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The Journal of Chemical Physics

Single molecule photon emission statistics in the slow modulation limit

Yujun Zheng, Frank L. H. Brown

Published Oct 12, 2004• 30 citations
Publisher

Abstract

A framework for calculating photon emission statistics for single chromophores perturbed by slow environmental fluctuations is introduced. When internal chromophore dynamics are significantly faster than time scales for environmental modulation it becomes possible to invoke a type of adiabatic approximation, allowing for straightforward calculation of photon counting moments including explicitly quantum effects. Unlike previous exact treatments, the present methodology involves calculation of dynamics reflecting only the modulation characteristics of the environment and quantum dynamics of an isolated chromophore separately, i.e., the complicated intermingling of chromophore quantum dynamics and the environmental modulation are suppressed via the adiabatic approximation. This leads to significant conceptual and computational simplifications. Within its regime of applicability, the present approximation reproduces exact calculations quantitatively. We demonstrate this accuracy explicitly for the case of a two-level chromophore modulated by a number of different stochastic models.

DOI
10.1063/1.1795671

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A framework for calculating photon emission statistics for single chromophores perturbed by slow environmental fluctuations is introduced. When internal chromophore dynamics are significantly faster than time scales for environmental modulation it becomes possible to invoke a type of adiabatic approximation, allowing for straightforward calculation of photon counting moments including explicitly quantum effects. Unlike previous exact treatments, the present methodology involves calculation of dynamics reflecting only the modulation characteristics of the environment and quantum dynamics of an isolated chromophore separately, i.e., the complicated intermingling of chromophore quantum dynamics and the environmental modulation are suppressed via the adiabatic approximation. This leads to significant conceptual and computational simplifications. Within its regime of applicability, the present approximation reproduces exact calculations quantitatively. We demonstrate this accuracy explicitly for the case of a two-level chromophore modulated by a number of different stochastic models.
We extend the generating function technique for calculation of single molecule photon emission statistics (Zheng, Y.; Brown, F. L. H. Phys. Rev. Lett. 2003, 90, 238305) to systems governed by multi-level quantum dynamics. This opens up the possibility to study phenomena that are outside the realm of purely stochastic and mixed quantum-stochastic models. In particular, the present methodology allows for calculation of photon statistics that are spectrally resolved and subject to quantum coherence. Several model calculations illustrate the generality of the technique and highlight quantitative and qualitative differences between quantum mechanical models and related stochastic approximations when they arise. Calculations suggest that studying photon statistics as a function of photon frequency has the potential to reveal more about system dynamics than the usual broadband detection schemes.
We derive the generating function for single molecule photon emission events within the context of the stochastically modulated optical Bloch equations. Statistical properties of single molecule photon counting experiments are deduced from a set of coupled differential equations only slightly more complicated than the Bloch equations themselves. This formulation allows for the study of photon bunching and antibunching within a single theoretical framework and provides a description of these behaviors that emphasizes the connection with single molecule experiments. Application is made to the spectroscopy of a chromophore coupled to a single two level system.
The detection of photons emitted from a single molecule under the influence of electromagnetic radiation is considered. Utilizing a generating function formalism, we derive several exact results for the statistics of such emitted photons within the framework of the temporally modulated optical Bloch equations. Additionally, it is shown how these results reduce to previously obtained limiting behaviors. An appealing feature of this formulation is the inclusion of both photon bunching and anti-bunching effects within a single theoretical framework that is well suited for numerical analysis. Several examples are considered to demonstrate the feasibility of the approach in calculations. In most cases, these results verify known phenomena. In one case, we report a result that was missed by prior approximate treatments. This new effect centers around the fact that a chromophore will display anti-bunching behavior in the limit of fast modulation of the resonant absorption frequency.
We theoretically investigate the statistical properties of photon emission of a driven two-level single-molecule system undergoing spectral diffusion induced by nonequilibrium environmental fluctuations. Within the framework of the generating function method and the stochastic Liouville equation, we analyze the influence of the nonequilibrium characteristics of environmental fluctuations respectively governed by nonstationary Ornstein-Uhlenbeck noise and random telegraph noise on the photon counting statistics of the driven single-molecule system. In the slow modulation limit of spectral diffusion, the intensity and statistical fluctuations of photon emission depend on the environmental nonequilibrium characteristics at short time scales, whereas they become independent of the nonequilibrium characteristics of environmental fluctuations in the steady state. In the fast modulation limit of spectral diffusion, neither the line shape nor the Mandel's parameter depends on the environmental nonequilibrium characteristics owing to the rapid relaxation of environmental fluctuations. These findings not only shed light on the role of nonequilibrium environmental fluctuations in shaping the photon emission properties of single-molecule systems but also provide a basis for distinguishing between equilibrium and nonequilibrium characteristics of environmental fluctuations in experimental measurements.
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.
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.