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.
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.
An inhomogeneously broadened spectral line, for a spatially random site distribution with dipole-dipole coupling to first-neighbor sites, is studied using a Monte Carlo Technique. The site-site coupling is chosen at random according to the probability density function for first-neighbor dipolar interactions $\frac{[\mathrm{exp}(\ensuremath{-}\frac{1}{{X}^{\frac{1}{2}}})]}{{X}^{\frac{3}{2}}}$; where X is the transfer rate proportional to r61, and r is the spatial separation of the interacting sites. This form is appropriate for a dilute concentration of sites on a three-dimensional random lattice. We arrange the sites in a linear array, and the spectral diffusion equation (including back transfer) is solved iteratively. We show that this is an adequate description of the three dimensional system at short and intermediate times for the dilute concentrations considered in this study. A group of sites with a specific energy is excited at (time) t=0. The time development of the excitation profile is then studied. For very short times, the decay of the initial excitation is linear in time, appropriate to the largest value of X (near neighbor site). For longer times, but over a fairly narrow time interval, the decay proceeds as $\mathrm{exp}(\ensuremath{-}a{t}^{\frac{1}{2}})$, as calculated by Inokuti and Hirayama for dipolar coupling in the absence of back transfer. At the longest times, the decay follows the one-dimensional diffusion result, ${t}^{\ensuremath{-}\frac{1}{2}}$. The intermediate-time regime, covering most of the decay of the initial excitation profile, does not seem to follow any simple time dependence.
We develop a systematic description of spectral diffusion of ideal chromophores interacting with incoherently relaxing two-state, localized environmental degrees of freedom ("spins") for general initial environment configurations. We remedy the existing, incomplete treatments by formulating the problem in terms of the proper correlation function and by obtaining an accurate solution for generic aperiodic arrangements of environmental spins, nearly free of the customary simplifying assumptions on the multiparticle spin coordinate distribution. We report and estimate, for the first time, the effects of the drift and distortion of a narrow spectral line that arise when the line is not in the center of the inhomogeneous band. While the drift turns out to be modest in most ensemble measurements, accounting for its effects is imperative in analyzing single chromophore spectral jumps, to which end the authors propose a novel experiment. Further, we argue that by employing a sufficiently large chromophore one can decouple the concentration of the fluctuating centers from the strength of their interaction with the chromophore. Finally, the additional line broadening, owing to a distribution of the central chromophore frequencies, is evaluated. Upper estimates for an analogous broadening stemming from a nonequilibrium environment are made.
A dynamic theory of two-photon correlators for a single impurity center developed recently by the author [I. S. Osad’ko, Zh. Éksp. Teor. Fiz. 113 , 1606 (1998) [JETP 86 , 875 (1998)] has been generalized to the case in which the center interacts with nonequilibrium two-level systems (TLSs) in polymers and glasses. Quantum tunneling transitions in TLS manifest themselves as random jumps of a spectral line of an impurity center. These jumps can be either spontaneous or light-induced. Interaction between the impurity center and many nonequilibrium TLSs, which exist in polymers, results in a time dependence of the optical dephasing rate 1/ T 2 of an impurity molecule, i.e., in spectral diffusion. This paper describes how the jumps of the spectral line manifest themselves in the two-photon correlator, which can be measured in experiments.
A two level model of a single molecule undergoing spectral diffusion dynamics and interacting with a sequence of two short laser pulses is investigated. Analytical solution for the probability of n=0,1,2 photon emission events for the telegraph and Gaussian processes is obtained. We examine under what circumstances the photon statistics emerging from such pump-probe setup provides new information on the stochastic process parameters and what are the measurement limitations of this technique. The impulsive and selective limits, the semiclassical approximation, and the fast modulation limit exhibit general behaviors of this new type of spectroscopy. We show that in the fast modulation limit, where one has to use impulsive pulses in order to obtain meaningful results, the information on the photon statistics is contained in the molecule's dipole correlation function, equivalently to continuous wave experiments. In contrast, the photon statistics obtained within the selective limit depends on the both spectral shifts and rates and exhibits oscillations, which are not found in the corresponding line shape.
Picosecond photon echo (PE) experiments are used to examine the optical dephasing and intermolecular interactions of organic chromophores in a variety of glassy organic and polymer hosts. The inherent disorder in these systems leads to dynamics which span times from picoseconds to hours and longer. The temperature dependent optical dephasing as measured by the PE follows a power law characteristic of tunnelling twolevel systems (TLS) at low temperatures (1 °K to 4 °K) and is exponentially activated at higher temperatures due to dephasing by additional modes. The nature of these modes will be addressed. These data and those from other line narrowing experiments such as hole burning allow the study both of fast dephasing processes and spectral diffusion due to slow fluctuations in the host. The observed lineshapes and holewidths require dipolar coupling between the chromophore and TLS and offer information on the distribution of relaxation rates. Furthermore, techniques sensitive to different timescales are used to examine host dynamics by removing the guest-host coupling from consideration. The data are discussed in conjunction with recently developed theoretical treatments of dephasing in disordered systems.