**2. Preliminaries**

#### **2.1 Density operator**

Let's use *e*<sup>1</sup> and *e*<sup>2</sup> to symbolize the two freely moving electrons whose entanglement characteristics we are interested in. The *i*(=1, 2)th electron's propagation vector is *k* ! *<sup>i</sup>* ¼ *ki*, *θi*, *ϕ<sup>i</sup>* ð Þ, and its kinetic energy is provided by *<sup>ε</sup><sup>i</sup>* <sup>¼</sup> <sup>ℏ</sup><sup>2</sup> *k*2 *<sup>i</sup> =*2*m:* These two electrons are a crucial component of atom *A* and are supposed to be simultaneously released from it after the absorption of a single photon. *μi*ð Þ ¼ �1*=*2 denotes the projection of the spin angular momentum of the *<sup>i</sup>*th electron along its spin quantization direction *<sup>u</sup>*^*<sup>i</sup>* <sup>¼</sup> *<sup>α</sup>i*, *<sup>β</sup><sup>i</sup>* ð Þ. If *<sup>A</sup>*2+ denotes the residual dication, then our process can schematically be represented by

*Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

$$h\nu\_r(|l\_r| = 1, m\_r) + A|\mathbf{0}\rangle \to A^{2+}|f\rangle + e\_1(\overrightarrow{k}\_1; \mu\_1\hat{u}\_1) + e\_2(\overrightarrow{k}\_2; \mu\_2\hat{u}\_2). \tag{1}$$

Here, in the electric dipole (*E*1) approximation, *Er* = *hν<sup>r</sup>* and *lr* j j¼ 1 represent, respectively, the energy and angular momentum of the photon absorbed by atom *A*. The parameter *mr* stands for the photon's state of polarization, where 0j i and j i*f* , respectively, represent the bound electronic states of *A* with energy *E*<sup>0</sup> and the remaining doubly charged photoion *A*2+ with energy *Ef*. The direction of the electric vector of the linearly polarized (*mr* = 0) radiation involved in the process (1) defines the quantization axis of our space (or photon) frame of reference; if the ionizing radiation is circularly polarized (*mr* = �1) or unpolarized, the direction of incidence then determines the polar axis of the photon frame.

Let us use *ρ<sup>r</sup>* ¼ *mr* j ih j *mr* and *ρ*<sup>0</sup> ¼ j i 0 h j 0 to represent the density operators of the ionizing radiation and the unpolarized atom A before DPI, respectively. Before the interaction between the two occurs, the incident photon and the atom are uncorrelated. This indicates that the direct product determines the density operator for the combined (atom + photon) system of Eq. (1).

$$
\rho\_i = \rho\_0 \otimes \rho\_r \tag{2}
$$

Let us denote by *Fp* the photoionization operator in the *E*<sup>1</sup> approximation. Then the density operator of the combined (*A*2+ + *e*<sup>1</sup> + *e*2) system in Eq. (1) after DPI becomes

$$
\rho\_f = K\_p F\_p \rho\_i F\_p^+. \tag{3}
$$

The *E*<sup>1</sup> photoionization operator *Fp* has been defined in Appendix A. Here, *Kp* <sup>¼</sup> <sup>3</sup>*<sup>π</sup> <sup>e</sup>*<sup>2</sup> ð Þ *<sup>=</sup>α*0*Er* <sup>2</sup> with *α*<sup>0</sup> the dimensionless fine structure constant [20].

### **2.2 Definitions of** *E***<sup>1</sup> photoionization operator and concurrence**

In the present case, *E*<sup>1</sup> photoionization operator *Fp* for *ne*-electron system can be defined as [20]:

$$F\_p = \sqrt{\frac{4\pi a^3 E\_r^3 m\_e}{4\epsilon^4 \hbar^2}} \sum\_{i=1}^{n\_e} \hat{\xi}\_{m\_r} \cdot \vec{r}\_i,\tag{4}$$

and

$$F\_p = \sqrt{\frac{4\pi\alpha^3 E\_r^3 a\_0^2 m\_e}{3\hbar^2}} \sum\_{i=1}^{n\_e} \hat{\xi}\_{m\_r} \vec{\nabla}\_i. \tag{5}$$

The operators of Eqs. (4) and (5) represent the interaction of the atomic electrons (their number being *ne*) with the incident electromagnetic radiation in the *E*<sup>1</sup> length and velocity approximations, respectively. In Eq. (4), *r* ! *<sup>i</sup>* is the position vector and in Eq. (5), ∇ ! *<sup>i</sup>* ¼ �ð Þ<sup>1</sup> <sup>1</sup>*=*<sup>2</sup> *p* ! *i =*ℏ h i is the linear momentum of *<sup>i</sup>*th bound atomic electron. Here ^*ξmr* is the spherical unit vector [21] in the direction of polarization of the incident.

