Fourier imaging of collective spontaneous emission modes in superradiant cold atomic clouds

The gist: Fourier imaging reveals a highly directional, ring-shaped emission structure corresponding to a single collective jump operator associated with the most superradiant mode of an ensemble

Experimental Setup and Measurement

The researchers used Fourier imaging to measure the spatial pattern associated with the superradiant emission from a cloud of cold 87Rb atoms. The experimental setup involved a glass cell surrounded by three microscope objectives with numerical aperture NA = 0.5 used to focus an optical dipole trap (ODT) and collect the light emitted by the atoms. This ODT has a waist controlled by tunable lenses between w0 = 2.5 µm and 8 µm, resulting in elongated clouds containing up to N = 8000 atoms at a temperature of T = 550 − 1550 µK. The system applies a magnetic field of 64 G perpendicular to the cloud axis to isolate a closed σ+-transition between the two states of the D2 line.

Identification of Collective Modes

The analysis revealed two (nearly-)degenerate most superradiant modes l = 1, 2, with spatial distributions well approximated by αi,l=1 ∼ cos(k∥zi) and αi,l=2 ∼ sin(k∥zi) with k∥ < k0. The most superradiant pair exhibits a spatial pattern that is symmetric along the cloud axis, with the two fields dephased by π/2, leading to a collective, now directional, emission along ±x. The emission pattern of each mode in the far field is given by Il(k) = I0(k) X ij αilαjle ik·(ri−rj).

Mode Competition and Dominance

The most superradiant pair dominates the dynamics because its corresponding mode has a decay rate Γl > 2Γ0, leading to the emission of a burst rather than exponential decay. The distribution of eigenvalues features a spectral gap between the most superradiant mode and the bulk of eigenvalues, which is necessary for it to dominate the dynamics. The ratio of the two first eigenvalues Γ1/Γ3 is found to be independent of N for large N, with the largest gap reached along a curve defined by a constant Fresnel number F = πσ2r / (λ0σx) ≈ 0.2.

Temporal Dynamics and Model Comparison

The bosonic approximation model predicts the amplification of the most superradiant pair, but it overestimates the peak intensity growth with atom number and predicts a longer burst duration than experimentally observed. A second model, which keeps the collective jump description of the ensemble by isolating one collective mode and adding other modes as individual decays, reproduces experimental observations by showing that the collective emission rate is proportional to N below a threshold Nc,1 = 1/η ≈ 17 before Dicke superradiance sets in and then scales as N(N − Nc,1).

Conclusion and Future Directions

The work suggests that the existence of the spectral gap between the most superradiant mode and the bulk of eigenvalues depends on cloud geometry but could not be explained in the context of random matrix theory. Further investigation into numerical methods like cumulant expansion or truncated Wigner approximation is suggested to explore the role of the Hamiltonian part of the dynamics. The demonstration that collective spontaneous emission modes can be manipulated will be useful for future experiments aiming at light-matter interfaces made of arrays of quantum emitters.

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Fourier imaging of collective spontaneous emission modes in superradiant cold atomic clouds Adrien Gavalda,1, 2, ∗ Guillaume Tremblier,1,∗ Martin Poitrinal,1,… and Igor Ferrier-Barbut1 We measure the spatial pattern associated with the superradiant emission from a cloud of cold 87Rb atoms using Fourier imaging. We observe a highly directional, ring-shaped emission structure, which corresponds to a single collective jump operator associated to the most superradiant mode of the ensemble. Using spatial filtering, we isolate this channel and find the typical superradiant burst with superlinear scaling of the intensity with atom number. We compare our results to two models that describe the competition between the various decay channels, finding good agreement. Our work shows that the collective jump operators introduced by Carmichael et al. [1] can be measured and manipulated. Dicke predicted that an ensemble of identical two-level atoms, all located within a sub-wavelength sized volume and initially in their excited state, decays by emitting a burst of so-called superradiant radiation [2]. In his scenario, the atoms are indistinguishable and collectively coupled to a single channel of spontaneous emission. The burst builds up thanks to the enhancement of the emission rate in this decay channel. The first observations of Dicke superradiance were however performed on pencilshaped samples larger than the transition wavelength, the emission occurring along the main axis [3–7]. These observations were explained by a mean-field treatment using the Maxwell-Bloch equations, relying on an atomic spin-wave written along the main axis after the emission of the first photons: the dynamics then follows closely that of Dicke, replacing the single atom decay rate Γ0 by a collective one µΓ0, where µ is the cloud form factor [8, 9]. As a consequence, superradiance occurs above a critical atom number Nc ∼ 1/µ. However the spin-wave ansatz still restricts the evolution of the system to a single decay channel, while the breaking of indistinguishability of the atoms due to the finite sample size should involve many channels for collective spontaneous emission. To explore this question, new generation experiments in the optical domain, using cold atoms [10–14], were paralleled by intense theoretical investigations of the role of the geometry and size of the ensembles on collective spontaneous emission [1, 15–21]. In this context, Carmichael et al. [1, 15] introduced the concept of collective jump operators to describe the superradiant emission by finite size ensembles. These operators, and their associated electromagnetic (EM) modes, could be useful concepts to devise efficient light-matter interfaces based on free-space atomic ensembles [22–25]. Here we demonstrate experimentally that, although these collective modes have so far remained a theoretical tool, they can in fact be measured and isolated. To do so, we record the spatial pattern of the light emitted by an elongated atomic sample using Fourier imaging, revealing in particular the most superradiant EM mode. Furthermore, using spatial filtering, we separate the contributions of various EM modes in the emission and explain our observations by a condition for superradiance to occur in a given mode. We also compare our findings to two models describing the competition between the various modes, finding good agreement. Our experimental setup [26] relies on a glass cell, surrounded by three microscope objectives with numerical aperture NA = 0.5 used to focus an optical dipole trap (ODT) and collect the light emitted by the atoms. It produces clouds of 87Rb atoms loaded in the ODT from a magneto-optical trap using grey molasses on the D1 transition [27]. The ODT (λ = 935 nm, power up to 1W) has a waist controlled by tunable lenses between w0 = 2.5 µm and 8 µm. The resulting elongated clouds contain up to N = 8000 atoms, at a temperature of T = 550 − 1550 µK, with calculated Gaussian rms axial (radial) sizes ranging from σx = 23 µm to 80 µm (σr = 1.3 µm to 2.2 µm).

1. textbf Fundamental Mode Isolation for Quantum Simulation Improvements: The AI system can now isolate the most superradiant mode from complex collective emission by applying spatial filtering, which is described as a method to isolate this channel. This allows the AI to focus simulations on the dominant physical process rather than averaging over all decay channels.

2. textbf Predictive Modeling of Collective Dynamics: The system can now predict whether superradiance will occur based on atom number and cloud geometry by using the derived condition: "a mode with Γl > 2Γ0 leads to the emission of a burst rather than to an exponential decay."

3. textbf Tailored Quantum Trajectory Simulation: By incorporating the second model that keeps the collective jump description of the ensemble, the AI can simulate time-dependent emission using a Lindblad operator that includes both collective and individual decays, allowing it to reproduce the exact time dependencies observed in experiments.

4. textbf Geometric Dependence Quantification: The system can quantify how cloud dimensions affect superradiance by analyzing the spectral gap between the most superradiant mode and the bulk of eigenvalues, which is found to be necessary for it [the mode] to dominate the dynamics.

5. textbf Optimized Light-Matter Interface Design: The AI can devise efficient light-matter interfaces by utilizing the collective jump operators, as they can be useful concepts to devise efficient light-matter interfaces based on free-space atomic ensembles.