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Physical review. B, Condensed matter

Spectral-diffusion decay in echo experiments

Pan Hu, L. R. Walker

Published Aug 1, 1978• 85 citations
Publisher

Abstract

We calculate the decay of phase memory due to spectral diffusion in certain echo experiments. Local-field fluctuations, at the site of resonant two-level systems, controlled by the spin-lattice relaxation of neighboring spins, cause the spin diffusion. The results generalize those previously obtained which were valid only at high temperature.

DOI
10.1103/physrevb.18.1300

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Chinese Journal of Luminescence•Jan 1, 2001•0 citations
We calculate the decay of phase memory due to spectral diffusion in certain echo experiments. Local-field fluctuations, at the site of resonant two-level systems, controlled by the spin-lattice relaxation of neighboring spins, cause the spin diffusion. The results generalize those previously obtained which were valid only at high temperature.
We report a quantitative study of spectral diffusion in a solid-state system by measuring the stimulated echo relaxation. The spectral diffusion process in Pr3+{\mathrm{Pr}}^{3+}Pr3+:YAlO3{\mathrm{YAlO}}_{3}YAlO3​ at low temperatures (2.0 K) is found to be dominated by the relaxation of the ``frozen core'' of nearest neighboring Al spins. The decay of the echo signal is found to be fully described by a simple spin-flip diffusion model and by a first-principle calculation. The complete understanding of the spectral diffusion process in this simple model system establishes a quantitative basis for using coherent optical techniques to study slow dynamics in complex systems.
The two-pulse echo sequence is examined for the case in which the frequency modulation time τm of the transition is intermediate between the well known limiting cases of very fast modulation (motional narrowing) and very slow or static modulation (inhomogeneous broadening). Within this spectral diffusion regime, the interpretation of the echo decay differs markedly from standard treatments. If the frequency-frequency correlation function initially decays as 1−tβ, the echo decay time TE is proportional to τmβ/(β+2). These results reduce to those of Yan and Mukamel [J. Chem. Phys. 94, 179 (1991)] for β=1. Drawing on a viscoelastic model, the theoretical results are compared to viscosity and temperature dependent vibrational echo experiments on myoglobin–CO. A τm1/3 dependence is observed, as is predicted for an exponential decay of the frequency–frequency correlation function.
We have measured the spectral content of the echo radiation emitted by a two-level spin system ([AlO4{\mathrm{AlO}}_{4}AlO4​]0{]}^{0}]0 centers in quartz), after a two-pulse excitation sequence, ${\mathrm{\ensuremath{\theta}}}_{1}$-\ensuremath{\tau}-${\mathrm{\ensuremath{\theta}}}_{2}$. In general, the experimental spectral shapes reproduce qualitatively those calculated for an inhomogeneous system of noninteracting spins. However, we found that in certain ranges of the sequence parameters (${\mathrm{\ensuremath{\theta}}}_{1}$, ${\mathrm{\ensuremath{\theta}}}_{2}$, and \ensuremath{\tau}) the widths of the experimental spectra are less than expected by a factor that amounts up to 0.5. The observed narrowing effect involves the spin packets located in a narrow spectral bandwidth (of the order of the Rabi frequency) around the excitation frequency and occurs whenever the excitation sequence induces a transverse oscillating moment of the strictly resonant packets. The spectral narrowing is a transient effect, in the sense that the spectral widths tend to recover their theoretical values for long interpulse distances \ensuremath{\tau}. We suggest a tentative interpretation which relates the narrowing to the coherence properties of the local field generated by precessional motion of the echo-active spins. \textcopyright{} 1996 The American Physical Society.
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.
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.