The *concurrence* (*C*) is a very successful and widely used measure for quantifying quantum entanglement between two qubits. It is an additive and operational measure of entanglement. Additivity is a very desirable property that can reduce calculational complexity of entanglement [4, 17, 22]. For any state j i *<sup>ϕ</sup> AB* in a *<sup>d</sup>* <sup>⊗</sup> *<sup>d</sup>*<sup>0</sup> *<sup>d</sup>*≤*d*<sup>0</sup> � � quantized system, it can be written as:

$$\mathbf{C} = \sqrt{2(\mathbf{1} - \mathbf{T} \mathbf{r} \mathbf{e} \boldsymbol{\sigma}\_A^2)},\tag{6}$$

where *<sup>ρ</sup><sup>A</sup>* is the reduced density matrix defined as *<sup>ρ</sup><sup>A</sup>* <sup>¼</sup> *traceB* j i *<sup>ϕ</sup> AB*h j *<sup>ϕ</sup>* � �. When 0≤*C*≤1 always; *C* = 0 for a separable (unentangled) state; *C* > 0 for a nonseparable (entangled) state.

#### **3. Entanglement between two electronic qubits for DPI**

Here, we calculate the DM for the angle- and spin-resolved DPI of an atom without considering SOI into account in either the bound electronic states of *A* and *A*2+ or the continua of the two photoelectrons (*e*1, *e*2) ejected in the process (1). Here LS-coupling is applicable. Therefore, the total orbital angular momenta (*L* ! 0, *L* ! *<sup>f</sup>* ) and the spin angular momenta (*S* ! 0, *S* ! *<sup>f</sup>* ) of *A* and *A*2+ are conserved quantities. If the orbital angular momentum of the photoelectron *e*<sup>1</sup> is *l* ! <sup>1</sup> and that of *e*<sup>2</sup> is *l* ! <sup>2</sup> with their respective spin angular momenta <sup>1</sup> 2 � � <sup>1</sup> and <sup>1</sup> 2 � � <sup>2</sup>, we then have

$$
\overrightarrow{L}\_0 + \overrightarrow{l}\_r = \overrightarrow{L}\_f + \overrightarrow{l} \left(= \overrightarrow{l}\_1 + \overrightarrow{l}\_2\right) \tag{7}
$$

and

$$
\overrightarrow{\mathcal{S}}\_0 = \overrightarrow{\mathcal{S}}\_f + \overrightarrow{s}\_t \left(= \left(\frac{1}{2}\right)\_1 + \left(\frac{1}{2}\right)\_2\right). \tag{8}
$$

Here *ML*<sup>0</sup> , *MS*<sup>0</sup> , *MLf* , and *MSf* are the respective projections of *L* ! 0, *S* ! 0, *L* ! *<sup>f</sup>* , and *S* ! *f* along the polar axis of the space frame, then, in Eq. (1), the bound electronic state of atom *<sup>A</sup>* is 0j i � *<sup>L</sup>*0*S*0*ML*0*MS*<sup>0</sup> j i and that of the dication *<sup>A</sup>*2+ is j i*<sup>f</sup>* � *Lf SfMLf MSf* � � � E . The density operator (2) for the combined (atom + photon) system becomes

$$\rho\_i = \frac{1}{(2L\_0 + 1)(2S\_0 + 1)} \sum\_{M\_{\mathbb{N}\_0}M\_{\mathbb{S}\_0}} |0; 1m\_r\rangle\langle 0; 1m\_r|,\tag{9}$$

here we have defined 0; 1*mr* j i � j i 0 1*mr* j i*:*

In order to calculate the DM for the (*A*2++*e*1+*e*2) system in process (1), we now calculate the matrix elements *ρ<sup>i</sup>* and *ρ<sup>f</sup>* . Following the procedures given in Ref. [15], the matrix elements in the present case are given by

*Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

$$
\left\langle \vec{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1, \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| \rho\_f \middle| \hat{f}; \vec{k}\_1, \mu\_1' \hat{u}\_1, \vec{k}\_2, \mu\_2' \hat{u}\_2 \right\rangle = \frac{K\_p}{(2L\_0 + 1)(2S\_0 + 1)}
$$

$$
\sum\_{M\_{\rm M\_0} M\_{\rm S\_0}} \left\langle \hat{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1, \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| F\_p \middle| 0; 1 \mathfrak{m}\_r \right\rangle \ge \left\langle \hat{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1, \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| F\_p \middle| 0; 1 \mathfrak{m}\_r \right\rangle^\*. \tag{10}
$$

As the DM in Eq. (10) is Hermitian, we can write

$$\begin{aligned} & \left< \vec{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1; \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| \rho\_f \middle| f; \vec{k}\_1, \mu\_1' \hat{u}\_1; \vec{k}\_2 \mu\_2' \hat{u}\_2 \right> \\ &= \left< \vec{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1; \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| \rho\_f \middle| f; \vec{k}\_1, \mu\_1 \hat{u}\_1; \vec{k}\_2 \mu\_2 \hat{u}\_2 \right>^\* \end{aligned} \tag{11}$$

Next, we evaluate the matrix elements of *E1* photoionization operator *Fp* occurring on the right-hand side of Eq. (10). To this end, we first introduce the coupling suggested by Eqs. (7) and (8) in each of the bras and kets used to calculate the matrix elements of *Fp*. We therefore can write

$$|0,1m\_r\rangle = \sum\_{LM\_L} (-1)^{1-L\_0-M\_L} \sqrt{2L+1} \begin{pmatrix} L\_0 & 1 & L \\ M\_{L\_0} & m\_r & -M\_L \end{pmatrix} |(L\_0 \mathbf{1})LM\_L; \mathbf{S}\_0 M\_{\mathbf{S}\_0}\rangle \tag{12}$$

and

*f*; ^ *<sup>k</sup>*1, *<sup>μ</sup>*1*u*^1; ^ *k*2*μ*2*u*^<sup>2</sup> � � � E ¼ �ð Þ<sup>1</sup> �*Lf* �*Sf* X *ℓ*1*ℓ*2*ℓ n*1*n*2*N* X *ν*1*ν*<sup>2</sup> *stν* X *LTMLT SMS i <sup>ℓ</sup>*1þ*ℓ*2*ℯ*�*<sup>i</sup> <sup>σ</sup>ℓ*1þ*σℓ*<sup>2</sup> ð Þð Þ �<sup>1</sup> �*ℓ*1þ*ℓ*2þ*ℓ*þ*st*�*N*�*ν*�*MLT* �*MS x* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup>*<sup>ℓ</sup>* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*st* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*LT* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*<sup>S</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *<sup>ℓ</sup>*<sup>1</sup> *<sup>ℓ</sup>*<sup>2</sup> *<sup>ℓ</sup> m*<sup>1</sup> *m*<sup>2</sup> � *m* � � 1 2 1 <sup>2</sup> *st ν*<sup>1</sup> *ν*<sup>2</sup> � *ν* ! *Lf ℓ LT MLf N* � *MLT* ! *x Sf st S MSf ν* � *MS* !*Yn*∗ 1 *ℓ*1 ^ *k*1 � �*Y<sup>n</sup>*<sup>∗</sup> 2 *ℓ*2 ^ *k*1 � �*<sup>D</sup>* 1 2 ∗ *<sup>μ</sup>*1*ν*<sup>1</sup> ð Þ *ω*<sup>1</sup> *D* 1 2 ∗ *<sup>μ</sup>*2*ν*<sup>2</sup> ð Þ *<sup>ω</sup>*<sup>2</sup> *Lf <sup>ℓ</sup>* � �*LTMLT* ; *Sf st* � �*SMS* � � � *:* (13)

Here, *σ<sup>ℓ</sup>*<sup>1</sup> and *σ<sup>ℓ</sup>*<sup>2</sup> are the Coulomb phases for *l*1th and *l*2th partial waves of the photoelectrons, respectively; *Ds* are the well-known rotational harmonics [23] with *ω*<sup>1</sup> *α*1, *β*<sup>1</sup> ð Þ , 0 and *ω*<sup>2</sup> *α*2, *β*<sup>2</sup> ð Þ , 0 , the Euler angles that rotate the axis of the space-frame into the spin-polarization directions *u*^<sup>1</sup> and *u*^<sup>2</sup> (**Figure 1**), respectively. Furthermore, in (13), the properly antisymmetrized and asymptotically normalized [24] ket *Lfℓ* � �*LTMLT* ; *Sf sy* � �*SMS* � � � represents the photoion A2+ in its electronic state and the two photoelectrons with their total orbitaland spin angular momenta coupled according to the scheme expressed in Eqs. (7) and (8).

