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arXiv:2410.08652 · Physical Review Letters

Emergence of Second-Order Coherence in the Superradiant Emission from a Free-Space Atomic Ensemble

Giovanni Ferioli, Igor Ferrier-Barbut, Antoine Browaeys

Published Apr 15, 2025• 19 citations
arXivHTMLPDF

Abstract

We investigate the evolution of the second-order temporal coherence during the emission of a superradiant pulse by an elongated cloud of cold Rb atoms in free space. To do so, we measure the two-times intensity correlation function g_{N}^{(2)}(t_{1},t_{2}) following the pulsed excitation of the cloud. By monitoring g_{N}^{(2)}(t,t) during the pulse, we observe the establishment of second-order coherence, and contrast it with the situation where the cloud is initially prepared in a steady state. We compare our findings to the predictions of the Dicke model, using an effective atom number to account for finite size effects, finding that the model reproduces the observed trend at early time. For longer times, we observe a subradiant decay, a feature that goes beyond Dicke's model. Finally, we measure the g_{N}^{(2)}(t_{1},t_{2}) at different times and observe the appearance of anticorrelations during the pulse that are not present when starting from a steady state.

DOI
10.1103/physrevlett.134.153602

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We investigate the evolution of the second-order temporal coherence during the emission of a superradiant pulse by an elongated cloud of cold Rb atoms in free space. To do so, we measure the two-times intensity correlation function g_{N}^{(2)}(t_{1},t_{2}) following the pulsed excitation of the cloud. By monitoring g_{N}^{(2)}(t,t) during the pulse, we observe the establishment of second-order coherence, and contrast it with the situation where the cloud is initially prepared in a steady state. We compare our findings to the predictions of the Dicke model, using an effective atom number to account for finite size effects, finding that the model reproduces the observed trend at early time. For longer times, we observe a subradiant decay, a feature that goes beyond Dicke's model. Finally, we measure the g_{N}^{(2)}(t_{1},t_{2}) at different times and observe the appearance of anticorrelations during the pulse that are not present when starting from a steady state.
We observe a non-equilibrium phase transition in a driven dissipative quantum system consisting of an pencil-shape cloud of up to N≈2000N\approx 2000N≈2000 laser-cooled atoms in free space, optically excited along its main axis. We find that our data are well reproduced by the Driven Dicke model, which assumes a sub-wavelength sample volume, by simply using an effective atom number. By measuring the excited state population of the atoms and the light emitted in the superradiant mode, we characterize the dynamics of the system and its steady-state properties. In particular, we observe the characteristic N2N^2N2 scaling of the photon emission rate in the superradiant phase, thus demonstrating steady-state superradiance in free space. Finally, we observe a modification of the statistics of the superradiant light as we cross the phase transition.
Since Dicke's seminal paper on coherence in spontaneous radiation by atomic ensembles, superradiance has been extensively studied. Subradiance, on the contrary, has remained elusive, mainly because subradiant states are weakly coupled to the environment and are very sensitive to nonradiative decoherence processes. Here, we report the experimental observation of subradiance in an extended and dilute cold-atom sample containing a large number of particles. We use a far detuned laser to avoid multiple scattering and observe the temporal decay after a sudden switch-off of the laser beam. After the fast decay of most of the fluorescence, we detect a very slow decay, with time constants as long as 100 times the natural lifetime of the excited state of individual atoms. This subradiant time constant scales linearly with the cooperativity parameter, corresponding to the on-resonance optical depth of the sample, and is independent of the laser detuning, as expected from a coupled-dipole model.
We experimentally investigate the second-order quantum coherence function of a superradiant burst in a cascaded quantum system. We chirally (i.e., direction dependently) couple about 900 cesium atoms to the forward-propagating mode of an optical nanofiber. We then prepare the ensemble close to the maximally excited state, where the subsequent collective emission of a burst is known as superfluorescence. Here, we observe that second-order coherence emerges in the course of the decay. This is a clear signature of the underlying collective dynamics that is also at the origin of the superradiant burst itself. We furthermore study the dynamics of the second-order coherence function of the emission in relation to the initial average dipole moment of the ensemble. In addition, by investigating the correlation of early and late photon emission events, we obtain evidence for fundamental shot-to-shot fluctuations in the delay of the emission of the burst. Our findings thus reveal surprising similarities between superradiance in cascaded and symmetrically coupled systems, despite the fact that they are described by fundamentally different coupling Hamiltonians.
