Phase Dynamics of Self-Accelerating Bose-Einstein Condensates

TL;DR

This work investigates the cubic-in-time Kennard phase governing self-accelerating Airy Bose–Einstein condensates by analyzing interference between Airy and reference wave packets. Using a quasi-1D Gross–Pitaevskii framework with a linear potential, the authors derive explicit phase dynamics for Airy and Gaussian propagation and for Ai–Ai and Ai–G collisions, identifying the leading cubic contributions. They compare heterodyne-based and density-based phase extraction, assessing robustness to fit-window choices via simulated robustness maps; the Airy–Gaussian geometry with HPE yields the most precise and window-insensitive access to the Kennard phase. Additionally, in the weakly nonlinear regime, the cubic coefficient $c_3$ varies linearly with the effective 1D interaction strength $g/\hbar$, enabling the cubic phase to function as a practical probe of weak mean-field nonlinearities. The results provide a concrete pathway for experimental observation of the Kennard phase in self-accelerating condensates and establish Ai–G as a robust interferometric benchmark for nonlinearities in ultracold quantum fluids.

Abstract

Self-accelerating Airy matter waves offer a clean setting to access the cubic Kennard phase. Here we reconstruct the relative phase of simulated Airy-shaped Bose-Einstein condensates in free space, a regime approached in microgravity, from interference fringes. The cubic phase dynamics are quantified via windowed polynomial fits with systematics-aware uncertainty estimates that account for window-induced correlations. We compare two experimentally feasible phase-extraction methods - heterodyne-based and density-based - and show that an Airy-Gaussian geometry yields substantially improved robustness to fit-window selection relative to an Airy-Airy collision. In the weakly interacting regime, the extracted cubic coefficient responds linearly to the effective one-dimensional interaction strength. Our approach turns cubic phase dynamics into a practical probe of weak mean-field nonlinearities in self-accelerating condensates.

Figures (9)

• Figure 1: Time evolution of density $|\Psi(x,t)|^2$: Interference of two counterpropagating Airy-Bose-Einstein Condensates (Ai-BECs) in free space. A slightly asymmetric scenario of the Airy scales $x_{0,1}$, $x_{0,2}=1.3 \cdot x_{0,1}$ is chosen, otherwise the cubic phase dynamics cancel out. Inset: Schematic of a possible experimental implementation in which an Airy-type, laser-induced potential by a spatial light modulator (SLM) is imprinted onto a Gaussian-like BEC ground state, producing an Ai-BEC.
• Figure 2: Deviation of $\Delta c_3 = c_3 - c_{3,\text{theo}}$ across $[t_{\text{start}}, t_{\text{end}}]$ for the Ai-Ai collision scenario, representing the robustness of the phase extraction visualizing the sensitivity of the cubic coefficient $c_3$ on the fitted region based on the heterodyne phase extraction. Two white, drawn contour lines mark the relative error of 5% (dotted) and 10% (dashed). The highlighted data points show the best combinations among these discrete pairs due to the finite resolution. The pink dot represents the closest value to $c_{\mathrm{3,theo}} = -3.472800\times 10^{10} \ \mathrm{rad/s^3}$. The colour indicates the signed, nonlinearly compressed deviation $\Delta c_3 = c_3 - c_{\mathrm{3,theo}}$, using $T = \operatorname{sgn}\![\tanh(k\,\Delta c_3 / 2s)]\ |\tanh(k\,\Delta c_3 / 2s)|^{\gamma}$ with $k = 2.5$, $\gamma = 0.7$, and $s$ given by the 90th percentile of $|\Delta c_3|$, which provides a symmetric, bounded, contrast-enhanced colour scale for $\Delta c_3$ and compresses outliers.
• Figure 3: Relative phase of Airy main lobe interference fringe from the demodulated signal of the HPE – continuing the Ai-Ai collision scenario. The best cubic fit window choice, corresponding to the pink dot of Fig. \ref{['fig:ai-ai-heatmap']}, is drawn in red relying on the highlighted data in black.
• Figure 4: Deviation of $\Delta c_3 = c_3 - c_{\mathrm{3,theo}}$ across $[t_{\mathrm{start}}, t_{\mathrm{end}}]$ for the Ai–Ai collision scenario, based on the DPE. As in Fig. \ref{['fig:ai-ai-heatmap']}, the axes parameterize the fit window, the white contour lines indicate relative errors of 5% (dotted) and 10% (dashed), and the highlighted points mark the best discrete window combinations, with the pink dot denoting the closest value to $c_{\mathrm{3,theo}}$. The region of small fit interval choices oscillates and diverges due to fit instability. The colour scale uses the same signed, nonlinearly compressed mapping of $\Delta c_3$ as in Fig. \ref{['fig:ai-ai-heatmap']}.
• Figure 5: Relative phase of the Airy main lobe interference fringe from the demodulated DPE signal for the Ai–Ai collision scenario. The best cubic fit window is shown in red and is based on the highlighted data points in black. This interval falls within the cubic regime, where the local slope reverses relative to the global trend.