Demonstration of Fold and Cusp Catastrophes in an Atomic Cloud Reflected from an Optical Barrier in the Presence of Gravity

We experimentally demonstrate first-order (fold) and second-order (cusp) catastrophes in the density of an atomic cloud reflected from an optical barrier in the presence of gravity and show their corresponding universal asymptotic behavior. These catastrophes, arising from classical dynamics, enable robust, field-free refocusing of an expanding atomic cloud with a wide velocity distribution. Specifically, the density attained at the cusp point in our experiment reached 65% of the peak density of the atoms in the trap prior to their release. We thereby add caustics to the various phenomena with parallels in optics that can be harnessed for manipulation of cold atoms. The structural stability of catastrophes provides inherent robustness against variations in the system’s dynamics and initial conditions, making them suitable for manipulation of atoms under imperfect conditions and limited controllability.

Figure 1

(a) Sketch of the experimental setup, showing a cloud of $\sim {10}^8$ ${}^{87}\mathrm{Rb}$ atoms at $10\mu \mathrm{K}$ above a floor created by a blue-detuned beam that generates a repelling potential. Immediately after turning off the trapping beams (black arrows), the cloud’s position and velocity distributions are tailored by the depump, removal, and heating beams (see the Supplemental Material [35]). (b) Measurement of the atomic density along the $z$ axis as a function of time for a bouncing atomic cloud with small initial spatial spread (standard deviation $\sim 0.1\mathrm{mm}$) and small vertical velocity spread (standard deviation $\sim 0.03\mathrm{m}/\mathrm{s}$). The cloud follows the same parabolic trajectory expected of a rigid ball (dashed line).

Figure 2

Simplified illustration of two Lagrangian manifolds $M$ spanned in the same system by the phase space flow of atoms starting with different sets of initial conditions ($h_0$). The critical set $C$ (red dashed line) is a smooth curve on $M$, defining the singularity points in the projection of $M$ on the $\{z,t\}$ measurement plane. In (a) the projection of $C$ on the measurement plane forms a fold line (blue dashed line). In (b) $C$ exhibits a parabolic turning point, as evident from its projection on the $\{v,t\}$ plane (purple dashed line). Accordingly, its projection on $\{z,t\}$ forms two fold lines that meet at a cusp point (blue dashed lines). The green arrows show examples of individual atom trajectories that end at time $t$ at the same $z$, with the one drawn in (b) being the same as the upper trajectory in (a), thus marking the intersection between the two manifolds.

Figure 3

Calculated, simulated, and experimental results. (a) Calculated trajectories of atoms released from a cold ($10\mu \mathrm{K}$) but large atomic cloud (standard deviation of $\sim 1\mathrm{mm}$, compared to release height of $\sim 2\mathrm{mm}$). (b) Calculated trajectories of atoms released from a hot ($\sim 400\mu \mathrm{K}$) pancake-shaped atomic cloud (standard deviation of $\sim 0.1\mathrm{mm}$, compared to release height of $\sim 2.8\mathrm{mm}$), demonstrating refocusing of the cloud after $\sim 48\mathrm{ms}$, followed by splitting into two high-density peaks, one moving upwards and the other downwards. [(c),(d)] Simulated results taking into account the experimental initial conditions and the temporal and spatial resolution of the imaging setup (2 ms and 0.15 mm, respectively), clearly showing the emergence of fold lines in (c) and a cusp splitting into two folds in (d). [(e),(f)] The experimentally measured vertical density distributions as a function of time, together with the analytically derived location of the catastrophe lines (dashed lines). The density attained at the cusp point is $\sim 65\%$ of the initial density of the expanding atomic cloud at $t=0$.

Figure 4

(a) For fold-type singularities, the measured density distribution at a given time (dots) is in excellent agreement with the theoretical distribution (solid line) using a divergence rate of $\alpha =1/2$. In comparison, divergence rates of $\alpha =1/3$ (green dashed line) and of $\alpha =2/3$ (red dashed line) are inconsistent with the measured results. (b) For a cusp-type singularity, the measured density distribution (dots) agrees with the theoretical distribution (solid line) using $\alpha =2/3$ and is inconsistent with divergence rates of $\alpha =1/2$ (green dashed line) and of $\alpha =1$ (red dashed line).