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Physical Review A

Phase-matched emission from an optically thin medium following one-photon pulse excitation: Energy considerations

P. R. Berman, J.-L. Le Gouët

Published Mar 15, 2011• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.83.035804

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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.
A collection of NNN 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 NNN 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 NNN atoms will be followed by spontaneous emission in the same direction. Furthermore, phase matched emission is found when one photon is absorbed by NNN atoms followed by two-photon down-conversion.
A theory of phase-matched cascade correlated emission is developed based on a source field approach. An ensemble of three-level atoms is prepared in a phased state for which the probability to have two excitations is negligibly small. The field intensity radiated on each of the transitions is calculated. The radiation on the upper transition is isotropic and unpolarized, but the phase-matched component on the lower transition can be directional and polarized. Moreover, for sufficiently high optical densities, the emission on the lower transition can be superradiant. In addition to the field intensities, the joint probability for emission on both transitions is calculated, exhibiting correlations in both the directions of emission and polarization of the fields.
Nuclear γ-ray-resonant superradiance is discussed, comparing Dicke's original approach 1 Dicke, RH. 1954. Phys. Rev., 93: 99–110. [Crossref], [Web of Science ®] , [Google Scholar] and recent extensions 2 Scully, MO, Fry, ES, Ooi, CHR and Wódkiewicz, K. 2006. Phys. Rev. Lett., 96: 010501[Crossref], [Web of Science ®] , [Google Scholar],3 Scully, MO. 2007. Laser Phys., 17: 635–646. [Crossref], [Web of Science ®] , [Google Scholar] of Scully et al. 2–5 Scully, MO, Fry, ES, Ooi, CHR and Wódkiewicz, K. 2006. Phys. Rev. Lett., 96: 010501 Scully, MO. 2007. Laser Phys., 17: 635–646. Svidzinsky, A and Chang, J-T. 2008. Phys. Rev. A, 77: 043833 Svidzinsky, AA, Chang, J-T and Scully, MO. 2008. Phys. Rev. Lett., 100: 160504 involving conditional excitation to the classical resonant-scattering 6–10 Lynch, FJ, Holland, RE and Hamermesh, M. 1960. Phys. Rev., 120: 513–520. Kagan, Y and Afanas'ev, AM. 1973. Z. Naturforsch Pt. A, 28: 1351–1360. Kagan, Y, Afanas'ev, AM and Kohn, VG. 1979. J. Phys. C, 12: 615–631. Kagan, Y. 1999. Hyperfine Interact., 123–124: 83–126. Smirnov, GV. 1999. Hyperfine Interact., 123–124: 31–77. and the Hannon–Trammell 11–14 Trammell, GT. 1962. Phys. Rev., 126: 1045–1054. Hannon, JP and Trammell, GT. 1968. Phys. Rev., 169: 315–329. Hannon, JP and Trammell, GT. 1969. Phys. Rev., 186: 306–325. Hannon, JP and Trammell, GT. 1999. Hyperfine Interact., 123–124: 127–274. quantum theories. Relevant questions are whether the sample is optically thin or thick in the excitation and re-radiation processes, the role of superpositions between ground and excited states, Fermi's golden rule, and decoherence due to spatially-dependent near-field interactions 15–17 Friedberg, R and Hartmann, SR. 1974. Phys. Rev. A, 10: 1728–1739. Friedberg, R, Hartmann, SR and Manassah, JT. 1972. Phys. Lett. A, 40: 365–366. Friedberg, R and Manassah, JT. 2008. Phys. Lett. A, 372: 2514–2521. . The relation of these considerations to experimental results is discussed.
We present a numerical study of the spatial coherence of light that is radiated from a dilute ensemble of atoms. The spatial coherence is established as a result of the collective (cooperative) coupling of the atoms to the light and is qualitatively different from the coherence of a laser. Specifically, the coherence in collective spontaneous emission does not rely on population inversion and stimulated emission, is governed by antiphasing of the dipoles (subradiance), and the key figure of merit for the observed coherence is $\ensuremath{\approx}N/(L/\ensuremath{\lambda})$. Here, NNN is the number of atoms in the ensemble, LLL is the size of the sample, and $\ensuremath{\lambda}$ is the wavelength of the emitted light.
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=0t=0t=0. The positions ${\mathrm{X}}_{1}\ensuremath{\cdots}{\mathrm{X}}_{N}$ 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.