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Optics and Spectroscopy

Fluctuations in the absorption coefficient of a single molecule at different signal acquisition times

Игорь Сергеевич Осадько, A. N. Lobanov

Published Aug 1, 2005• 3 citations
Publisher

Abstract

The relation between the absorption coefficient of a single molecule calculated within the framework of a dynamical theory and the fluctuations in the absorption coefficient measured in experiments on scanning the laser frequency is analyzed in relation to the acquisition time of fluorescence photons. The analysis is performed for the case where the optical band of an impurity center consists of two well-resolved lines. A method of processing the fluctuating fluorescence is proposed that makes it possible to obtain the averaged absorption coefficients, which can be compared with the coefficients calculated theoretically.

DOI
10.1134/1.2034617

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The relation between the absorption coefficient of a single molecule calculated within the framework of a dynamical theory and the fluctuations in the absorption coefficient measured in experiments on scanning the laser frequency is analyzed in relation to the acquisition time of fluorescence photons. The analysis is performed for the case where the optical band of an impurity center consists of two well-resolved lines. A method of processing the fluctuating fluorescence is proposed that makes it possible to obtain the averaged absorption coefficients, which can be compared with the coefficients calculated theoretically.
Various theories that use the sudden jump model to explain the time dependence of the optical band of a single molecule in a fluctuating environment are analyzed. It is shown that the dynamic theory of the optical band, based on the Hamiltonian of the system, should necessarily include the initial conditions for the quantum system. We show how these initial conditions should be chosen for the dynamic theory to properly describe the quantum jumps of the optical band of a single molecule. Unlike stochastic theories based on the sudden jump model, the dynamic theory predicts several absorptances of a single molecule and directly connects the absorptance fluctuations with this fact. The theory is used to account for the time dependence of the broadening of experimentally measured lines of individual molecules.
Abstract The fluorescence of single molecules coupled to a thermal bath is studied both experimentally and theoretically. The effect of different fluctuations on the coherence properties of resonance fluorescence is considered first. Coherence is measured in an interference experiment where a single molecule is used as a light source. A standard approach based on the optical Bloch equations apparently provides quite an accurate description of the interference experiment. Systems with long correlation times (where spectra are time dependent on any timescale) are considered next. It is shown that intensity—time—frequency correlation spectroscopy, which provides both high signal‐to‐noise ratio and high time resolution, is very suitable for such a case. The Bloch equations are further tested in an experiment where the shape of an excitation spectral line of a single molecule is accurately measured over six orders of magnitude of the exciting laser power. Significant deviations from the predictions of the Bloch equations are found. The role of critical parameters—the correlation time of the bath, the Rabi oscillation period, and the coupling constant between the bath and the molecule—is discussed. The paper also includes a short general introduction to the methodology of single‐molecule studies.
A theory is formulated for time dependent fluctuations of the spectrum of a single molecule in a dynamic environment. In particular, we investigate the photon counting statistics of a single molecule undergoing a spectral diffusion process. Based on the stochastic optical Bloch equation, fluctuations are characterized by Mandel's QQQ parameter yielding the variance of number of emitted photons and the second order intensity correlation function, g(2)(t)g^{(2)}(t)g(2)(t). Using a semi-classical approach and linear response theory, we show that the QQQ parameter can be described by a three-time dipole correlation function. This approach gene ralizes the Wiener-Khintchine formula that gives the average number of fluorescent photons in terms of a one-time dipole correlation function. We classify the time ordering properties of the three-time dipole correlation function, and show that it can be represented by three different pulse shape functions similar to those used in the context of nonlinear spectroscopy. An exact solution is found for a single molecule whose absorption frequency undergoes a two state random telegraph process (i.e., the Kubo-Anderson sudden jump process.) Simple expressions are obtained from the exact solution in the slow and fast modulation regimes based on appropriate approximations for each case. In the slow modulation regime QQQ can be large even in the long time limit, while in the fast modulation regime it becomes small.
By severely reducing the number of solute molecules in the illuminated sample, the optical resonances of individual molecules can be resolved in a fluorescence excitation spectrum. Single molecule lines can be studied as a function of time and temperature: sudden jumps of their resonance frequencies are due to spectral diffusion processes. The signal from a single molecule displays specific correlations which makes time-resolved studies possible. Here, emphasis is put on photon-bunching arising from intersystem crossing (ISC). ISC rates are deduced from the observed decay rates of the correlation and are found to differ from molecule to molecule. A single molecule is a truly local probe of its environment by means of which fundamental studies of the matrix dynamics as well as nanophysics experiments may be undertaken.
We derive the moment generating function for photon emissions from a single molecule driven by laser excitation. The frequencies of the fluoresced photons are explicitly considered. Calculations are performed for the case of a two-level dye molecule, showing that measured photon statistics will display a strong and nonintuitive dependence on detector bandwidth. Moreover, it is demonstrated that the antibunching phenomenon, associated with negative values of Mandel's Q parameter, results from correlations between photons with well separated frequencies.
A method of calculation of multiple-time correlation functions of the resonance fluorescence light from a two-level atom driven by a realistic laser field with phase and amplitude fluctuations is developed. Both kinds of fluctuations are assumed to be Gaussian. The phase fluctuations are treated in the phase-diffusion model. The averaging over the amplitude fluctuations can be performed in a closed way when the relative mean-square amplitude fluctuation is small compared with the ratio of the amplitude correlation decay rate to the characteristic atomic relaxation rate. There is no restriction to Markovian amplitude fluctuations. Arbitrary values for the mean Rabi frequency are allowed. Results are presented for the intensity, the intensity correlation function, and the spectrum of the fluorescent light. It is shown that even in the case when the relative strength of the amplitude fluctuations is small, significant effects can occur due to the finite correlation length of the fluctuations. The results for fast amplitude correlation decay correspond to the situation without amplitude fluctuations. When the correlation time of the amplitude fluctuations becomes comparable with the atomic relaxation times, the Rabi oscillations of the intensity and the intensity correlation function are nearly washed out, and non-Lorentzian line shapes for the side peaks in the spectrum are observed. It is found that the ratio of the heights of the central peak to the side peak increases with increasing correlation time.