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Physical Review A

Nonperturbative approach to multimode photodetection

Michael Fleischhauer, D.‐G. Welsch

Published Jul 1, 1991• 11 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.44.747

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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.
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.
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.
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.
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.
A new technique is introduced in which a canonical transformation is used to transform the atom-radiation Hamiltonian to one without counter-rotating terms. As well as allowing an exact treatment of a one-mode dipole-coupled photodetection model, the theory predicts the existence of squeezing induced by the detector coupling. A feature of the canonical transformation used here is that the resulting interaction Hamiltonian has a symmetric combination of the p\ensuremath{\cdot}A and d\ensuremath{\cdot}E interactions. This allows the resolution of long-standing questions about the correct interaction Hamiltonian to use in photodetector theory. The new theory is able to handle dipole-coupled radiative transitions over ultrafast time scales of the order of a single oscillation period, where the rotating-wave approximation is invalid. The coupling of the detector to the radiation field is predicted to vary as ${f}^{\mathrm{\ensuremath{-}}1/2}$ at high frequencies, and as f+1/2{f}^{+1/2}f+1/2 at low frequencies.
Photodetection converts optical quantum states into measurement events, but the usual electric-field response model becomes restrictive when the detector response is shaped by cavity, superconducting, or metamaterial engineering. We develop a generalized quantum photodetection framework in which electric and magnetic field amplitudes contribute coherently to the detection operator, and analyze it in a far-field two-source geometry, a two-mode single-photon setting, and a lossy resonant detector model. The far-field reference case exhibits complete detector-amplitude cancellation, absent in the electric-only Glauber response, while the single-photon model shows that the detector continuously rotates the effective measurement basis and controls the first-order visibility via an exact closed-form law. In the resonant realization, a monitored radiative output channel can be dark while the detector remains internally excited and absorptive, with unit absorption of the matched input mode at critical coupling. These results identify basis-selective readout and channel-selective absorption as experimentally relevant signatures of engineered electric-magnetic photodetection.