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Journal of Physics A Mathematical and General

Quantum-theory of photodetection without the rotating wave approximation

Michael Fleischhauer

Published Jan 16, 1998• 15 citations
Publisher

Abstract

A perturbative description of photodetection without the rotating wave approximation is presented. With the help of Green function techniques on an extended time contour renormalized expressions for the single- and multitime counting probabilities are derived, which explicitly reflect the causal propagation of signals. In the case of classical light fields the moments of the counting statistics can be expressed in terms of field correlation functions in agreement with the result of Bykov and Tatarskii. For non-classical radiation fields such as squeezed states the Bykov-Tatarskii formula is shown to be incorrect.

DOI
10.1088/0305-4470/31/2/007

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A perturbative description of photodetection without the rotating wave approximation is presented. With the help of Green function techniques on an extended time contour renormalized expressions for the single- and multitime counting probabilities are derived, which explicitly reflect the causal propagation of signals. In the case of classical light fields the moments of the counting statistics can be expressed in terms of field correlation functions in agreement with the result of Bykov and Tatarskii. For non-classical radiation fields such as squeezed states the Bykov-Tatarskii formula is shown to be incorrect.
A multimode photodetection theory is developed that includes both the presence of radiation sources and the damping of the field due to the measurement. Making use of Green's-function techniques and the rotating-wave approximation, the counting statistics is calculated in terms of input field quantities. The treatment of the detector-field interaction yields expressions that essentially differ from the results of nonperturbative single-mode theories but are very similar to those of the well-known perturbative photodetection theories. The field damping due to the detection of photons is found to lead to a renormalized detector response function that approaches a maximum value if the number of detector atoms tends to infinity. It is worth noting that our result may be closely related to those of Mandel's heuristic description of photodetection based on the concept of configuration-space photon number operators.
The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcell's explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The representation of the classical wave amplitude, of the light by Gabor's complex analytic signal appears naturally in this treatment. It is shown that the correlation of counts registered by N separate photodetectors at N points in space is determined by a 2 N th order correlation function of the complex classical field. The variance of the individual counts is shown to be expressible as the sum of terms representing the effects of classical particles and classical waves, in analogy to a well-known result of Einstein relating to black-body radiation. Since the theory applies to correlation effects obtained with any type of light it applies, in particular, to the output of an optical maser, although, for a maser operating on one mode, correlation effects are likely to be very small.
A nonequilibrium open-system theory for continuous photodetection processes is developed using a probability-density functional combined with the path-integral technique. Nonunitary time evolution of the system-environment density operator due to measurement back action and continuous measurement is exactly described for three familiar photodetection processes. New exact photocounting formulas are obtained for these three processes; in particular, Mollow's photocounting formula is generalized for a nonequilibrium open system. Distinctive features between a closed system and an open system are presented; for example, the probability distribution for the number of counts is shown to contain complete information about photoelectron statistics for a closed system, whereas it does not for an open system. The obtained formulas are applied to photodetection of a single-mode photon field that interacts linearly with a pump source consisting of a single harmonic oscillator and a reservoir consisting of an infinite number of harmonic oscillators. In the former case, increasing the ratio of the source-field coupling constant to the field-detector coupling constant causes the photon field to cross over from a closed attenuating field to an open stationary field. In the latter case, a quantum-mechanical fluctuation-dissipation theorem in the continuous photodetection context is discussed.
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.
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 Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T) that nnn photo-ionizations are observed in the time interval ttt to t+Tt+Tt+T, and (2) quantities related to Pn(t,t+T){P}_{n}(t, t+T)Pn​(t,t+T), such as its generating function and various moments. Pn(t,t+T){P}_{n}(t, t+T)Pn​(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 PPP representation. Using this result, the character of Pn(t,t+T){P}_{n}(t, t+T)Pn​(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.
We reconsider various approaches to the quantum theory of photodetection from the point of view of the quantum theory of measurement, and show that important differences between them depend upon the manner in which they take into account the modification of the field distribution produced by the presence of the detector. We show that the Mandel photon counting formula may lead to unphysical results, such as negative probabilities, in some situations just because this modification is not incorporated into the model. We also show that the recently developed quantum theory of continuous measurements provides a completely satisfactory framework for discussing photon counting experiments. As an illustration, an exact derivation of the photon statistics of a single-mode free field is presented, and some of the results are shown to be identical to those derived earlier by Mollow and others. We also indicate how our approach can be extended to discuss the photon statistics of fields in interaction with sources and reservoirs.