Optical techniques in single molecule imaging rely heavily on photon counting for data acquisition. Extraction of information from the recorded readings is often done by means of statistical signal processing, however this requires a full knowledge of the photoelectron statistics. In addition to counting statistics we include a specific form of random signal variations namely reorientational dynamics, or wobble to derive the general probability density function of the number of detected photons. The relative importance of the two factors is dependent upon the total number of photons in the system and results are given in all regimes.
Optical techniques in single molecule imaging rely heavily on photon counting for data acquisition. Extraction of information from the recorded readings is often done by means of statistical signal processing, however this requires a full knowledge of the photoelectron statistics. In addition to counting statistics we include a specific form of random signal variations namely reorientational dynamics, or wobble to derive the general probability density function of the number of detected photons. The relative importance of the two factors is dependent upon the total number of photons in the system and results are given in all regimes.
The statistical properties of photon sequences originating from single-molecule emitters differ from those produced by generic light sources. Let us start with the latter. A typical light source (such as a lightbulb) is composed of numerous independent lightemitting atoms or molecules. A sufficiently sensitive photodetector may be used to record the arrival time of each individual photon. The simplest assumption one could adopt to describe this kind of measurement is that the arrival times for different photons are completely uncorrelated (see, however, the discussion at the end of this section). If the properties of the light source (such as, e.g., its brightness) do not depend on time then we should expect that the probability of detecting a photon during an infinitesimal time interval dt is proportional to dt , with the proportionality constant being independent of time or the arrival times of other photons. Let us call this probability λdt , where λ is the probability of detecting a photon per unit time (which will be referred to as the photon count rate). To estimate λ we could measure the number of photons n detected over a certain period of time t and divide it by t : λ ≈ n/t. (7.1) But, since photon arrival times are random, the exact number of photons found during any finite time interval t will fluctuate around its mean value λt , as illustrated in Fig.7.1. A complete description of this process thus requires the probability wn(t) that a specified number of photons, n, is observed during the time interval t . To obtain this probability, it is helpful to think of the number of detected photons, n, as the “state” of the system at time t . At t = 0 we have n = 0. Each time a new photon is detected, the system undergoes a first-order irreversible “reaction” n → n + 1. The overall process thus can be described by the kinetic scheme: 0 λ→ 1 λ→ 2 λ→ 3 λ→ 4 → · · · . Accordingly, wn(t) is described by a system of differential equations of the form dwn(t)/dt = λwn−1(t) − λwn(t), n > 0 (7.2) and dw0(t)/dt = −λw0(t), (7.3) with the initial conditions wn(0) = 0, n > 0 and w0(0) = 1. The solution to Eq. 7.3 is w0(t) = e−λt . (7.4) Substituting this into the differential equation for w1 (Eq. 7.2) we can find the time dependence of w1(t), then substitute the result into the equation for w2(t) and so on. Suppose we have found wn−1(t) and now wish to solve for wn(t). It is convenient to use the ansatz wn(t) = ρn(t)e−λt .
Counting the emitters within a diffraction-limited region is a prerequisite for localizing overlapping molecules. Under pulsed excitation, each molecule emits at most one photon per pulse, so photon-number-resolved detection can count emitters without resolving them. But the required number of pulses grows steeply with the emitter number. Here we show that polarization, conventionally used to measure the orientations of dipole emitters, also carries information on their number. Because mean polarization-resolved signals are unchanged when every emitter is replaced by multiple dimmer copies, this information resides in the photon statistics within each pulse, which polarization reshapes in two ways. Linearly polarized excitation at three angles increases the sub-Poissonian deficit of the photon count by a factor of 1.5 for any emitter orientation, and detection in four polarization channels reveals the emitter number through how the photons of multiphoton pulses are distributed among the channels. Maximum-likelihood simulations for two to five emitters of unknown orientations and brightnesses show that three-angle excitation reduces the required number of pulses by about 2.3-fold, close to the predicted 1.52=2.25, and that adding four-channel detection raises the reduction to as much as 4.8-fold. These gains require only polarization optics added to photon-number-resolved detection.
We suggest an approach to detect the conformation of single molecule by using the photon counting statistics. The generalized Smoluchoswki equation is employed to describe the dynamical process of conformational change of single molecule. The resonant trajectories of the emission photon numbers $$ and the Mandel's Q parameter, in the space of conformational coordinates X and frequency ωL of external field (X−ωL space), can be used to rebuild the conformation of the single molecule. As an example, we consider Thioflavin T molecule. It demonstrates that the results of conformations extracted by employing the photon counting statistics is excellent agreement with the results of {\it ab initio} computation.
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.
The fundamental understanding of biological pathways requires minimally invasive nanoscopic optical resolution imaging. Many approaches to high-resolution imaging rely on localization of single emitters, such as fluorescent molecules or quantum dots. Additionally, the exact determination of the number of such emitters in an imaging volume is essential for a number of applications; however, in standard intensity-based microscopy it is not possible to determine the number of individual emitters within a diffraction limited spot without initial knowledge of system parameters. Here we explore how quantum measurements of the emitted photons using photon number resolving detectors can be used to address this challenging task. In the proposed new approach, the problem of counting emitters reduces to the task of determining differences between the emitted photon distribution and the Poisson limit. We show that quantum measurements of the number of photons emitted from an ensemble of emitters enable the determination of both the number of emitters and the probability of emission. This method can be applied for any type of single-photon emitters. The scaling laws of this new approach are presented by the Cramer-Rao Lower Bounds, and this technique has great potential in quantum optical imaging with nanoscopic resolution.
Single-molecule fluorescence microscopy has been established in life science as an essential tool for studying the characteristics and dynamics of individual fluorescent emitters both in vitro as well as in vivo. Acquiring quantitative information from the confocal observation volume is still a challenging task and knowing the absolute number or concentration of proteins in, e.g., cellular structures can significantly improve our understanding of cell biology, which is an important step towards quantitative microscopy. In this talk, a new quantitative analytical tool called Counting by Photon Statistics (CoPS) will be presented. The approach relies on a statistical analysis of detected photon coincidences to estimate the number of independent fluorescent labels in the observation volume [1]. CoPS does exploit the photon antibunching effect: a single photon emitter can only generate one photon at a time. Originally developed for point measurements, CoPS has recently been extended to an imaging scheme [2]. Using a confocal fluorescence microscopy setup with pulsed excitation, four single-photon, detectors and parallellized time-correlated single photon counting electronics (MicroTime 200, PicoQuant), we prove the applicability of the method with artificial model systems (immobilized DNA origami) and present first steps towards biological samples. [1] Ta, H., Wolfrum, J., Herten, D.-P., An extended scheme for counting fluorescent molecules by photon-antibunching Laser Phys. 20:119 (2010). [2] Ta, H. et al., Mapping molecules in scanning far-field fluorescence nanoscopy. Nat. Commun. 6:7977 (2015).