The functional formalism, providing a unified treatment of the optical coherence theory is developed for arbitrarily ordered field operators. Multimode ordered expansions of operators in terms of boson amplitude operators as well as integral expansions are discussed. They form a basis of associating a class of s-ordered probability functionals with the density operator of the field. As an application, the distributions of photons and photoelectrons are studied. The generalized photo-detection equation of Perina and Horak (1970) is rederived by using the functional approach. A formula expressing the photocount generating function through the characteristic functional of the field is derived.
The functional formalism, providing a unified treatment of the optical coherence theory is developed for arbitrarily ordered field operators. Multimode ordered expansions of operators in terms of boson amplitude operators as well as integral expansions are discussed. They form a basis of associating a class of s-ordered probability functionals with the density operator of the field. As an application, the distributions of photons and photoelectrons are studied. The generalized photo-detection equation of Perina and Horak (1970) is rederived by using the functional approach. A formula expressing the photocount generating function through the characteristic functional of the field is derived.
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 general formula is derived relating the generating function of the N-fold photoelectron counting distribution to the characteristic functional of the optical field. For a stationary Gaussian radiation field the generating function is expressed as an infinite product involving eigenvalues of a set of integral equations. The results of Bédard, Dialetis, and Cantrell are shown to hold as special cases. Addition of photoelectric counts from correlated Gaussian light is analyzed and the appropriate generating function is given.
Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functionals of phase space fields, leading to expressions for quantum correlation functions in terms of phase space functional integrals, where the field functions—complex fields for bosons, Grassmann fields for fermions—replace field annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functionals, with distinct fields for annihilation and creation field operators; however, symmetrically ordered (Wigner type) distribution functionals are also considered, along with unnormalised B distribution functionals, which lead to phase space functional integrals for Fock position state populations and coherences. Characteristic functionals are first defined and then shown to be related to distribution functionals via phase space functional integrals. The existence of the distribution functionals is demonstrated from previous results for equivalent distribution functions by using mode expansions.
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]
We express the exact probability density distribution function as the product of a gamma distribution and a series of associated Laguerre polynomials, with the expansion coefficients determined by moments of the integrated intensity. Orthogonal polynomials with respect to the exact probability distribution function are then expanded in similar fashion. These polynomials are then used to construct an expansion of the joint probability distribution function in the second-order photoelectron statistics. Since the polynomials are identical with the corresponding Laguerre polynomials when the exact probability distribution function is the gamma distribution, the new polynomials are generalized versions of the associated Laguerre polynomials. The joint photoelectron statistics may be studied with these new polynomials.
The "diagonal" P representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P(\ensuremath{\alpha})$ required to represent all density operators by means of the P representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the P representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the P representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.