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Laser Physics

The single photon superradiance from the eigenmode analysis

Jamal T. Manassah

Published Sep 23, 2009• 12 citations
Publisher

Abstract

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%.

DOI
10.1134/s1054660x09210087

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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%.
We calculate the field eigenmodes of the superradiant emission from an ensemble of NNN two-level atoms and confirm the results of other authors that Dicke superradiant emission is restricted to a small number of modes, with more than 90% of the photons emitted in a single dominant mode. While numerical techniques are effective due to the symmetry of the problem, we develop also an analytical method to approximate the modes in the limit of a large number of emitters.
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.
Using the eigenmodes of the scalar-photon kernel of interacting NNN two-level atoms in a cylindrical configuration, I obtain the time dependence of the superradiant emission for this geometry in two cases of interest. In particular, I consider first the case when one or two eigenmodes dominate the system's dynamics for an inverted system which has been initially prepared in an isotropic state and show that for certain values of the radius of the cylinder ($r=m\ensuremath{\lambda},m \mathrm{integer}$) one obtains temporal modulation in the state function. In the second case of interest, I show that the angular distribution of the state function of a system initially prepared in a phased state appears to be chaotic; however, I show that this behavior is due to the high multiplicity of modes having close values for their amplification coefficients.
The collective Lamb shift and associated radiative decay of a large cloud of radius R containing N atoms uniformly excited by one photon of wavelength lambda is analyzed. It is shown that the time evolution of the symmetric state prepared by single photon absorption in the limit R>>lambda is similar to that encountered in the Dicke limit of small sample (R<<lambda) superradiance. The theory includes virtual (counterrotating) terms naturally and thus provides a simple calculation of the collective Lamb shift of a single Dicke state.
Previous numerical work on the spatiotemporal distributions of the atomic polarization and of the difference in levels population of a one-dimensional system of two-level atoms initially inverted strongly suggested that the dynamics of the slab atomic ensemble in the linear regime of superradiance may best be treated by an eigenmode analysis. This method proved effective in obtaining both the time dependence of the atomic polarization and the spectral distribution of the emitted radiation from the slab in a number of physically interesting problems in the linear regime. In the present paper, I use the eigenfunction expansion technique to reduce the Maxwell-Bloch system of partial differential equations into a system of coupled first-order ordinary differential equations whose solutions clearly detail the contributions of the different modes to the system dynamics in the nonlinear regime of superradiant emission. I compute also the delay in time of the peak of the superradiant burst and its magnitude as functions of the ratio of the longitudinal decay rate to the transverse decay rate of the system.