Probability Densities for Fluorescent Photons Emitted by a Two-State Atom Driven by a Laser

Fluorescent photons emitted by a two-state atom in a laser beam are correlated. We have obtained the probability density for the emission of the th photon after a random initial time . It is shown that the correlations between the photons lead to a deviation from the poissonian value for this function (the probability density for independent events), although the deviation is not as significant as one may expect. 1. Introduction When a two-state atom, with energy level separation , is immersed in a laser beam with an angular frequency , then photons are exchanged between the atom and the field in stimulated absorption and emission, provided that is near . We shall consider the case of resonance, so . In addition, photons are emitted in spontaneous transitions from the excited state to the ground state, and these photons are emitted in all direction as electric dipole radiation. The emission of these fluorescent photons can be considered as a random event process on the time axis, and we assume that the duration of each event is negligible (dots on the time axis). This interpretation can be justified with the theory of photon detection from an electromagnetic field [1, 2]. Let be the Einstein coefficient for spontaneous transitions from the excited state to the ground state [3] and let be the population of the excited state. Then, the number of emitted photons per second is equal to and this is the intensity of the random process. We shall assume that the atom is in the steady state, so that is independent of time. The temporal statistics of photon emissions can be represented by the probability densities , with , 2,…. Let be the time at which the th photon is emitted, after an initial time . Since photons are emitted randomly, is a random variable, and its probability density is : We shall evaluate for the emission of fluorescent photons. If photons were emitted independently of each other, then the emission process would be a Poisson process, and the probability for the emission of photons in would be The probability for the emission of the th photon in is . If this emission would be independent of previous emissions, then so that 2. Photon Correlations in Resonance Fluorescence Correlations between random events are expressed through the intensity correlation functions [4, 5]: For resonance fluorescence, these correlation functions take the form [6, 7] involving the function . This function equals the population of the excited state at time , under the condition that the atom is in the ground state at . Therefore, From (7), we then see that an