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Physical Review A

Delay-time statistics of cooperative emission in the presence of homogeneous line broadening

Kazimierz Rza̧ewski, M. G. Raymer, Robert W. Boyd

Published Jun 1, 1989• 3 citations
Publisher

Abstract

We have developed a theoretical model that allows us to study the delay-time statistics of superfluorescent pulses for the case of collisional broadening with a Lorentzian line shape. Our model is based upon a more robust definition of delay time, based upon an energy rather than an intensity reference. We calculate the probability distribution of delay times in the initiation regime of superfluorescence, where the inversion is essentially constant. We find that, as the collisional dephasing rate is increased, the mean delay time increases and the distribution broadens. We compare our results with those for the case of inhomogeneous broadening with a Lorentzian line shape and find the unexpected result that the statistical behavior is identical.

DOI
10.1103/physreva.39.5785

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We have developed a theoretical model that allows us to study the delay-time statistics of superfluorescent pulses for the case of collisional broadening with a Lorentzian line shape. Our model is based upon a more robust definition of delay time, based upon an energy rather than an intensity reference. We calculate the probability distribution of delay times in the initiation regime of superfluorescence, where the inversion is essentially constant. We find that, as the collisional dephasing rate is increased, the mean delay time increases and the distribution broadens. We compare our results with those for the case of inhomogeneous broadening with a Lorentzian line shape and find the unexpected result that the statistical behavior is identical.
A probability distribution W(t)W(t)W(t) for the delay time ttt of superfluorescent pulses is defined in terms of a functional integral, which is evaluated exactly. Although the linearized early-stage dynamics is used, full agreement is obtained with previous numerical and experimental results for the variance of ttt and, more generally, for the shape of W(t)W(t)W(t). The effect of inhomogeneous broadening of the atomic levels, showing that decrease of the dephasing time T2∗{{T}_{2}}^{*}T2​∗ leads to substantial increase of the mean and variance of ttt, is also analyzed.
We discuss the delay-time statistics of superfluorescent pulses by introducing the concept of a passage time at which the field intensity reaches a specified value. Such passage times can be estimated accurately from the early-stage linear dynamics of the radiating system. The distribution function for the passage times, we show, can be expressed as a functional integral over the random zero point fluctuations of the atomic polarization. The functional integral can be evaluated exactly and reduced to an elementary closed form. Our theory applies both to the sharp and the broadened atomic line. The results we obtain agree with experimental data and numerical calculations already in existence for the unbroadened line. For the broadened line we predict substantial changes of the mean value and the variance of the passage time as the width of the atomic line increases. We also investigate the influence of the shape of the spectral distribution of atomic frequencies and show that the delay statistics depend significantly on the way in which the spectral density decreases in the wings of the atomic line.
We present a quantum-mechanical treatment of the influence of collisional dephasing processes on the statistical properties of superfluorescence (SF). The theory, which treats nonlinear propagation effects as well as quantum noise, shows how the nature of the cooperative emission process changes from that of SF to that of amplified spontaneous emission as the collisional dephasing rate is varied. The predictions of how the SF delay time varies with the collisional dephasing rate are in good agreement with the results of a recent experiment [M. S. Malcuit, J. J. Maki, D. J. Simkin, and R. W. Boyd, Phys. Rev. Lett. 59, 1189 (1987)].
The inverse scattering method enables a generic nonlinear theory of superfluorescence. In the semiclassical theory used here the process starts from a fluctuating polarization source. The source function is expanded with respect to an appropriate function system, and in this way it can be replaced by a set of normalized random numbers. The Gelfand-Levitan-Marchenko equations are solved approximately. Delay statistics and the pulse shape are studied, and the influence of inhomogeneous broadening is discussed. Results are compared with other theories as well as with experiment.
The dynamics of the pulse shape and spectra of superfluorescence, stimulated superradiance, and coherent amplification of ultrashort light pulses is studied for a system of four-level atoms under incoherent cw pumping taking into account homogeneous and inhomogeneous broadening of the luminescence lines. It is found that incoherent pumping at a high level of excitation may significantly change the shape of the emission pulse in cooperative coherent processes. Also, it is found that, depending on the ratio of parameters of the pump, nonradiative transitions, and relaxation, different regimes of superfluorescence and stimulated superradiance may be realized.
The time-dependent spectral redistribution in fluorescence line narrowing in an inhomogeneously broadened system is investigated by means of simple theoretical models and Monte Carlo calculations. The two-site phonon-assisted energy transfer rate is assumed to be site-symmetric and a Lorentzian function of energy mismatch. For the Monte Carlo calculations initial excitations are made at the wing, shoulder, and center of the inhomogeneously broadened Gaussian band. For the wing excitation it is argued that, because of the above Lorentzian factor, the intensity in the initial site of excitation decays with a much smaller back-transfer rate than for the center excitation, simplifying the theoretical treatment. This effect is demonstrated by the Monte Carlo study. A refinement of this model is developed for a general situation, and its predictions are compared with the Monte Carlo results.