Now we substitute in Eq. (9) the normalized condition of reference [25]

$$\begin{aligned} & \left< \left( \mathbf{L}\_f \boldsymbol{\varepsilon} \right) \mathbf{L}\_T \mathbf{M}\_{L\_T} ; \left( \mathbf{S}\_{\boldsymbol{f}} \mathbf{s}\_t \right) \mathbf{S} \mathbf{M}\_S \left| \boldsymbol{F}\_p \right| \left( \mathbf{L}\_0 \mathbf{1} \right) \mathbf{L} \mathbf{M}\_L ; \mathbf{S}\_0 \mathbf{M}\_{\mathbf{S}\_0} \right> \\ & \quad = \delta\_{\mathrm{L} \boldsymbol{L}\_T} \delta\_{\mathrm{M}\_{\mathrm{L} \boldsymbol{M} \boldsymbol{L}\_T}} \delta\_{\mathrm{S}\_0 \boldsymbol{S}} \delta\_{\mathrm{M}\_{\mathbf{S}\_0} \mathbf{M}\_S} \left< \left( \mathbf{L}\_{\boldsymbol{f}} \boldsymbol{\varepsilon} \right) \mathbf{L} \left| \boldsymbol{F}\_p \right| (\mathbf{L}\_0 \mathbf{1}) \boldsymbol{L} \right>. \end{aligned}$$

*Quantum Dots - Recent Advances, New Perspectives and Contemporary Applications*

This result arises from the conservation circumstances in (7) and (8).

By using Racah algebra to analytically evaluate as many of the sums that are contained there, the following subsequent equation is made simpler. It necessitates, for instance, the application of (a) the addition theorems (i.e., Eqs. (4.3.2) and (4.6.5) from [23] for rotational and spherical harmonics, (b) Eq. (6.2.5) [23] for transforming a single sum of the product of three 3-j symbols into a product of one 3-j and one 6-j symbols, (c) identity (5) given on page 453 in Ref. [26] for converting a product of two 3-j symbols and one 6-j symbol summed over two variables into a product of two 3-j and one 6-j symbols, (d) Eq. (14.42) from [27] that transforms a quadruple sum of the product of four 3-j symbols into a double sum containing two 3-j and one 9-j symbols, (e) Eq. (3.7.9) [23] to convert a phase factor into a 3-j symbol, (f) the orthogonality of 3-j symbols (3.7.7) [23], (g) relation (6.4.14) [23] to write a 9-j symbol in terms of a 6-j symbol, and (h) relation (6.4.14) [23] to turn the sum of the products of two 6-j symbols into one 6-j symbol. Due to these and other simplifications, the DM (10) is written as follows:

*Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

*f*; *k* ! 1, *μ*1*u*^1; *k* ! 2, *μ*2*u*^<sup>2</sup> *ρ<sup>f</sup>* � � � � � �*f*; *<sup>k</sup>* ! 1, *μ*<sup>0</sup> <sup>1</sup>*u*^1; *k* ! 2*μ*0 <sup>2</sup>*u*^<sup>2</sup> D E <sup>¼</sup> ¼ *Kp* 4*π*ð Þ 2*L*<sup>0</sup> þ 1 X *l*1*l*2*l l* 0 1*l* 0 2*l* 0 X *L*1*L*2*M LL*0 *Lr* X *sQm*1*m*2*n* ð Þ �<sup>1</sup> *<sup>l</sup>*01þ*l*02þ*l*0þQ�*n*þ*μ*01þ*μ*02þ*M*0*Sf* <sup>2</sup>*Sf* <sup>þ</sup> <sup>1</sup> � �ð Þ 2L*<sup>r</sup>* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*<sup>s</sup>* <sup>þ</sup> <sup>1</sup> x 2ð Þ *<sup>Q</sup>* <sup>þ</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup>*L*<sup>1</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*L*<sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> *l*<sup>1</sup> *l* 0 <sup>1</sup> *L*<sup>1</sup> 00 0 0 @ 1 A *l*<sup>2</sup> *l* 0 <sup>2</sup> *L*<sup>2</sup> 00 0 0 @ 1 A 1 1 *Lr mr* � *mr* 0 0 @ 1 A *L*<sup>1</sup> *L*<sup>2</sup> *Lr M* � *M* 0 0 @ 1 A x 1 1 *Lr* L<sup>0</sup> L L0 8 < : 9 = ; *l l*<sup>0</sup> *Lr L*<sup>0</sup> *L Lf* 8 < : 9 = ; *l*<sup>1</sup> *l*<sup>2</sup> *l l* 0 <sup>1</sup> *l* 0 <sup>2</sup> *l* 0 *L*<sup>1</sup> *L*<sup>2</sup> *Lr* 8 >>>>< >>>>: 9 >>>>= >>>>; 1 2 1 <sup>2</sup> *<sup>s</sup> μ*<sup>1</sup> � *μ*<sup>0</sup> <sup>1</sup> *m*<sup>1</sup> 0 BB@ 1 CCA 1 2 1 <sup>2</sup> *<sup>s</sup> μ*<sup>2</sup> � *μ*<sup>0</sup> <sup>2</sup> *m*<sup>2</sup> 0 BB@ 1 CCA *Sf Sf Q MSf* � *M*<sup>0</sup> *Sf n* 0 B@ 1 CA x *s sQ m*<sup>1</sup> *m*<sup>2</sup> *n* 0 @ 1 A 1 2 1 <sup>2</sup> *<sup>s</sup> Sf Sf Q* 8 >>< >>: 9 >>= >>; *Y<sup>M</sup> L*1 ^ *k*1 � �*Y*�*<sup>M</sup> L*2 ^ *k*2 � �*Ds* <sup>∗</sup> *<sup>m</sup>*1*<sup>n</sup>*ð Þ *<sup>ω</sup>*<sup>1</sup> *<sup>D</sup><sup>s</sup>* <sup>∗</sup> *<sup>m</sup>*2�*<sup>n</sup>*ð Þ *<sup>ω</sup>*<sup>2</sup> <sup>x</sup> *Lf lF L* j j ð Þ *<sup>L</sup>*<sup>01</sup> � � *Lf <sup>l</sup>* <sup>0</sup> *F L*<sup>0</sup> j j ð Þ *<sup>L</sup>*<sup>01</sup> � � <sup>∗</sup> (14)

with

$$\sqrt{L\_{\hat{f}}}l|F(\mathbf{L})|L\_{\mathbf{0}}\mathbf{1}\rangle = (-i)^{l\_{1}+l\_{2}}e^{-i\left(\sigma\_{l\_{1}}+\sigma\_{l\_{2}}\right)}(2L+1)\sqrt{(2l\_{1}+1)(2l\_{2}+1)(2l+1)}\langle\{L\_{\hat{f}}l\}\mathbf{L}|F\_{\mathbf{p}}|(L\_{\mathbf{0}}\mathbf{1})L\_{\mathbf{0}}\mathbf{1}\rangle.$$

From Eq. (12), the DM for the angle- and spin-resolved DPI process (1) in the absence of SOI can be written in the following form:

$$\left\langle \vec{f}; \vec{k}\_1, \mu\_1 \hat{u}\_1, \vec{k}\_2, \mu\_2 \hat{u}\_2 \middle| \rho\_{\hat{f}} \middle| \hat{f}; \vec{k}\_1, \mu\_1' \hat{u}\_1, \vec{k}\_2, \mu\_2' \hat{u}\_2 \right\rangle = \frac{d^3 \sigma(m\_r)}{d \varepsilon\_1 d \hat{k}\_1 d \hat{k}\_2} \sigma \left( \mathbf{S}\_0; \mathbf{S}\_{\hat{f}}; \hat{u}\_1, \hat{u}\_2 \right)\_{\mathbf{M}\_{\hat{f}} \mu\_1 \mu\_2; M\_{\mathbb{S}\_{\hat{f}}}' \mu\_1' \hat{u}\_1} \tag{15}$$

