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Journal of Modern Optics

Waiting times, probabilities and theQfactor of fluorescent photons

Henk F. Arnoldus, Robertsen A. Riehle

Published Jun 12, 2012• 4 citations
Publisher

Abstract

Photons in resonance fluorescence from a single atom are correlated, and this affects the photon detection statistics. We consider the time dependence of the variance for the number of counts in a time interval [0, T], and we show that there exist Poisson points for certain values of the atomic parameters. At these points, the variance equals the average, like for a Poisson process. We also show that the conditional probability density for the detection of the nth photon after a detection of a photon at time zero can be obtained from the conditional probability density for the first photon, and we illustrate this by computing several conditional probability densities for a special case of interest. The unconditional probability density for the detection of the nth photon can be obtained from the conditional probability densities and the probability for detecting n photons in [0, T] can subsequently be found from the unconditional probability densities.

DOI
10.1080/09500340.2012.693634

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Photons in resonance fluorescence from a single atom are correlated, and this affects the photon detection statistics. We consider the time dependence of the variance for the number of counts in a time interval [0, T], and we show that there exist Poisson points for certain values of the atomic parameters. At these points, the variance equals the average, like for a Poisson process. We also show that the conditional probability density for the detection of the nth photon after a detection of a photon at time zero can be obtained from the conditional probability density for the first photon, and we illustrate this by computing several conditional probability densities for a special case of interest. The unconditional probability density for the detection of the nth photon can be obtained from the conditional probability densities and the probability for detecting n photons in [0, T] can subsequently be found from the unconditional probability densities.
We study the general properties of the photon statistics of fluorescence radiation, emitted by a two-level atom in a strong laser field and a perturber bath. We investigate the deviation of the factorial moments from a Poisson distribution, and we show that for long counting times the lowest-order correction can be expressed entirely by the quantity Q, which represents the deviation of the variance from the average. We introduce and evaluate the quantities gn(t), which serve as a measure for the deviation of the higher-order statistics from Poissonian statistics. Subsequently we obtain explicit expressions for the average waiting time for the appearance of the n th photon, after an arbitrary initialization of the counting process. It turns out that the average time delay is again determined by Q. Hence this parameter can be measured in a photon-counting experiment, which involves only the observation of a single photon, rather than an (in principle) infinite number of photons.
We have studied the statistics of fluorescent photons emitted by a two-state atom in a laser beam in the limit where either the detuning or the Rabi frequency is large. For this case, the spectrum of resonance fluorescence has three separated lines. We have obtained closed form expressions for the conditional probability density for the emission of the nth photon and for the probability for emission of n photons in a time interval [0,T]. Our solutions are complementary to the known solutions for the case of perfect resonance.
Detection of photons from electromagnetic radiation can be considered as the appearance of random events on the time axis. When an attenuator is placed in front of the detector, which attenuates the intensity by a factor of α, the statistical properties of the detected photons are altered. We show that simple relations exist between the statistical functions of the photons detected from the attenuated field and the same functions for the photons that would be detected from the unattenuated field. We also derive several recurrence relations for the statistical functions involving their dependence on the parameter α. For photon detection from resonance fluorescence, the parameter α appears naturally as the probability that an emitted photon is detected. In this case, there is no attenuator, but the parameter α appears in the same way. We show that the probability for the emission (α=1) of n photons in a given time interval can easily be computed, and with the general theory we can then obtain the result for the detection of n photons (α<1).
In recent papers by Cohen-Tannoudji and Dalibard1 and Zoller et al.2 the statistics of quantum jumps in a three-level atom are analyzed in terms of the joint probability density W(t + τ;t) for a photon emission at time t followed by the next photon emission at time t+ τ. W(t+ τ;t) differs from the second-order correlation function G(2) (t + τ;t) often used to characterize photon statistics; the latter gives the joint emission probability without the condition that there be no emissions in the interval t to t + τ. Alter dividing by the density for single-photon emission, W(t + τ;t) gives the probability density for a delay τ between photon emissions.