Using optical detection, we have observed rf nutation, free-induction decay, and echoes, on the 16.70-MHz nuclear hyperfine transition of dilute
Pr3+ in La
F3 at 2 K. From a comparison of free-induction decay and echo decay we find that the previously observed linewidth of 200 kHz is inhomogeneous and that the homogeneous width is 19 kHz. On the basis of these measurements it is suggested that optical dephasing times for the $^{1}D_{2}\ensuremath{\leftrightarrow}^{3}H_{4}$ and $^{3}P_{0}\ensuremath{\leftrightarrow}^{3}H_{4}$ transitions are as long as \ensuremath{\sim}20 \ensuremath{\mu} sec.
Recent optical free-induction-decay (FID) measurements of the impurity ion
Pr3+ in La
F3 at 2 \ifmmode^\circ\else\textdegree\fi{}K reveal optical homogeneous linewidths of only a few kilohertz, considerably narrower than the inhomogeneous broadening due to crystalline strains (5 GHz) or the static local magnetic fields of the
19F nuclei (100 kHz). In this regime, the homogeneous broadening arises from local field fluctuations, as in NMR, and is due to the
19F nuclei which undergo resonant flip-flops and modulate the
Pr3+ optical transition frequency. We treat the optical response of a two-level quantum system to an intense coherent field and to fluctuating perturbations using a Monte Carlo computer routine that assumes (1) the La
F3 crystal structure and (2) a sudden fluorine spin-flip model. This procedure avoids many of the approximations of previous analytic theories of spectral diffusion in magnetic resonance and extends the calculation specifically to optical FID. The decay behavior is obtained by sampling statistically the
Pr3+ phase history as subgroups of
19F spins flip randomly in space and time. These fluctuations modify the Bloch equations where the solutions for the preparative and post-preparative periods are obtained by numerical integration. In spite of the large lattice size assumed (2250 fluorines), only a few
19F spins contribute substantially to the homogeneous width, a result which shows for the first time that spin-flip correlations are not significant. Furthermore, a
Pr3+ ion polarizes and detunes the nearest fluorines forming a frozen core that is incapable of resonant spin flipping with the bulk fluorines. We demonstrate that the core grows radially as the
141Pr(I=25) magnetic moment increases with
Iz, but the Pr optical linewidth changes little, producing essentially one rather than three linewidths. Our calculations utilize no free parameters and predict a Lorentzian line shape of 8.4 kHz half-width at half maximum which compares to the optical FID observation of a 10.1-kHz Lorentzian. The Monte Carlo algorithm is verified further by the static local magnetic broadening of a Pr quadrupole transition which is found to be Gaussian, 82 kHz full width at half maximum, in agreement with a second-moment calculation and current observations.
The dephasing time of coherently prepared
Pr3+ impurity ions in a La
F3 crystal is measured by optical free-induction decay. Above 4\ifmmode^\circ\else\textdegree\fi{}K, phonon processes severely shorten the decay time. Below 4\ifmmode^\circ\else\textdegree\fi{}K, we observe a temperature-independent decay time of 0.38 \ensuremath{\mu}sec. This is surprisingly short considering that there are no contributions from first-order hyperfine, phonon, or
Pr3+-
Pr3+ interactions.
The sizable ground-state nuclear polarization which accompanies the optical pumping of the $^{3}H_{4}\ensuremath{-}^{1}D_{2}$ (16 872-${\mathrm{cm}}^{\ensuremath{-}1}$) $\ensuremath{\sigma}$ transition of
Pr3+ in La
F3 was utilized to observe enhanced and saturated absorptions within the inhomogeneously broadened spectral line. The observed hyperfine splittings of the lowest
1D2 state are 3.7 and 4.7 MHz at 2.0\ifmmode^\circ\else\textdegree\fi{}K. These splittings were fitted by a spin Hamiltonian $\mathcal{H}=D[{I}_{z}^{2}\ensuremath{-}\frac{I(I+1)}{3}]+(\frac{E}{2})({I}_{+}^{2}+{I}_{\ensuremath{-}}^{2})$, where $D=\ensuremath{-}1.10\ifmmode\pm\else\textpm\fi{}0.05$ and $E=\ensuremath{-}0.48\ifmmode\pm\else\textpm\fi{}0.05$ MHz. The observed linewidths (including the exciting laser linewidth) were 0.8 MHz. The deconvoluted optical transition linewidths (full widths at half-maximum) are 0.20\ifmmode\pm\else\textpm\fi{}0.05 MHz.
By using a new, delayed-heterodyne photon-echo technique, we have obtained a spectral resolving power of 1.2\ifmmode\times\else\texttimes\fi{}
1011, measuring a 2.0-kHz linewidth (half width at half maximum) for the 6105-\AA{} $^{1}D_{2}\ensuremath{\leftrightarrow}^{3}H_{4}$ transition of YAl
O3:
Pr3+ in an external magnetic field. This appears to be the highest resolution ever obtained in optical spectroscopy. Measurements were also made on the corresponding transition of La
F3:
Pr3+ giving a linewidth of 5 kHz. Optical free induction decay gave incorrect results (i.e., significantly broader linewidths).
Using two-pulse photon echo excitation a strong resonance in the measured homogeneous dephasing time as a function of magnetic field is observed in the
3H4${\mathrm{\ensuremath{-}}}^{1}$
D2 transition of
Pr3+:
YAlO3. The resonance could only be observed at low excitation energies since it otherwise was masked by excitation intensity dependent extra dephasing processes in the crystal. Thus intensity dependent dephasing processes may not only affect the apparent dephasing time of a system but may also mask other more subtle interactions in the materials studied. It is reasonable to assume that the type of resonance observed could be caused by Zeeman induced level crossings in the active ion or in the host crystal ions or, possibly, that it could be due to coincidences between Zeeman level transition frequencies. However, based on current data of the Pr:
YAlO3 system the observed resonance could not be unambiguously tied to such effects. \textcopyright{} 1996 The American Physical Society.