The first term, that is, the triple differential cross section (TDCS, i.e., *d*3 *<sup>σ</sup>*ð Þ *mr <sup>=</sup>dε*1*d*^ *k*1*d*^ *k*2) on the right-hand side of (15) depends upon the orbital angular momenta of *A* and *A*2+; phase shifts, energies (*ε*1, *ε*2), and the directions (^ *k*1, ^ *k*2) of the emitted electrons (*e*1, *e*2); the state of polarization (*mr*) of the ionizing radiation; and the photoionization dynamics. It does not include spins of the photoelectrons or the target atom or the residual dication. Thus, *d*<sup>3</sup> *<sup>σ</sup>*ð Þ *mr <sup>=</sup>dε*1*d*^ *k*1*d*^ *k*<sup>2</sup> in the DM (15) describes purely the angular correlation between the photoelectrons in the L-S coupling scheme for the angular momenta of the particles involved in DPI (1). Its value is always positive. Here,

*d*3 *σ*ð Þ *mr dε*1*d*^ *k*1*d*^ *k*2 ¼ �ð Þ<sup>1</sup> *mr*þ*L*0þ*Lf Kp* 4*π*ð Þ 2*L*<sup>0</sup> þ 1 X *l*1*l*2*l l* 0 1*l* 0 2*l* 0 X *L*1*L*2*M LL*0 *Lr* ð Þ �<sup>1</sup> *<sup>l</sup>*01þ*l*02þ*l*<sup>0</sup> ð Þ 2L*<sup>r</sup>* þ 1 x ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup>*L*<sup>1</sup> <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*L*<sup>2</sup> <sup>þ</sup> <sup>1</sup> <sup>p</sup> *<sup>l</sup>*<sup>1</sup> *<sup>l</sup>* 0 <sup>1</sup> *L*<sup>1</sup> 00 0 ! *<sup>l</sup>*<sup>2</sup> *<sup>l</sup>* 0 <sup>2</sup> *L*<sup>2</sup> 00 0 ! 1 1 *Lr mr* � *mr* 0 ! *L*<sup>1</sup> *L*<sup>2</sup> *Lr M* � *M* 0 ! 1 1 *Lr* L<sup>0</sup> L L0 ( ) <sup>x</sup> *l l*<sup>0</sup> *Lr L*<sup>0</sup> *L Lf* ( ) *l*<sup>1</sup> *l*<sup>2</sup> *l l* 0 <sup>1</sup> *l* 0 <sup>2</sup> *l* 0 *L*<sup>1</sup> *L*<sup>2</sup> *Lr* 8 >>< >>: 9 >>= >>; *Y<sup>M</sup> L*1 ^ *k*1 � �*Y*�*<sup>M</sup> L*2 ^ *k*2 � � *Lf lF L* j j ð Þ *<sup>L</sup>*<sup>01</sup> � � *Lf <sup>l</sup>* <sup>0</sup> *<sup>F</sup>* <sup>L</sup><sup>0</sup> j j ð Þ *<sup>L</sup>*<sup>01</sup> � � <sup>∗</sup> *:* (16)

The second term (i.e., *σ S*0; *Sf* ; *u*^ � � *MSf μ*; *M*<sup>0</sup> *Sf <sup>μ</sup>*0) present on the right-hand side of Eq. (15) is the spin-correlation density matrix (SCDM), which completely determines the entanglement properties among electronic qubit (*ep*) and ionic qudit *M*<sup>þ</sup> ð Þ, can be written as:

$$
\sigma\_{\star\_{1r\_2}} = (-1)^{\mu\_1 + \mu\_2} \sum\_{\substack{s, m, m \neq n}} (-1)^{\mu\_1 - n} (2s+1)
$$

$$
\mathbf{x} \begin{pmatrix} \mathbf{1} & \mathbf{1} & \mathbf{s} \\ \mathbf{2} & \mathbf{2} & \mathbf{s} \\ \mu\_1 - \mu\_1' & m\_1 \end{pmatrix} \begin{pmatrix} \mathbf{1} & \mathbf{1} & \mathbf{s} \\ \mathbf{2} & \mathbf{2} & \mathbf{s} \\ \mu\_2 - \mu\_2' & m\_2 \end{pmatrix} \begin{Bmatrix} \mathbf{1} & \mathbf{1} & \mathbf{s} \\ \mathbf{2} & \mathbf{2} & \mathbf{s} \\ \mathbf{1} & \mathbf{1} & s\_l \end{Bmatrix} D\_{m,n}^{\mathbf{v}^\*}(o\_1) D\_{m,-n}^{\mathbf{v}^\*}(o\_2) \; . \tag{17}
$$

