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Journal of Physics B Atomic and Molecular Physics

Correlation functions and the AC Stark effect

B. Renaud, R. M. Whitley, Carlos R. Stroud

Published Feb 21, 1976• 33 citations
Publisher

Abstract

A Heisenberg picture QED treatment is presented of the resonance fluorescence of a two-level atom. It is shown that the equations of motion for first-order correlation functions have an asymmetry with respect to the two time arguments. This asymmetry arises in a correct and careful evaluation of the unequal-time commutator between free-field and atomic operators. General solutions of these equations are found without going to the stationary limit.

DOI
10.1088/0022-3700/9/3/001

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A Heisenberg picture QED treatment is presented of the resonance fluorescence of a two-level atom. It is shown that the equations of motion for first-order correlation functions have an asymmetry with respect to the two time arguments. This asymmetry arises in a correct and careful evaluation of the unequal-time commutator between free-field and atomic operators. General solutions of these equations are found without going to the stationary limit.
The photon statistics of the two-photon fluorescence of a two-level atom in a squeezed vacuum is investigated by using the second-order correlation function g (2) ( t ). Based on its analytical solution, g (2) ( t ) is found to be an oscillatory function of time when the average photon number of the squeezed vacuum light field is above a threshold. The origin of the oscillatory behavior of g (2) ( t ), which generally results from the external driving field for resonance fluorescence, is the driving effect of the squeezed vacuum light field on the atom. It is also found that the oscillatory behavior of g (2) ( t ) can provide a measurement of the ac Stark effect.
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 problem of the interaction between a two-level atom and a quantum electromagnetic field is treated without the use of perturbation theory, without introduction of classical fields or factorization conditions for the states, and without assumptions about loss of memory. The calculation is carried out in the Heisenberg picture, without mode decomposition, and the conclusions all refer to physically measurable quantities, such as the fluorescence detected in the far field of the atom. It is shown that in a coherent field of constant amplitude the system always settles down to a quasistationary state, and that the stationarity is a manifestation of the quantum fluctuations. A solution for the growth of the fluorescent light intensity is presented that holds for any coherent exciting field. The two-time correlation function and the spectral density of the fluorescence are calculated, and are found to agree in the long-time limit with earlier results of Mollow. The two-time intensity correlation function of the field is derived, which corresponds to measurable photoelectric pair correlations, and it is found that this reflects several quantum features of the field. It is shown that quantum fluctuations are manifest more explicitly in two-time correlations in the steady state than in transient effects, like spontaneous emission in the vacuum. The measurement of such correlations therefore presents an opportunity for further experimental tests of quantum electrodynamics.
The authors give a theoretical description of the spectrally resolved intensity correlations of an electromagnetic field, in analogy with the definition of the physical time-dependent spectrum. The results are applied to evaluate the time correlations between the three lines in the spectrum of resonance fluorescence of a two-level atom. The authors include the possibility of a finite bandwidth of the incident radiation, and they allow for collisions with perturber atoms. Photons in the central Rayleigh line are emitted in a fully random fashion, without any correlation with previous or subsequent emissions. Two photons from the same sideband display antibunching in time, whereas two photons from opposite sidebands tend to bunch with a strong asymmetry in time. The effect of collisions and of a finite bandwidth is to diminish this asymmetry.
A Green's function formulation of the problem of resonance fluorescence from a system of identical two level atoms is presented. The general formulation is suitable for the investigation of a variety of nonlinear phenomena, e.g., propagation of electromagnetic waves inside the medium, fluorescence from a finite medium, optical bistability, atomic correlations, and additional side bands. Translational motion, collisions, Coulomb dipole–dipole interaction, excitation transfer, and local field corrections are also included in the basic equations. The effect of atom–atom correlations in the relaxation mechanism (radiative or collisional) is also taken into account. Finally, the results are also presented in the "generalized dressed atom" (GDA) representation. Applications of the general theory will be given in a future paper.