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Physical Review Letters

Directed Spontaneous Emission from an Extended Ensemble of N Atoms: Timing Is Everything

Marlan O. Scully, Edward S. Fry, C. H. Raymond Ooi, Krzysztof Wódkiewicz

Published Jan 3, 2006• 399 citations

Abstract

A collection of $N$ static atoms is fixed in a crystal at a low temperature and prepared by a pulse of incident radiation of wave vector ${\stackrel{\ensuremath{\rightarrow}}{k}}_{0}$. The $N$ atoms are well described by an entangled Dicke-like state, in which each atom carries a characteristic phase factor $\mathrm{exp}(i{\stackrel{\ensuremath{\rightarrow}}{k}}_{0}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{r}}_{j})$, where ${\stackrel{\ensuremath{\rightarrow}}{r}}_{j}$ is the atomic position in the crystal. It is shown that a single photon absorbed by the $N$ atoms will be followed by spontaneous emission in the same direction. Furthermore, phase matched emission is found when one photon is absorbed by $N$ atoms followed by two-photon down-conversion.

DOI: 10.1103/physrevlett.96.010501

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A collection of

N

static atoms is fixed in a crystal at a low temperature and prepared by a pulse of incident radiation of wave vector

{\stackrel{\ensuremath{\rightarrow}}{k}}_{0}

. The

N

atoms are well described by an entangled Dicke-like state, in which each atom carries a characteristic phase factor

\mathrm{exp}﻿(i{\stackrel{\ensuremath{\rightarrow}}{k}}_{0}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{\ensuremath{\rightarrow}}{r}}_{j})

, where

{\stackrel{\ensuremath{\rightarrow}}{r}}_{j}

is the atomic position in the crystal. It is shown that a single photon absorbed by the

N

atoms will be followed by spontaneous emission in the same direction. Furthermore, phase matched emission is found when one photon is absorbed by

N

atoms followed by two-photon down-conversion.

Scully and coworkers [M. O. Scully, E. S. Fry, C. H. R. Oii, and K. W\'odkiewicz, Phys. Rev. Lett. 96, 010501 (2006)] demonstrated that there is directional, phase-matched emission following the excitation of an ensemble of atoms by a single-photon pulse. While the phase-matched emission intensity is proportional to the the number of atoms, for optically thin samples the total energy emitted in the phase-matched direction is much less than that radiated in other directions. Moreover, even for optically thin samples, it is necessary to take into account effects related to cooperative decay if energy is to be conserved in the overall emission process. An analytic calculation is presented to show explicitly how cooperative decay reduces the incoherent emission and restores energy conservation in this low-density limit.

The spontaneous emission of a set of

N

identical two-level atoms interacting with a quantized electromagnetic field is studied. The atoms are assumed to be close together compared to the mean wavelength of the emitted radiation, and their initial state is either a Dicke state or a superposition of the ground and excited state for each atom. Approximate expressions are obtained for transition probabilities, expectation values, and correlations as functions of time. The spontaneous emission for a very large separation of the atoms is also studied. In this case the directional properties of the emitted radiation, as well as the time within which the atoms decay to their ground state, depend on the position of each atom.

We study the collective spontaneous emission of a linear atomic chain excited by a single photon. The interaction between the atoms and the common vacuum field can significantly change the eigenenergy and the spontaneous emission rate of the system. Due to the dipole-dipole interactions, the system prepared in a single-photon timed Dicke state is the superposition of superradiant and subradiant eigenstates that can have a nonexponential decay dynamics. We can tune the frequency and linewidth of the superradiant and subradiant emission from a timed Dicke state by changing the direction of the atomic dipole moment or the atomic separation. In addition, the emission direction of the superradiant and subradiant photons also depends on the polarization of the atoms.

Recent works have shown that collective single photon spontaneous emission from an ensemble of N resonant two-level atoms is a rich field of study. Superradiance describes emission from a completely symmetric state of N atoms, with a single excited atom prepared with a given phase, for instance imprinted by an external laser. Instead, subradiance is associated with the emission from the remaining N−1 asymmetric states, with a collective decay rate less than the single-atom value. Here, we discuss the properties of the orthonormal basis of symmetric and asymmetric states and the entanglement properties of superradiant and subradiant states.