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Physical Review A

Cooperative Lamb shift and the cooperative decay rate for an initially detuned phased state

Richard M. Friedberg, Jamal T. Manassah

Published Apr 30, 2010• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.81.043845

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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 k1\ensuremath≅k0\ifmmode±\else\textpm\fiO(1R){k}_{1}\ensuremath{\cong}{k}_{0}\ifmmode\pm\else\textpm\fi{}O(\frac{1}{R})k1​\ensuremath≅k0​\ifmmode±\else\textpm\fiO(R1​) (where k0=\ensuremathω0/c{k}_{0}={\ensuremath{\omega}}_{0}/ck0​=\ensuremathω0​/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.
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.
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.
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.