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Annual Meeting Optical Society of America

Quantum measurement theory of photoelectric detection

H. J. Carmichael

Published Jan 1, 1990• 2 citations
Publisher

Abstract

The criterion used to distinguish classical from nonclassical effects in quantum optics is based on the theory of photoelectric detection. In the semiclassical theory of photoelectric detection, the optical field enters as a stochastic intensity. Results obtained from the quantum theory can be reproduced within this stochastic formalism whenever the quantum state of the field is a statistical mixture of coherent states. When the quantum state of the field is not of this form, certain statistical properties of the observed photocurrents and photopulse sequences cannot be reproduced within the semiclassical formalism.

DOI
10.1364/oam.1990.mc2

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The criterion used to distinguish classical from nonclassical effects in quantum optics is based on the theory of photoelectric detection. In the semiclassical theory of photoelectric detection, the optical field enters as a stochastic intensity. Results obtained from the quantum theory can be reproduced within this stochastic formalism whenever the quantum state of the field is a statistical mixture of coherent states. When the quantum state of the field is not of this form, certain statistical properties of the observed photocurrents and photopulse sequences cannot be reproduced within the semiclassical formalism.
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.
It was found possible to develop a mathematical theory of communication which includes quantum effects, and as is also important, reduces to the mathematics of classical communication theory when quantum effects can be neglected. Because photoelectric detection is used universally at optical frequencies, and because it lends itself to a simpler theoretical description, earlier publications have considered the statistics of photo detection - how they differ for coherent and incoherent signals, and the consequences of these differences for information transmission. The work was concentrated in two major areas: (a) statistics of photodetection, and (b) quantum statistical analysis of communication.
In view of the misunderstanding which surrounds the use of semiclassical methods in the treatment of optical coherence problems, a simple analysis is given which illustrates the generality of these methods. In particular, it is pointed out that, under commonly occurring circumstances, these methods apply at arbitrarily low light levels, and give correct results for the photoelectric counting of photons. The conclusions are illustrated by treating the interference between laser light and thermal light to demonstrate that this nontrivial case, recently treated by quantum field theory, is correctly described by semiclassical methods.
It is pointed out that it is impossible to distinguish between photons and classical wave packets by measurement of photoelectric counting statistics for any state of a light beam for which a classical description exists. Therefore, an experiment with a thermal light source recently proposed by Karp as a test for the existence of photons can yield no information on this question, even if it should be able to distinguish between two different statistical models of a light beam. The quantum nature of the optical field can, however, be inferred from other photoelectric measurements, some of which are briefly discussed.
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Based on the quantum postulates of Planck and Einstein, without the use of quantum formalism, a quantum description of optical signals in the form of discrete sequences of instantaneous power and instantaneous values of the electrical component of optical radiation is proposed. The classical approximation of the quantum representation of optical signals is carried out using the methods of signal theory and communication theory. A quantum description of photodetection of optical signals is proposed, within the framework of which the photocurrent is given in the form of a discrete sequence, and its classical description is carried out.