The problem of the interaction of N atoms with a laser beam and vacuum modes can give rise to many interesting phenomena concerning the spontaneous emission of light and its propagation in the medium. The cooperative effects, for example, such as superadiance and subradiance, are effects related to the coherence created between the atoms when a photon is emitted spontaneously by a single excited atom. Superradiance can be defined as the enhancement of the spontaneous emission due to constructive interference of the scattered light. Its counterpart, subradiance, is the trapping of some remaining light due to destructive interference. In cold atoms, some previous theoretical works predict and characterize these two cooperative effects in a large and diluted atomic cloud, in the regime of low intensities and large detunings of the incident laser. The theoretical model is a coupled-dipole model for two-level atoms driven by a low-intensity field and in the scalar approach. The experiment consists in measuring the super- and subradiant decay rates from the temporal emitted intensity after the switch off of the incident laser in the steady state. Our experimental setup consists in a magneto-optical trap of rubidium 87 atoms at large resonant optical thicknesses. A probe beam excites the atoms close to the D2 line. The intensity emitted is detected by a single photon detector with no afterpulsing and a calibration procedure allows us to determine the resonant optical thickness of the cloud and its temperature. In this work, we report the experimental observation of super- and subradiance in a large cloud of cold atoms. For subradiance, the main result is the linear evolution of the characteristic time with the resonant optical thickness of the cloud and its independence of the detuning. For superradiance, we observe superradiance out of the forward direction. We verify the validity of our interpretations with the predictions of the coupled-dipole model. Finally, we discuss the interplay of subradiance and radiation trapping, as well as theoretical predictions for: a setup of a phased cloud, to control the subradiant amplitude emission; and temperature effects, where subradiance is shown to be robust in a large range of temperatures.
Fully excited two-level atoms separated by less than the transition wavelength cooperatively emit light in a short burst, a phenomenon called superradiance by R. Dicke in 1954 [Phys. Rev. 93, 99 (1954)]. The burst is characterized by a maximum intensity scaling with the square of the number of atoms NNN and a temporal width reduced by NNN compared to the single atom spontaneous decay time. Both effects are usually attributed to a synchronization of the electric dipole moments of the atoms occurring during the process of light emission. Contrary to this explanation, it was recently shown by use of a quantum path description that the peak intensity results from the quantum correlations among the atoms when occupying symmetric Dicke states. Here we investigate from this perspective the temporal evolution of the ensemble, starting in the small sample limit, i.e., when the atoms have mutual separations much smaller than the transition wavelength $\ensuremath{\lambda}$ and pass down the ladder of symmetric Dicke states. In addition, we explore the temporal evolution for the case of distant noninteracting atoms with mutual separations much larger than $\ensuremath{\lambda}$. We show that in this case a similar superradiant burst of the emitted radiation is observed if the quantum correlations of the atoms are generated by conditional photon measurements retaining the atomic ensemble within or close to the symmetric subspace.
In this work we present a numerical and experimental investigation of the collective early-time decay rates of a strongly driven and optically dense cold atomic cloud. We prepare the atomic ensemble by driving the system to its steady state with varying Rabi frequencies ΩΩΩ that go from the weak Ω≪ΓΩ\ll ΓΩ≪Γ to the strong driving regime Ω≫ΓΩ\gg ΓΩ≫Γ, where ΓΓΓ is the single-atom decay rate. We investigate the early-time dynamics in the transition between the strong and weak driving regimes using: i) angular-dependent observables such as the light emitted by the cloud, and ii) global observables, i.e., the excited state population. When driving the cloud on-resonance, we find that as a function of the driving frequency, the behavior of the collected light at certain angles transitions from the single-photon subradiant regime to a superradiant regime while the behavior of the excited state population does not show superradiance. The experiment shows good agreement with numerical predictions in the regime of parameters under study.