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Physical Review Letters

Spontaneous Emission from an Excited Atom in the Presence of N Atoms and M Modes

F.W. Cummings

Published May 27, 1985• 21 citations
Publisher

Abstract

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 $N$ unexcited identical atoms and $M$ field modes. The radiation trapping which occurs with only one mode disappears when the number of modes $M$ approaches the total number of atoms in the continuum mode approximation.

DOI
10.1103/physrevlett.54.2329

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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.
NNN two-level "atoms" are considered in interaction with a single-mode resonant electromagnetic field. The exact solution is given nonrelativistically for all times for the case of spontaneous emission, when only one atom is initially excited. The solution is given for the general case of the NNN atoms in inequivalent mode positions.
The problem of the spontaneous emission from a single excited two-level atom in the presence of N unexcited similar atoms is discussed. A formalism is presented for evaluation of the time-development operator (and thus the density matrix) which has certain desirable features for time-dependent problems in the area, e.g. of atom-field interactions, when the off-diagonal elements are large compared to the diagonal elements of the Hamiltonian. By use of this formalism, it is shown that the spontaneous emission from a single atom is often drastically altered in the presence of unexcited atoms with similar resonances. When the number of atoms present becomes very large relative to the number of modes of the electromagnetic field, the excited atom becomes trapped in its excited state.>
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}r21​ can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}O2​ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2{A}_{2}A2​ and A2{A}_{2}A2​ radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
The quantum-electrodynamical problem of N identical two-level atoms interacting with M field modes is considered, and an exact solution for the case of spontaneous emission from a single atom is obtained. The behavior of this system is shown to be quite different from the exponential decay of a single excited atom radiating into free space. A ``ringing'' behavior at the enigmatic frequency \ensuremath{\sim} \ensuremath{\surd}N which occurs when only one mode is accessible to the field persists when many modes are present. When a continuum of modes is accessible, the long-time limit shows that the energy of the originally excited atom is equally likely to be shared by the N-1 originally unexcited atoms and the electromagnetic field when M/N\ensuremath{\rightarrow}1. ``Radiation suppression'' occurs whenever N\ensuremath{\gg}1 and M\ensuremath{\ll}N, when the originally excited atom does not emit its energy. When M=N\ensuremath{\gg}1, the single atom decays monotonically to zero (and approximately exponentially) but with a decay rate that is \ensuremath{\surd}N times the Rabi frequency. Such unexpected effects are presently within experimental range.
A quantum model of N two-level atoms coupled by a single-mode radiation field under a restricted rotating-wave approximation is studied, assuming a dipole-interaction approximation and that the total number of excitations of the system is conserved. A solution is found for the case of spontaneous emission when the atoms are initially prepared in the state of complete inversion. This solution is valid for all times. A general solution of the model is also found for any initial configuration of the system, in the cases of both spontaneous and stimulated emission. This solution is in a closed form and valid for times tau < tau max .