Theory of Photoelectric Detection

The problem of a radiation field interacting with a photodetector from time 0 to $t$ is formulated in terms of quantum transport theory. The counting distribution $P_n(0, t)$ is related to a set of $n$-electron correlation functions ${〈{\Psi }^{†}\left({x_1}^{\prime }\right)\cdots {\Psi }^{†}\left({x_m}^{\prime }\right)\Psi \left(x_m\right)\cdots \Psi \left(x_1\right)〉}_D$, where the subscript $D$ indicates an average over detector variables. These are similar to the correlation functions used in many-body theory, excepting that they contain radiation operators. It is shown that if the photoelectrons do not interact with one another at any time, either before or after leaving cathode $C$, then the correlation functions factorize into products of single-electron functions ${〈{\Psi }^{†}\left({x_i}^{\prime }\right)\Psi \left(x_j\right)〉}_D$ with all radiation operators combined in normal order. No restriction is placed on the interaction of an omission electron with the large background of ions, phonons, or nonemitted electrons which it encounters on its way out of $C$, nor is there any restriction to single-photon excitation processes. The resulting expression for $P_n(0, t)$ contains terms which arise from electron correlation effects outside $C$. Under normal experimental conditions, where these are negligible, $P_n(0, t)$ reduces to a compound Poisson formula, expressed in terms of a photoelectron number operator averaged over detector variables. The effects of the electron detection process are considered briefly, and the compound Poisson formula is generalized to the interval ($t, t+\Delta t$).