Photons statistics

The photon statistical weight is g = 2, corresponding to the two directions of polarization of the photon. The photon energy E is related to its momentum p and wavenumber k and to the ionization energy of the... 

Keywords quantum optics, one photon state, photon statistics... 

The acquisition throughput of a microscope is often determined by photon statistics, but depends also on many parameters including instrumental limitations, for example, the read-out and dead-time of the detector and electronics [40],... 

For instance, a time-gated SPC microscope with four gates exhibits a better maximal (FTG4 = 1.3) photon-economy than a two-gated system (FTGi = 1.5). The latter setup will thus require (F (,2//- if,4)2 = 1.7-fold longer acquisition time than the former because of photon-statistics alone. 

A closer look at the data shows the lifetime distributions are comparatively broad, about 0.25 ns for both distributions. This is in fact much broader than what one would expect from photon statistics alone. Based on realistic / -values (1.2-1.5) lifetime images recorded with this many counts are expected to yield distributions with widths on the order of 0.1 ns. The broadening is therefore not because of photon statistics. Variations in the microenvironment of the GFP are the most likely source of the lifetime heterogeneities. Importantly, such sensitivity for local microenvironment may be the source of apparent FRET signals. In this particular FRET-FLIM experiment, we found that the presence of CTB itself without the acceptor dye already introduced a noticeable shift of the donor lifetime. Therefore, in this experiment the donor-only lifetime image was recorded after unlabeled CTB was added to the cells. The low FRET efficiency and broadened lifetime distribution call for careful control experiments and repeatability checks. 

Each photon that flows from the entrance slit may exist only in such a mode. However, more than one may crowd in. This is a property of Bose-Einstein particles, to which photons belong as a class. Photons are also known to be microscopically indistinguishable, so that a given configuration of them within the modes cannot be distinguished from the same configuration where some of the photons have interchanged mode positions. As we shall see, the ML solution that we seek will depend vitally on this Bose-Einstein aspect of photon statistics. 

Since the photon version of the reservoir correlation functions Cuv includes the photon statistical operator which is defined by the projector on the photon vacuum the correlation functions simply read... 

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... 

G. Zumofen, J. Hohlbein, and C. G. Hiibner, Recurrence and photon statistics in fluorescence fluctuation spectroscopy. Phys. Rev. Lett. 93 260601 (2004). 

R. Verberk and M. Orrit, Photon statistics in the fluorescence of single molecules and nanocrystals Correlation functions versus distributions of on- and off-times. J. Chem. Phys. 119(4) 2214-2222 (2003). 

A final requirement arises if the stroboscopic technique is to be used, where the experiment is repeated many times and the resulting data summed to provide enough photon statistics. In this case the detector must be capable of being synchronised to the experiment. 

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.

Another nonclassical effect is referred to as sub-Poissonian photon statistics (see, e.g., Refs. 7 and 8 and papers cited therein). It is well known that in a coherent state dehned as an inhnite superposition of the number states... 

If the variance of the number of photons is smaller than its mean value, the held is said to exhibit the sub-Poissonian photon statistics. This effect is related to the second-order intensity correlation function... 

Negative values of this parameter indicate sub-Poissonian photon statistics, namely, nonclassical character of the field. One obvious example of the nonclassical field is a field in a number state n) for which the photon number variance is zero, and we have g 2 (0) = 1 — 1 /n and q = — 1. For coherent states, g (0) = 1 and q = 0. In this context, coherent states draw a somewhat arbitrary line between the quantum states that have classical analogs and the states that do not have them. The coherent states belong to the former category, while the states for which g (0) < 1 or q < 0 belong to the latter category. This distinction is better understood when the Glauber-Sudarshan quasidistribution function P(ct) is used to describe the field. 

Again, ((An)2) < (ii) only if P(a) is not positive definite, and thus sub-Poissonian photon statistics is a nonclassical feature. 

We can thus expect from the short-time approximation that quantum noise does not significantly affect the classical solutions when the initial pump field is strong. We will return to this point later on, but now let us try to find the short-time solutions for the evolution of the quantum noise itself—let us take a look at the quadrature noise variances and the photon statistics. Using the operator solutions (94) and (95), one can find the solutions for the quadrature operators Q and P as well as for Q2 and P2. It is, however, more convenient to use the computer program to calculate the evolution of these quantities directly. Let us consider the purely SHG process, we drop the terms containing b and b+ after performing the normal ordering and take the expectation value in the coherent... 

---