Developing Gravitational Caustics in an Atom Laser

By Laura ThomsonDec 21 2021

Caustics play an important role in the way optics appears in nature, from light refracted by a sheet of glass to light patterns seen on the ocean floor. This article looks at a recent study in Nature Communications that applied various flow-visualization techniques and simulated three-dimensional folded sheets to achieve quantitative agreement between theory and experimental results.

Prototypical caustics mostly visible in optics can be easily observed with polarized, coherent light. The phenomenon of caustics, as well as the underlying catastrophe theory, has sparked widespread interest.

Catastrophe atom optics has previously discussed the formation of caustics by atom trajectories in specific settings. An atom laser is a particularly powerful tool and natural setting for studying caustics in matter waves.

Even though caustics and catastrophe theory were utilized to characterize the atom laser itself, particularly in terms of its transverse beam profile, the researchers in the current study use the atom laser as a source of flow interacting with external potentials to create a broad variety of caustic features.

To examine and categorize the generated flow patterns, scientists apply a variety of flow-visualization techniques, including experimental fluid-flow tracing based on internal-state manipulation of the atoms, and simulated three-dimensional folded sheets.

The use of an atom laser in conjunction with external potentials and adjustable effective gravity offers a general platform for catastrophe atom optics research that can lead to rich noticeable dynamics with applications to cosmological effects. The ability to track down atomic flow, direct imaging of sharply delineated features, and a wealth of observable phenomena dependent on the shape, strength, and sign of the potential make this setting a very powerful platform.

Figure 1. Caustic networks from two repulsive potentials in an atom laser. (a) Experimental data averaged over 15 runs. The atom laser propagates from top (z = 0) to bottom. Each barrier is Gaussian-shaped with U₀/k_B = 23.1 μK and σ = 10.3 μm. The left barrier is positioned h = 85 μm below the injection point of the atoms (ε = 0.95). The right barrier is offset from this by (Δx, Δz) = (39, −42) μm (ε = 0.64). (b) A numerical rendering of the classical trajectories as a sheet (x_i, t_i) ↦ (x, z, t_i − √−2z/a_z), the singularities of which appear as caustics when projected down into the (x, z) imaging plane where the experimental data is shown again. Here, t is the vertical direction, while x increases to the left, and z decreases into the image. Red shading denotes regions where detJ⁻¹ is large, corresponding to the caustics when projected onto the imaging plane. (c) A numerical rendering of the classical trajectories as a sheet (x_i, t_i) ↦ (x, z, t_i), which now includes the free-fall background. Image Credit: Mossman, et al., 2021

Results

Figure 2 illustrates the experimental setup for the analysis. The x-axis is oriented along the trap’s weakly confined direction, the y-axis along the imaging axis, and the z-axis vertically in this setup. Imaging is carried out along the negative y-axis, with a dichroic mirror used to add the potential beam path with the imaging laser.

Figure 2. Experimental setup. Atoms from a ⁸⁷Rb BEC (red) are coherently out-coupled from a dipole trap (maroon) to form an atom laser. The atoms are accelerated downward by the combined action of gravity and an external magnetic gradient. The accelerated flow encounters a potential created by an additional laser (blue). In this depiction, the potential is repulsive to the atoms. The flow profile is imaged by a CCD camera using absorption imaging. Image Credit: Mossman, et al., 2021

Figure 3 depicts typical results. The top row (see Figure 3a–d) was produced by introducing a repulsive potential (ε > 0) in the atom laser, whereas the bottom row (see Figure 3e–h) was produced by inserting an attractive potential (ε < 0).

Figure 3. Experimental observation of flow with repulsive or attractive potentials. (a–d) A repulsive potential is centered h = 78(1) μm below the trapped BEC. (e–h) An attractive potential is centered h = 46(1) μm below the trapped BEC. Individual panels are labeled by the energy ratio ϵ as defined in the main text. All images have been averaged over 6 independent experimental images with the same parameters. A faint grid is overlaid at 100 μm increments. The scale for square root of the density has been set so that pure white corresponds to the 99.95^th percentile over all images to emphasize the caustic structure. The slight fringing seen in the images is due to optical effects in the imaging system, not matter-wave interference. Image Credit: Mossman, et al., 2021

The results shown in Figures 3a–d were obtained by employing a repulsive Gaussian potential. The steep edges are identified as fold caustics in Figure 4.

