Lr + 1) (2L1 + 1)(2L2 + 1) p0 pf 1 1 2 2 jt jt L1 ML1 L2 pe pe j1 j1 j2 j2 J1 J2 Lr Q ML2      

2 2 L2 L1 S1 J1 L2 S2 J2 1 1 Lr

1 1 L1 0 0 0 0 0 0 ML1 NS1 Q ML2 NS2 −Q mr −mr 0         1 1 L1  2 2 L2 j1 j  J1  1 J1 J2 Lr Je Je J2 Je Je J2 1 1 1 1 j j J S S Q −Q 0 jt jt J0 j2 j2 Jf  2 2 1  2 2 2  t t 2  1 1 Lr j1 j1 J1 j2 j2 J2  ∗  ∗ M M ∗ YL2 L2 (kˆ2 ) pe |Je ; 1 j1 |F (jt )|p0 J0 ; 1pe Je ; 1 j1 |F (jt )|p0 J0 ; 1 YL1 L1 (kˆ1 ) (a)

(a)  (pf 2 j2

A2 j2 (pf Λf Σf Ωf Jf ; pe Λe Σe Ωe Je ) A

Λf Σf Ωf Jf ; pe Λe Σe Ωe Je )∗

(4b)

The photoionization matrix elements pe Je ; 1 j1 |F (jt )|p0 J0 ; 1 and the Auger decay matrix (a) elements A2 j2 (pf Λf Σf Ωf Jf ; pe Λe Σe Ωe Je ), needed in (4b), are defined by the expressions [(A4), (A5)] in Reference [13] and Eq. (8) in [13], respectively. The QE of (e1 , e2 ), ejected during the processes (2), is to be determined by using the density matrix (DM) (4). This (4 × 4) DM has, obviously, a very complicated structure. The inclusion of SDIs makes the spin-entanglement between (e1 , e2 ) to be very much dependent on both the kinematics and dynamics of the two-step process in Eq. (2). That is, now the QE is affected, among other things, by the energy γr and polarization mr of light used for ionizing AB in the step (1a) and by the linear momenta (k1 , k2 ) of the ejected (e1 , e2 ). More importantly, the entanglement properties of a photo-Auger electron pair in the presence of SDIs can not be studied at all without a knowledge of the photoionization and Auger matrix elements present in Eqs. (4b). These properties of (4) are completely different from those of the DM derived in [9] without taking any SDIs into account. Among the most significant consequences of the differences in the structure of the DM (4) obtained herein and that derived in [9] are that inclusion of the SDIs in the present case makes it impossible to (i) a priory predict the QE between (e1 , e2 ) and (ii) detect it in a laboratory simply by measuring energies of the photoelectron e1 and of the Auger electron e2 , without using any of the protocols

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(e.g., entanglement witness [14]) hitherto developed in the quantum information theory. Now one has to have, among other things, the complete dynamical information about both of the steps of the process (2) before being able to say any thing at all about the QE properties of (e1 , e2 ); whereas, none of such information is needed in the case of CQE [9] generated between (e1 , e2 ) in the absence of SDIs. 2.2 Applications Let us consider an application of the theoretical frame-work developed herein to the following generic example ∗

γr + AB (|p0 , J0 = 0, M0 ) −→ AB + (|pe , Je = 1/2, Me ) + e1 with

∗

AB + (|pe , Je = 1/2, Me ) −→ AB ++ (|pf , Jf = 0, Mf ) + e2

(5a) (5b)

