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.
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.
Transient hole-burning and free induction decay (FID) in low-temperature ion-doped crystals are studied consistently using the model of two jump processes Ut equals Ut1 + Ut2 characterized by essentially different jump rates v1 >> v2 and distribution widths (sigma) 1 << (sigma) 2 for the impurity ion optical transition frequency fluctuations Ut due to the bulk and the frozen core spin flipping. At short pump/probe pulsewidth T and short delay time (tau) d ((tau) d, T << v2-1) the calculated hole shape is narrow with the hole width determined by Ut1 process characteristics only. The hole width in this case coincides with the calculated FID rate. At long delay (tau) d >= v2-1 the hole becomes a wide lorentzian with the halfwidth 2(sigma) 2. The theory fits well the experimental data by A. Szabo et. al. on narrow hole burning and FID in ruby under low and high magnetic fields supposing the fluctuations Ut1 to be rather slow ((sigma) 21/v21 equals 0.5).
The mechanisms and effects of spectral diffusion for optical transitions of paramagnetic ions have been explored using the inhomogeneously broadened 1536 nm $^{4}\mathrm{I}_{15∕2}\ensuremath{\rightarrow}^{4}\mathrm{I}_{13∕2}$ transition in
Er3+:Y2SiO5. Using photon echo spectroscopy, spectral diffusion was measured by observing the evolution of the effective coherence lifetimes over time scales from $1\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}\mathrm{s}$ to 20 ms for magnetic-field strengths from 0.3 to 6.0 T, temperatures from 1.6 to 6.5 K, and nominal
Er3+ concentrations of 0.0015%, 0.005%, and 0.02%. To understand the effect of spectral diffusion on material decoherence for different environmental conditions and material compositions, data and models were compared to identify spectral diffusion mechanisms and microscopic spin dynamics. Observations were successfully modeled by ${\mathrm{Er}}^{3+}\ensuremath{-}{\mathrm{Er}}^{3+}$ magnetic dipole interactions and
Er3+ electron spin flips driven by the one-phonon direct process. At temperatures of 4.2 K and higher, spectral diffusion due to
89Y nuclear spin flips was also observed. The success in describing our extensive experimental results using simple models provides an important capability for exploring larger parameter spaces, accelerating the design and optimization of materials for spatial-spectral holography, and spectral hole-burning devices. The broad insight into spectral diffusion mechanisms and dynamics is applicable to other paramagnetic materials, such as those containing
Yb3+ or
Nd3+.
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.
We consider the spectral diffusion of a chromophore coupled through dipolar interactions to a regular lattice of flipping two-level systems. In particular, we calculate the spectral diffusion kernel, P(ω,t‖ω0), which is the conditional probability density that the chromophore will have transition frequency ω at time t, given that it had frequency ω0 at time 0. At very short times we find that the spectral diffusion kernel is Lorentzian, for any value of the two-level system excitation probability, p. For longer times the form of the spectral diffusion kernel depends on the value of p. We derive several approximate expressions for the spectral diffusion kernel, all of which go to the correct equilibrium distribution of frequencies for long times. For p≂1/2, when the frequency distribution is nearly Gaussian, we find that the spectral diffusion kernel is not at all Gaussian for short times. We compare all of our approximate expressions with numerically exact results. Motivated by certain optical spectral diffusion experiments on individual molecules in solids, we also calculate P(Δ;t), the distribution of spectral jumps, which is the probability density that the chromophore’s frequency will change by an amount Δ in time t. In a subsequent paper we will analyze these spectral diffusion experiments with our results.
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.