Near-UV excited-state absorption spectra obtained with a pulsed pump–probe technique were recorded in LiYF4:Pr3+ at 77 and at 8 K and revealed a detailed structure of the 4f2 1D2→4f5d and 4f2 (3P0+1I6)→4f5d optical transitions. These experimental results, together with those previously obtained at room temperature, are compared with the numerical predictions of a full calculation of the 4f5d sublevels and of the electric dipole transitions toward these sublevels from different states of the 4f2 configuration. The agreement is good, provided that the four adjustable parameters of the theory are taken as B20(5d)=7290 cm-1,B40(5d)=-14 900 cm-1,F0=60 557 cm-1, and 〈r〉fd=0.245 Å. The level structure of the 4f5d configuration of LiYF4:Pr3+ is shown to be dominated, as expected, by the crystal field coupling of the 5d electron but also, to a noticeable extent, by the spin–orbit coupling of the 4f electron and the Coulomb repulsion of both electrons. The former of these two interactions is somewhat more efficient than the latter in the low-energy part of the 4f5d configuration, i.e., in the spectral region where the largest amount of experimental data is currently available.
A comprehensive report and analysis of our observations of numerous parity-allowed direct two-photon transitions within the
4f7 configuration of the
Gd3+ ion is presented. The experimental technique is ultraviolet fluorescence detection following optical excitation with a dye-laser beam. Previous results and analyses are extended in three ways. (1) The intensities and polarization dependence of individual Stark components, in addition to integrated multiplet intensities, are reported and analyzed for
Gd3+:La
F3. In those cases ($^{8}S_{\frac{7}{2}}\ensuremath{\rightarrow}^{6}P_{\frac{5}{2}},^{6}D_{\frac{9}{2},\frac{3}{2},\frac{5}{2}}$) where a second-order theory of two-photon absorption adequately explains the integrated intensity, it also (with the exception of
6D29) explains the Stark-component intensities. The remaining transitions are anomalously strong, and in some cases violate the angular momentum selection rules $\ensuremath{\Delta}L,\ensuremath{\Delta}J\ensuremath{\le}2$ and exhibit strong anisotropies, in contradiction to the second-order theory. It is shown that introduction of third- and fourth-order contributions involving spin-orbit and/or crystal-field interactions among levels of the
4f65d configuration, which serve as intermediate states, can account for both the integrated and Stark-component intensities of these transitions. (2) The strong anisotropy observed for the transitions $^{8}S_{\frac{7}{2}}\ensuremath{\rightarrow}^{6}P_{\frac{3}{2}},^{6}I_{J},^{6}D_{\frac{1}{2}}$ in
Gd3+:La
F3 is explained quantitatively as a polarization-dependent interference between contributions to the intensity which are comparable in magnitude. Interference is destructive for $\stackrel{\ensuremath{\rightarrow}}{E}\ensuremath{\parallel}\stackrel{^}{z}$ and constructive for $\stackrel{\ensuremath{\rightarrow}}{E}\ensuremath{\perp}\stackrel{^}{z}$, creating an order-of-magnitude contrast in the line strength for the two polarizations. (3) A two-photon excitation spectrum of
Gd3+ in aqueous solution is reported. Linewidths comparable to those of room-temperature solids and Stark splittings which suggest a low-symmetry quasistatic environment are observed.
The method of correlated basis functions provides a means of systematically exploiting our intuitive understanding of real physical systems. Here we seek especially to develop a formalism for describing systems of strongly interacting fermions, for example, nuclei and liquid
He3, using a complete set of correlated functions ${\ensuremath{\Psi}}_{\mathrm{m}}=\frac{F{\ensuremath{\Phi}}_{\mathrm{m}}}{{({\ensuremath{\Phi}}_{\mathrm{m}}, {F}^{2}{\ensuremath{\Phi}}_{\mathrm{m}})}^{\frac{1}{2}}}$.
