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Physical Review

Emission of Radiation from a System of Many Excited Atoms

V. Ernst, P. Stehle

Published Dec 25, 1968• 183 citations
Publisher

Abstract

The Weisskopf-Wigner theory of the natural linewidth of a single isolated atom is extended to a system of $N(\ensuremath{\gg}1)$ identical nonoverlapping atoms which are all in the same excited state at time $t=0$. The positions ${\mathrm{X}}_{1}\ensuremath{\cdots}{\mathrm{X}}_{N}$ of the atoms are assumed to fill a volume $\mathcal{U}$ of given shape and size with macroscopically constant density. Emission of radiation from this system takes place only in the form of one narrow, but nonzero-width, bundle of nearly equal photons, which contains all the emitted radiation. If the density of atoms within $\mathcal{U}$ exceeds a certain threshold, the rate of emission of photons has the form of a typical spike. All effects depend sensitively on the shape and size of $\mathcal{U}$ and on the density of atoms within $\mathcal{U}$, and cannot be explained in conventional terms of spontaneous or stimulated emission of radiation.

DOI
10.1103/physrev.176.1456

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The Weisskopf-Wigner theory of the natural linewidth of a single isolated atom is extended to a system of N(\ensuremath≫1)N(\ensuremath{\gg}1)N(\ensuremath≫1) identical nonoverlapping atoms which are all in the same excited state at time t=0t=0t=0. The positions X1\ensuremath⋯XN{\mathrm{X}}_{1}\ensuremath{\cdots}{\mathrm{X}}_{N}X1​\ensuremath⋯XN​ of the atoms are assumed to fill a volume U\mathcal{U}U of given shape and size with macroscopically constant density. Emission of radiation from this system takes place only in the form of one narrow, but nonzero-width, bundle of nearly equal photons, which contains all the emitted radiation. If the density of atoms within U\mathcal{U}U exceeds a certain threshold, the rate of emission of photons has the form of a typical spike. All effects depend sensitively on the shape and size of U\mathcal{U}U and on the density of atoms within U\mathcal{U}U, and cannot be explained in conventional terms of spontaneous or stimulated emission of radiation.
Spontaneous decay is a phenomenon that has been described for the first time by Dirac, Wigner and Weisskopf, at the early ages of the development of quantum electrodynamics. The theory is applicable to a multi-body system only when atoms don’t interact during their decay. Dicke introduces in his seminal paper the case where this assumption doesn’t hold. The results he obtains not only tell that the system emits pulses, but the pulse duration now depends on 1/N, where N is the number of participating atoms to the process. The pulses temporal profile is described by a squared hyperbolic secant function. The resulting emission intensity scales as N^2. The atomic system radiates more intensively than the ordinary fluorescent emission, this is the reason why the process is called superradiance, or by a generalization to extended systems, superfluorescence. The superradiant emission has preferential escape directions, depending on the atomic sample shape. Superradiance is a collective process and it takes place when the group of excited atoms is left unper- turbed during the formation of of the macroscopic dipole. A fundamental difference compared to the spontaneous and stimulated emission processes is that all atoms act together as a single entity and it isn’t possible to identify single atoms involved. This remarkable behaviour is manifest when the atomic coherence of the system, while influenced by environmental interactions that cause dephasing in atoms, persists for a sufficient interval of time. Typical dephasing mechanisms, such as collisions in gases or lattice phonon interactions in crystals, make its experimental observation difficult. A possible solution to this problem is the use of optical crystals doped with rare-earths that are characterized by long and narrow levels lifetimes.
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Interaction with a multimode quantized radiation field of an assembly of NNN two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state ∣r,m>|r,m>∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are r+∣m∣r+|m|r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
Laser amplifiers when pumped to a significant degree of population inversion undergo decay via (1) natural spontaneous emission of radiation from individual excited atoms, and (2) a size-dependent decay due to self-amplification of the spontaneously emitted radiation which passes through a large fraction of the amplifier length before escaping. This latter effect has a definite size threshold near which the decay rate changes several orders of magnitude. The decay rate is roughly independent of size except near the threshold.
The long-time behavior of the number of photons in a mode of the electromagnetic field is given in closed form in terms of elementary functions when there is initially one (two-level) atom excited in the presence of NNN unexcited identical atoms and MMM field modes. The radiation trapping which occurs with only one mode disappears when the number of modes MMM approaches the total number of atoms in the continuum mode approximation.