Considering the condition given by Eq. (8), we see that *Sf* � *S*<sup>0</sup> � � � � can take only two values: 0 and 1. Let us consider the entanglement between two electrons *e*<sup>1</sup> and *e*<sup>2</sup> in both cases. In order to calculate the SCDMs, we have to consider the real part of the outgoing wave function [28]. Before writing the DM, we first consider

$$a = a\_1 - a\_2,\\ c\_1 = \cos \beta\_1, \ s\_1 = \sin \beta\_1, \ c\_2 = \cos \beta\_2, \ s\_2 = \sin \beta\_2, \ \text{and } c = \cos a$$

i. Let us first consider the case *Sf* � *S*<sup>0</sup> � � � � = 0. We obtain the following SCDM from (17):

$$
\sigma\_{\circ\_{\circ\_2}} = 
$$


#### **Table 1.**

*SCDM of qubit-qubit system for the case Sf* � *S*<sup>0</sup> � � � � *= 0.*

We calculate from **Table 1** that the det (*σ<sup>e</sup>*<sup>1</sup> *<sup>e</sup>*<sup>2</sup> ) = � <sup>1</sup> <sup>16</sup> cos <sup>2</sup>*α*. In order to quantifying entanglement, we have obtained concurrence, using SCDM of **Table 1**, according to the definition given in Eq. (6), which yields

*Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

#### **Figure 2.**

*Variation of concurrence for the case Sf* � *S*<sup>0</sup> � � � � *= 0 with respect to spin quantization directions* α*<sup>1</sup> and* α*2.*

$$C = \sqrt{1 - \frac{1 + \left(\cos a\_1 \cos a\_2 + \cos a \sin a\_1 \sin a\_2\right)^2}{16384}}.\tag{18}$$

Eq. (18) means [4, 17, 22] that the values of *C* are positive and so the spin state *Sf* � *S*<sup>0</sup> � � � � = 0 is entangled, depending on *α*<sup>1</sup> and *α*2*:* This variation of concurrence is shown in **Figure 2**.

We can see from **Figure 2** that the states are entangled (the values of concurrence are positive) and separable (for the zero value of concurrence) depending on spin quantization directions.

ii. Considering the case of *Sf* � *S*<sup>0</sup> � � � � = 1, we procure the following SCDM from Eq. (17):

$$
\sigma\_{\circ\_{\mathfrak{a}\_2}} = 
$$


#### **Table 2.**

*SCDM of qubit-qubit system for the case Sf* � *S*<sup>0</sup> � � � � *= 1.*

#### According to **Table 2**, we thus obtain

$$\begin{aligned} \det\left(\sigma\_{\epsilon\_{1c\_2}}\right) &= \frac{1}{576} \left( 140 + 8\cos 2a\_1 - 4\cos 4a\_1 + \cos\left(a\_1 - 5a\_2\right) - 8\cos 2(a\_1 - 2a\_2) \right) \\ &+ 8\cos\left(4a\_1 - 2a\_2\right) + \cos\left(5a\_1 - a\_2\right) - 8\cos a - 120\cos 2a - 7\cos 3a - 4\cos 4a - \cos 5a \\ &- 8\cos 2a\_2 - 4\cos 4a\_2 - 8\cos 2(a\_1 + a\_2) + 7\cos\left(3a\_1 + a\_2\right) + 7\cos\left(a\_1 + 3a\_2\right) \end{aligned}$$

#### **Figure 3.**

*Variation of concurrence for the case Sf* � *S*<sup>0</sup> � � � � *= 1 with respect to spin quantization directions* α*<sup>1</sup> and* α*2.*

In order to quantifying entanglement, we obtain the concurrence, using SCDM of **Table 2**. According to the definition given in Eq. (6), which yields

$$C = \sqrt{1 - \frac{9 + \left(\cos a\_1 \cos a\_2 + \cos a \sin a\_1 \sin a\_2\right)^2}{5184}}.\tag{19}$$

Eq. (19) indicates [4, 17, 22] that the values of *C* are positive and so the spin state *Sf* � *S*<sup>0</sup> � � � � = 1 is entangled, depending on *α*1and *α*2*:* This variation of concurrence is given in **Figure 3**.

