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Physical Review A

Coherence and correlation in doubly excited heliumlike atoms

Joachim Burgdörfer, Reinhard Morgenstern

Published Dec 1, 1988• 8 citations
Publisher

Abstract

We analyze properties of the density matrix of doubly excited two-electron systems formed in inelastic collisions. Formulas for the two-particle joint angular probability density, the angular correlation function, and the reduced single-particle density are derived. Of particular interest is the interplay between the intrinsic correlations of the stationary two-electron state and collisionally induced coherences. We focus on its effects on the correlated and single-particle motion of the electrons. If one chooses approximate stationary wave functions reflecting the approximate O(4)\ifmmode\times\else\texttimes\fi{}O(4)\ensuremath{\supset}O(4) dynamical symmetry, a simple quasiclassical interpretation of coherence and correlation in terms of shapes and modes of the relative motion of Kepler orbits can be given. The present description is applied to recent experimental results by Van der Straten and Morgenstern [Comments At. Mol. Phys. 19, 243 (1986)].

DOI
10.1103/physreva.38.5520

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We analyze properties of the density matrix of doubly excited two-electron systems formed in inelastic collisions. Formulas for the two-particle joint angular probability density, the angular correlation function, and the reduced single-particle density are derived. Of particular interest is the interplay between the intrinsic correlations of the stationary two-electron state and collisionally induced coherences. We focus on its effects on the correlated and single-particle motion of the electrons. If one chooses approximate stationary wave functions reflecting the approximate O(4)\ifmmode\times\else\texttimes\fi{}O(4)\ensuremath{\supset}O(4) dynamical symmetry, a simple quasiclassical interpretation of coherence and correlation in terms of shapes and modes of the relative motion of Kepler orbits can be given. The present description is applied to recent experimental results by Van der Straten and Morgenstern [Comments At. Mol. Phys. 19, 243 (1986)].
The physical origin of the feature of strong correlation between electrons is mainly ascribed to doubly occupied atomic orbitals. Two-particle wave functions are believed necessary to describe strongly correlated systems. We emphasize that the on-site Coulomb interaction, which is important for strong correlation, should appear in equations as a natural term. Hartree-Fock-like self-consistent equations are derived with wave functions including two-particle wave functions for such systems. In the self-consistent equations, a strong correlation term is separated from the mean field. To our knowledge, this is the first attempt to formulate self-consistent equations in which the strong correlation effect is directly embodied. Some features of the equations are discussed. Koopmans' theorem and Brillouin's theorem are applied to the equations. The former is valid for singly occupied orbitals, and the latter is entirely available. By separating an angular correlation factor which is believed important for a strong correlation effect, an approximate form of a two-particle wave function is suggested. This kind of wave function causes the six multiple integrals to degrade to triple integrals. Under such forms of two-particle wave functions, corresponding Roothaan equations are established. Through these steps, it is believed that the equations become solvable within present computational limits.
We have calculated quadrupole moments, Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​, of helium in several of its doubly excited states and in two of its singly excited Rydberg states, and of the alkaline-earth atoms Be, Mg, Ca, Sr, and Ba in their ground and low-lying excited states. The calculations use well-converged, frozen-core configuration-interaction (CI) wave functions and, for interpretive purposes, Hartree-Fock (HF) atomic wave functions and single-term, optimized, molecular rotor-vibrator (RV) wave functions. The quadrupole moments calculated using RV wave functions serve as a test of the validity of the correlated, moleculelike model, which has been used to describe the effects of electron correlation in these two-electron and pseudo-two-electron atoms. Likewise, the quadrupole moments calculated with HF wave functions test the validity of the independent-particle model. In addition to their predictive use and their application to testing simple models, the quadrupole moments calculated with CI wave functions reveal previously unavailable information about the electronic structure of these atoms. Experimental methods by which these quadrupole moments might be measured are also discussed. The quadrupole moments computed from CI wave functions are presented as predictions; measurements of Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​ have been made for only two singly excited Rydberg states of He, and a value of Qzz{\mathit{Q}}_{\mathit{z}\mathit{z}}Qzz​ has been computed previously for only one of the states reported here. We present these results in the hope of stimulating others to measure some of these quadrupole moments.
Explicit asymptotic expressions are derived for the reduced one-electron density matrices (the 1-matrices) of strongly correlated two- and three-electron harmonium atoms in the ground and first excited states. These expressions, which are valid at the limit of small confinement strength ω, yield electron densities and kinetic energies in agreement with the published values. In addition, they reveal the ω(5/6) asymptotic scaling of the exchange components of the electron-electron repulsion energies that differs from the ω(2/3) scaling of their Coulomb and correlation counterparts. The natural orbitals of the totally symmetric ground state of the two-electron harmonium atom are found to possess collective occupancies that follow a mixed power/Gaussian dependence on the angular momentum in variance with the simple power-law prediction of Hill's asymptotics. Providing rigorous constraints on energies as functionals of 1-matrices, these results are expected to facilitate development of approximate implementations of the density matrix functional theory and ensure their proper description of strongly correlated systems.
We study the Born-Oppenheimer dynamics within a model for a coupled electron-nuclear motion. Differential Shannon entropies are calculated from the time-dependent probability densities of the combined system and, using single particle densities, entropies for the electronic and nuclear degrees of freedom are derived. These functions provide information on details of the wave packet motion. From the entropies, we determine the mutual information which characterizes particle correlations. This quantity is compared to other measures of electron-nuclear entanglement. Numerical results are interpreted within an analytically solvable approach, and it is documented how these functions depend on properties of the Born-Oppenheimer wave function and, in particular, how dynamical effects like wave packet focusing and dispersion influence the correlation between the particles.
During a collision of Bose-Einstein condensates correlated pairs of atoms are emitted. The scattered massive particles, in analogy to photon pairs in quantum optics, might be used in the violation of Bell's inequalities, demonstration of Einstein-Podolsky-Rosen correlations, or sub-shot-noise atomic interferometry. Usually, a theoretical description of the collision relies either on stochastic numerical methods or on analytical treatments involving various approximations. Here, we investigate elastic scattering of atoms from colliding elongated Bose-Einstein condensates within the Bogoliubov method, carefully controlling performed approximations at every stage of the analysis. We derive expressions for the one- and two-particle correlation functions. The obtained formulas, which relate the correlation functions to the condensate wave function, are convenient for numerical calculations. We employ the variational approach for condensate wave functions to obtain analytical expressions for the correlation functions, whose properties we analyze in detail. We also present a useful semiclassical model of the process and compare its results with the quantum one. The results are relevant for recent experiments with excited helium atoms, as well as for planned experiments aimed at investigating the nonclassicality of the system.
This paper focusses on the description, classification and interpretation of coherent excitation of atomic or ionic systems with Coulombic two-body final state interactions. A group-theoretical approach is used to classify and interpret coherent excitation. The most significant result is that the state of excitation represented by a density operator can be mapped one to one onto expectation values of a set of operators. Examples are used to illustrate what can be learned about the collision process from investigations of coherent excitation. (JDH)