Experimental verification of the predicted statistical properties of superfluorescent pulses

Superfluorescence — the cooperative emission of a fully inverted, extended ensemble of two-level atoms, initiated by quantum fluctuations alone — produces single-shot pulses whose delay time, peak intensity, and shape fluctuate from shot to shot by macroscopic, experimentally observable amounts. Haake, King, Schröder, Haus, Glauber, and Hopf (Phys. Rev. Lett. 42, 1740 (1979)) computed these statistics by numerically integrating the nonlinear Maxwell–Bloch equations over an ensemble of random Gaussian initial polarizations and predicted, for $P=0.3$ and $N=4\times10^9$ (a cesium-beam-like sample), relative fluctuations of about 12% for the delay time and 18% for the first-peak intensity (root-mean-square spreads about the mean; the paper words them as variances of the respective mean values), pulse-shape fluctuations that cannot be reduced to a single random parameter such as a uniform tipping angle, and a tendency toward symmetric bidirectional emission; the paper closes by stating that experimental data on these statistics would be of great interest. In the subsequent literature, the delay-time statistics were measured in cesium in 1981 and found in reasonable agreement with the one-dimensional theory at Fresnel number near unity (with known deviations at larger Fresnel numbers), and an exact analytical functional-integral theory of the delay-time distribution appeared in 1980. What remains open is the direct experimental measurement of the peak-intensity fluctuations, of the conditional pulse-shape statistics that would rule out single-parameter descriptions, of the right/left intensity-difference distribution, and of the predicted dephasing-time dependence.

Background

Superfluorescence (SF) is a form of cooperative spontaneous emission from an extended sample of $N$ two-level atoms (atoms with two relevant quantum states, all prepared in the excited state with no initial coherence). Whereas ordinary spontaneous emission proceeds independently and incoherently, in SF the atoms radiate through a common field: starting from quantum vacuum fluctuations alone, a macroscopic polarization builds up and is released as a short, intense pulse whose peak intensity grows roughly with $N^2$. It is the extended-sample analogue of Dicke superradiance, the cooperative emission of a sample small compared with the wavelength. For a pencil-shaped sample with transition wavelength $\lambda$, diameter $d$, and length $l$, with $\lambda\ll d\ll l$ and Fresnel number (the ratio of the emitting cross-section to $\lambda l$, roughly counting transverse modes) near unity, the radiated fields are well approximated by weakly modulated plane waves traveling in both directions along the cylinder axis. The natural time unit is the superfluorescence time $\tau_R$ (inversely proportional to the atom density), and the cooperation length $l_c=c\tau_R$ is the length over which atomic cooperation establishes itself; the source paper's parameter $P=l/l_c$ measures the sample length in cooperation lengths, and its analysis is restricted to $P\lesssim1$. The dynamics are described by the Maxwell–Bloch equations, the coupled propagation equations for the slowly varying envelopes of the electric field and of the atomic polarization and inversion densities. Quantum mechanics enters through the initial conditions: the collective Bloch vector (the representation in which the collective polarization and inversion form a vector on a unit sphere) starts exactly at the pole, and quantum fluctuations of the initial polarization act as a small random tipping angle — the angle by which the collective Bloch vector is displaced from the fully inverted state — with root-mean-square value $2/\sqrt{N}$ per cooperation volume. Each experimental shot is then a classical trajectory launched from one random initial polarization, so the macroscopic pulse-to-pulse fluctuations are the amplified image of the initiating quantum uncertainty. This picture was established for the early, linear stage of the process by Glauber and Haake (Phys. Lett. A 68, 29 (1978)) and by Polder, Schuurmans, and Vrehen (Phys. Rev. A 19, 1192 (1979)), and the effective tipping angle $2/\sqrt{N}$ was confirmed by the direct measurement of Vrehen and Schuurmans (Phys. Rev. Lett. 42, 224 (1979)). The experiments that triggered this development — single-pulse SF in cesium by Gibbs, Vrehen, and Hikspoors (Phys. Rev. Lett. 39, 547 (1977)) — had shown pulses whose shapes fluctuate greatly from pulse to pulse even for the same delay time, and disagreed quantitatively with the theories then available. In this context Haake, King, Schröder, Haus, Glauber, and Hopf (Phys. Rev. Lett. 42, 1740 (1979)) gave the first complete account of the statistical properties of SF pulses through the nonlinear regime: they solved the Maxwell–Bloch equations numerically for several hundred random Gaussian initial polarizations at $P=0.3$, $N=4\times10^9$ (parameters matched to the cesium experiment) and found (i) relative fluctuations of the delay time of the first intensity peak and of the height of that peak of about 12% and 18% of the respective mean values (the paper's wording is 'variances' of the respective mean values; the subsequent experimental literature compares these figures as relative standard deviations, i.e., root-mean-square spreads about the mean); (ii) that the pulse-shape statistics cannot be described as fluctuations of a single parameter, such as a spatially uniform initial polarization or an equivalent uniform initial tipping angle — within subsets of pulses of equal delay time the peak intensities still spread by up to about 20% of their mean; and (iii) a tendency toward symmetric bidirectional emission, with right- and left-going first-peak intensities differing in shape but their differences clustering about zero. The paper closes with the statement that experimental data on these and other statistical properties of superfluorescent pulses would be of great interest. The statistics were obtained numerically; the nonlinear regime has no closed-form treatment in that paper.

