Efficient simulation of Bose-Einstein condensates in nontrivial topologies

Abstract

Bubble-shaped Bose-Einstein condensates (BECs) constitute a unique class of quantum fluids with a hollow, thin-shell geometry that supports a wide variety of phenomena that are distinct from those of compact condensates. Numerical simulation of such systems is particularly challenging due to their inherently three-dimensional structure and extreme aspect ratios. We present an efficient finite-difference simulation framework designed for solving partial differential equations in such nontrivial topologies with a focus on the static and dynamical modeling of bubble-shaped BECs. By employing selective spatial sampling on a semi-structured grid, our method substantially reduces memory usage and achieves more than an order-of-magnitude improvement in computational performance compared to conventional split-step Fourier solvers. The algorithm is naturally extendable for highly parallel execution on GPUs, enabling large-scale, time-dependent simulations of thin-shell condensates. We apply this framework to simulate the formation of bubble BECs through a controlled hollowing-out protocol using ab initio trapping potentials relevant to the Cold Atom Laboratory aboard the International Space Station. From these simulations, we identify characteristic timescales and parameter ramps required to achieve adiabatic evolution, thereby assessing the feasibility of experimentally realizing bubble-shaped condensates in microgravity environments.

Figures (5)

• Figure 1: Illustration of the semi-structured grid using a Thomas-Fermi trial solution corresponding to $\mu/h=450Hz$ in a typical bubble trap potential, corresponding to $\sim9\times10^5$ particles. For visualization purposes, a quarter of the grid is cut out and the resulting boundary is colored orange. (a): General scheme resulting in fine-grained post-selection of grid points. The local structure of the grid is shown in the inset, with only the nearest neighbors along $x$ at deterministic offsets ($\pm1$) from the central point. The axes are labeled in grid-point units. (b): Parallel, GPU-optimized scheme resulting in coarse-grained post-selection of $8^3$ blocks of grid points supporting storage in shared memory. The axes are labeled in block units. The inset show the structure of a single block, where the halo is shown with orange shading. Due to the coarser graining, the local structure of the grid is highly ordered, with all nearest neighbors at deterministic offsets ($\pm1$, $\pm10$, $\pm100$) in shared memory.
• Figure 2: Verification of the structured grid method with GPE ground states. (a): GPE ground state density in a typical bubble potential, along the $y$-$z$ and $x$-$y$ planes. The solid black line shows the boundary of the region of interest. (b): Simulation errors for the structured grid method, shown along the $y$-$z$ and $x$-$y$ planes. (c): Scaling of simulation error with grid selection cutoff for the rougher grid with $\delta r=0.3µm$ (top) and the finer grid with $\delta r=0.15µm$ (bottom), for various Laplacian stencils. The grey dashed line shows the choice of chemical potential cutoff used in this work.
• Figure 3: Benchmark of GPE ground state solvers using the conventional split-step Fourier method on CPU and GPU as well as the structured grid methods proposed in this work, highlighted with the hatched fill. The runtimes are averages from 7 runs. The uncertainties of the mean values are negligible. The FFT-based algorithms did not demonstrate a significant speedup for similar size radix-2 grids. The simulations for the general algorithm were run on an A2338 MacBook Pro equipped with an M1 processor and 16 GB memory. The GPU simulations were run on a Tesla V100 with 16 GB memory.
• Figure 4: Benchmarking the time-evolution algorithms and comparison with conventional split-step Fourier solvers for various grid sizes. The runtimes are averages from 7 runs, with negligible variance. The hatched fill highlights the results from the algorithms from this work. (a)-(b): CPU algorithm runtimes for a single-stage ramp between two potentials which correspond to inflating bubble-shaped clouds by $\approx\!10$ percent. (c)-(d): GPU algorithm runtimes for a ten-stage ramp, which simulated the hollowing-out dynamics and subsequent inflation.
• Figure 5: Inflation dynamics of bubble-shaped BECs. (a): Evaluation of adiabaticity of various RF ramps with different durations. The lines connecting the markers serve as a guide to the eye. (b): Optimal RF detuning ramps of duration 200 ms and 400 ms, compared to linear and smoothstep ($S_1$) ramps. (c)-(d): Column density and projected density difference between the ground state and the dynamically loaded bubble state for the 400 ms smoothstep ramp. (e)-(f): Column density and projected density difference between ground state and dynamically loaded bubble state for the 400 ms optimal ramp. For the projected images, the area-preserving Equal Earth projection was used.