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Journal of the Optical Society of America B

Distribution of photoelectron separation times and photoelectron counting probabilities for resonance fluorescence

H. J. Carmichael

Published Feb 1, 1987• 0 citations
OpenAlex

Abstract

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.

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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.
The question of the temporal correlation of photons arising from a three-level atomic cascade is considered by calculating the joint probability of photoelectric detection for two detectors placed in the far field of the atomic emission. The analysis is carried out in the Heisenberg picture and involves the use of dipole, adiabatic (Weisskopf-Wigner), and rotating-wave approximations. Except for a small interval about zero time delay, which is however too short to be detected, it is found that the temporal correlation between emitted photons is one of sequential but otherwise statistically independent events, even when Landau's generalized decay and frequency-shift terms are included. This result is compared with a well-known expression for the joint spectral density of the emitted photons.
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 ${T}^{\ensuremath{-}\frac{1}{2}}$. 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.
We consider a three-level atomic system driven strongly on one transition and weakly on the other. The excited state on the weak transition is assumed to be metastable. We give an analysis of the fluorescence from the strong transition in terms of the elementary probability density p[0{p}_{[0}p[0​,t)(t1{\mathrm{t}}_{1}t1​,t2{t}_{2}t2​,..., tn{t}_{n}tn​) which gives us the probability density that exactly n photons are emitted at times t1{t}_{1}t1​,t2{t}_{2}t2​,...,tn{t}_{n}tn​ by the atom in the time interval [0,t). We show that p[0{p}_{[0}p[0​,t) essentially factorizes into products of conditional densities c\ifmmode \tilde{}\else \~{}\fi{}(\ensuremath{\tau}) that, given a photon is emitted at time zero, the next photon emission occurs at time \ensuremath{\tau}. This enables a simulation of the individual photon emissions to be given which shows directly the existence of prolonged dark windows in the fluorescence corresponding to the shelving of the electron in the metastable state or ``quantum jumps.''
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 present a microscopic model for the so-called quantum jumps in three-level atoms with one transition between two levels being strongly driven and the third level being metastable. Our model is based on calculating the elementary probability density P(0,t)(t1, … , t n ) which gives the probability density that exactly n photons are emitted at times t1, … , t n by the atom in the time interval (0, t) on the strong transition. This enables us to calculate the conditional probabilities c ˜ ( τ ) that given a photon has emitted at τ = 0 the next photon emission occurs at time τ. This quantity is shown to be the sum of two exponentials which for sufficient differetial in the time scales describe the periods of light and darkness. Using this conditional probability it is possible to numerically simulate the individual photon emissions.