Abstract
Photoionization (1.1) is, probably, the simplest among the basic processes in atomic and molecular physics for generating an entangled state of two flying particles one of which is an electronic qubit. It is simplest for two reasons: First, the whole process is completed in a single step; second, it is based on the well-understood physics of photoelectric effect [1] taking place due to the absorption of a single photon γ r in a target \(\mathfrak{T}\) in the E1 approximation. Here, it is obvious from (1.1), the constituents of a bipartite state are the residual photoion \({\mbox{ $\mathfrak{T}$}}^{1+}\) and the photoelectron e p. Unlike the processes (1.2)-(1.8), the incident photon in (1.1) is absorbed by one of the electrons in the outer most shell of \(\mathfrak{T}\) so that the residual photoion \({\mbox{ $\mathfrak{T}$}}^{1+}\) is left in its ground electronic state \(\vert {1}^{+}\rangle\), which is stable against any decay.
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Notes
- 1.
For a definition of a qudit, see, for example, pages 22 and 26.
- 2.
If the spin of the residual photoion happens to be zero (i.e.,\(S_{{1}^{+}}\) = 0), the non-local correlation in a \((e_{p},\,{\mbox{ $\mathfrak{T}$}}^{1+})\) pair is then never possible as \({\mbox{ $\mathfrak{T}$}}^{1+}\) can now exists only in a single spin state corresponding to \(M_{S_{{ 1}^{+}}},M_{S_{{1}^{+}}}^{\,{\prime}} = 0\).
- 3.
See, appendix C on pages 269–271.
- 4.
See, appendix C on pages 269–271.
- 5.
See, footnote (2) on page 98.
- 6.
Spin \(\boldsymbol{S}_{{1}^{+}}\) of the photoion \({\mbox{ $\mathfrak{T}$}}^{1+}\) (as well as S 0 of the atom \(\mathfrak{T}\)) in the 1-SPI process (1.1) is quantized along the polar (i.e., OZ-) axis in Fig. 4.1. It is for this reason that no angles corresponding to the direction of quantization of the \(\boldsymbol{S}_{{1}^{+}}\) and/or S 0 appear in the density matrix (4.7), or in the subsequent related discussions in the present Chap. 4 of this monograph.
- 7.
In view of the discussion given on pages 40 and 41, the partial transpose\({\sigma }^{T_{{1}^{+}}}(S_{ 0} = 0; S_{1+} = \frac{1} {2}; \hat{\boldsymbol{u}}_{p})\), obtained using (4.11b), of the \((e_{p},\,{\mbox{ $\mathfrak{T}$}}^{1+})\) state (4.13) with respect to the photoion \({\mbox{ $\mathfrak{T}$}}^{1+}\) may, in general, be different from that given in (4.14); but its eigenvalues are necessarily identical to those of \({\sigma }^{T_{p}}\)(S 0 = 0; \(S_{1+} = \frac{1} {2}; \hat{\boldsymbol{u}}_{p}\)) obtained herein.
- 8.
See footnotes (8) and (9) on page 58, respectively.
- 9.
- 10.
See footnote (2) on page 98.
- 11.
The partial transpose of a diagonal matrix is the matrix itself.
- 12.
See discussion on page 40.
- 13.
See footnote (9) on page 40.
- 14.
- 15.
In the present case, each of these eigenvectors will, obviously, be a column matrix consisting of 2(2S 1 + + 1) rows. Although eigenvectors belonging to non-degenerate eigenvalues of a Hermitian matrix are always orthogonal; however, it is not necessarily the case with those belonging to degenerate eigenvalues. Consequently, before using in the expression (4.22) for range, the degenerate eigenvectors need to be orthonormalized using the well-known Gram–Schmidt procedure (explained, for example, on page 62 in [110]).
- 16.
For LOCC, see footnotes (12) and (13) on page 42.
- 17.
Henceforth, unless stated otherwise, distillation of entanglement means its either concentration or purification, as the case may be.
- 18.
See footnote (17).
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Chandra, N., Ghosh, R. (2013). Coulombic Entanglement: One-Step Single Photoionization of Atoms. In: Quantum Entanglement in Electron Optics. Springer Series on Atomic, Optical, and Plasma Physics, vol 67. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24070-6_4
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