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Journal of Physics B Atomic and Molecular Physics

Superradiance and subradiance. IV. Atomic cascades between degenerate levels

A. Crubellier, D Pavolini

Published Apr 14, 1987• 21 citations
Publisher

Abstract

For pt.III see ibid., vol.20, p.971 (1987). This paper is the fourth in a series devoted to the theoretical study of subradiance, i.e. of the cooperative inhibition of spontaneous emission. It first describes how the main results of the first paper have been developed in the general case of j to j' transitions between degenerate levels. In particular, the link between statistical mixing and symmetry properties is now derived in the general case of n-level atoms. The authors also explain how these results have allowed them to conceive the experiment which has recently provided the first evidence for the subradiance phenomenon, in the case of the 3 / 2 to 1 / 2 to 1 / 2 cascade of atomic gallium. The specific influence of the degenerate transitions which are associated with the Zeeman transitions of same polarisation is then discussed in a simplified example. Numerical simulations corresponding to the case of the experiment and to similar cases are finally presented and analysed.

DOI
10.1088/0022-3700/20/7/014

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•Jan 1, 1986•0 citations
For pt.III see ibid., vol.20, p.971 (1987). This paper is the fourth in a series devoted to the theoretical study of subradiance, i.e. of the cooperative inhibition of spontaneous emission. It first describes how the main results of the first paper have been developed in the general case of j to j' transitions between degenerate levels. In particular, the link between statistical mixing and symmetry properties is now derived in the general case of n-level atoms. The authors also explain how these results have allowed them to conceive the experiment which has recently provided the first evidence for the subradiance phenomenon, in the case of the 3 / 2 to 1 / 2 to 1 / 2 cascade of atomic gallium. The specific influence of the degenerate transitions which are associated with the Zeeman transitions of same polarisation is then discussed in a simplified example. Numerical simulations corresponding to the case of the experiment and to similar cases are finally presented and analysed.
For pt.I see ibid., vol.18, p.3811 (1985). This paper is the second of a theoretical study of subradiance, i.e. of the cooperative inhibition of spontaneous emission by a destructive interatomic interference, a phenomenon which has recently been observed. It is devoted to the particular case of atomic systems with degenerate transitions, i.e. with several transitions of the same frequency and the same polarisation. The interatomic interference which then occurs between the different transitions is responsible, when destructive, for a new sort of subradiance. The phenomenon is studied for a few typical level configurations, using both quantum mechanical and semiclassical models. The first example concerns a four-level configuration with two transitions of the same frequency and polarisation. The other examples concern three-level configurations with a degenerate transition; in the case of the 'degenerate' cascade, new superradiant beats and subradiance effects are predicted.
This paper is the first of a sequence of four papers devoted to the theoretical study of subradiance, i.e. of the cooperative inhibition of spontaneous emission, a phenomenon which has been recently observed. It is shown, in the typical example of a pencil-shaped sample of three-level atoms in the 'V' configuration (two transitions sharing a common lower level), that the system can spontaneously reach a state in which the emission in the so-called 'end-fire' superradiant field modes is cooperatively inhibited; this occurs for initially uncorrelated atoms when the initial state is a statistical mixing of the two upper states. The whole study is centered on the concept of destructive interatomic interference and it is therefore based on the symmetry properties of the collective atomic state with respect to the permutations of locally indistinguishable atoms. Semiclassical and fully quantum-mechanical models are used in turn. The obstacles to the interatomic interference and thus to subradiance are finally discussed.
For pt.II see ibid., vol.19, p.2109 (1986). This paper is the third in a theoretical study of subradiance, i.e. the cooperative inhibition of spontaneous emission by a destructive interatomic interference, a phenomenon which has recently been observed. The paper is devoted to the problem of the cooperative spontaneous emission of atomic samples with linear dimensions much smaller than the wavelength of the emitted light. The treatment, which is based on a fully quantum mechanical description of the problem, corroborates the results obtained in a semiclassical treatment by Friedberg and Hartmann (1974). Starting from a master equation describing both the cooperative spontaneous emission and the dipole-dipole interaction, a statistical treatment of the latter is introduced and the corresponding symmetry relaxation is studied. The evolution of the atoms is described in three typical cases that correspond to different sample geometries and/or atomic distributions. In one of these cases, in particular, the symmetry is first conserved and the atoms follow Dicke's predictions for a time. A 'retarded' symmetry breaking then occurs suddenly, near the zero value of the collective energy, as in Friedberg and Hartmann's calculations for a small sphere. With regard to the further evolution of the atoms in this case, which cannot be studied in a semiclassical model, it is shown that the decay rate quickly becomes much smaller than the incoherent decay rate. An unquestionable subradiance phenomenon is thus predicted in this case.
Up to now, the problem of the influence of level degeneracy on superradiance has been neglected. We describe here a first approach to this problem, in the frame of "small system" quantum model, generalized to the case of a collection of atoms with two degenerate levels of angular momenta jjj and ${j}^{\ensuremath{'}}$ connected by an electric or magnetic dipole transition. The differences between this case and the well-known case of two (nondegenerate) level atoms are of two types. The first type is due to the competition between different transitions sharing a common state: superradiant emission on a transition inhibits or initiates superradiance on transitions with smaller transition probability, depending upon which state, upper or lower, is the common one. This effect does not come from the equality of the different wavelengths and would also appear with the same states even if the degeneracy were removed; its qualitative influence on the features of the superradiant pulse can be easily estimated. The second type of differences is quite specific to level degeneracy and consists in interferences between the transitions of the collective system having the same frequency and polarization. In analogy with the description of collections of two-level atoms as spin-\textonehalf{} systems, a group-theoretical description of the considered collection of atoms is found suitable; the problem of the determination of a basis of collective states and its group-theoretical implications are discussed; in particular, it is found that the isotropic character of spontaneous emission allows us to use tensorial formalism and simplifies the formal description of the problem.
The first observation of the subradiance phenomenon is reported. As previously predicted, destructive interatomic interference may prevent cooperative deexcitation of a collection of atoms from being complete. In the case of a $j=\frac{3}{2}\ensuremath{\rightarrow}{j}^{\ensuremath{'}}=\frac{1}{2}$ transition, for example, half the atoms remain excited when the initial atomic state is a full statistical mixture with equal weights of all Zeeman sublevels of the upper level. The experiment has been realized with gallium atoms. The results agree with semiclassical calculations.
L'interference entre les atomes emetteurs qui caracterise l'emission spontanee cooperative peut etre constructive ou destructive. Dans le second cas, on prevoit une inhibition de l'emission, qu'on appelle subradiance. Cette these est consacree a l'etude theorique de ce phenomene, ainsi qu'a sa premiere demonstration experimentale. La partie theorique s'interesse notamment a la recherche des conditions necessaires a l'apparition de la subradiance pour des systemes macroscopiques. On y etudie les configurations a plusieurs niveaux, et, plus particulierement les transitions entre niveaux degeneres. La mise en evidence a ete realisee sur la cascade 4d₃/₂ → 5p₁/₂ → 5s₁/₂ du gallium atomique.