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The Journal of Chemical Physics

Manifestation of single macromolecule quantum dynamics in photon distribution function of blinking fluorescence

Игорь Сергеевич Осадько, V. V. Fedyanin

Published Feb 10, 2009• 22 citations
Publisher

Abstract

Distribution function w(N)(T) for photons created by three-level nanoparticle in time interval T under cw laser excitation is calculated for various methods of photon counting. It is found that each exponential process exp(-lambda(i)t) in quantum dynamics of three-level nanoparticle manifests itself via Poissonian function P(N)(lambda(i)t)=(lambda(i)t)(N) exp(-lambda(i)t)/N! in the photon distribution function w(N)(T). The distribution function w(N)(T) is expressed via two or three integrals of two or three Poissonian functions P(N)(lambda(i)t). The simple mathematical expression for w(N)(T) enables one to calculate photon distribution in blinking fluorescence with on and off intervals. A scaling between photon distribution function w(N)(T) and photoelectric pulse distribution function w(n)(T) is found. Comparison of the theoretical distribution w(n)(T) and the distribution measured in blinking fluorescence of single polymer molecule dPPV-PPyV and complex organic molecule 1,1(')-didodecyl-3,3,3('),3(')-tetramethylindocarbocyanine perchlorate (DiI) is carried out. The theoretical distributions are able to describe those found in an experiment.

DOI
10.1063/1.3055287

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Distribution function w(N)(T) for photons created by three-level nanoparticle in time interval T under cw laser excitation is calculated for various methods of photon counting. It is found that each exponential process exp(-lambda(i)t) in quantum dynamics of three-level nanoparticle manifests itself via Poissonian function P(N)(lambda(i)t)=(lambda(i)t)(N) exp(-lambda(i)t)/N! in the photon distribution function w(N)(T). The distribution function w(N)(T) is expressed via two or three integrals of two or three Poissonian functions P(N)(lambda(i)t). The simple mathematical expression for w(N)(T) enables one to calculate photon distribution in blinking fluorescence with on and off intervals. A scaling between photon distribution function w(N)(T) and photoelectric pulse distribution function w(n)(T) is found. Comparison of the theoretical distribution w(n)(T) and the distribution measured in blinking fluorescence of single polymer molecule dPPV-PPyV and complex organic molecule 1,1(')-didodecyl-3,3,3('),3(')-tetramethylindocarbocyanine perchlorate (DiI) is carried out. The theoretical distributions are able to describe those found in an experiment.
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.
Simple expressions have been derived for three photon distribution functions w (T), w (T), and w (T) corresponding to three different methods for counting fluorescence photons from a single nanoparticle excited by continuous laser radiation. In contrast to the previously derived expressions represented in the form of N multiple integrals, the new expressions contain only single or double integrals of Poisson functions, which makes it possible to easily perform the numerical calculation of the photon distribution. The simplest photon counting method corresponds to the lengthiest function w (T); on the contrary, the simplest function w (T) corresponds to the most complex photon counting method. The functions w (T), w (T), and w (T) are noticeably different in short time intervals T; however, the distributions calculated using these functions are almost indistinguishable from each other in long T intervals. This circumstance makes it possible to use the simplest function w (T) to consider the photon statistics measured by the simplest method. This possibility is particularly important for investigating the fluorescence photon statistics, where the intensity fluctuates.
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.
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.