The problem of resonance fluorescence from a two-level atom that is driven near resonance by a laser beam of finite bandwidth is tackled via the Heisenberg equations of motion, by a simple generalization of our previous treatment. It is found that the two-time intensity correlation function is again factorizable into a product of the mean light intensities at the two times, each of which is slightly modified under broad-band excitation from its value under monochromatic excitation. The spectrum of the fluorescence is affected rather more drastically, and, in general, becomes asymmetric under broad-band excitation.
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.
Exact closed-form expressions for the probability
p(n) that
n 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.
The photon-number statistics from resonance fluorescence of a two-level atom near a metal nanosphere driven by a laser field with finite bandwidth is studied theoretically. Our analysis shows that all interesting physics here takes place in a small area around the nanosphere where the near field and the atom-nanosphere coupling essentially affect the radiative properties of the atom. Computer modeling estimates this area roughly as $r\ensuremath{\le}2a$
(r is the distance from the center of the nanosphere to the atom), with
a being the radius of the nanosphere. At the larger distances, the influence of the nanoparticle vanishes and the atom tends to behave similarly to that in free space. It is shown that the distribution function
p(n,T) of the emission probability of
n photons in a given time interval
T in steady-state resonance fluorescence drastically depends on the atom location around the nanosphere for $r\ensuremath{\le}2a$, featuring a characteristic twist in the ridgelike dependence and a convergence time of up to $9\ensuremath{\mu}\text{s}$, two orders of magnitude slower than for the atom in free space. At large distances, the distribution converges to a Gaussian one, as for the atom in free space. The typical convergence time scale at large distances
r>2a tends to the convergence time of the atom in free space. There are also two areas symmetrical around the nanosphere in which $\ensuremath{\Omega}\ensuremath{\sim}\ensuremath{\gamma}$ and the convergence time goes to zero. This behavior is determined by the interplay of the radiative and nonradiative decay rates of the atom due to the coupling with the metal nanosphere and by the near-field intensity. Additional parameters are the normalized laser frequency detuning from the atomic resonance and the bandwidth of the incoming laser field.
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.
We derive the moment generating function for photon emissions from a single molecule driven by laser excitation. The frequencies of the fluoresced photons are explicitly considered. Calculations are performed for the case of a two-level dye molecule, showing that measured photon statistics will display a strong and nonintuitive dependence on detector bandwidth. Moreover, it is demonstrated that the antibunching phenomenon, associated with negative values of Mandel's Q parameter, results from correlations between photons with well separated frequencies.