Optical free-induction dephasing times as long as 16 \ensuremath{\mu}sec, corresponding to an optical homogenous linewidth of 10 kHz, have been observed for the $^{3}H_{4}\ensuremath{\leftrightarrow}^{1}D_{2}$ transition of
Pr3+ ions in La
F3 at 2\ifmmode^\circ\else\textdegree\fi{}K. Measurements are facilitated by a frequency-locked cw dye laser and a new form of laser frequency switching. Zeeman studies reveal a Pr-F dipole-dipole dephasing mechanism where the Pr nuclear moment is enhanced in both
1D2 and
3H4.
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.
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.
This paper provides the first experimental test of the Bloch equations for an optical transition in a low-temperature impurity-ion solid,
Pr3+: La
F3, where the condition ${T}_{1}\ensuremath{\gg}{T}_{2}$ applies. Intensity-dependent optical free-induction-decay measurements are performed by use of an ultrastable cw dye laser (linewidth, 300 Hz) and reveal an $\mathrm{exp}(\ensuremath{-}\frac{2t}{{T}_{2}})$ decay law at low Rabi frequency ($\ensuremath{\chi}$), as predicted by the Bloch equations, and an $\mathrm{exp}(\ensuremath{-}\ensuremath{\chi}t)$ decay at high $\ensuremath{\chi}$, which violates the Bloch prediction but agrees with Redfield's characterization of NMR saturation in solids.
In a recent experiment of DEVOE and BREWER [1], it was concluded that the optical Bloch equations are incapable of describing the saturation phenomena observed. Optical free-induction decay (FID) measurements of the impurity ion crystal Pr3+:LaF3 were conducted,where the Pr3+ ions are cohertntly prepared by a laser field under steady-state conditions,and then freely precess when the driving field is removed. At low optical fields, the observed Pr3+ optical line width is dominated by magnetic fluctuations.arising from pairs of fluorine nuclear flip-flops. At high optical fields, this nuclear broadening mechanism is quenched.and the Bloch equations are seriously violated. On physical grounds, this failure is due to a time-averaging of the magnetic interaction as the optical nutation frequency increases [2]. The phenomenological dipole dephasing time T2 of the Bloch equations is therefore not a true constant,but lengthens with increasing field strength.
We have measured the optical dephasing time
T2 of the $^{1}A_{1}\ensuremath{\leftrightarrow}^{1}E$ zero-phonon transition of the
F3+ color center in NaF by optical free-induction decay. In this way we eliminate the effect of the 54-GHz inhomogeneous broadening, and find a homogeneous linewidth of 20 \ifmmode\pm\else\textpm\fi{} 5 MHz ${T}_{2}=16\ifmmode\pm\else\textpm\fi{}4$ nsec). We have also measured a spontaneous emission lifetime of ${T}_{1}=10\ifmmode\pm\else\textpm\fi{}2$ nsec. In the range 1.5-4 K, our results are consistent with the relation
T2=2T1, indicating that optical dephasing is limited by population-decay processes. This relation was even more clearly demonstrated by performing the same experiment on the
D lines of sodium vapor at 3 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}5}$ Torr, where the effects of collisions are negligible. We find that the dephasing is again limited by radiative decay, and that ${T}_{2}(32\ifmmode\pm\else\textpm\fi{}3 \mathrm{nsec})=2{T}_{1}(16.3\ifmmode\pm\else\textpm\fi{}0.4 \mathrm{nsec})$.