Probability distribution of photoelectric currents in photodetection processes and its connection to the measurement of a quantum state

Abstract

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.