Problem Statement

Determine, experimentally, the statistical properties of superfluorescent pulses emitted by identically prepared inverted samples of two-level atoms, as predicted by the stochastic semiclassical theory of Haake, King, Schröder, Haus, Glauber, and Hopf (Phys. Rev. Lett. 42, 1740 (1979)). Specifically: (1) Measure, from single-shot ensembles of superfluorescent pulses under fixed initial conditions, the fluctuations of the peak intensity of the first intensity peak — predicted to be about 18% of the mean value for $P=l/l_c=0.3$ (sample length in units of the cooperation length $l_c=c\tau_R$, with $\tau_R$ the superfluorescence time) and $N=4\times10^9$ atoms; this figure is a root-mean-square relative fluctuation, i.e., a standard deviation of about 18% of the mean, though the source paper words it as a variance. The companion delay-time statistics of the same paper (relative fluctuation about 12%, together with the full delay-time distribution) were measured in cesium in 1981 in reasonable agreement with the one-dimensional theory at Fresnel number near unity, with known deviations at larger Fresnel numbers. A remaining untested prediction is the dependence of the mean and variance of this delay-time distribution on the inhomogeneous dephasing time $T_2^*$ (the decay time of atomic phase coherence caused by the spread of atomic transition frequencies across the sample, e.g., Doppler broadening in a vapor), which the analytical delay-time theory of Haake, Haus, King, Schröder, and Glauber (Phys. Rev. Lett. 45, 558 (1980)) — not the 1979 source paper, which assumes dephasing absent — predicts to increase as $T_2^*$ shortens. (2) Test whether the pulse-shape fluctuations can be described by the uncertainty of a single parameter, such as a spatially uniform initial polarization or an equivalent uniform initial tipping angle (the angle by which the collective Bloch vector is displaced from the fully inverted state): the prediction is that they cannot, with peak intensities spreading by up to about 20% of their mean even within subsets of pulses having equal delay times. (3) Measure the statistics of bidirectional emission: the prediction is a tendency toward symmetric emission along the two axial directions, with the right- and left-going first-peak intensity differences clustering about zero. These measurements respond to the source paper's own closing request that experimental data on these and other statistical properties of superfluorescent pulses would be of great interest. The measured spread must be attributable to the quantum initiation, at the stated sample parameters ($N$, Fresnel number, $P$, dephasing times); the three components are jointly posed, with component (1) covering both the peak-intensity fluctuation and the dependence on the inhomogeneous dephasing time $T_2^*$ of the delay-time distribution.

References

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