Recent experiments on time-resolved hole-burning (TRHB) [1-3] and nonexponential photon-echo decays [4,5] in low-temperature ion-doped crystals have demonstrated the multi-timescale dynamics of host nuclei whose spin flips provide the main source of stochastic fluctuations Ut of an impurity ion transition frequency ωab(ωabt=ωab+Ut) leading to optical dephasing and spectral diffusion. Such complicated spin dynamics results from the existence around the impurity ion of the "frozen core" of nuclear spins with their flip rates to be greatly slowed compared to the unperturbed bulk spins.
Flip-flop processes due to magnetic dipole-dipole interaction between neighboring ions in crystals doped by rare-earth ions are one of the mechanisms of relaxation between hyperfine levels. Modeling of this mechanism has so far been macroscopic, characterized by an average rate describing the relaxation of all ions. Here, however, we present a microscopic model of flip-flop interactions between individual nuclear spins of dopant ions. Every ion is situated in a unique local environment in the crystal, where each ion has different distances and a unique orientation relative to its nearest neighbors, as determined by the lattice structure. Thus each ion has a unique flip-flop rate and the collective relaxation dynamics of all ions in a bulk crystal is a sum of many exponential decays, giving rise to a distribution of rates rather than a single average decay rate. We employ this model to calculate flip-flop rates in
Pr3+:
Y2SiO5 and show experimental measurements of population decay of the ground state hyperfine levels at $\ensuremath{\sim}2$ K. We also present a method to measure rates of individual transitions from hole burning spectra that requires significantly fewer fitting parameters in theoretical rate equations compared to earlier work. Furthermore, we measure the effect of external magnetic field on the flip-flop rates and observe that the rates slow down by two orders of magnitude in a field of 5--10 mT.
Frequency-domain optical hole burning of the
R1 line
[4A2(-1/2)\ensuremath{\leftrightarrows}E\ifmmode\bar\else\textasciimacron\fi{}(-1/2) transition] in dilute ruby is studied at 2 K and with a field of 3.6 kG applied along the c axis. A pump-probe (PP) technique is used with precise probe-frequency shifting obtained by an acousto-optic modulator. For sufficiently short PP pulses (50 \ensuremath{\mu}s), two holes (narrow and wide) are observed. The narrow (\ensuremath{\sim}20 kHz half width at half maximum at low power) hole-width dependence on Rabi frequency approximately agrees with that inferred from free-induction-decay measurements. For the present concentration (0.0034 wt. %
Cr2O3) and field, the narrow hole width is determined by fluctuating fields produced by Cr-Cr spin flips [A. Szabo, T. Muramoto, and R. Kaarli, Phys. Rev. B 42, 7769 (1990)]. When the PP delay time is varied, the amplitude of the narrow hole decays with a 1/e time constant of \ensuremath{\sim}300 \ensuremath{\mu}s compared to the E\ifmmode\bar\else\textasciimacron\fi{} fluorescence lifetime of 4 ms. The wide hole (\ensuremath{\sim}500 kHz half width at half maximum) is due to Al-Cr superhyperfine interactions and is created by a spectral diffusion mechanism driven by Al-spin flipping in the host lattice.
Random frequency modulations exist in many systems, such as magnetic system and glasses at low temperature, dye solution. In magnetic systems, spin flips of doped ions or nuclei in the host lattice can cause the magnetic fluctuations at the impurity site, and hence the stochastic change of the transition frequency of the impurity ions. It is the main source of doped ions' optical dephasing at low temperature. Because the interactions have various time scales and strengths, they can lead to a time dependent broadening spectral line (heterogeneous line), the process is socalled spectral diffusion. Several theoretical methods have been used to explain this phenomenon. They all divided the interactions into only two time scales, the fast one results from spin flipping of the bulk nuclei, and the slow one comes from the contributions of the nearby nuclear spins in a frozen core. However, to describe the complete dynamic process in practical systems, these two time scales are obviously not enough. In addition, spectral diffusion is a typical frequency domain process, but almost no theoretical work started from the pure frequency domain point.In this paper , the spectral diffusion in paramagnetic ions doped system at high field, in which has the modulation frequency dependent jump rates, is studied in both frequency and time domains. In general, the spectral diffusion goes as [1-exp(- WT w )] x . Without frozen core, the values of x are 1/2 and 1 for frequency and time domains, respectively. Considering frozen core effect, the spectral diffusions are slow down by the factor of ( ω eff / ω 0) 2, where ω 0 is the frequency characterized frozen core, and ω eff are ω max and H ∞ for frequency and time domains, respectively. When the frozen core is large enough, the spectral diffusions in two domains tend to have the same form as x ≈0 22.
Following a brief review of laser frequency and time domain techniques for overcoming inhomogeneous broadening of optical transitions in solids, recent studies on the failure of the optical Bloch equations (OBE) in describing saturation and coherent transient phenomena in ruby will be discussed. Experiments using free induction decay (FID), as in the original work on the system Pr3+:LaF3 by DeVoe and Brewer1, as well as extensions to rotary echos2 and hole-burning3 will be described. At low fields (~ 0.4 T), the time and frequency domain linewidth variation with Rabi frequency agree provided care is taken to avoid line broadening caused by spectral diffusion4. Recent work at high fields (~ 5 T), where Cr-Cr electron spin flipping is suppressed5, shows that spectral diffusion decreases and the dephasing time increases. Observed frequency domain hole widths as narrow as 18 kHz (FWHM) have been obtained for the 4 A2(−1/2)→E¯(−1/2) optical transition in dilute ruby using a pump-probe technique4. Work with such narrow holes emphasizes the fundamental limits imposed by the uncertainty principle and as an aside, a thought experiment where single photons are transmitted through "frequency slits" will be discussed.
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.