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Israel Journal of Chemistry

Fluctuations and single molecules

Taras Plakhotnik

Published Dec 1, 2004• 5 citations
Publisher

Abstract

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.

DOI
10.1560/m671-xh1n-7pf7-ttgf

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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.
We describe a method to probe the spectral fluctuations of a transition over broad ranges of frequencies and timescales with the high spectral resolution of Fourier spectroscopy, and a temporal resolution as high as the excited state lifetime, even in the limit of very low photocounting rates. The method derives from a simple relation between the fluorescence spectral dynamics of a single radi-ating dipole and its fluorescence intensity correlations at the outputs of a continuously scanning Michelson interferometer. These findings define an approach to investigate the fast fluorescence spectral dynamics of single molecules and other faint light sources beyond the time-resolution capabilities of standard spectroscopy experiments.
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.
The effects of amplitude and phase fluctuations of the pump laser on the resonance fluorescence under two-photon resonant excitation are studied. The averaging of the stochastic Bloch equations is carried out within the approximation scheme that has recently been applied to the case of one-photon resonant excitation. In the case of two-photon resonant excitation the stochastic pump field gives rise to multiplicative nonlinear Gaussian fluctuation effects. The phase fluctuations are treated in the phase-diffusion model, and the averaging over the amplitude fluctuations is performed by means of generating-function methods. Both the intensity and the intensity correlation of the fluorescent light are studied. When the amplitude fluctuations are weak, the cases of one- and two-photon resonant excitations are similar. The difference between both cases consists of a modification of the Rabi frequency and a faster damping of the Rabi oscillations in the two-photon case. For sufficiently fast correlation decay of the amplitude fluctuations their effect can simply be described by a rate. In the general case of n-photon resonant excitation this rate is larger by a factor of n2{n}^{2}n2 than that in the case of one-photon resonant excitation. When the amplitude fluctuations are sufficiently strong, the Rabi oscillations are ``washed out.'' In the case of one-photon resonant excitation both the intensity and the intensity correlation function of the scattered light develop into step functions over time. In the two-photon case the dynamics of both quantities is determined by a new time constant which acts as both the oscillation time and the damping time. Consequently an overshoot peak occurs in both the intensity and the intensity correlation instead of the tendency to become step functions.
Abstract We discuss the statistics of emission of photons by a single atom or ion illuminated by a laser beam at the frequency of quasi-resonance between two energy levels, a situation that corresponds to real experiments. We extend this to the case of two laser beams resonant with the energy differences between two excited levels and the ground state (three level atom in V-configuration). We use a novel approach to this type of problem by considering a Kolmogorov equation for the probability distribution of the atomic state, which takes into account first the deterministic evolution of this state under the effect of the incoming laser beam and second the random emission of photons during the spontaneous decay of the excited state(s) to the ground state. This approach yields solvable equations in the two level atom case. For the three level atom case we set the problem and clearly define its frame. The results obtained are valid in both opposite limits of rare and frequent spontaneous decay, compared to the period of the optical Rabi oscillations due to the interaction between resonant excitation and atomic levels. Our analysis gives access to various statistical properties of the fluorescence light, including one showing that its fluctuations in time are not invariants under time reversal. This result makes evident the fundamentally irreversible character of quantum measurements, represented here by the emission of photons of fluorescence.
Abstract In recent years different microscopic and spectroscopic techniques were successfully employed to detect the fluorescence signals of single dye molecules in the condensed phase. In low temperature solids the fluorescence excitation line of a single dopant molecule is a high resolution optical probe of its local nanoenvironment. As inhomogeneous contributions and ensemble averaging are absent, many effects hidden when looking at large populations of molecules become observable. Two important research areas in single molecule spectroscopy deal with the investigation of quantum optical effects and the spectral jumping behaviour of single molecule excitation lines caused by fluctuations of two‐level systems inherent to disordered solids. Rather than delivering a complete presentation, we will focus on some selected examples which emphasize the unique features of spectroscopy with a single absorber.
Classical time-resolved optical spectroscopy experiments are performed using sequences of ultrashort light pulses, with photon fluxes incident on the sample which are many orders of magnitude higher than real-world conditions corresponding to sunlight illumination. Spectroscopy and microscopy schemes that use quantum states of light have been widely described theoretically with fewer experimental demonstrations that typically require very long measurements that can extend for hours or more. Here, we show that time-resolved spectroscopy with quantum light can be performed without compromising measurement time or wavelength tunability, recording a fluorescence lifetime trace in biological samples in less than a second with acceptable signal-to-noise ratio. Starting from spontaneous parametric down-conversion driven by a continuous-wave laser, we exploit the temporal correlation between randomly generated signal/idler pairs to obtain temporal resolution, and their spectral correlation to select the excitation frequency. We also add spectral resolution in detection, using a 'photon-efficient' Fourier transform approach which employs a common-path interferometer. We demonstrate the potential of this approach by resolving, at the single-photon level, excitation energy transfer cascades from LH2 to LH1 in the photosynthetic membrane and disentangling the lifetimes of two dyes in a mixture. Our results provide a new approach to ultrafast optical spectroscopy, where experiments are performed under illumination intensity conditions comparable to real-world sunlight illumination.