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The Journal of Physical Chemistry A

Photon Statistics of Resonance Fluorescence in the Limit of Separated Spectral Lines

Henk F. Arnoldus, Franco Battaglia, Thomas F. George

Published Apr 1, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1021/jp5029728

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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.
Abstract We discuss the statistics of emission of photons by a single atom or ion illuminated by a laser beam at the frequency of quasi-resonance between two energy levels, a situation that corresponds to real experiments. We extend this to the case of two laser beams resonant with the energy differences between two excited levels and the ground state (three level atom in V-configuration). We use a novel approach to this type of problem by considering a Kolmogorov equation for the probability distribution of the atomic state, which takes into account first the deterministic evolution of this state under the effect of the incoming laser beam and second the random emission of photons during the spontaneous decay of the excited state(s) to the ground state. This approach yields solvable equations in the two level atom case. For the three level atom case we set the problem and clearly define its frame. The results obtained are valid in both opposite limits of rare and frequent spontaneous decay, compared to the period of the optical Rabi oscillations due to the interaction between resonant excitation and atomic levels. Our analysis gives access to various statistical properties of the fluorescence light, including one showing that its fluctuations in time are not invariants under time reversal. This result makes evident the fundamentally irreversible character of quantum measurements, represented here by the emission of photons of fluorescence.
We study the conditions for sub-poissonian photon statistics and squeezed states in the field of resonance fluorescence of a two-state atom. These conditions as a function of the detuning from resonance, the linewidth and the Rabi frequency have some overlap, but they are largely complementary. Super-poissonian statistics arise for small linewidths and large detunings, irrespective of the Rabi frequency. Squeezed states require small linewidths and either low or moderate Rabi frequencies, or large detunings from resonance.
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.
Summary form only given. Spectral detection of resonance fluorescence allows to regard atomic spectra as a sequence of elementary scattering events accompanied by emissions of photons with different frequencies. What the reduced atomic state (RAS) is after such a photon detection represents fundamental interest and appears important in understanding some correlation experiments. We report about derivation of analytic expression for the RAS for the important particular case of the well separated spectral lines of the resonance fluorescence spectrum.
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.