Theory of Electromagnetic Field Measurement and Photoelectron Counting

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 $P_n(t, t+T)$ that $n$ photo-ionizations are observed in the time interval $t$ to $t+T$, and (2) quantities related to $P_n(t, t+T)$, such as its generating function and various moments. $P_n(t, t+T)$ 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 $P_n(t, t+T)$ 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.