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Physical Review A

Liouville-space pathways for spectral diffusion in photon statistics from single molecules

František Šanda, Shaul Mukamel

Published Mar 11, 2005• 25 citations
Publisher

Abstract

The factorial moments of photon counting statistics from a single molecule coupled to a quantum bath are expressed in terms of multipoint quantum correlation functions and represented by double-sided Feynman diagrams, in close formal analogy with nonlinear spectroscopy. At infinite temperature we recover the results of stochastic models of spectral diffusion where the bath dynamics is independent on the state of the system and the moments are described by lower-order correlation functions.

DOI
10.1103/physreva.71.033807

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The factorial moments of photon counting statistics from a single molecule coupled to a quantum bath are expressed in terms of multipoint quantum correlation functions and represented by double-sided Feynman diagrams, in close formal analogy with nonlinear spectroscopy. At infinite temperature we recover the results of stochastic models of spectral diffusion where the bath dynamics is independent on the state of the system and the moments are described by lower-order correlation functions.
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.
We present a general relation between Mandel's QQQ parameter describing variance of number of photons emitted from a single molecule and a three time correlation function describing a spectral diffusion process the molecule is undergoing. For a prototype of such a stochastic process an exact solution is found which exhibits rich types of physical behaviors.
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.
A general theory and calculation framework for the prediction of frequency-resolved single molecule photon counting statistics is presented. Expressions for the generating function of photon counts are derived, both for the case of naive "detection" based solely on photon emission from the molecule and also for experimentally realizable detection of emitted photons, and are used to explicitly calculate low-order photon-counting moments. The two cases of naive detection versus physical detection are compared to one another and it is demonstrated that the physical detection scheme resolves certain inconsistencies predicted via the naive detection approach. Applications to two different models for molecular dynamics are considered: a simple two-level system and a two-level absorber subject to spectral diffusion.