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Physical Review A

Cooperative Lamb shift for a cloud of two-level atoms located between two parallel-plate conductors

Jamal T. Manassah

Published Feb 18, 2011• 3 citations
Publisher

Abstract

I give the closed-form analytic expression for the cooperative Lamb shift (CLS) in the scalar photon theory for an ensemble of two-level atoms in the shape of a slab located between two parallel-plate perfect conductors. I compute this quantity for the cases when the atomic system is initially prepared in a phased state and in the uniform state.

DOI
10.1103/physreva.83.025801

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I give the closed-form analytic expression for the cooperative Lamb shift (CLS) in the scalar photon theory for an ensemble of two-level atoms in the shape of a slab located between two parallel-plate perfect conductors. I compute this quantity for the cases when the atomic system is initially prepared in a phased state and in the uniform state.
We describe how the interatomic potential, or equivalently the so-called cooperative Lamb shift for a system of two identical atoms, is modified when the system is situated near metallic surfaces. For a special configuration of the atomic dipoles at a fixed distance RRR, we obtain closed analytical expressions of the potential as a function of RRR as well as the distance ZZZ of the system from a perfectly conducting surface. The potential for the same configuration is also calculated with the perfect conductor replaced by a semi-infinite electron plasma. In this case the potential gains additional contributions from the surface plasmons as well as from photon modes partially transmitted into the plasma. The surface-plasmon contribution contains an exponential function decreasing with respect to ZZZ and is consequently significant only close to the surface of the plasma. The treatment is fully quantum mechanical.
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.