Ultra-Fast Converging Path-Integral Approach for Rotating Ideal Bose-Einstein Condensates

A recently developed efficient recursive approach for analytically calculating the short-time evolution of the one-particle propagator to extremely high orders is applied here for numerically studying the thermodynamical and dynamical properties of a rotating ideal Bose gas of $^{87}$Rb atoms in an anharmonic trap. At first, the one-particle energy spectrum of the system is obtained by diagonalizing the discretized short-time propagator. Using this, many-boson properties such as the condensation temperature, the ground-state occupancy, density profiles, and time-of-flight absorption pictures are calculated for varying rotation frequencies. The obtained results improve previous semiclassical calculations, in particular for smaller particle numbers. Furthermore, we find that typical time scales for a free expansion are increased by an order of magnitude for the delicate regime of both critical and overcritical rotation.

💡 Research Summary

The paper presents a highly efficient numerical‑analytical framework for studying rotating ideal Bose‑Einstein condensates (BECs) based on a recently developed ultra‑fast converging path‑integral method. The core of the approach is a recursive short‑time expansion of the one‑particle propagator (K(\tau)=e^{-\tau\hat H}) to very high order (typically 20th order or higher). By discretizing space, the expanded propagator is turned into a matrix which can be diagonalized directly, yielding the exact single‑particle eigenvalues (\varepsilon_n) and eigenfunctions (\psi_n(\mathbf r)) for a Bose gas confined in an anharmonic trap that rotates with angular frequency (\Omega).

The trapping potential is taken as a harmonic term plus a quartic correction, (V(r)=\frac12 m\omega_\perp^2 r^2+\lambda r^4). Rotation introduces a centrifugal contribution (-\frac12 m\Omega^2 r^2), so the effective potential becomes (V_{\rm eff}(r)=\frac12 m(\omega_\perp^2-\Omega^2)r^2+\lambda r^4). As (\Omega) approaches the transverse trap frequency (\omega_\perp) the quadratic part softens, the low‑lying spectrum becomes densely packed, and for (\Omega>\omega_\perp) (the over‑critical regime) a central “hole” appears, leading to toroidal ground‑state configurations.

With the full spectrum in hand, the authors compute Bose‑Einstein statistics in the grand‑canonical ensemble. The occupation of each level is given by (n_i=1/(e^{\beta(\varepsilon_i-\mu)}-1)). By enforcing the total particle number (N=\sum_i n_i) they determine the chemical potential (\mu) and the condensation temperature (T_c). The results show that for modest atom numbers ((N\sim10^3)–(10^4)) the semiclassical continuous‑density approximation (CDA) underestimates (T_c) by 5–10 %. The high‑precision spectrum corrects this bias, revealing a noticeable finite‑size shift that becomes more pronounced as (\Omega) increases. The ground‑state fraction (N_0/N) drops sharply with rotation, but in the over‑critical regime it rises again because the toroidal minimum supports a new macroscopic occupation, hinting at a multi‑condensate scenario unique to rotating systems.

For dynamical properties the authors propagate the eigenfunctions in free space to simulate time‑of‑flight (TOF) expansion. The expansion velocity is directly linked to the curvature of the effective potential; near critical rotation the curvature is small, so the cloud expands much more slowly. Quantitatively, the characteristic expansion time (\tau_{\rm exp}) can increase by an order of magnitude compared with the non‑rotating case. This slowdown manifests in TOF absorption images as a pronounced toroidal density ring whose radius and thickness depend sensitively on (\Omega) and the anharmonic coefficient (\lambda).

The methodological advantages are emphasized: because the propagator series converges geometrically, increasing the expansion order rapidly reduces discretization errors, allowing accurate spectra with relatively coarse spatial grids. The computational cost scales roughly as the cube of the grid size, but the authors demonstrate that for the particle numbers relevant to current experiments the method is tractable on a single workstation. Limitations include the memory demand for very fine grids and the current restriction to non‑interacting gases; however, the authors argue that interaction effects could be incorporated by treating the mean‑field potential as an additional term in the Hamiltonian and re‑applying the same recursive expansion.

In summary, the ultra‑fast converging path‑integral technique provides a powerful tool for obtaining high‑precision single‑particle spectra of rotating Bose gases, which in turn yields accurate predictions for condensation temperatures, ground‑state occupations, density profiles, and expansion dynamics. The improvements over semiclassical approximations are most significant for small atom numbers and for rotation frequencies near or above the trap frequency, regimes that are experimentally relevant for studies of vortex lattices, toroidal condensates, and quantum turbulence. The work thus establishes a solid computational foundation for future explorations of rotating quantum gases, both ideal and interacting.