Information and entanglement measures applied to the analysis of complexity in doubly excited states of helium

J. P. Restrepo Cuartas and J. L. Sanz-Vicario

Phys. Rev. A 91, 052301 – Published 1 May 2015

Abstract

Shannon entropy and Fisher information calculated from one-particle density distributions and von Neumann and linear entropies (the latter two as measures of entanglement) computed from the reduced one-particle density matrix are analyzed for the ${}^{1, 3}S^{e}, {}^{1, 3}P^{o}$ , and ${}^{1, 3}D^{e}$ Rydberg series of He doubly excited states below the second ionization threshold. In contrast with the Shannon entropy, we find that both the Fisher information and entanglement measures are able to discriminate low-energy resonances pertaining to different ${}_{2}{(K, T)}_{n_{2}}^{A}$ series according to the Herrick-Sinanoğlu-Lin classification. Contrary to bound states, which show a clear and unique asymptotic value for both Fisher information and entanglement measures in their Rydberg series $1 s n ℓ$ for $n \to \infty$ (which implies a loss of spatial entanglement), the variety of behaviors and asymptotic values of entanglement above the noninteracting limit value in the Rydberg series of doubly excited states ${}_{2}{(K, T)}_{n_{2}}^{A}$ indicates a signature of the intrinsic complexity and remnant entanglement in these high-lying resonances even with infinite excitation $n_{2} \to \infty$ , for which all known attempts of resonance classifications fail in helium.

Received 21 October 2014
Revised 26 February 2015

DOI:https://doi.org/10.1103/PhysRevA.91.052301

Authors & Affiliations

J. P. Restrepo Cuartas and J. L. Sanz-Vicario^*

Grupo de Física Atómica y Molecular, Instituto de Física, Universidad de Antioquia, Medellín, Colombia

^*Corresponding author: jose.sanz@udea.edu.co; present address: Departamento de Química, Módulo 13, Universidad Autónoma de Madrid, 28049 Madrid, Spain.

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Vol. 91, Iss. 5 — May 2015

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Images

Figure 1
Two-electron radial density $ρ (r_{1}, r_{2}) r_{1}^{2} r_{2}^{2}$ for the lowest nine $^{1} P^{o}$ Feshbach resonance states in He, located below the second ionization threshold. Resonances are labeled according to the classification proposed by Lin [46] using $N^{1} P^{o}_{n_{1}} {(K, T)}_{n_{2}}^{A}$ . Each column contains the three lowest resonances corresponding to a different ${(K, T)}^{A}$ series. The energy ordering indicated with the label $N$ follows a zigzag pattern in the graphical array, according to the energies quoted in Table 2. Inside each series, the increasing excitation energy is accompanied by an increasing number of radial nodes along $r_{1}$ (or $r_{2}$ ) for a given value of $r_{2}$ ( $r_{1}$ ). Clearly the series with $A = - 1$ shows a node in the bisection line $r_{1} = r_{2}$ .
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Figure 2
Fisher information $I [ρ (r)]$ for the ground state and the lowest members in the Rydberg series of singly excited $^{1} S^{e}$ states, singly excited $^{1} P^{o}$ states, and singly excited $^{1} D^{e}$ states in helium (left panel); and for the lowest members of the Rydberg series of doubly excited states organized according to series with different $(K, T)$ labels: $^{3} S^{e}$ with two $(K, T)$ series, $^{3} P^{o}$ with three series, and $^{3} D^{e}$ with three series (right panel). Solid lines connect points corresponding to states pertaining to the same Rydberg $(K, T)$ series. Vertical dashed lines indicate the energy position for the thresholds ${He}^{+}$ ( $n_{1} = 1$ ) (left panel) and ${He}^{+}$ ( $n_{1} = 2$ ) (right panel).
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Figure 3
Same as Fig. 2 but for the spin triplets, for bound (left) and doubly excited states (right).
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Figure 4
Von Neumann entropy $S_{V N}$ for the ground state and the lowest members in the Rydberg series of singly excited $^{1} S^{e}$ states, singly excited $^{1} P^{o}$ states, and singly excited $^{1} D^{e}$ states in helium (left panel); and for the lowest members of the Rydberg series of doubly excited states organized according to series with different $(K, T)$ labels: $^{3} S^{e}$ with two series, $^{3} P^{o}$ with three series, and $^{3} D^{e}$ with three series (right panel). Solid lines connect points corresponding to states pertaining to the same Rydberg $(K, T)$ series. Vertical dashed lines indicate the energy position for the thresholds ${He}^{+}$ ( $n_{1} = 1$ ) (left panel) and ${He}^{+}$ ( $n_{1} = 2$ ) (right panel).
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Figure 5
Same as Fig. 4 but for the spin triplets, for bound (left) and doubly excited states (right).
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Figure 6
Von Neumann entropy $S_{V N}$ for the lowest members in the Rydberg series of $^{3} P^{o}$ DES; blue squares for the $(1, 0)$ series, red circles for the $(0, 1)$ series, and green diamonds for $(- 1, 0)$ . Solid line: full CI ab initio results; dashed line: $3 \times 3$ Hamiltonian results from Eq. (27) and $Z \to \infty$ ; dotted line: results from the DESB combinations using Eqs. (28) and (29). The von Neumann entropy values from the three different approaches are set up with the same ab initio DES energies for the sake of comparison. The vertical dashed line indicates the energy position for the threshold ${He}^{+}$ ( $n_{1} = 2$ ).
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