F is taken as a product of two-body factors, $\ensuremath{\Pi}{i<j}^{}[f({r}_{\mathrm{ij}})]$, the
f(rij) embodying at least the correlations arising from strong short-range repulsions; and ${{\ensuremath{\Phi}}_{\mathrm{m}}}$, a set of independent-particle-model functions which might serve as a basis in the absence of highly singular interactions. The matrix elements
Hmn,
Nmn of the Hamiltonian and identity operators with respect to the correlated basis ${{\ensuremath{\Psi}}_{\mathrm{m}}}$ are evaluated by generalization of cluster-expansion techniques developed originally for the calculation of the partition function of a classical imperfect gas and adapted to the quantum-mechanical many-body problem by Iwamoto and Yamada. Contributions to our expansions for the
Hmn are classified according to order of magnitude in the quantity $\ensuremath{\rho}\ensuremath{\omega}$, where $\ensuremath{\omega}=\ensuremath{\int}({f}^{2}(r)\ensuremath{-}1)d\mathrm{r}$ is a "correlation parameter" and $\ensuremath{\rho}$ the average particle density. To the extent that $|\ensuremath{\rho}\ensuremath{\omega}|$ is small, it provides a true expansion parameter. A detailed technical discussion of the cluster expansions is given, with special emphasis on their symmetry properties upon truncation. Although the many-body problem is soluble by simultaneous diagonalization of the matrices (
Hmn) and (
Nmn), it is preferable first to transform to an orthonormal basis of "unperturbed" correlated functions ${{\ensuremath{\Theta}}_{\mathrm{m}}}$ with respect to which the Hamiltonian and identity operators have the matrix representations (
Hmn) and $({N}_{\mathrm{mn}})=({\ensuremath{\delta}}_{\mathrm{mn}})$, and then to attempt the separate diagonalization of (
Hmn). In general, such a transformation is expedited by the L\"owdin procedure carried out to the desired order in $\ensuremath{\rho}\ensuremath{\omega}$. However, a Schmidt orthogonalization scheme might prove more advantageous in specific cases. Field-theoretic techniques (of both perturbative and nonperturbative nature) may be brought to bear in the diagonalization of (
Hmn). The inverse of the unitary transformation which takes the basis ${{\ensuremath{\Phi}}_{\mathrm{m}}}$ into the basis ${{\ensuremath{\Theta}}_{\mathrm{m}}}$, applied to the given singular Hamiltonian, leads---for any
F in the class considered---to a non-singular "effective Hamiltonian." Following this formal cue, we set out to construct explicitly a second-quantized effective Hamiltonian which generates the same matrix elements, to prescribed order in $\ensuremath{\rho}\ensuremath{\omega}$, when operating between independent-particle kets of the model occupation-number representation, as does the given Hamiltonian in the configuration-space representation, between the corresponding correlated functions of the L\"owdin basis. The quasiparticles described by this effective Hamiltonian interact via well-behaved two-body, three-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},
Q-body, \ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{} potentials, in contrast to the highly singular two-body interactions of the real particles. For small $|\ensuremath{\rho}\ensuremath{\omega}|$, the effective two-body forces predominate. An avenue to deeper understanding of the applicability of the nuclear-shell model and the dressed-quasiparticle theory of liquid
He3 has been opened. This article treats only the formalism for a uniform extended medium. Numerical calculations, as well as the extension of the method to finite systems, will follow.
Magic wavelengths (${\ensuremath{\lambda}}_{\mathrm{magic}}$) for the $7{S}_{1/2}\ensuremath{-}7{P}_{1/2,3/2}$ transitions (D lines) in Fr were reported by Dammalapati et al. [U. Dammalapati, K. Harada, and Y. Sakemi, Phys. Rev. A 93, 043407 (2016)]. These ${\ensuremath{\lambda}}_{\mathrm{magic}}$ were determined by plotting dynamic polarizabilities ($\ensuremath{\alpha}$) of the involved states with the above transitions against a desired range of wavelengths. Electric dipole (E1) matrix elements listed in [J. E. Sansonetti, J. Phys. Chem. Ref. Data 36, 497 (2007)], from the measured lifetimes of the
7P1/2,3/2 states and from the calculations considering core-polarization effects in the relativistic Hartree-Fock (HFR) method, were used to determine $\ensuremath{\alpha}$. However, contributions from core correlation effects and from the E1 matrix elements of the $7P\ensuremath{-}7S, 7P\ensuremath{-}8S$, and $7P\ensuremath{-}6D$ transitions to $\ensuremath{\alpha}$ of the
7P states were ignored. In this work, we demonstrate importance of these contributions and improve accuracies of $\ensuremath{\alpha}$ further by replacing the E1 matrix elements taken from the HFR method by the values obtained employing relativistic coupled-cluster theory. Our static $\ensuremath{\alpha}$ are found to be in excellent agreement with the other available theoretical results, whereas substituting the E1 matrix elements used by Dammalapati et al. gives very small $\ensuremath{\alpha}$ values for the
7P states. Owing to this, we find disagreement in ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported by Dammalapati et al. for linearly polarized light, especially at wavelengths close to the D lines and in the infrared region. As a consequence, a ${\ensuremath{\lambda}}_{\mathrm{magic}}$ reported at 797.75 nm which was seen supporting a blue detuned trap in their work is now estimated at 771.03 nm and is supporting a red detuned trap. Also, none of our results match with the earlier results for circularly polarized light. Moreover, our static values of $\ensuremath{\alpha}$ will be very useful for guiding experiments to carry out their measurements.
The fluorescence of a forbidden transition from the ground state
(5I, J''=4) to the 18 249.275-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ (J=6) level in Nd i was directly observed by atomic-beam laser spectroscopy. This transition was found to be caused by the second-order hyperfine perturbation from the close-lying 18 249.784-${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ (J'=5) level. Thus the fluorescence peaks were observed only for odd isotopes of
143Nd and
145Nd, and no peaks for even isotopes. Both the relative frequencies and the relative intensities of all the observed forbidden transitions agree well with the theoretical calculation.
Resonant enhancement by a factor of over 20 of direct two-photon absorption from the ground state
7F6 to the excited state
5G6 of the 4
f8 configuration of the
Tb3+ ion at 40 200 ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$ has been observed in time-resolved experiments with two separate pulsed lasers. The results provide clear evidence for resonant enhancement of two-photon absorption in rare-earth compounds and imply a similar enhancement for Raman scattering and nonlinear-optical mixing. Two separate contributions to the direct transition moment were observed. When a single laser frequency was used, the intermediate states which made the largest contribution were from excited configurations of opposite parity; those intermediate states were far from resonance. Detailed two-frequency experiments showed, however, that near a single photon
7F6${\mathrm{\ensuremath{-}}}^{5}$
D4 resonance, a much stronger contribution arose from the 4
f8 configuration
5D4 intermediate state. Two-photon absorption was clearly isolated from two-step (incoherent) excitation by varying the timing of two separately triggered lasers.