A concept 'probability-density functional' combined with the path integral technique is shown to provide a new framework for quantum photodetection theory. It replaces the conventional picture of unitary evolution of the total density operator followed by von Neumann's projection postulate for the case in which the effects of nonunitary state reduction and continuous measurements must be taken into account. This paper shows that an expression of the probability-density functional which incorporates both these effects provides complete quantum-statistical information about photoelectron statistics. Simple procedures are given for calculating the ensemble average and generating the functional of a quantity distributed in time.
A concept 'probability-density functional' combined with the path integral technique is shown to provide a new framework for quantum photodetection theory. It replaces the conventional picture of unitary evolution of the total density operator followed by von Neumann's projection postulate for the case in which the effects of nonunitary state reduction and continuous measurements must be taken into account. This paper shows that an expression of the probability-density functional which incorporates both these effects provides complete quantum-statistical information about photoelectron statistics. Simple procedures are given for calculating the ensemble average and generating the functional of a quantity distributed in time.
A nonequilibrium open-system theory for continuous photodetection processes is developed using a probability-density functional combined with the path-integral technique. Nonunitary time evolution of the system-environment density operator due to measurement back action and continuous measurement is exactly described for three familiar photodetection processes. New exact photocounting formulas are obtained for these three processes; in particular, Mollow's photocounting formula is generalized for a nonequilibrium open system. Distinctive features between a closed system and an open system are presented; for example, the probability distribution for the number of counts is shown to contain complete information about photoelectron statistics for a closed system, whereas it does not for an open system. The obtained formulas are applied to photodetection of a single-mode photon field that interacts linearly with a pump source consisting of a single harmonic oscillator and a reservoir consisting of an infinite number of harmonic oscillators. In the former case, increasing the ratio of the source-field coupling constant to the field-detector coupling constant causes the photon field to cross over from a closed attenuating field to an open stationary field. In the latter case, a quantum-mechanical fluctuation-dissipation theorem in the continuous photodetection context is discussed.
We show that time distributions of photoelectrons can best be described by a generalized probability-density function which we call the probability-density functional. The probability-density functional gives complete information about the statistical properties of any random-point process, and provides the most natural definitions of the ensemble average and generating functional of a quantity distributed in time. Using this functional, we develop a new approach to photoelectron statistics that systematically relates various joint probability distributions of photoelectrons to one another and to the corresponding generating functionals. We demonstrate that this method can be used to derive previously obtained results and we use it to obtain new results. In particular, the amount of information about the statistical properties of photoelectrons that is contained in a probability distribution can be obtained by comparing the corresponding generating functional with the probability-density functional. Further, we examine time derivatives of the probability distribution for the number of counts and develop new relationships for probability distributions of triggered counting. Finally, we apply our results to obtain an ``inverse'' relationship between photon bunching (antibunching) and super-Poisson (sub-Poisson) statistics.
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 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]
This paper attempts to explain both the basic concepts and the mathematical formalism of von Neumann's operator method in quantum statistics, in terms that should be understandable by anyone who has mastered first courses in quantum mechanics and statistical mechanics. A simple model is used throughout the paper to illustrate the various theorems and techniques. The discussion begins with the statistical interpretation of the Heisenberg uncertainty principle, the definition of mixed ensembles, and the meaning of measurement in quantum mechanics. The von Neumann projection operator is explained, and is used to set up the ensemble density matrix and the von Neumann entropy operator. In the second part of the paper, to be published later, the quantum-mechanical equation of change with time is applied to the von Neumann operators.
A general theory is developed that describes continuous state reduction of two quantum-mechanically correlated photon fields by continuous photodetection of one of the fields. The effects of measurement back action and quantum correlation on the state reduction of the other field are highlighted from the viewpoint of continuous photodetection. Nonunitary time evolutions of the density operators and photon statistics are exactly described for both fields. The obtained formulas are applied to parametrically generated photon twins, i.e., signal and idler fields. The continuous state reduction of the signal field by photodetection of the idler field is examined. It is shown that the Manley-Rowe relation can be generalized to the intermediate state under the photodetection process; that is, the initial quantum correlation remains intact throughout the ``destructive'' photodetection process if we retain the information about the idler readout. The obtained results are compared to the case in which the idler readout is discarded. In this case, the Manley-Rowe relation can no longer hold. This fact demonstrates that retaining the readout information is essential for preserving the established quantum correlation.