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Journal of Physics B Atomic Molecular and Optical Physics

On the origin of divergences in the coincidence probabilities in cavity photodetection experiments

Teppo Häyrynen, Jani Oksanen, Jukka Tulkki

Published Jun 17, 2009• 8 citations
Publisher

Abstract

The theory of photon correlation is an established part of quantum electronics. However, recently reported divergences in the theory of time correlated detection of photons show that important details of cavity photon statistics are still incompletely understood. The quantum jump superoperators of the SD photon counting model given by Srinivas and Davies (1981 J. Mod. Opt. 28 981–96 ) do not fulfil the assumption of the bounded interaction rate. This has raised doubts about the consistency of the SD photon counting model and especially about the existence of coincidence probability density (CPD) functions (Dodonov et al 2005 J. Opt. B: Quantum Semiclass. Opt. 7 99–108). In this work, we start from the first principles of the quantum trajectory theory and show how the different coincidence probability densities and coincidence probabilities (CPs) have to be calculated. CPDs derived by us are well defined, and CPs are finite and correctly normalized for all fields with finite photon number expectation value. Furthermore, we show that the SD model reproduces photon bunching and antibunching phenomena when consistent derivation for the second-order coherence degree is used.

DOI
10.1088/0953-4075/42/14/145506

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The theory of photon correlation is an established part of quantum electronics. However, recently reported divergences in the theory of time correlated detection of photons show that important details of cavity photon statistics are still incompletely understood. The quantum jump superoperators of the SD photon counting model given by Srinivas and Davies (1981 J. Mod. Opt. 28 981–96 ) do not fulfil the assumption of the bounded interaction rate. This has raised doubts about the consistency of the SD photon counting model and especially about the existence of coincidence probability density (CPD) functions (Dodonov et al 2005 J. Opt. B: Quantum Semiclass. Opt. 7 99–108). In this work, we start from the first principles of the quantum trajectory theory and show how the different coincidence probability densities and coincidence probabilities (CPs) have to be calculated. CPDs derived by us are well defined, and CPs are finite and correctly normalized for all fields with finite photon number expectation value. Furthermore, we show that the SD model reproduces photon bunching and antibunching phenomena when consistent derivation for the second-order coherence degree is used.
Time development of electromagnetic fields in closed cavities under continuous detection of photons continues to be a subject of confusing controversy. Recently Dodonov et al. [Phys. Rev. A, 75, 013806, 2007] argued that their model of quantum superoperators (E model) invalidates some of the predictions of the previously introduced photon counting model of Srinivas and Davies [J. Mod. Optic. 28, 981, 1981] (SD model). Both the SD and the E models are based on two postulated quantum jump superoperators: (1) the one-count operator corresponding to the absorption of a single photon and (2) the no-count operator. In this work we develop a stochastic difference equation that describes the dissipative coupling of the cavity field and the detector. The difference equation is based on non-perturbative treatment of the cavity-detector coupling. In spite of being non-integrable due to the coupling of the detector with an external reservoir it can be used to derive the exact forms of the quantum jump superoperators. When applied to a particular photon counting measurement our theory gives predictions identical with those of the SD model which should be considered a non-perturbative and ab-initio result. It is pointed out that available experimental results coincide with the results given by the ab-initio SD model. We summarize some of the key characteristics of cavity fields and photon counting processes to demonstrate that the results given by the SD model are consistent with the principles of quantum mechanics while those given by the E model are not.
We address the issue of the detection and counting of photons. We assume an electromagnetic field enclosed in an ideal cavity, which together with the detector constitute a closed system, so no photon is lost to the environment. Basing ourselves on a microscopic model consisting of a set of two-level atoms (the detector) interacting with the field, we derive a 'nonlinear' jump superoperator. We compare the count statistics calculated within our model with those obtained from previous models, such as the coincidence probability density , two-count conditional probability density , waiting times and the second order correlation function , for several field states.
The external photodetection of cavity radiation is considered as a continuous quantum measurement in the sense of Srinivas and Davies [Opt. Acta 28, 981 (1981)]. It is analyzed in terms of three elementary processes: the no-leak process, the one-leak-undetected process, and the one-leak-detected process. The first two form a no-count process and the last one, a one-count process. Superoperators for these elementary processes are postulated, following the examples of Srinivas and Davies. The superoperators for the one-count process, the no-count process, the m-count process, and the nonreferring process are then derived based on the postulates. The following implications of these superoperators are found: (1) the change of average photon number due to the one-count process is identical to the well-known Q parameter introduced by Mandel [Opt. Lett. 4, 205 (1979)]; (2) the no-count process is governed by a pseudomaster equation; (3) The probability for the m-count process is the Poisson transform of the initial P function of the cavity radiation; and (4) the nonreferring measurement process is governed by a master equation.
We modify some aspects of the continuous photodetection theory proposed by Srinivas and Davies (SD) (1981 Opt. Acta 28 981), which describes the non-unitary evolution of a quantum field state subjected to a continuous photocount measurement. In order to remedy inconsistencies that appear in their approach, we redefine the ‘annihilation’ and ‘creation’ operators that enter in the photocount super-operators. We show that this new approach not only still satisfies all the requirements for a consistent photocount theory according to SD precepts, but also avoids some weird result appearing when previous definitions are used.
The existence of a correlation between the arrival times of photons has been confirmed by measurements with a coincidence counter having a resolving time of 3·5 X 10-9 sec in three different experiments. In the first experiment it was found that the number of coincidence counts recorded from two photomultipliers, the apertures of which were optically superimposed, was significantly greater than when the light beams were incoherent. Furthermore, the number of these correlated counts was in satisfactory agreement with that predicted by theory. In the second experiment the change in the number of excess coincidences was measured as the degree of coherence of the light was altered by increasing the apparent separation of the photocathodes, and in this case also there was reasonable agreement between theory and experiment. In the final experiment it was shown that there was a significant difference between the number of coincidences observed when the light beams were in identical as opposed to orthogonal polarizations, and this last result especially makes it extremely improbable that the correlation could be caused by some spurious effect, such as plasma oscillations in the source, since the light source itself was found to be completely unpolarized.
Cavity-QED systems have recently reached a regime where the light-matter interaction strength amounts to a non-negligible fraction of the resonance frequencies of the bare subsystems. In this regime, it is known that the usual normal-order correlation functions for the cavity-photon operators fail to describe both the rate and the statistics of emitted photons. Following Glauber's original approach, we derive a simple and general quantum theory of photodetection, valid for arbitrary light-matter interaction strengths. Our derivation uses Fermi's golden rule, together with an expansion of system operators in the eigenbasis of the interacting light-matter system, to arrive at the correct photodetection probabilities. We consider both narrow- and wide-band photodetectors. Our description is also valid for point-like detectors placed inside the optical cavity. As an application, we propose a gedanken experiment confirming the virtual nature of the bare excitations that enrich the ground state of the quantum Rabi model.