We present in this paper results of a nonlinear optical spectroscopy approach to measurement of excited-state transition matrix elements. Recent advances in the quality of excited-state transition matrix elements have permitted renormalization of earlier measurements of transition amplitudes associated with the 5s 2 S 1/2 → 5p 2 P j → 5d 2 D 3/2 two-photon transitions in atomic 87 Rb. Previous measurements were made to high precision, but further improvement of the accuracy was limited by uncertainties in data describing the influence of energetically distant transitions. Availability of more reliable matrix elements, including relativistic all-order calculations of transition matrix elements in alkali atoms, has since significantly improved the situation. In the present paper, we show that theoretical relative transition amplitudes for the excited state 5p 2 P j → 5d 2 D 3/2 doublet (ratio = 1.098(9)) are now in excellent agreement with experiment (ratio = 1.090(6)). This result, combined with our recent work on caesium, shows that it is possible to determine relative line strengths, for transitions connecting atomic excited states, with precision previously found only in state-of-the-art measurements of alkali resonance doublets. Detailed discussion of the experimental technique and supporting data, including polarization measurements on the 5s 2 S 1/2 → 5p 2 P j → 5d 2 D 5/2 transitions, is also presented.
Summary form only given. A number of sophisticated techniques have been applied to precise measurements of the strength of resonance transitions in heavier alkali atoms. These include refined lifetime measurements in atomic beams, photoassociative spectroscopy of cold atoms confined to atomic traps, and direct measurements of the natural width of an atomic transition. We have made a precise study of the 5s-5p-5d transition in atomic Rb. Measurements of the non-resonant two-photon, two-color linear depolarization spectrum show remarkable polarization-dependent interference structure. Using this novel approach, it was possible to extract the ratio of the transition matrix elements for the two contributing pathways.
Precise measurements are reported on the linear polarization spectrum associated with two-photon, two-color spectroscopy of the 3
s2 S1/2 \ensuremath{\rightarrow}5
s2 S1/2 transition in atomic Na. Measurements are made in an approximately 40-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ range in the vicinity of the 3
p2 Pj (j=1/2,3/2) levels, which serve as the principal intermediate levels for the two-photon transitions. Because of interference between the transition amplitudes through each fine-structure multiplet component, the polarization varies from nearly 100% to -100% within this spectral range. Modeling the data to include the effects of more energetic p levels and allowing for a j-dependent ratio of transition amplitudes R results in an excellent fit to the data. Estimation of the contribution of higher p levels allows extraction of a value and sign for R=+1.0012(12). This result is combined with recent measurements of the lifetime of the resonance transitions in Na to obtain a value for the ratio of excited-state transition-matrix elements for the 3
p2 Pj \ensuremath{\rightarrow}5
s2 S1/2 transition (j=3/2 compared to j=1/2) of 1.0013(15).
Precision measurements on the 5s
2S1/2\ensuremath{\rightarrow}5p
2Pj\ensuremath{\rightarrow}8s
2S1/2 two-photon, two-color polarization spectrum in atomic Rb are reported. As a result of interference between the transition amplitudes for the process, the spectrum is strongly frequency and linear polarization dependent in the vicinity of the 5s
2S1/2\ensuremath{\rightarrow}5p
2P1/2 and 5s
2S1/2\ensuremath{\rightarrow}5p
2P3/2 resonance transitions, with the linear polarization degree varying from nearly +100% to -100% over the approximate 500-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ range of the data. For absorption of two collinearly polarized photons, complete destructive interference for the process occurs at a critical detuning from the 5s
2S1/2\ensuremath{\rightarrow}5p
2P3/2 resonance transition given by ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{c}}$=-155.83(3) ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. The location of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{c}}$ is determined primarily by the relative transition matrix elements through the 5p
2Pj multiplet, but also depends significantly on contributions from more energetic np
2P levels. The null in the excitation occurring at the critical frequency allows experimental determination of a type of sum rule that depends only on the atomic matrix elements and the value of ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{c}}$. Such sum rules are defined and presented for the present experiment in Rb and for a recent similar experiment in Na.
A combined experimental and theoretical study of transition matrix elements of the $6p{}^{2}{P}_{j}\ensuremath{\rightarrow}8s{}^{2}{S}_{1/2}$ transition in atomic Cs is reported. Measurements of the polarization-dependent two-photon excitation spectrum associated with the transition were made in an $\ensuremath{\approx}200{\mathrm{cm}}^{\ensuremath{-}1}$ range on the low-frequency side of the $6s{}^{2}{S}_{1/2}\ensuremath{\rightarrow}6p{}^{2}{P}_{3/2}$ resonance. The measurements depend parametrically on the relative transition matrix elements, but also are sensitive to far-off-resonance $6s{}^{2}{S}_{1/2}\ensuremath{\rightarrow}\mathrm{np}{}^{2}{P}_{j}\ensuremath{\rightarrow}8s{}^{2}{S}_{1/2}$ transitions. In addition, as the measured quantities are ratios of polarization-dependent intensities at a single-excitation frequency, they are quite insensitive to a variety of common-mode systematic effects; matrix-element ratios may then be determined to high accuracy. In the past, the matrix-element dependence has yielded a generalized sum rule, the value of which is dependent on sums of relative two-photon transition matrix elements. In the present case, best available determinations from other experiments are combined with theoretical matrix elements to extract the ratio of transition matrix elements for the $6p{}^{2}{P}_{j}\ensuremath{\rightarrow}8s{}^{2}{S}_{1/2}(j=1/2,3/2)$ transition. The resulting experimental value of 1.423(2) is in excellent agreement with the theoretical value, calculated using a relativistic all-order method, of 1.425(2).
The estimation of the adequacy of theoretical calculations on the atomic structure requires availability of the precise experimental data on radiative properties of the atoms. Such data is also required in astronomy and some important areas of technology. The lack of precision of traditional spectroscopic studies of atom presents a fundamental obstacle for progress in these areas. For example, in atomic rubidium, the best precision of the traditional spectroscopic results is on the order of about 1 - 5%, which does not allow for clear assessment of the latest sophisticated theoretical calculations on atomic rubidium structure, with emphasis on different, in nature, effects. This situation is typical for atomic physics in general. The purpose of present study is obtaining the experimental data on the radiative properties of atomic rubidium with precision considerably higher than that of the traditional spectroscopic methods. This is accomplished by means of the two-photon quantum interference polarization spectroscopy. A two-photon polarization spectrum of the rubidium atom is obtained in the range of detunings -417 cm-1 to +99 cm-1 from atomic 5s2S1/2-5p 2P3/2-*8s2S1/2resonance. From analysis of the spectra the relativistic and many body effects on the wavefiinctions are revealed in the form of a uniquely defined parameter q = 2 x 10-6 (5) cm and an exact relation between parameters R and p which quantitatively describes the process: R = 1.01756 (57) + 81.466 (15) p where R is dimensionless and p is in cm. The obtained results can be thought of as specific experimentally established two-photon sum rules and can be used for testing the accuracy of the theoretical wavefiinctions. The experimental technique has important advantages comparing to some traditional spectroscopic methods and is essentially free of systematic effects.
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.