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No Cooperative Lamb Shift in Response of Thin Slab to cw Beam of Resonant Light

Richard M. Friedberg, Jamal T. Manassah

Published May 9, 2022• 3 citations

Abstract

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.

DOI: 10.3390/atoms10020049

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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.

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].

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.

The cooperative Lamb shift (CLS) is hard to measure because in samples much larger than a resonant wavelength it is much smaller, for an initially prepared resonantly phased state, than the cooperative decay rate (CDR). We show, however, that if the phasing of the initial state is detuned so that the spatial wave vector is ${k}_{1}\ensuremath{\cong}{k}_{0}\ifmmode\pm\else\textpm\fi{}O(\frac{1}{R})$ (where ${k}_{0}={\ensuremath{\omega}}_{0}/c$ is the resonant frequency), the CLS grows to ``giant'' magnitudes making it comparable to the CDR. Moreover, for certain controlled values of detuning, the initial CDR becomes small so that the dynamical Lamb shift (DLS) can be measured over a considerable period of time.