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Physical Review A

Quantum beats in correlation functions of spontaneously emitted radiation and the transition rates of a multilevel atom

G. S. Agarwal

Published Jun 1, 1977• 19 citations
Publisher

Abstract

Spontaneous emission from a multilevel atom is studied from the viewpoint of quantum beats. The master equation describing the atomic dynamics is shown to have some parameters that oscillate at the beat frequencies, and because of this a Pauli type of master equation for the diagonal elements of the density matrix is not obtained. As a result, the transition rates in the atomic system show the characteristic beats. The relation of the rate of change of the energy of the atomic system to the total power radiated in the far zone is also established. The normally and antinormally ordered correlation functions of the spontaneously emitted radiation are calculated under the approximation that the level widths are much smaller than the beat frequencies. The contribution arising from the interference of the free field and the source terms to antinormally ordered correlations is carefully evaluated. The nature of beats present in normally and antinormally ordered correlations is discussed, and their detection from the viewpoint of causing transitions in another atom is considered. The change in the nature of beats due to atomic collective effects is studied. Finally, it is shown that because of quantum effects, intensity correlations do not show any beats.

DOI
10.1103/physreva.15.2380

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Soviet Journal of Quantum Electronics•Jun 30, 1988•1 citation
Spontaneous emission from a multilevel atom is studied from the viewpoint of quantum beats. The master equation describing the atomic dynamics is shown to have some parameters that oscillate at the beat frequencies, and because of this a Pauli type of master equation for the diagonal elements of the density matrix is not obtained. As a result, the transition rates in the atomic system show the characteristic beats. The relation of the rate of change of the energy of the atomic system to the total power radiated in the far zone is also established. The normally and antinormally ordered correlation functions of the spontaneously emitted radiation are calculated under the approximation that the level widths are much smaller than the beat frequencies. The contribution arising from the interference of the free field and the source terms to antinormally ordered correlations is carefully evaluated. The nature of beats present in normally and antinormally ordered correlations is discussed, and their detection from the viewpoint of causing transitions in another atom is considered. The change in the nature of beats due to atomic collective effects is studied. Finally, it is shown that because of quantum effects, intensity correlations do not show any beats.
The two-time second-order correlation function of the electric field spontaneously emitted by a system of two non-identical two-level atoms with different transition frequencies and different natural linewidths is calculated from the viewpoint of quantum beats and photon anticorrelation effects. Our analytical solution shows that the intensity correlation exhibits quantum beats. The quantum beats are shown to appear even if the two atoms are independent. For identical atoms, beats appear only if dipole–dipole interaction between the atoms is included, and they are then strongly dependent on the direction of observation with respect to the line connecting the two atoms. The anticorrelation effect between the photons emitted in different directions is also discussed. It is shown that for a single-time correlation function the condition for anticorrelation is independent of interatomic interactions. For a two-time correlation function, interatomic interactions reduce the anticorrelation effect.
Spontaneous emission from two non-identical two-level atoms is examined from the viewpoint of quantum beats and super-radiant effects. Our analytical solution, illustrated graphically, shows that the total radiation rate exhibits temporal sinusoidal modulation (beats). The beats result from the fact that the atoms have different transition frequencies and different natural linewidths. For small interatomic separations, the total radiation rate is greater than its initial value. This ‘super-radiant’ property is absent in the case of noninteracting atoms, as well as for two identical atoms. In the case when the atoms are separated by distances much smaller than the resonant wavelength, the population trapping effect can take place in such a system.
The cooperative spontaneous emission from three identical, initially excited two-level atoms separated by distances much smaller than the resonance wavelength is studied taking into account the dipole-dipole near-field interaction. Approximate expressions are obtained for the time evolution of the atomic system, the radiated intensity and the one-time intensity correlation function. As a new result the authors find that the near-field interaction may lead to the occurrence of beats in the intensity. Moreover, the destructive effect of the near-field interaction on the coherence properties of the emitted radiation is discussed.
Even with incoherent driving, coherence effects may show up in the spontaneous emissions of an atom. We derive a general master equation and predict that for both the three-level V and \ensuremath{\Lambda} systems the intensity correlation function g(2){\mathit{g}}^{(2)}g(2)(\ensuremath{\tau}) may exhibit pronounced oscillations with frequency \ensuremath{\delta}\ensuremath{\omega} if the level separation \ensuremath{\Elzxh}\ensuremath{\delta}\ensuremath{\omega} is small. We obtain a marked difference in the beat amplitude for a single \ensuremath{\Lambda} system as compared to a beam or gas. If \ensuremath{\delta}\ensuremath{\omega} is much smaller than the natural linewidth we find dark and light periods in the spontaneous emission.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
The appearance of quantum beats in a two-time second-order correlation function is discussed for the case of spontaneous emission from two nonidentical atoms. The atoms are separated by distances comparable with the resonance wavelength and have different transition frequencies and natural line widths. It is shown that quantum beats appear for nonidentical and identical atoms. In the latter case the dipole–dipole interaction between the atoms is a necessary condition for the appearance of quantum beats. Two mechanisms lead to quantum beats in the case of nonidentical atoms and such beats may appear even for independent atoms. The effect is similar to that of interference between independent beams.