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Physics Letters A

Conditional probability densities for photon emission in resonance fluorescence

Henk F. Arnoldus, Robertsen A. Riehle

Published Jul 13, 2012• 4 citations
Publisher
DOI
10.1016/j.physleta.2012.06.038

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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.
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.
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.