We find from **Figure 3** that the states for *Sf* � *S*<sup>0</sup> � � � � = 1 are entangled (the values of concurrence are positive) and separable at valley points (for the zero value of concurrence) depending on spin quantization directions.

#### **4. Example for Entanglement in DPI for He**

As an application qubit-qubit entanglement, we consider helium atom where DPI in its ground electronic state can be represented by

#### **Figure 4.**

*Variation of concurrence for the case Sf* � *S*<sup>0</sup> � � � � *= 0 with respect to directions of ejection of photoelectrons.*

*Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

**Figure 5.** *Variation of concurrences for the case Sf S*<sup>0</sup> *= 0 with respect to spin quantization directions and directions of ejection of photoelectron.*

**Figure 6.**

*Variation of concurrence for the case Sf* � *S*<sup>0</sup> *= 1 with respect to directions of ejection of photoelectrons.*

$$\hat{h}\,h\nu\_r + He\left(\mathbf{1}\mathbf{s}^{21}\mathbf{S}\_0\right) \to He^{2+}\left(\mathbf{1}\mathbf{s}^{01}\mathbf{S}\_0\right) + e\_1\left(\overrightarrow{k}\_1;\mu\_1\hat{u}\_1\right) + e\_2\left(\overrightarrow{k}\_2;\mu\_2\hat{u}\_2\right) \quad . \tag{20}$$

We use the values of TDCS of helium given in Ref. [29] for photon energy 99 eV along with linear polarization of photon and for equal energy sharing between the two electrons. The TDCS is

$$\left. \frac{d^3 \sigma}{de\_1 d\hat{k}\_1 d\hat{k}\_2} \right|\_{c\_1 = c\_2} = a \left( \cos \theta\_1 + \cos \theta\_2 \right)^2 \exp \left\{ -\frac{1}{2} \left[ \left( \theta\_{12} - 180^0 \right) / \gamma \right]^2 \right\},\tag{21}$$

where *θ*<sup>12</sup> is the angle between two electrons, *γ* named Gaussian half-width of value 90*:*<sup>2</sup> � 20. The value of normalization factor *<sup>a</sup>* <sup>¼</sup> <sup>107</sup> � <sup>16</sup> *b eV*�<sup>1</sup> *sr*�2.

i. In case of *Sf* � *S*<sup>0</sup> = 0, the variations of concurrence with respect to the direction of ejection and spin polarization of the photoelectrons are shown in **Figures 4** and **5**.

From **Figures 4** and **5**, we see that the values of concurrence are either zero or positive depending on the directions of ejection and spin quantization of *e*<sup>1</sup> and *e*2. So we can conclude that depending on the values of *θ*1, *θ*2, *α*1, and *α*2, most of the states for *Sf* � *S*<sup>0</sup> = 0 of helium atom are entangled.

ii. In case of *Sf* � *S*<sup>0</sup> = 1, the variations of concurrence with respect to the direction of ejection and spin polarization of the photoelectrons are shown in **Figures 6** and **7**.

From **Figures 6** and **7**, we see that for the states for *Sf* � *S*<sup>0</sup> = 1 of a helium atom, the nature of variations as well as magnitudes of concurrence (i.e., entanglement) depend on the directions of ejection of photoelectrons along with their spin polarization.

#### **5. Conclusion**

For concurrently creating two electrons in continuum in a single step, DPI is the most natural approach. It is the most obvious example of electron-electron correlation *Determination of Qubit Entanglement in One-step Double Photoionization of Helium Atom DOI: http://dx.doi.org/10.5772/intechopen.106047*

#### **Figure 7.**

*Variation of concurrences for the case Sf S*<sup>0</sup> *= 1 with respect to spin quantization directions and directions of ejection of photoelectron.*

in an atom because if the independent particle model were true, only one photon would have been absorbed before two electrons would have emerged at the same time. Thus, the presence of correlation effects between them could lead to the simultaneous ejection of two electrons from an atom following the absorption of a single photon. In this article, we have tried to demonstrate how effective the DPI method is for creating different types of entanglement between two qubits. A quantitative application for this case is studied for DPI in helium atom. For helium atom, we have studied the states for *Sf S*<sup>0</sup> = 0 and *Sf S*<sup>0</sup> = 1, we have shown that depending on the direction of ejection, as well as spin polarization of the ejected photoelectrons, the states are totally entangled, partially entangled, and separate.
