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Physical Review A

Cone emission from laser-pumped two-level atoms. I. Quantum theory of resonant light propagation

Li You, Jan Mostowski, J. Cooper

Published Sep 1, 1992• 20 citations
Publisher

Abstract

This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission. These radiation sources enter into the equations for the fields in the same form as the source generated by four-wave mixing and therefore may be considered as a ``spontaneous four-wave mixing,'' since the photons emitted in the Rabi sidebands are correlated. This has far-reaching consequences on the physics of cone emission and also sheds some new light on the fluoresence from strongly driven atomic systems.

DOI
10.1103/physreva.46.2903

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In spite of the fact that cone emission was first observed over a decade ago, there is still no generally accepted theory of the effect. Here we give an outline of a unified approach and point out a mechanism that supports Cherenkov-type radiation due to vacuum fluctuations as a possible source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing, and initial encoding and follow-up refraction effects. We compare our treatment to previously considered models.

Physical Review A•Dec 1, 1991•26 citations
This is the first of a series of papers on the theory of cone emission from a laser-pumped two-level atomic medium. Starting from microscopic quantum theory for both the active medium and the field, equations for the slowly varying macroscopic quantities are derived. In the steady-state limit, which experimentally corresponds to cw excitation, we find that the sources for the paraxial fields of interest have a long coherence length despite the random nature of spontaneous emission. These radiation sources enter into the equations for the fields in the same form as the source generated by four-wave mixing and therefore may be considered as a ``spontaneous four-wave mixing,'' since the photons emitted in the Rabi sidebands are correlated. This has far-reaching consequences on the physics of cone emission and also sheds some new light on the fluoresence from strongly driven atomic systems.
We establish a mechanism, which supports Cherenkov-type radiation, as a source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing (4WM) and initial encoding and follow-up refraction effects. The radiation source is provided by spontaneous emission of the driven atoms at the frequencies of interest. Due to the medium dispersion, the pump field propagates faster than the phase velocity of the field that is generated at the Rabi sideband at lower frequency. Cherenkov radiation is then possible if the source has a long correlation length, which is indeed the case despite the random nature of spontaneous emission. Coherent superposition of the source for the lower-frequency field is in an off-axis direction given by the Cherenkov radiation condition. Differences between 4WM and Cherenkov-type models are consequently resolved.
The predictions of a quantum-mechanical theory of forward four-wave mixing in a homogeneously broadened system of two-level atoms are presented. In the limit of a very short interaction region, the predictions of this theory reproduce those of well-known theories for the spontaneous-emission spectrum of an atom in the presence of an intense laser field. More generally, the theory predicts how the emission spectrum is modified due to propagation effects for a medium of arbitrary length. For long propagation path lengths, the emitted radiation can be quite intense and has a spectrum that is strongly peaked at the Rabi sidebands of the incident laser frequency. The theory shows that Rabi sideband generation in the forward direction can be understood as parametric amplification of weak radiation emitted spontaneously at the Rabi sidebands. The quantum noise that initiates the four-wave-mixing process has contributions both from fluctuations in the incident vacuum radiation field and from fluctuations in the polarization of the atomic dipoles. Both contributions are important for the case of a radiatively broadened medium, although the material fluctuations make the dominant contribution for the case of a medium in which the broadening is largely collisional. Under certain conditions large amounts of squeezing in the radiated field are predicted.
A quantum theory of two-photon correlated-spontaneous-emission lasers (CEL's) is developed, starting from the exact atom-field interaction Hamiltonian for cascade three-level atoms interacting with a single-mode radiation field. We consider the situation where the active atoms are prepared initially in a coherent superposition of three atomic levels and derive a master equation for the field-density operator by using a quantum theory for coherently pumped lasers. The master equation is transformed into a Fokker-Planck equation for the antinormal-ordering Q function. The drift coefficients of the Fokker-Planck equation enable us to study the steady-state operation of the two-photon CEL's analytically. We have studied both resonant two-photon CEL for which there is no threshold, and off-resonant two-photon CEL for which there exists a threshold. In both cases the initial atomic coherences provide phase locking, and squeezing in the phase quadrature of the field is found. The off-resonant two-photon CEL can build up from a vacuum when its linear gain is larger than the cavity loss (even without population inversion). Maximum squeezing is found in the no-population-inversion region with the laser intensities far below saturation in both cases, which are more than 90% for the resonant two-photon CEL and nearly 50% for the off-resonant one. Approximate steady-state Q functions are obtained for the resonant two-photon CEL and, in certain circumstances, for the off-resonant one.
A time-dependent theory for conical emission during near-resonant propagation of laser light in an atomic vapor, which includes full propagation for the laser and frequency sidebands in a nonlinear two-level medium is presented. The density-matrix equations for the dipole moment and population are solved in the dressed atomic frame. The polarization source terms are accurate to order \ensuremath{\gamma}/R, where \ensuremath{\gamma} is a damping constant and R is the generalized Rabi frequency. Analytical plane-wave solutions and numerical, cylindrically symmetric propagation simulations including diffraction are presented. It is shown that the calculations with cylindrically symmetric fields and atomic excitation profiles are incapable of accounting for the high levels of optical gain that are responsible for the intense conical emission observed in experiments. This result is at first surprising, since the model accounts rigorously for all of the physical phenomena that have been previously proposed as being responsible for generating large gains, and the calculation matches the symmetry of the observations. The lack of large calculated gain seems to imply the existence of higher-order (m>0)(m>0)(m>0) radial modes in the field for the experimental conditions that give rise to cone emission. In the simulations, however, the cylindrically symmetric fields do produce weak red-detuned cones with angular-frequency distributions similar to those seen in experiments.
The cone emission obtained from pumped two-level atoms is related, by a perturbation theory, to time-dependent terms in the macroscopic Bloch equations. It is described as a cooperative superfluorescence radiation from collections of spatially distributed atoms, including in the theory the far-zone dipole-dipole interactions, in the presence of the strong monochromatic field. The distribution of the cone emission as a function of frequencies and cone angles is analyzed. A phase-matching condition is obtained for the wave vectors of the pump and conical beams that is similar to the Cherenkov condition. The frequency shift for the maximum of the cone light is found to be equal to the generalized Rabi frequency. The broadening of the cone emission is related to inhomogeneity in Rabi frequencies and decoherence effects. An approximate result is obtained for the ratio of intensities of the blue-shifted cone relative to the red-shifted cone so that the blue-shifted cone can appear only for extremely strong laser intensities.
In spite of the fact that cone emission was first observed over a decade ago, there is still no generally accepted theory of the effect. Here we give an outline of a unified approach and point out a mechanism that supports Cherenkov-type radiation due to vacuum fluctuations as a possible source of cone emission. This can be considered as an additional mechanism to the previously discussed four-wave mixing, and initial encoding and follow-up refraction effects. We compare our treatment to previously considered models.