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International Conference on Atomic Physics (ICAP)

COOPERATIVE EMISSION OF LIGHT QUANTA: A THEORY OF COHERENT RADIATION DAMPING

Roy J. Glauber

Published Mar 1, 2009• 0 citations
Publisher

Abstract

A quantum emitted by any of a collection of identical atoms may be absorbed and re-emitted by other atoms many times before it eventually emerges. The radiation process is thus best described as collective or cooperative in nature. The atomic excitations are shown to attenuate as linear combinations of certain characteristic decay modes that lend a complex structure to the spectrum radiated. Instead of a single line, it becomes a closely-spaced multiplet of lines, the elements of which have a variety of lifetimes, line-shifts and line-widths. We calculate these quantities, first with an abstract two-state model for the atoms and then with an isotropic four-state model that accommodates the full polarization dependence of the radiation.

DOI
10.1142/9789814273008_0004

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A quantum emitted by any of a collection of identical atoms may be absorbed and re-emitted by other atoms many times before it eventually emerges. The radiation process is thus best described as collective or cooperative in nature. The atomic excitations are shown to attenuate as linear combinations of certain characteristic decay modes that lend a complex structure to the spectrum radiated. Instead of a single line, it becomes a closely-spaced multiplet of lines, the elements of which have a variety of lifetimes, line-shifts and line-widths. We calculate these quantities, first with an abstract two-state model for the atoms and then with an isotropic four-state model that accommodates the full polarization dependence of the radiation.
The absorbed/emitted radiation will be treated classically, as a monochromatic plane wave, whereas the atomic system that can absorb/emit such radiation will be described quantum-mechanically. However, no incident radiation beam is needed, and the atom “falls down” after a certain time interval to the ground state without any external reason. In the spontaneous emission processes the emitting atom can be thought of as a classical oscillating dipole, whose amplitude decreases due to the energy loss through the radiation. If an ensemble of identical atoms is considered and if, for simplicity, each atom emits radiation with the same spectral shape and full width at half maximum, it is impossible to ascribe a definite spectral component with intensity to a definite atom. In this case, such a broadening is identical for all atoms in the ensemble. On the microscopic scale, absorption of radiation is reduced to the transitions between pairs of energy levels of a discrete atomic spectrum.
We discuss from first principles the cooperative decay of a system of two-level atoms, initially prepared in an uncorrelated excited state with population inversion NNN, and we give the conditions under which the superfluorescence effect occurs. Describing the atomic system in terms of collective variables, we derive a master equation for the reduced atomic density operator, which gives rise both to a damping and to a time-dependent frequency shift in the dynamics of collective modes. The coupled equations of motion are solved with a self-consistent approach. It is found that the system goes through a nonexponential decay if the maximum length of the active volume is smaller than a "cooperation range" and larger than a "threshold length," in agreement with the one-mode theory. The radiation burst has a time width proportional to ${N}^{\ensuremath{-}1}$, and its intensity is proportional to N2{N}^{2}N2. Specializing to a pencil-shaped volume, we find that only two atomic modes need to be considered; in this case, the average emitted radiation is all condensed in the two diffraction patterns of the opposite axial modes.
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.
The processes of absorption and emission of radiation by a system of two interacting atoms is investigated. Separation between the atoms is assumed to be much larger than the atomic diameter, and interactions considered are those between the individual atoms and the radiation field and the electrostatic interaction between the atoms. Calculations include those for the time-dependent probability amplitude of the intermediate and final states as well as that of the emission line shape. The line shape of the emitted radiation consists of two peaks whose amplitude, separation, and half-width are functions of both the internuclear separation R of the atoms and the energy difference of the excited states of the two atoms. For a large energy difference, the peaks are distinct; for small energy differences, the peaks are distinct at smaller separations, but merge toward the line shape characteristic of a single, isolated atom as R increases.
In the absence of incoherent relaxation processes, the emission, absorption, and transport of resonant radiation by an extended medium are all cooperative processes in which the medium participates as a whole. Such radiative processes depend sensitively on the geometry of the source, on its size and density, and on the interplay of electric polarization resonances with the ambient electromagnetic field. By making a succession of simple approximations, we reduce the radiative interaction problem to one of solving a linear integral equation for the time-dependent polarization of the radiating medium. For the example of one-dimensional geometry, we show that the polarization possesses a sequence of exponentially decaying eigenmodes that oscillate at frequencies slightly shifted from that of the fundamental resonance. Only a few of these modes, for any radiating medium, are strongly coupled to the electromagnetic field, and thereby radiate efficiently. All of the other modes, infinitely many of them, are found to radiate quite slowly, or not at all. We consider in detail the radiation emitted by an arbitrary initial polarization distribution when the electromagnetic field is initially empty. Analytical and numerical results are presented for the time-dependent fields that follow from a wide variety of initial excitations. A locally oscillatory exchange of energy is found to take place between the electric and polarization fields, which gives rise to a ringing of the radiated intensity. The spectra of the emitted light are shown accordingly to be complex in structure, and to have a frequency gap which becomes independent of the size of the medium and its initial polarization distribution for media that are sufficiently large. The gap is caused by the interference and interaction of oppositely propagating excitations.