Here, (p0 , pe , pf ) and (J0 , Je , Jf ) are, respectively, the parities and total angular momenta ∗ of the states (|0, |e, |f ) of (AB, AB + , AB ++ ) participating in the two-step process of ˆ Each of the Eq. (5); whereas, (M0 , Me , Mf ) are the projections of (J0 , Je , Jf ) along R. photoelectron e1 and the Auger electron e2 can come out from their respective parents AB ∗ and AB + in an infinite number of possible directions (kˆ1 , kˆ2 ). For simplicity, let us select an experimental configuration for the diametric emission of (e1 , e2 ), i.e., photoelectron and Auger electron are receding from the residual dication AB ++ in Eqs. (5) in opposite directions [i.e., ˆ φ) with (θ1 = θ, φ1 = φ), then kˆ2 (θ2 , φ2 )  −k(θ, ˆ φ) with (θ2 = π − θ, if kˆ1 (θ1 , φ1 )  k(θ, φ2 = π + φ)]. Further, the spins of each of the photoelectron e1 and the Auger electron e2 can be quantized in all possible directions in space, quite independently of each other. But, entanglement is well known [3] to be independent of the local choice of the basis and coordinate system. Hence the choice of the spin quantization directions (ˆ u1 , u ˆ2 ) of (e1 , e2 ) should not affect the QE between these two electrons. In the present study, we have considered, for simplicity, that each of the photoelectron and the Auger electron has its spin quantized longitudinally ˆ i.e., (ϑ1 = θ, ϕ1 = to its respective direction of free motion in space. Namely, u ˆ1  kˆ1  k, ˆ ˆ φ); u ˆ2  k2  −k, i.e., (ϑ2 = π − θ, ϕ2 = π + φ). It, in turn, implies, ω1 = (φ, θ, 0), and ω2 = (π + φ, π − θ, 0) for the two sets of Euler angles [12] to be used in (4a). The choice of this experimental geometry for (ki , u ˆi ) means that the final form of DM (4) will contain only two ˆ φ). The ionizing radiation γr in (5a) has been considered by angles specifying the direction k(θ, us [11] to be lineally polarized (LP), circularly polarized (CP) with positive or negative helicity, or unpolarized (UP). Herein, we present our results for LP incident light only. The three measures of bipartite entanglement, currently in vogue, are [14] negativity N , concurrence C, and entanglement of formation EF . Each of these three measures can take values between zero and one: N , C, EF = 0, if (e1 , e2 ) are not entangled; N , C, EF = 1 for maximum possible entanglement of (e1 , e2 ). Fig. 1 shows our results when the ionizing radiation γr in the generic example (5) is LP. The entanglement here is, obviously, highly anisotropic: A photoAuger electron pair is in a maximally entangled state when (e1 , e2 ) are observed in opposite directions either parallel (i.e., θ = 0, π) or perpendicular (i.e., θ = π/2) to the electric field vector of the LP ionizing radiation; (e1 , e2 ), on the other hand, are barely entangled in the directions θ  π/4, 3π/4. An analysis of the DM obtained for this example showed that (e1 , e2 ) are always in a pure state whatever may be their direction of motion. It has been shown [14] that N = C for pure bipartite states. Results in the Fig. 1 verifies this theoretical prediction. Our calculations [11] for CP or UP ionizing radiation showed that the variations of N , C, EF with θ were very different not only from that shown in Fig. 1 herein for LP light, but also from each other. However, each of these three measures was found to have same values for CP radiation with positive or negative helicity, i.e., entanglement in the present example of process (5) does not possess any circular dichroism. Also, while (e1 , e2 ) formed a pure state for γr to be either LP or CP, but these two electrons are in a mixed state when the ionizing radiation in (5) is taken to be UP.

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Fig. 1. Variation of negativity N , concurrence C, and entanglement of formation EF with the angle θ which the line joining oppositely moving photoelectron and Auger electron makes with the electric field vector of the linearly polarized ionizing radiation used in the two-step generic process (5).

3 Conclusions Photo-Auger electron entanglement provides an alternative approach with a new observable for studying the fundamental process of inner-shell ionization followed by the non-radiative decay of the consequent vacancy in an atom [15] or molecule. A recent experiment [5] has shown that QE between (e1 , e2 ) is needed for studies not only in the field of quantum information but also for probing the fundamental properties of atoms and molecules. The present study, combined with our earlier investigation [9], develops a complete frame-work and provides a methodology for theoretical calculations and experimental measurements of spin-entanglement in a photoAuger electron pair both without or with SDIs taken into account. These studies show, among many other things, that one can produce (e1 , e2 ) with a tunable degree of spin-entanglement. This work was supported, in part, by the Council of Scientific & Industrial Research, New Delhi, India, under grant number 03(1033)/05/EMR-II.

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