Nonlinear stage of modulational instability in repulsive two-component Bose-Einstein condensates

TL;DR

This work establishes the nonlinear stage of modulational instability in a repulsive two-component BEC by combining barrier-driven dam-break initial conditions with a rigorous linear-stability framework for a coupled NLS system. It derives a general edge-velocity formula for the MI wedge, $V_{2c} = 4\sqrt{-g_+ + \sqrt{g_-^2 + 4Q_1^2Q_2^2 g_{12}^2}}$, valid for arbitrary component ratios and interaction strengths, and confirms it against full 3D and effective 1D simulations as well as experimental data. The study experimentally demonstrates MI and dispersive shock wave formation, validates the 1D reduction, and extends the phenomenology to barrier-induced dam-break interference that produces Peregrine-soliton–like structures, highlighting the versatility of ultracold-atom platforms for studying DSWs and rogue waves in multi-component defocusing media. The results provide quantitative tools for characterizing MI in multicomponent BECs and open pathways to controlled collisions and PS generation in atomic quantum fluids.

Abstract

Modulational instability (MI) is a fundamental phenomenon in the study of nonlinear dynamics, spanning diverse areas such as shallow water waves, optics, and ultracold atomic gases. In particular, the nonlinear stage of MI has recently been a topic of intense exploration, and has been shown to manifest, in many cases, in the generation of dispersive shock waves (DSWs). In this work, we experimentally probe the MI dynamics in an immiscible two-component ultracold atomic gas with exclusively repulsive interactions, catalyzed by a hard-wall-like boundary produced by a repulsive optical barrier. We analytically describe the expansion rate of the DSWs in this system, generalized to arbitrary inter-component interaction strengths and species ratios. We observe excellent agreement among the analytical results, an effective 1D numerical model, full 3D numerical simulations, and experimental data. Additionally, we extend this scenario to the interaction between two counterpropagating DSWs, which leads to the production of Peregrine soliton structures. These results further demonstrate the versatility of atomic platforms towards the controlled realization of DSWs and rogue waves.

Figures (7)

• Figure 1: Experimental and numerical demonstration of the nonlinear stage of MI from a repulsive barrier for an initially 50:50 mixture of $\lvert2,0\rangle$ and $\lvert1,0\rangle$ atoms. (a) Single-shot absorption image taken after 60 ms of evolution time showing the $\lvert 2,0\rangle$ state above the $\lvert 1,0\rangle$ state. (b) Integrated cross sections averaging over 20 independent experimental realizations corresponding to the absorption image in panel (a). (c) Spacetime evolution of the average integrated cross sections for the $\lvert2,0\rangle$ state. The blue dashed lines show the experimentally determined speed of sound and the red lines depict the theoretically predicted MI expansion rate given in Eq. \ref{['eq:edge_gino']}. Density evolution of the (d) integrated cross sections from 3D numerical simulations and (e) 1D simulations of the experimental conditions overlaid with the same lines as in panel (c).
• Figure 2: Numerical 1D results of the nonlinear stage of MI of the $\lvert 2,0\rangle$ component in a (a)-(b) 50:50, (c)-(d) 85:15, and (e)-(f) 96:4 mixture. (a), (c), (e), Spatiotemporal evolution of the minority component's wavefunction magnitude, $|q_2(x,t)|$. (b), (d), (f), Snapshot of $|q_2(x,t)|$ at $t=564$ ms. Red lines mark the edge of the modulated region described by Eq. \ref{['eq:edge_gino']}. Dashed green lines depict the envelope of the modulations dictated by Eq. \ref{['eq:edge_gino_limit']}. Due to symmetry, only the positive $x$-axis is shown.
• Figure 3: Experimental images depicting DSW interaction produced by two barriers and subsequent PS formation in a 50:50 immiscible mixture. (a) Absorption image taken after 80 ms of evolution time with $\lvert 2,0\rangle$ appearing above $\lvert 1,0\rangle$. Spacetime evolution of the (b) $\lvert 2,0 \rangle$ and (c) $\lvert 1,0 \rangle$ component featuring complementary density profiles. A PS (red circle) appears at the center of the interference region.
• Figure S1: Fractional populations of the two spin populations over time corresponding to the data shown in Fig. 1 of the main text. An exponential fit to the decay in the $\ket{2,0}$ population is shown to have a time constant of $\tau=220$ ms.
• Figure S2: Experimental and numerical determination of the speed of sound. An equal spin mixture of (a) $\lvert 2,0\rangle$ and (b) $\lvert 1,0\rangle$ is prepared in the presence of an optical barrier. A sudden increase of the barrier height is used to induce sound waves. Each panel is an average over 10 experimental realizations for every 5 ms. Blue lines mark the linear fit to the traveling sound pulses from the experiment used to determine the speed of sound, averaging at $v_s = 2.0(1)\ \mu$m/ms. (c) 1D GPE simulations of the experimental procedure showing a consistent result for the speed of sound compared to the average over the experimental results, shown as blue lines.