Photons, statistical fluctuations

Generally, inaccuracies can also be expected at low photon counts (N < 100). Besides comparatively large statistical fluctuations, also a bias in lifetime is introduced by the data fitting procedure [37],... 

The signal to noise ratio is limited in any physical intensity measurement, however, by the statistical fluctuations in the photon flux (photon shot noise). This limit can be reached with Fourier transform infrared spectrometers. 

Intrinsic X-Ray Absorption. The maximum amount of information coded in a radiologic image is limited, in part, by the statistical fluctuations of the x-ray photons that form the radiologic image (I). These fluctuations are often referred to as "quantum nois e and as "quantum mottle" when viewed on films since the films have a grainy appearance (I7 ). These fluctuations can be expressed as a signal to noise ratios (1. ... 

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

An important limitation that is sometimes encountered is due to the particulate nature of electricity (electrons, ions) and of radiation (photons). The measurement of radiation intensities is in certain cases (e.g., X rays) performed by counting particles or photons one at a time. The number A counted in a time interval of given magnitude is subject to statistical fluctuations a count of A is subject to an estimated standard error given by... 

For each absorbed x-ray photon or neutron, the proportional or scintillation counter produces a discrete electric pulse. The flux J of the beam of x-rays or neutrons is measured as the number of counts of such pulses observed per second. If measurements are made repeatedly with a beam of constant flux, the number of counts observed during a fixed time period is not exactly the same, but is rather subject to statistical fluctuations. The arrival time of any one particle (x-ray photon or neutron) is totally uncorrelated with the arrival time of the next particle. The flux J of the particles,... 

We investigate the overall effect of the bath fluctuation on the photon statistics for the steady-state case as the fluctuation rate R is varied from slow to fast modulation regime. To characterize the overall fluctuation behavior of the photon statistics, we define an order parameter q. 

When the bath fluctuation becomes extremely fast such that R v7r, the splitting behavior of Q is observed, as discussed in Eq. (4.79) for Ey E, and then q cc l/R similar to the slow modulation regime. The approximate value of q based on fast modulation approximation, Eq. (4.77) (dotted line) shows good agreement with the exact calculation found using Eqs. (A.47) and (A.48). Finally, when R CO, <2 = 0. As mentioned before, this is expected since the molecule cannot interact with a very fast bath hence the photon statistics becomes Poissonian. 

This nonlinear interaction of photons with the absorbing and the amplifying media leads under favorable conditions to mode-locked laser operation starting from a statistically fluctuating, unstable threshold situation. After this short unstable transient state the laser emission consists of a stable, regular train of short pulses with the time separation T = Idfc, as long as the pump power remains above threshold (which is now lower than at the beginning because the absorption is saturated). 

The one-atom maser can be used to investigate the statistical properties of non-classical light [1298, 1299]. If the cavity resonator is cooled down to T < 0.5 K, the number of thermal photons becomes very small and can be neglected. The number of photons induced by the atomic fluorescence can be measured via the fluctuations in the number of atoms leaving the cavity in the lower level n — 1). It turns out that the statistical distribution does not follow Poisson statistics, as in the output of a laser with many photons per mode, but shows a sub-Poisson distribution with photon number fluctuations 70 % below the vacuum-state limit [1300]. In cavities with low losses, pure photon number states of the radiation field (Fock states) can be observed (Fig. 9.77) [1301], with photon lifetimes as high as 0.2 s At very low... 

For very low light intensities the quantum structure of light becomes evident by statistical fluctuations of the number of detected photons, which lead to corresponding fluctuations of the measured photoelectron rate (Sect. 7.8). This photon noise, which is proportional to -/N at a measured rate of N photoelectrons per second, imposes a principal detection limit for experiments with low-level light detection [1330]. Additionally, the frequency stabilization of lasers on the millihertz scale is limited by photon noise of the detector that activates the electronic feedback loop [1331]. 

The energy resolution of the proportional counter is determined by the statistical fluctuations of the number of ion-electron pairs generated by a photon of specific energy. Full-width at half-maximum (FWHM) of the distribution determines the resolution of the detector, which is of the order of 900 eV for a proportional counter. For this reason in X-ray spectrometers this type of detector is always used in conjunction with a dispersive crystal or equivalent. 

When light strikes the photocathode of a photomultiplier, photoelectrons are continuously produced but the instantaneous rate of production shows a statistical fluctuation because the photons of incident light arrive at random. This fluctuation of photoelectron formation is the reason for the fluctuations known as shot noise. The root mean square fluctuation of the cathode current, A4, is given as... 

If the modulation frequency X2 is chosen sufficiently high Q > 1000 MHz), the technical noise may drop below the quantum-noise limit set by the statistical fluctuations of detected photons. In this case, the detection limit is mainly due to the quantum limit [6.4]. Since lock-in detectors cannot handle such high frequencies, the signal input has to be downconverted in a mixer, where the difference frequency between a local oscillator and the signal is generated. 

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