Figure 4. Numerical simulation and visualization. (a–d) Numerical results of 50 trajectories from x_i ∈ 0–80 μm uniformly spaced in x_i^1/2 to emphasize the scattering, overlaid on the experimental data. For clarity, a reversed color scheme has been used to plot the experimental images. (e–h) Corresponding map (xi, ti) ↦ (x, z, t_i − √−2z/az) with the vertical time axis scaled by a factor of 5 for frame (g). These numerics are generated for a single repulsive (a, b, e, f) or attractive (c, d, g, h) potential with the energy ratio noted at the top of each panel. Image Credit: Mossman, et al., 2021

Figures 4a, b illustrate a comparison to numerical simulation for two repulsive potential strengths, with corresponding visualizations in the form of folded sheets, to provide a clearer picture of the physics behind the observed features. Figure 5 depicts the maximal scattering angle with the average and estimated errors.

Figure 5. Maximal scattering angle ∣θ_c(ε)∣ vs. ε. Angle obtained by fitting the parabolic trajectory to the outer left (orange squares) and right (green circles) caustics in the experimental images. The standard deviation at each ϵ provides an estimate of the uncertainty based on three data points. The solid line is the classical result θ_c(ε) for a particle with kinetic energy K = mv²/2 = εU₀ scattering off of a Gaussian potential. The small differences between the left and right θ_c above ε ≈ 0.5 are likely due to slight asymmetries in optical potential. Image Credit: Mossman, et al., 2021

This attractive potential functions similarly to a lens that focuses the atom laser. The focusing is weak at very low potential depths (see Figure 3e). Figure 6 illustrates the flow tracing in the presence of an attractive or repulsive potential.

Figure 6. Fluid-flow tracing in the presence of a repulsive or attractive potential. (a) A repulsive potential with waist σ ≈ 12 μm is located h = 68.0(5) μm below the trapped BEC. A small horizontal stripe of atoms (located in the region between the trapped BEC and the potential) is converted back to the |1,−1⟩ state by a series of brief 100 μs mw pulses in 0.5 ms intervals, leaving a void of atoms in the imaged |2,−2⟩ state. Overlaying the experimental image are the corresponding classical trajectories at these fixed t_i, shown as dotted (red) curves for positive initial impact parameters 0 < x₀ < 100 μm. These correspond to slices through the embedded sheet (x_i, t_i) ↦ (x, z, t_i) (see for example Figure 1c), demonstrating that these fluid-flow tracers provide a direct way to visualize the folding of this sheet which gives rise to the caustics. (b) Same as (a), but for an attractive potential with waist σ ≈ 25 μm located h = 50(5) μm below the trapped BEC. Both of the experimental images here have been averaged over 101 runs. Image Credit: Mossman, et al., 2021

Methodology

The dynamics of an atom laser scattered by a Gaussian potential were investigated with a ⁸⁷Rb Bose–Einstein condensate. The repulsive (attractive) Gaussian potential is created. Absorption imaging was carried out along the −y direction.

The classical trajectories were determined by integrating the classical equations of motion with the physical potential.

Discussion

The experiments detailed expand on and contribute to the field of catastrophe atom optics by providing a powerful experimental setting for investigations.

The results presented here are based on classical trajectories, and they provide a concrete demonstration of the principles underlying classical catastrophe theory in a context other than optics. The experimental platform presented in this article serves as a versatile model system for investigating the effects of advanced beam shaping techniques.

Journal Reference:

Mossman, M. E., Bersano, T. M., Forbes, M. M., Engels, P. (2021) Gravitational caustics in an atom laser. Nature Communications, 12(7266). Available online: https://www.nature.com/articles/s41467-021-27555-3.

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