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Physical Review Letters

Measurement of a Quadrupole Transition Moment by Interference of Quadrupole and dc-Field-Induced Sum-Frequency Generation

Donald S. Bethune, Robert W. Smith, Y. R. Shen

Published Mar 21, 1977• 23 citations
Publisher

Abstract

We show that, by measuring the interference between quadrupole and dc-field-induced sum-frequency generation, both the magnitude and the sign of a quadrupole matrix element can be obtained relative to the dipole matrix elements in atomic transitions. We use the measurement of $〈3s|{z}^{2}|4d〉$ of sodium to demonstrate the technique.

DOI
10.1103/physrevlett.38.647

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arXiv:1811.10177•Physical Review A•Feb 25, 2019•6 citations
We show that, by measuring the interference between quadrupole and dc-field-induced sum-frequency generation, both the magnitude and the sign of a quadrupole matrix element can be obtained relative to the dipole matrix elements in atomic transitions. We use the measurement of 〈3s∣z2∣4d〉〈3s|{z}^{2}|4d〉〈3s∣z2∣4d〉 of sodium to demonstrate the technique.
Relative dipole matrix elements are measured by means of a new spectroscopic technique called polarization interference spectroscopy. It is a two-color two-photon nonresonant process that maps the matrix elements into the energy domain by means of quantum mechanical interference. The technique measures the relative magnitude of the reduced matrix elements, particularly those involving excited states, to accuracies that are an order of magnitude better than previously available. Also, for the first time the relative phase of the reduced matrix elements is measured. The ratio of reduced matrix elements for the transitions 3s 2S1/2 →3P 2P3/2 →5s 2S1/2 to 3s 2S1/2 →3P 2P1/2 →5s 2S1/2 in sodium is +1.0012(12). This result is in agreement with the result of +1 for J independent matrices.
Two flashlamp-pumped, tunable, dye lasers have been utilized to study three-photon ionization in atomic sodium. The resulting ion yield shows large peaks at laser frequencies which correspond to electric quadrupole transitions. This is the first direct observation of electric quadrupole effects in multiphoton ionization. The $3p^{2}P_{\frac{3}{2}}\ensuremath{\rightarrow}4f$ matrix element is determined, and the fine-structure splitting of the 5p5p5p state is measured.
We show that second-order nonlinear optical processes such as sum and difference generation can be observed in isotropic media, despite the fact that such processes are forbidden in the electric-dipole approximation. A theoretical treatment of quadrupole sum- and difference-frequency generation is given, which includes the effects of spin-orbit and Zeeman splittings. The quadrupole nonlinear susceptibility is evaluated and expressions for phase matching and output power are found. We have generally confirmed the theoretical predictions by observing quadrupole sum-frequency generation of ultraviolet light in sodium vapor. Deviations of the data from the simple theory can be satisfactorily accounted for by the effects of two-photon saturation, induced index of refraction changes, and single- and multi-photon absorption. Generalization of the dipole forbidden susceptibilities to the case of quadrupole-pumped processes is discussed, with difference-frequency generation in Cs vapor considered as an example.
Quantum interference between an electric-quadrupole transition and a Stark--induced-dipole-transition amplitude leads to a net orientation of an atomic excited state. When the barium 6s2{\mathit{s}}^{2}s2 1S-6s5d 1^{1}1D transition is excited in the presence of an electric field, we find the induced orientation of the 6s5d 1^{1}1D state to be (2.86\ifmmode\pm\else\textpm\fi{}0.59)\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}6}$(V/cm${)}^{\mathrm{\ensuremath{-}}1}$.
The influence of oscillating quadrupole fields on atomic energy levels is examined theoretically, and general expressions for the quadrupole matrix elements are given. The results are relevant to any ion-based clock in which one of the clock states supports a quadrupole moment. Clock shifts are estimated for 176Lu+^{176}\mathrm{Lu}^{+}176Lu+ and indicate that coupling to the quadrupole field would not be a limitation to clock accuracy at the $\ensuremath{\lesssim}{10}^{\ensuremath{-}19}$ level. Nevertheless, a method is suggested that would allow this shift to be calibrated. This method utilizes a resonant quadrupole coupling that enables the quadrupole moment of the atom to be measured. A proof-of-principle demonstration is given using 138Ba+^{138}\mathrm{Ba}^{+}138Ba+, in which the quadrupole moment of the D5/2{D}_{5/2}D5/2​ state is estimated to be $\mathrm{\ensuremath{\Theta}}=3.229(89)e{a}_{0}^{2}$.