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Physical Review A

Cooperative Lamb shift in an ellipsoid

Richard M. Friedberg, Jamal T. Manassah

Published Jun 23, 2010• 21 citations
Publisher

Abstract

It has been long known that the global cooperative Lamb shift (CLS) in a large superradiantly emitting sphere has equal magnitude but opposite sign to that of a slab. This result was obtained from QED in 1973 for samples of uniform density. This change of sign holds as well for a Gaussian density distribution. The same result is also obtained for either density in the scalar simplification of QED now in frequent use. Since the CLS must be a continuous function of shape, and the slab resembles a limiting case of oblate ellipsoid, there must be some shape of ellipsoid for which the CLS is zero. We report a calculation of CLS for a Gaussian distribution of general azimuthally symmetric ellipsoidal shape. The CLS is found to vanish when the mean square radius is twice as great transversely as longitudinally.

DOI
10.1103/physreva.81.063822

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It has been long known that the global cooperative Lamb shift (CLS) in a large superradiantly emitting sphere has equal magnitude but opposite sign to that of a slab. This result was obtained from QED in 1973 for samples of uniform density. This change of sign holds as well for a Gaussian density distribution. The same result is also obtained for either density in the scalar simplification of QED now in frequent use. Since the CLS must be a continuous function of shape, and the slab resembles a limiting case of oblate ellipsoid, there must be some shape of ellipsoid for which the CLS is zero. We report a calculation of CLS for a Gaussian distribution of general azimuthally symmetric ellipsoidal shape. The CLS is found to vanish when the mean square radius is twice as great transversely as longitudinally.
We review the definition of cooperative Lamb shift originally introduced by ourselves and S. R. Hartmann in 1973. We point out that the definition specified the preparation of a sample of identical two-level atoms prepared with partial excitation by a short pulse. We spell out in some detail the reasoning behind our assertion that the CLS does not enter into the dielectric constant, which determines the transmission of cw radiation through a sample. We give a prescription, using the transfer matrix formalism, for determining the transmission coefficient through a slab, given the thickness in wavelengths and the dielectric constant. We explore the possibility of achieving a true measurement of the CLS in a gaseous cold-atom cloud, with the help of a large foreign gas broadening of the resonant line.
The electromagnetic transitions of Mössbauer nuclei provide ideal two-level systems to study the cooperative effects of X-rays with matter. Recently, we have observed the collective Lamb shift (CLS) in nuclear γ-ray superradiance [Röhlsberger, R.; Schlage, K.; Sahoo, B.; Couet, S.; Rüffer, R. Science 2010, 328, 1248–1251]. An ensemble of resonant 57Fe nuclei was embedded in the center of a planar cavity and was collectively excited by synchrotron radiation into a purely superradiant state. The decay of the superradiant state can be almost two orders of magnitude faster than the single-atom nuclear decay and can be accompanied by a CLS of more than 10 times the natural linewidth. Here we review the basic features of cooperative effects that occur in nuclear γ-ray superradiance in comparison with similar effects observed in the optical regime of the electromagnetic spectrum. We show that nuclear resonant scattering of synchrotron radiation opens new ways for systematic studies of cooperative effects in the interaction of X-rays with matter.
A landmark in the development of quantum electrodynamics was the discovery that emission–reabsorption of virtual photons modifies the value of energy levels in an atom from those computed by using Dirac’s equation (Lamb shift). An early result of statistical quantum electrodynamics was that the exchange of virtual photons in an ensemble of identical atoms leads as well to a change in the frequency of the radiation emitted from this system (cooperative Lamb shift). Dicke’s discovery that coherence effects lead to the shortening of the emission lifetime from a small sample by a factor equal to the number of atoms in the ensemble (superradiance or cooperative decay rate) was an early landmark in quantum optics. Both cooperative decay rate and cooperative Lamb shift were shown to have the same physical origin—the exchange of virtual photons, a process described by the Lienard–Wiechert dipole–dipole interaction. This effective potential is the kernel of the integral equation describing the dynamics of the system. This complex long-range kernel gives, for both cooperative quantities, strong dependence on the geometry of the atomic cloud. I summarize the known expressions for the initial cooperative decay rate and the cooperative Lamb shift in different geometries. The results for both the scalar photon and the vector photon (electrodynamics) theories for experimentally realizable systems of either uniform or phased polarization are given.
When many light emitters are subjected to an electromagnetic field with a wavelength λ, they may react collectively to the field. The resonant resulting, light-induced dipole-dipole interactions between atoms should lead to a collective frequency shift of the atomic lines. This shift, unfortunately named the cooperative or collective Lamb shift (CLS) despite its classical nature, depends on the shape of the sample. I will present a new investigation of the origin and validity of the CLS. To do so, we have measured the transmission resonance line shape of a dense hot vapor of potassium atoms confined in a slab with nanometer thickness. Using a complete model, we extract a density-dependent shift and width of the bulk atomic medium resonance (See Fig. 1(a)), deconvolved from the cavity effect [1].