Poisson photon statistics

Table 1. Scaling for the anti-Stokes pulse Q-parameter and Fock state fidelity F. n refers to the mean number of excitations in the rubidium cell, nfG is the mean photon number of background counts in the write channel ( we assume they are mainly due to leak of the write drive and so follow Poisson statistics), as is the Stokes detection efficiency and ns is the number of Stokes photons on which we condition. The mean number of atomic excitations is calculated via (nsw) = Tr (pas nsw), similarly (n2sw = Tr (pas h2sw). The subscript T (P) refers to thermal (Poisson) photon statistics of the unconditional Stokes light.

Eberly et al. have shown [18] that the physical reason for the collapse is the Poisson photon statistics of coherent states, while the revivals find their origin in the discrete nature of the photon number distribution, leading to a kind of "granularity" of the quantized radiation field present even at high average photon numbers. 

One of the main advantages of the photon-counting system over lock-in detection is the fact that, if the instrument is working properly, i.e. obeying Poisson photon statistics, then it can be shown that the distribution of measured gi , values should obey a Gaussian distribution. It can also be shown that the standard deviation in g y , is equal to the inverse square root of the total number of photon counts ... 

Fig. 7.7. Effects of Poisson photon noise on calculated SE and FRET values. (A) Statistical distribution of number of incoming photons for the mean fluorescence intensities of 5,10, 20, 50, and 100 photons/pixel, respectively. For n = 100 (rightmost curve), the SD is 10 thus the relative coefficient of variation (RCV this is SD/mean) is 10 %. In this case, 95% of observations are between 80 and 120. For example, n — 10 the RCY has increased to 33%. (B) To visualize the spread in s.e. caused by the Poisson distribution of pixel intensities that averaged 100 photons for each A, D, and S (right-most curve), s.e. was calculated repeatedly using a Monte Carlo simulation approach. Realistic correction factors were used (a = 0.0023,/ = 0.59, y = 0.15, <5 = 0.0015) that determine 25% FRET efficiency. Note that spread in s.e. based on a population of pixels with RCY = 10 % amounts to RCV = 60 % for these particular settings Other curves for photon counts decreasing as in (A), the uncertainty further grows and an increasing fraction of calculated s.e. values are actually below zero. (C) Spread in Ed values for photon counts as in (A). Note that whereas the value of the mean remains the same, the spread (RCV) increases to several hundred percent. (D) Spread depends not only on photon counts but also on values of the correction...

Another advantage of the differential photon counting method of detection is that the measurement uncertainties can be estimated directly from photon statistics. Schippers [11] has demonstrated that the standard deviation, a, of gium detected by the differential photon counting method can be related through Poisson statistics to the total number of photon counts, N, by the simple formula... 

In (98), Q is the Mandel factor [14], describing the deviation of the photon statistics from the Poisson distribution for the total intensity ... 

Of fundamental interest are measurements of the photon statistics in a three-level system, which can be performed by observing the statistics of quantum jumps. While the durations Af/ of the on-phases or the off-phases show an exponential distribution, the probability P m) of m quantum jumps per second exhibits a Poisson distribution (Fig. 9.48). In a two-level system the situation is different. Here, a second fluorescence photon can be emitted after a first emission only, when the upper state has been reexcited by absorption of a photon. The distribution P(AP) of the time intervals AT between successive emission of fluorescence photons shows a sub-Poisson distribution that tends to zero for AT 0 photon antibunching), because at least half of a Rabi period has to pass after the emission of a photon before a second photon can be emitted [1234]. 

One way of studying temporal coherence in laser systems is by measuring photon statistics [224]. In this technique the transient laser emission properties are measured using pulsed excitation and a time-resolved setup [225], The transient emission curve generated by each pulse above the laser threshold intensity is divided into time intervals that are smaller than the emission coherence time. The number of photons is then measured in each time interval and for each pulse, and a photon number histogram is calculated to obtain the probability distribution function (PDF) of the photons for each time interval. Photon statistics is achieved separately for each time interval, and correlation between different time intervals or between different wavelengths of the emission spectrum can be also studied. It is expected that for coherent radiation the Poisson distribution determines the PDF, whereas for noncoherent light... 

The generation of photons obeys Poisson statistics where the variance is N and the deviation or noise is. The noise spectral density, N/, is obtained by a Fourier transform of the deviation yielding the following at sampling frequency,... 

In single-photon counting experiments, the statistics obey a Poisson distribution and the expected deviation [Pg.182]

In the experiment, the transmission intensities for the excited and the dark sample are determined by the number of x-ray photons (/t) recorded on the detector behind the sample, and we typically accumulate for several pump-probe shots. In the absence of external noise sources the accuracy of such a measurement is governed by the shot noise distribution, which is given by Poisson statistics of the transmitted pulse intensity. Indeed, we have demonstrated that we can suppress the majority of electronic noise in experiment, which validates this rather idealistic treatment [13,14]. Applying the error propagation formula to eq. (1) then delivers the experimental noise of the measurement, and we can thus calculate the signal-to-noise ratio S/N as a function of the input parameters. Most important is hereby the sample concentration nsam at the chosen sample thickness d. Via the occasionally very different absorption cross sections in the optical (pump) and the x-ray (probe) domains it will determine the fraction of excited state species as a function of laser fluence. 

The following problem is in a certain sense the inverse of the one treated in the two preceding sections. Consider a photoconductor in which the electrons are excited into the conduction band by a beam of incoming photons. The arrival times of the incident photons constitute a set of random events, described by distribution functions/ or correlation functions gm. If they are independent (Poisson process or shot noise) they merely give rise to a constant probability per unit time for an electron to be excited, and (VI.9.1) applies. For any other stochastic distribution of the arrival events, however, successive excitations are no longer independent and therefore the number of excited electrons is not a Markov process and does not obey an M-equation. The problem is then to find how the statistics of the number of charge carriers is affected by the statistics of the incident photon beam. Their statistical properties are supposed to be known and furthermore it is supposed that they have the cluster property, i.e., their correlation functions gm obey (II.5.8). The problem was solved by Ubbink ) in the form of a... 

To determine the unconditional probability distribution for the spin-wave excitations Psw(n), we must find the effective number of transverse modes which contribute to the Raman processes. We identify two extreme regimes which permit analytic treatment a single mode regime where the number of excitations in the 87Rb cell follows Bose-Einstein (thermal) statistics and a multimode regime where it follows Poisson statistics. We find in both cases that the quantities F and Q depend on two experimental parameters 0 ( number of lost Stokes photons) and v ( noise to signal ratio), which are defined in Tab. 1. 

The shot noise is associated with random arrival of photons on a detector and corresponding random production of photoelectrons. Conventional hght sources produce photon flux that obeys Poisson statistics, which produces a shot noise as directly proportional to the square root of the detected hght intensity as = where al is relative (intensity-independent) stan-... 

The DQE is a useful concept because it describes the noise characteristics of the input. Consider, for example, a photon source which obeys Poisson statistics. Suppose that a measurement is required of the output to an accuracy q, i.e. 

For a Poisson process such as photon absorption, the statistical variation in the number of absorbed photons cttv is equal to the square root of the number of absorbed photons, N, and is expressed as... 

Figure 5.111 shows the fluctuations of the photon flux integrated in 1 ms intervals over an interval of 1 second. Due to the diffusion of the GFP moleeules, the intensity fluctuations are clearly larger than the Poisson statistics of the average photon number. Figure 5.112 shows the fluorescence decay function over several laser periods and the FCS function. 

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