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Laser Physics

The dynamical cooperative lamb shift in a system of two-level atoms in a sphere in the scalar photon theory

Jamal T. Manassah

Published Dec 7, 2009• 21 citations
Publisher

Abstract

Using the eigenmode analysis, we show that the frequency shift of the radiation emitted from a spherical cloud of two-level atoms in a scalar photon theory model, the dynamical Cooperative Lamb Shift ( CLS ), is dependent on the size of the sphere, the angle of emission, the instant of observation, the degree of initial excitation and the form of the initial state function.

DOI
10.1134/s1054660x10010159

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Using the eigenmode analysis, we show that the frequency shift of the radiation emitted from a spherical cloud of two-level atoms in a scalar photon theory model, the dynamical Cooperative Lamb Shift ( CLS ), is dependent on the size of the sphere, the angle of emission, the instant of observation, the degree of initial excitation and the form of the initial state function.
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.
Using the eigenmode analysis of the scalar photon theory, I compute the probability of the atoms remaining excited and the probability for the atoms remaining in the initial quantum state of a system of two-level atoms cloud in a sphere initially prepared to radiate in the forward direction, i.e., the single photon superradiance problem. The convergence in the results obtained for increasingly larger radii for the sphere suggests that the asymptotic limits for these quantities are obtained for a sphere with a radius equal to six times the resonant wavelength. I predict the maximal value of the probability of secondary excited states from large spheres at 17.1%.
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.
Radiation by the atoms of a resonant medium is a cooperative process in which the medium participates as a whole. In two previous papers we treated this problem for the case of a medium having slab geometry, which, under plane-wave excitation, supports coherent waves that propagate in one dimension. We extend the treatment here to the three-dimensional problem, focusing principally on the case of spherical geometry. By regarding the radiation field as a superposition of electric and magnetic multipole fields of different orders, we express it in terms of suitably defined scalar fields. The latter fields possess a sequence of exponentially decaying eigenmodes corresponding to each multipole order. We consider several examples of spherically symmetric initial excitations of a sphere. Small uniformly excited spheres, we find, tend to radiate superradiantly, while the radiation from a large sphere with an initially excited inner core exhibits temporal oscillations that result from the participation of a large number of coherently excited amplitudes in different modes. The frequency spectrum of the emitted radiation possesses a rich structure, including a frequency gap for large spheres and sharply defined and closely spaced peaks caused by the small frequency shifts and even smaller decay rates characteristic of the majority of eigenmodes.
The eigenmodes of a nanosphere configuration where the core is an ensemble of two-level atoms with resonance frequency ${\ensuremath{\omega}}_{C}$, and the shell is an ensemble of two-level atoms but with shifted resonant frequency ${\ensuremath{\omega}}_{B}$ are obtained for different values of the ratio of the core radius to the shell outer radius as function of $({\ensuremath{\omega}}_{C}\ensuremath{-}{\ensuremath{\omega}}_{B})$. I show that the eigenmodes belong to one of two branches. The dominant decay mode for superradiant emission may belong to either branch, depending on the system's parameters. In instances of configurations where the cooperative decay rate values for the two branches become equal, the cooperative exponential amplification (decay) is accompanied by temporal oscillations.