We report the observation of optically resolved isotope, hyperfine, and superhyperfine structure in the spectrum of
Nd3+ ions in
YLiF4, isotopically pure in
7Li. This observation was made on the ${(4f}^{3}\ensuremath{-}{4f}^{3}),$
4I9/2(1)4F3/2(1) transition of
Nd3+ ions present as impurities at ppm levels. The optical linewidths of the transitions of the even mass number isotopes,
142Nd, 144Nd, 146Nd, 148Nd, and
150Nd, are only 45 MHz which makes it very easy to resolve the isotope shift of 115 MHz/unit mass. The hyperfine structure of the two odd mass number isotopes,
143Nd and
145Nd, are resolved and the hyperfine parameters are determined in both the ground and electronically excited states. Some of the transitions have linewidths as narrow as 10 MHz. These are the narrowest optical inhomogeneous linewidths observed in solids. In addition, each even isotope line is split by 30 MHz. The observed splitting is assigned to a superhyperfine coupling between
Nd3+ and its nearest-neighbor fluorine nuclei. Here, a superhyperfine splitting has been resolved in an optical spectrum. The optical line shapes for the even isotopes are calculated as the sum of all transitions between the superhyperfine components
(Nd3+ with its eight nearest-neighbor F nuclei) of the ground and excited electronic states, considering both magnetic dipole-dipole (including spin-transfer contributions) and contact interactions. The results are in quantitative agreement with experiment.
Remarkably narrow optical homogeneous linewidths of the order of 1 kHz have now been observed in low temperature zero‐phonon transitions of dilute impurity ion crystals, such as Pr3+ in LaF3. Novel nonlinear optical resonance techniques have been devised for this purpose using ultrastable phase locked cw dye lasers where the measurements are performed either in the frequency domain (hole burning) or in the time domain (coherent optical transients). These studies effectively bring the Mossbauer effect into the optical region. Hence, the observed linewidths are no longer limited by inhomogeneous strain broadening (∼5 GHz) or even by static local fields due to neighboring spins (∼100 kHz). However, weak magnetic field fluctuations from local spins are readily detected. As an example, spin decoupling and line narrowing, which are well known in NMR, are observed in an optical transition of Pr3+:LaF3 at 2 K where the 19F– 19F dipolar interaction is quenched and the optical linewidth drops from 10 to 2 kHz, clearly demonstrating the spin broadening mechanism. Results will be discussed in terms of a Monte Carlo line broadening theory.
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 optical Bloch equations which incorporate the phenomenological population
(T1) and dipole dephasing
(T2) times have been tested recently by optical free-induction-decay (FID) measurements on an impurity-ion crystal
Pr3+:
La3 at 1.6 K. At low optical fields, the observed
Pr3+ optical linewidth is dominated by magnetic fluctuations arising from pairs of fluorine nuclear flip-flops where the condition ${T}_{1}\ensuremath{\gg}{T}_{2}$ prevails. At elevated fields, this nuclear broadening mechanism is quenched and the Bloch equations are violated with ${T}_{2}\ensuremath{\rightarrow}{T}_{1}$. In this paper, a microscopic theory appropriate for a low-temperature impurity solid is presented which reveals the above features both for optical and radio frequencies, and a simple physical interpretation of this line narrowing phenomenon is given. Modified Bloch equations of a novel form are derived to second order and yield analytic FID solutions over the entire range of optical-field strength. A discussion of the earlier NMR theories is given, pointing out similarities and differences.
Photon echo measurements made at 2 °K on the 3H4 - 1D2 transition in Pr3+:LaF3 show that magnetic dipolar couplings between the Pr and F nuclei account for the 56 kHz homogeneous linewidth of this transition.1 The homogeneous broadening arises from the enhanced 141Pr nuclear moment (I=5/2) interacting with the local field fluctuations of the 19F nuclear moments undergoing mutual spin flip transitions. Such resonant fluctuations should, in the absence of a fluorine spin diffusion barrier, produce a homogeneous linewidth of about 200 kHz which is, in fact, roughly what is observed for the inhomogeneous broadening of the Pr3+ hyperfine levels and is considerably broader than that obtained by the photon echo measurements. Shelby et al2 proposed a simple model analogous to the spin diffusion barriers responsible for narrowing the homogeneous lines in certain electron paramagnetic resonance transitions3. In such systems, the field produced by the electron magnetic dipole moment (2-3 orders of magnitude larger than the enhanced nuclear moment associated with the ground state of Pr3+ in LaF3) de-tunes the nearest neighbors from each other, prohibiting mutual spin flips among them. Thus, the fields produced by the neighboring spins are static and their interaction with the paramagnetic ion contributes to the inhomogeneous linewidth and not to homogeneous broadening. The de-tuned neighbors are referred to as the "frozen core." Application of this picture to the case of Pr3+:LaF3 is supported by the calculation of Devoe et al.4 Their Monte Carlo calculation of the Pr3+ optical dephasing times shows that the observed homogeneous linewidth could be explained by deleting the nearest neighbor F spins from the lattice of rapidly fluctuating fluorine moments, in effect, treating them as part of a frozen core.
Narrow linewidth (high-Q) quantum discriminators are essential for realizing solid-state
229Th nuclear clocks, yet the observed transition linewidths remain at the tens-to-hundreds-of-kilohertz level, far exceeding the sub-Hertz linewidth set by the radiative lifetime of the nuclear isomer. Here, we investigate the origin of this broadening in
229Th:CaF
2 through systematic measurements of the linewidth as functions of temperature, thorium concentration, and quadrupole state, accompanied by theoretical analysis and first-principle simulations. We find that the linewidth is independent of temperature, increases linearly with thorium concentration, and depends strongly on the quadrupole state. Data analysis identifies variations of the local electric-field gradient (EFG) as the dominant source of inhomogeneous broadening and reveals that the EFG disorder increases linearly with thorium concentration. The observed Lorentzian line shapes and linewidth linear concentration dependence are explained naturally from the
1/r3 scaling of a random distribution of long-range dopant-induced perturbations. Complementary atomistic and \textit{ab initio} density-functional-theory calculations reproduce these characteristic features and provide insights into the microscopic defect configurations responsible for the broadening. These results establish the dominant broadening mechanism in solid-state
229Th and provide guidance for engineering nuclear-clock materials with reduced inhomogeneous broadening.