We consider the probability ${P}^{(n)}({t}_{0}, t)$ of counting $n$ photoelectrons during any time interval [${t}_{0}, t$] in a photodetection model built in the general framework of quantum field theory. Owing to an equilibrium assumption on the detector state, ${P}^{(n)}({t}_{0}, t)$ is evaluated from the incident field and some electron-correlation functions. The conditions for ${P}^{(n)}({t}_{0}, t)$ to be a compound Poisson distribution as expected are discussed. This leads to the definition of coherence properties for the electrons of the detector in connection with the reliability of the detection operation in counting experiments.
We consider the probability P(n)(t0,t) of counting n photoelectrons during any time interval [t0,t] in a photodetection model built in the general framework of quantum field theory. Owing to an equilibrium assumption on the detector state, P(n)(t0,t) is evaluated from the incident field and some electron-correlation functions. The conditions for P(n)(t0,t) to be a compound Poisson distribution as expected are discussed. This leads to the definition of coherence properties for the electrons of the detector in connection with the reliability of the detection operation in counting experiments.
A theory of electromagnetic field measurement by means of photoionization is developed and applied to photoelectron counting. A probability theory involving multitime joint probability functions for a sequence of photoionizations is formulated. A general quantum-theory definition is proposed for the nonexclusive probability function which occurs in the probability theory. Approximations are then introduced to derive expressions for this probability function which involve correlation functions of the photoionization detector and the electromagnetic field-plus-source. The general theory is used to derive quantities of interest in photo-ionization counting experiments. Expressions are derived for (1) the probability Pn(t,t+T) that n photo-ionizations are observed in the time interval t to t+T, and (2) quantities related to Pn(t,t+T), such as its generating function and various moments. Pn(t,t+T) is found to be a compound Poisson distribution determined by the density operator of the field when the latter is expressed in Glauber's P representation. Using this result, the character of Pn(t,t+T) is examined for several specific density operators. These correspond to a coherent state, various fields with the mode phases distributed independently of the mode amplitudes, and a "spread-out" coherent state.
A relation is derived between the probability distribution of the integrated intensity of light incident on a photoelectric detector and that of the emission of n photoelectrons in a time interval (t, t + T) from the properties of a quantized electromagnetic field. Both the direct and the inverse relations are derived, thereby enabling the statistical properties of light to be determined from photoelectric measurements. An explicit expression is given for the coefficients for calculating the moments ( W k) by means of the moments (n J) (] = 1, 2 ..... k). The correctness of the operations in which generalized functions occur is confirmed by some direct calculations. The significance of photoelectric detectors has been increasing of late. Since their response is proportional to the intensity of incident light (to the square of the amplitude of light) (more exactly, to the local mean time value of the intensity), one speaks of quadratic detectors. These detectors are of basic importance in the theory of the coherence of the electromagnetic field, since if we place N such detectors into N different points in space and if we lead the outputs of these detectors to the correlator we can measure the correlation (coherence) effects of the 2N-th order in the given field [1]. Such correlation measurements provide information on the energy distribution in the radiation spectrum [2], the angular dimensions of stars [3], and the state of polarization [4] and also enable the phase of physically important functions (e.g. the scattering amplitudes) [5] to be determined. Although in such experiments the properties of photoelectrons are studied rather than of photons, they reflect the basic properties of photons. Let the probability distribution Of the emission of n photoelectrons in a time interval (t, t + T) be p(n, T, t) - p(n) and let the probability distribution of the quantity W(T, t) = W = J'tt+r I(t') dr', where I(t) is the intensity of linearly polarized light incident on the photodetector (for simplicity we assume that the sensitivity of the photocathode is a = 1), be P(W). The quantity W can be regarded as a classical characteristic of the field, the quantity n represents the quantum characteristic of the field. Then, on the basis of such a semi-classical description, in which the field is described classically, but its interaction is described quantum-wise, one can derive the following relation [6]
Various probability distributions are calculated for photocurrent fluctuations for photoelectric detection and are expressed in terms of quantum averages of field operators. The calculation is based on an extended theory of Kelley and Kleiner for photoelectric counting [Phys. Rev. 136, A316 (1963)], which allows us to obtain the counting distributions for multiple time intervals and multiple detectors. The finite duration of the detector's response function is also considered in our formalism, as is the possibility of fluctuations in the response function itself. This generalized theory is applied to homodyne and heterodyne detection processes. We find that with ideal conditions (perfectly balanced detectors with unit quantum efficiency), the characteristic function of the photocurrent fluctuations from homodyne detection is connected to the Fourier transformation of the Wigner function of a single-mode field, while that from heterodyne detection is linked to the antinormally ordered characteristic function of the field. More specifically, if the two orthogonal quadratures of the photocurrent are recorded in the heterodyne detection, the joint probability distribution for the two quadratures is simply the Q function of the field without the need for optical tomography. Our formalism is applied to the multimode case and allows one to draw important conclusions about the possibility of measurements of the complete quantum state of a multimode field. In particular, the complete characterization of fields with intermode correlations is a nontrivial undertaking, as we demonstrate with our general formalism as well as by specific examples.
