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Physical Review

Antinormally Ordered Correlations and Quantum Counters

L. Mandeļ

Published Dec 2, 1966• 85 citations
Publisher

Abstract

The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers $n$ counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.

DOI
10.1103/physrev.152.438

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Physical Review•Apr 29, 1966•138 citations
The properties of a photon counter functioning by stimulated emission---rather than by absorption---of photons in an external field are examined. A possible scheme for such a quantum counter is described, and it is shown that correlation measurements performed with a number of quantum counters correspond to antinormally ordered products of field operators. These correlations, unlike the normally ordered ones, are always positive definite and depend explicitly on the number of radiation modes per unit volume to which the counter is coupled. It is shown that the antinormally ordered correlations carry useful information about the field only when the average photon occupation number per mode is large. A general expression for the probability that the quantum counter registers nnn counts in a certain time interval is derived and is shown to be related in an interesting way to the corresponding expression for the photoelectric detector. The variance of the probability distribution is evaluated for some simple states of the field.
When the electromagnetic field is detected by stimulated emission, rather than by absorption, antinormally ordered photodetection can be realized. One of the distinct features of this photodetection scheme is its sensitivity to zero-point fluctuation due to the existence of the spontaneous emission. We have recently succeeded in experimentally demonstrating the antinormally ordered photodetection by exploiting nondegenerate stimulated parametric down-conversion process. To properly account for the experiment, the detection process needs to be treated with time-dependent and continuous-mode operators because of the broadband nature of the parametric down-conversion process and the wide spectrum of the pump that we used. Here, we theoretically analyze the antinormally ordered intensity correlation of the continuous-mode fields by pursuing the detection process in the Heisenberg picture. It is shown that the excess positive correlation due to zero-point fluctuation reduces because of the frequency-distinguishability of the two emitted photon pairs.
We have measured antinormally ordered Hanbury Brown-Twiss correlations for coherent states of the electromagnetic field by using a stimulated parametric down-conversion process. Photons were detected by stimulated emission, rather than by absorption, so that the detection responded not only to actual photons but also to zero-point fluctuations via spontaneous emission. The observed correlations were distinct from normally ordered ones as they showed excess positive correlations, i.e., photon bunching effects, which arose from the thermal nature of zero-point fluctuations.
A quantum-mechanical theory of photon detection is presented which takes fully into account the attenuation of the field due to the detection process. The time evolution of the joint quantum state of the detector and the field is found, to all orders in perturbation theory. Formulas are derived for the probabilities of absorbing specified numbers of photons within a given time interval, and for the correlations in the positions and times at which absorptions take place. The results are free from the inconsistencies which arise in the conventional theory of photon detection, and remain valid even when the field becomes appreciably attenuated during the experiment. It is found that an initially coherent field state remains coherent during its interaction with a given detector, and that its amplitude becomes attenuated by an amount which is completely independent of the number of counts which the detector records. This independence is shown to be a simple consequence of the Poisson quantum-number distribution of coherent fields. The counting statistics for arbitrary fields are expressed in a way which shows explicitly the relationship between the attenuation of the field and the absorption of quanta by the detector. The analysis is performed by first treating the case in which the field is confined within a homogeneous detecting medium throughout the experiment, and then generalizing to the case in which the field spends a limited amount of time in a spatially localized detecting region. The detector is assumed to consist of harmonic oscillators, which are shown to represent a suitable formal model for the absorption of radiation by large numbers of conventional detecting atoms.
Introduction We have now prepared the way for a quantum treatment of problems that are of interest in optics. We have examined some basic properties of the quantized field and have explored the coherent-state representation and some of its virtues. Until now our discussions have been essentially formal; we have not concerned ourselves with what is actually measured in the laboratory, nor with questions concerning the interactions of the quantized field with other systems. In the following sections we shall turn our attention to some of the quantum field theoretic quantities that are of particular interest in quantum optics and are of significance in the measurement of the field. Our discussions in this chapter will still be somewhat formal, in that we shall not yet come to grips with the problem of the interaction of the field with the apparatus. That problem will be tackled in Chapter 14. For the moment we shall continue to oversimplify and treat the field as a free field. However, within the idealized context of the free field we shall examine some of the questions encountered in measurements in quantum optics. We shall see that normally ordered correlations play a dominant role and examine some of their properties. Anti-normally ordered correlations will be seen to arise in much less common and less useful measurements of the field. We shall also examine the sense in which a photon may be said to be localized in a measurement in which a photoelectric emission is registered.
Some properties of the operator ${\stackrel{^}{n}}_{V,t}$ representing the number of photons localized in a finite volume VVV at time ttt are investigated. To some extent these properties reflect the well-known difficulty of localizing photons in space-time. However, it is shown that, when the linear dimensions of VVV are large compared with the wave-length of any occupied mode of the field, the ${\stackrel{^}{n}}_{V,t}$ operator acquires some simple properties. The commutator of ${\stackrel{^}{n}}_{V,t}$ and the detection operator $\stackrel{^}{\mathrm{A}}(\mathrm{x}, t)$ is expressible in an interesting form. The commutator and relations derived from it become particularly simple for certain space-time regions which we label conjoint and disjoint. An orthogonal set of eigenstates of ${\stackrel{^}{n}}_{V,t}$ is found, together with the corresponding eigenvalues, and it is shown that an arbitrary state is expressible in terms of these eigenstates. Some NthN\mathrm{th}Nth-order correlations of the ${\stackrel{^}{n}}_{V,t}$ operators are evaluated, and the results are used to calculate the probability distribution of eigenvalues of ${\stackrel{^}{n}}_{V,t}$ for an arbitrary state of the field.