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Journal of Modern Optics

Fermi's golden rule does not adequately describe Dicke's superradiance

Anatoly A. Svidzinsky, Jun-Tao Chang, Harry J. Lipkin, Marlan O. Scully

Published Nov 10, 2008• 10 citations
Publisher

Abstract

We consider the application of Fermi's golden rule to the problem of N atoms, one of which is excited, in an extended medium with dimensions that are large compared with the wavelength. We find that, in contrast to an often-voiced opinion, the golden rule does not describe the physics of one photon absorbed and subsequently emitted by N atoms. The correct treatment includes many-body effects such as Fano interference. For a finite atomic cloud with size R, in the Markovian limit, the system will decay exponentially at a rate of around N(λ2/R 2) faster than the single-atom decay rate.

DOI
10.1080/09500340802334579

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We consider the application of Fermi's golden rule to the problem of N atoms, one of which is excited, in an extended medium with dimensions that are large compared with the wavelength. We find that, in contrast to an often-voiced opinion, the golden rule does not describe the physics of one photon absorbed and subsequently emitted by N atoms. The correct treatment includes many-body effects such as Fano interference. For a finite atomic cloud with size R, in the Markovian limit, the system will decay exponentially at a rate of around N(λ2/R 2) faster than the single-atom decay rate.
We study emission of a single photon from a spherically symmetric cloud of NNN atoms (one atom is excited; $N\ensuremath{-}1$ are in the ground state) and present an exact analytical expression for eigenvalues and eigenstates of this many-body problem. We found that some states decay much faster than the single-atom decay rate, while other states are trapped and undergo very slow decay. When the size of the atomic cloud is small compared with the radiation wavelength, we found that the radiation frequency undergoes a large shift.
The decay of isolated quantum systems is studied. In particular, a specific model for the radiative decay of an excited atomic state is considered. The atom is taken to have two levels and is described by a current distribution $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$. Both the case in which the modes of the electromagnetic field are continuous and the case in which they are discrete are considered. Conditions on $\stackrel{\ensuremath{\rightarrow}}{\mathrm{j}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{x}};1,2)$ are found under which the decay law for the excited state is approximately exponential with a lifetime given by Fermi's golden rule. The current should be well localized, not too strong, and not too singular. The $2p\ensuremath{\rightarrow}1s$ decay in hydrogen is examined in detail. It is also found that the existence of an approximately exponential decay law does not depend upon the ability to extend the resolvent operator of the system to a second Riemann sheet. The case in which the electromagnetic field is quantized in a cavity illustrates this point.
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.
We introduce a straightforward quantum-electrodynamic approach to the problem of superradiant spontaneous emission from a system of two-level atoms, and also discuss two classical superradiant systems. The same physical concepts underlie our treatment of both the quantum and classical cases. Explicit expressions are found which describe the time evolution and directional character of the radiated power. Our approach applies to an arbitrary number of atoms which are confined to a region with linear dimensions which may be large compared with the mean wavelength of the emitted radiation. The radiative decay half-life of a large many-atom system is found to be much shorter than the natural lifetime of a single atom, but not as short as the half-life of a small atomic system with the same number of atoms. The far-field radiation pattern appropriate to the case of a circular cylinder of emitters which have been excited by a plane-wave pulse is considered in detail. Polar plots are presented of the radiation patterns produced when the atoms are excited by a plane-wave pulse. Exciting pulses traveling in directions both parallel and at an angle of 10\ifmmode^\circ\else\textdegree\fi{} to the axis of the cylinder are considered. A discussion and generalization of Dicke's "coherence-brightening" criterion is given.
A quantum mechanical wave of a finite size moves like a classical particle and shows a unique decay probability. Because the wave function evolves according to the Schrodinger equation, it preserves the total energy but not the kinetic energy in the intermediate-time region of a decay process where those of the parent and daughters overlap. The decay rate computed with Fermi's golden rule requires corrections that vary with the distance between the initial and final states, and the energy distribution of the daughter is distorted from that of plane waves. The corrections have universal properties in relativistically invariant systems and reveal macroscopic quantum phenomena for light particles. The implications for precision experiments in beta decays and various radiative transitions are presented.
Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.