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Physical Review A

Photon-number statistics in resonance fluorescence

Daan Lenstra

Published Dec 1, 1982• 57 citations
Publisher

Abstract

The theory of photon-number statistics in resonance fluorescence is treated, starting with the general formula for the emission probability of $n$ photons during a given time interval $T$. The results fully confirm formerly obtained results by Cook that were based on the theory of atomic motion in a traveling wave. General expressions for the factorial moments are derived and explicit results for the mean and the variance are given. It is explicitly shown that the distribution function tends to a Gaussian when $T$ becomes much larger than the natural lifetime of the excited atom. The speed of convergence towards the Gaussian is found to be typically slow, that is, the third normalized central moment (or the skewness) is proportional to ${T}^{\ensuremath{-}\frac{1}{2}}$. However, numerical results illustrate that the overall features of the distribution function are already well represented by a Gaussian when $T$ is larger than a few natural lifetimes only, at least if the intensity of the exciting field is not too small and its detuning is not too large.

DOI
10.1103/physreva.26.3369

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The theory of photon-number statistics in resonance fluorescence is treated, starting with the general formula for the emission probability of nnn photons during a given time interval TTT. The results fully confirm formerly obtained results by Cook that were based on the theory of atomic motion in a traveling wave. General expressions for the factorial moments are derived and explicit results for the mean and the variance are given. It is explicitly shown that the distribution function tends to a Gaussian when TTT becomes much larger than the natural lifetime of the excited atom. The speed of convergence towards the Gaussian is found to be typically slow, that is, the third normalized central moment (or the skewness) is proportional to T\ensuremath−12{T}^{\ensuremath{-}\frac{1}{2}}T\ensuremath−21​. However, numerical results illustrate that the overall features of the distribution function are already well represented by a Gaussian when TTT is larger than a few natural lifetimes only, at least if the intensity of the exciting field is not too small and its detuning is not too large.
Exact closed form expressions are derived for the photon number distribution functions and the factorial moments in resonance fluorescence from a two-level atom that is driven on or near resonance by a laser beam of finite bandwidth. Derivation procedure that was used by Lenstra in the theory of photon number statistics for limited cases is applied to our generalized problem with help of the theory of resonance fluorescence with excitation of finite bandwidth. All of the results are presented in a fully suitable form for practical analyses of statistical features of resonance fluorescence.
Exact closed-form expressions for the probability p(n)p(n)p(n) that nnn photons are spontaneously emitted in a given time by a two-level atom in a coherent exciting field are derived using the theory of atomic motion in a traveling wave.
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.
The photon counting statistics of the fluorescence of a single atom driven by a resonant coherent field has been calculated and the results checked experimentally. The authors examine the effect of detuning the coherent field and/or the effect of an additional thermal field on the photon counting statistics. They work with a master equation that they have used previously and derive exact results for the probability distribution function P(n, tau ) for the emitted photons and for the normalised second factorial moment Q. The subPoissonian character of the photon statistics, for which Q<0, is most pronounced for weak thermal fields or weak detuned coherent fields. For short counting times tau the statistic is dominated by any spontaneous emission that occurs, as expected. Increasing numbers of black-body photons tend to increase the Q values, as might be expected-especially in the case of an initially excited atom-but no simple rule emerges. The recent experiments in the microwave region on Rydberg atoms interacting with thermal radiation in (low-Q) cavities offers the possibility of observing thermal-field effects on the photon counting statistics.
It is shown that the distribution of the momentum transfer to an atom in resonance fluorescence may be easily derived from the angular distribution of spontaneously emitted photons. Some general properties of the momentum distribution are found, and explicit forms of the distribution are derived for several special cases. It is shown that the distribution tends to become Gaussian after a large number n of spontaneous photon emissions, with variance proportional to n, but it is already close to the asymptotic form when n=3. When the appropriate averaging is performed to allow for the velocity spread of the atoms, the derived distributions account for the results of atomic beam deflection measurements, but much of the information about the original distributions is thereby lost. In a standing wave field both stimulated and spontaneous emissions contribute to the atomic momentum, in such a way that the variance of the momentum spread is again proportional to the number n of photon emissions.
Statistics of fluorescence photons emitted by a single two-level atom interacting with a continuous wave laser field are analyzed. The photon-counting distribution is calculated for the so-called intermediate fluorescence photons, i.e., those counted during the time interval between instants at which photons are emitted. The result is a sub-Poissonian (narrower than Poisson) distribution, which agrees with experimental observations. This intermediate-photon distribution is used to calculate the average number of fluorescence photons, the second factorial moment of the photon-counting distribution, and Mandel’s Q parameter commonly used to evaluate the deviation of photon statistics from the Poisson distribution. The theoretical expressions obtained for moments of the intermediate-photon distribution are different from well-known Mandel’s formulas.