One can attempt to derive some information of the field from the measured random properties of the photoelectron process: this is the object of photodetection theory developed first for classical fields. This chapter presents a review of photodetection theory, quantum theory and population monitoring model. It discusses two interesting applications to photostatistics. The photodetection equation is used to classify quantum and classical fields. The chapter presents an example of collective antibunching of photons. The double random character of the photodetection process prevents a simple correspondence between the statistical properties of the intensity and the statistics of the photocounts. However it is more instructive to connect the photoelectron process with the quantum nature of the electromagnetic field. Although the problem of photodetection appears to be satisfactorily solved for free fields interacting only with the detector, a heuristic model permits evolution of the detected field during the detection process.
We develop a quantum theory for continuous photodetection processes that describes nonunitary time development of the field under continuous measurement of photon number. Exact expressions are obtained for time evolutions of the photon-field density operator, average and variance of the photon number, and the Fano factor. These are applied to typical quantum states, i.e., number, coherent, thermal, and squeezed states. The continuous photodetection process is made up of two elementary processes in terms of the referring measurement process, that is, one-count and no-count processes. Just after the one-count process in which a photodetector registers one photoelectron, the average photon number 〈n(t)〉 of the remaining field is shown to increase for super-Poissonian states (e.g., thermal state) and decrease for sub-Poissonian states (e.g., number state); for the Poissonian state (e.g., coherent light), 〈n(t)〉 does not change. During the no-count process in which the photodetector registers no photoelectrons, on the other hand, 〈n(t)〉 decreases in time for all states except the number state. The physical origins for these results are clarified from the viewpoint of nonunitary state reduction by continuous measurement of photon number. Furthermore, we introduce a nonreferring measurement process in which the detector registers photocounts, but we discard all readout information. We discuss the difference in the way the photon field evolves in this process compared to the referring measurement process.
A quantum-mechanical theory of photon detection is presented which takes fully into account the attenuation of the field due to the detection process. The time evolution of the joint quantum state of the detector and the field is found, to all orders in perturbation theory. Formulas are derived for the probabilities of absorbing specified numbers of photons within a given time interval, and for the correlations in the positions and times at which absorptions take place. The results are free from the inconsistencies which arise in the conventional theory of photon detection, and remain valid even when the field becomes appreciably attenuated during the experiment. It is found that an initially coherent field state remains coherent during its interaction with a given detector, and that its amplitude becomes attenuated by an amount which is completely independent of the number of counts which the detector records. This independence is shown to be a simple consequence of the Poisson quantum-number distribution of coherent fields. The counting statistics for arbitrary fields are expressed in a way which shows explicitly the relationship between the attenuation of the field and the absorption of quanta by the detector. The analysis is performed by first treating the case in which the field is confined within a homogeneous detecting medium throughout the experiment, and then generalizing to the case in which the field spends a limited amount of time in a spatially localized detecting region. The detector is assumed to consist of harmonic oscillators, which are shown to represent a suitable formal model for the absorption of radiation by large numbers of conventional detecting atoms.