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Physical Review A

Topological angular momentum in electron exchange excitation of a single atom

Jim Williams, Luka Pravica, Sergey N. Samarin

Published Feb 6, 2012• 11 citations
Publisher

Abstract

In a single free two-valence-electron atom, the motion of the electron spin is a consequence of quantum statistics and the Pauli exclusion principle. Subsequently, during an electron impact exchange excitation from a ${}^{1}{S}_{0}\phantom{\rule{0.16em}{0ex}}{M}_{S}=0$ to a ${}^{3}{S}_{1}\phantom{\rule{0.16em}{0ex}}{M}_{S}=0$ state, the electron spin is ``parallel transported'' around a closed path with a geometrical Berry phase of \ensuremath{\pi} radians creating an aligned exchange spin angular momentum. This alignment is observed via the Stokes parameter ${P}_{2}$ of the photon decay into a ${}^{3}P$ state. The geometric phase is in addition to the dynamic phase. Measurements from zinc and mercury atoms in different laboratories show the effect close to the excitation threshold where there are no competing excitation processes. Similar effects are expected in other atomic and molecular quantum scattering processes where comparable geometrical or topological paths exist. Electron quantum scattering theories use antisymmetrized wave functions but none include this geometrical exchange angular momentum.

DOI
10.1103/physreva.85.022701

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In a single free two-valence-electron atom, the motion of the electron spin is a consequence of quantum statistics and the Pauli exclusion principle. Subsequently, during an electron impact exchange excitation from a 1S0MS=0{}^{1}{S}_{0}\phantom{\rule{0.16em}{0ex}}{M}_{S}=01S0​MS​=0 to a 3S1MS=0{}^{3}{S}_{1}\phantom{\rule{0.16em}{0ex}}{M}_{S}=03S1​MS​=0 state, the electron spin is ``parallel transported'' around a closed path with a geometrical Berry phase of \ensuremath{\pi} radians creating an aligned exchange spin angular momentum. This alignment is observed via the Stokes parameter P2{P}_{2}P2​ of the photon decay into a 3P{}^{3}P3P state. The geometric phase is in addition to the dynamic phase. Measurements from zinc and mercury atoms in different laboratories show the effect close to the excitation threshold where there are no competing excitation processes. Similar effects are expected in other atomic and molecular quantum scattering processes where comparable geometrical or topological paths exist. Electron quantum scattering theories use antisymmetrized wave functions but none include this geometrical exchange angular momentum.
The Fano-equivalent profiles of two negative-ion resonant states have been observed in the integrated Stokes parameters of 636.2 nm photons emitted from the 3d104s4d1D23{d}^{10}4s4{d\phantom{\rule{0.2em}{0ex}}}^{1}{D}_{2}3d104s4d1D2​ state of zinc. These profiles reflect the effects of electron exchange and the spin-orbit interaction via the angular-momentum-dependent properties of the alignment and orientation Stokes parameters. The resonant states have energies of $10.98\ifmmode\pm\else\textpm\fi{}0.02\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and $11.33\ifmmode\pm\else\textpm\fi{}0.02\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and widths of $0.25\ifmmode\pm\else\textpm\fi{}0.03\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ and $0.33\ifmmode\pm\else\textpm\fi{}0.05\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, respectively, and are bound to an excited atomic state with a 3d93{d}^{9}3d9 core hole. The lower-energy resonance shows a strong spin-orbit interaction whereas the higher-energy resonance shows negligible spin-orbit interaction.
The quantum mechanics of two identical particles with spin S in three dimensions is reformulated by employing not the usual fixed spin basis but a transported spin basis that exchanges the spins along with the positions. Such a basis, required to be smooth and parallel-transported, can be generated by an ‘exchange rotation’ operator resembling angular momentum. This is constructed from the four harmonic oscillators from which the two spins are made according to Schwinger's scheme. It emerges automatically that the phase factor accompanying spin exchange with the transported basis is just the Pauli sign, that is (−1)2S. Singlevaluedness of the total wavefunction, involving the transported basis, then implies the correct relation between spin and statistics. The Pauli sign is a geometric phase factor of topological origin, associated with non-contractible circuits in the doubly connected (and non-orientable) configuration space of relative positions with identified antipodes. The theory extends to more than two particles.
The Stokes parameters P1{P}_{1}P1​, P2{P}_{2}P2​, P3{P}_{3}P3​, and P4{P}_{4}P4​ have been measured in a scattered electron-polarized cascade photon correlation study of the 41F4^{1}F41F state of helium excited by 29.6 eV electron impact. This is the first detailed experimental study of electron impact excitation leading to the transfer of three units of angular momentum. The data are expected to be influenced by the breakdown of LS\mathrm{LS}LS coupling for $L\ensuremath{\ge}3$ states, even in a simple system like helium, providing a further challenge for theory.
A theory of electron-impact excitation of atoms in which fine structure is included as a perturbation is presented in detail. Hyperfine structure is included in the target atom but neglected in the electron-atom interaction. Expressions for electron-photon angular-correlation parameters and scattering cross sections are derived in terms of products of complex scattering amplitudes, with emphasis on electron-sodium excitation. Examples are given in which zero-field quantum beats may be observed in electron-photon angular-correlation and differential-cross-section measurements. Some of the interference terms are only observable in angular-correlation experiments. Examples are also given of partial cross sections for electron-impact excitation of state-to-state hyperfine levels in terms of direct- and exchange-scattering amplitudes plus additional terms due to the inclusion of fine-structure effects. The expressions given allow a comparison between the various possible scattering experiments.
Spin-orbit and exchange effects are separately identified through the measurement of integrated Stokes parameters P2{P}_{2}P2​ and P3{P}_{3}P3​ for the decay radiation from spin-polarized electron impact excited states. The observations are made for the radiations from the transitions ${3p}^{\ensuremath{'}}[\frac{1}{2}{]}_{1}\ensuremath{\rightarrow}{3s}^{\ensuremath{'}}[\frac{1}{2}{]}_{0}$ (616.4 nm) and $3p[\frac{1}{2}{]}_{1}\ensuremath{\rightarrow}3s[\frac{3}{2}{]}_{2}$ (703.3 nm) in neon just above threshold.