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Optics and Spectroscopy

From quantum light emitted by single molecule to classical light emitted by molecular ensemble

Игорь Сергеевич Осадько, Alexandra Shchukina

Published Nov 1, 2010• 1 citation
Publisher

Abstract

Fluorescence from an ensemble of noninteracting molecules excited by continuous laser radiation is considered as a glow of M individual molecules. The expression that governs the autocorrelation function (ACF) of fluorescence from M molecules is derived based on the theoretical expression that governs the ACF of fluorescence from a single-molecule. Fluctuations of blinking fluorescence and its statistical properties are analyzed using a computer simulation. The results of statistical analysis of the computer experiment are in good agreement with the derived formula that governs the ACF of fluorescence from several molecules.

DOI
10.1134/s0030400x10110184

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Fluorescence from an ensemble of noninteracting molecules excited by continuous laser radiation is considered as a glow of M individual molecules. The expression that governs the autocorrelation function (ACF) of fluorescence from M molecules is derived based on the theoretical expression that governs the ACF of fluorescence from a single-molecule. Fluctuations of blinking fluorescence and its statistical properties are analyzed using a computer simulation. The results of statistical analysis of the computer experiment are in good agreement with the derived formula that governs the ACF of fluorescence from several molecules.
The method previously used by the author for the measurement and calculation of the distribution function w N(T)of fluorescence photons from a single two-level atom that is continuously excited by laser light and has a unity fluorescence quantum yield is generalized to the case of a single molecule whose fluorescence quantum yield is smaller than unity and to the case of a three-level molecule whose fluorescence is blinking. The functions w N(T) calculated for these two cases demonstrate a sub-Poissonian distribution of fluorescence photons.
The stream of photons emitted by a single quantum system such as a molecule or a nanocrystal is often statistically characterized by the distribution of delays between consecutive photons, or by the autocorrelation function of the intensity, or by the distributions of on- and off-times. We derive and discuss general relations between their Laplace transforms, addressing the influence of detection yield and background. Our analytical treatment applies to any distribution of delays and to random telegraph signals, including nonexponential distributions. We examine the special case of systems switching between two states characterized by different distributions of delays, where the switching can obey various statistics. We show that the second-order autocorrelation function keeps track of long-time fluctuations which are obviously lost in averaging the distributions of delays. We apply our formalism to random telegraphs, in particular to those with power-law distributions of on- and/or of off-times, which are encountered in the blinking of single semiconductor nanocrystals.
A single complex molecule with conformational changes (conformations 0 and 2) is considered. When such a molecule is irradiated by cw-laser light it can randomly change the intensity or polarization of its fluorescence due to jumps from one conformation to another. In fact, the molecule manifests itself either like the 0-type or the 2-type emitter. An expression for the matrix ${s}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$ called the start-stop correlator (waiting time distribution) in which $\ensuremath{\alpha}=0,2$ and $\ensuremath{\beta}=0,2$ is derived. An expression for the matrix ${p}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$ called the full correlator is derived as well. It determines the density of the probability of finding an event of $\ensuremath{\alpha}$ type and an event of $\ensuremath{\beta}$ type separated by time interval ttt. A relation between matrices ${s}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$ and ${p}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$ is found. A mathematical expression for the distribution wN(T){w}_{N}(T)wN​(T) of events measured in time interval TTT is derived. It is expressed solely via the matrix ${s}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$. Numerical calculations of the event distribution function for various rates of intra- and interconformational jumps are carried out with the help of the formula for wN(T){w}_{N}(T)wN​(T) and by the Monte Carlo method. Both methods of the calculation yield identical distributions. Fluctuating fluorescence intensity III(ttt) for a bin time of 5 ms is calculated for slow and fast interconformational jumps. A relation is found between the autocorrelation function g(2)(t){g}^{(2)}(t)g(2)(t) of fluorescence measurable in experiments and the matrix ${p}_{\ensuremath{\alpha}\ensuremath{\beta}}(t)$ calculated theoretically.
Various methods of photon counting in single molecule blinking fluorescence are considered and photon distribution functions, , , and related to these methods are derived. It is shown that each exponential process exp(−λ j t) in quantum dynamics of a single molecule manifests itself in the mathematical expressions for the photon distribution function via Poissonian function PN (−λ j t) with the same rate constant. This relation enables one to avoid N-fold integration in the mathematical expressions for the functions, , , and derived in previous works. Numerical calculations of the functions , , and for blinking fluorescence are carried out. A relation between photon distribution and photo-electric pulse distribution is discussed. Thanks to relation found we can use photo-electric pulse distribution functions for studying photon distribution function and for studying quantum dynamics of single nanoparticle with blinking fluorescence.
An analysis of intermittent fluorescence is presented for a single molecule driven by a continuous-wave laser field. The interruptions of fluorescence are caused by transition of the molecule to a triplet state. A method previously developed to calculate photon distribution for continuous-wave fluorescence is applied to analyze photon statistics of fluorescence blinking. The probability w N (T) of counting N photons over a time interval T is derived for intermittent fluorescence. The photons counted over relatively short intervals are found to have a sub-Poissonian (narrower than Poisson) distribution. The photon distribution over intervals longer than the mean off time has a complicated form with two maxima; i.e., a super-Poissonian (wider than Poisson) distribution is obtained.