Quantitative theory of dark-soliton transmission through a domain wall in phase-separated two-component Bose–Einstein condensates

In an immiscible two-component Bose–Einstein condensate, the interface (domain wall) separating the two superfluid components acts as a nonlinear scattering interface for dark solitons. Building on the 2001 discovery by Öhberg and Santos that a dark soliton can be transferred between components when it exceeds a critical velocity, a 2019 study (Zheng, Zhang, and Liu) documented reflection, transmission, and critical transmission regimes in a harmonically trapped symmetry-breaking configuration, including the carrying of atoms of one component into the other, sound radiation, long-time decay of soliton depth and energy, and back-action of the wall — but only at a single parameter set. The open problem is a quantitative characterization of this transport: the critical transmission threshold as a function of trap strength, intra- and inter-species interactions, and atom numbers; quantitative decay laws for soliton depth and energy versus the mass entrained per crossing; the effects of transverse (snake) instability; and the feedback of domain-wall deformation on subsequent crossings. Later work (Arazo et al. 2021; Ye et al. 2026) mapped parts of the interaction-parameter dependence in 1D and 2D and confirmed that wall back-action invalidates a pure critical-velocity criterion, but the full quantitative theory remains open.

Background

A Bose–Einstein condensate (BEC) is a dilute ultracold gas in which all atoms occupy a single quantum state, described at mean-field level by the Gross–Pitaevskii (GP) equation — a nonlinear Schrödinger equation whose cubic term encodes repulsive contact interactions. In a one-dimensional repulsive BEC, the GP equation supports dark solitons: localized density notches with a phase step across them, stabilized by the balance of dispersion and nonlinearity, with width of order the healing length $\xi=\hbar/\sqrt{\mu m}$ (the distance over which the condensate density recovers from a perturbation, with $\mu$ the chemical potential and $m$ the atomic mass). A fast dark soliton is shallow and carries little intrinsic energy; a slow one is deep and energetic.

When two condensates coexist, the coupled GP equations contain intraspecies couplings $g_1,g_2$ and an interspecies coupling $g_{12}$ . For $g_{12}>\sqrt{g_1g_2}$ the mixture is immiscible: the ground state phase-separates, and in a symmetry-breaking configuration each component occupies one side of the trap, meeting at a domain wall — a narrow interface where the two densities overlap. Öhberg and Santos (2001) showed that a dark soliton launched in one component can be transferred into the other at this wall, provided its velocity exceeds a critical value, and that the wall undergoes a back-action (a positional jump) at the crossing.

Zheng, Zhang, and Liu (Chin. Phys. B 28, 116701 (2019)) revisited this in a harmonic trap with the symmetry-breaking state, where a soliton imprinted off-center accelerates toward the trap center where the wall sits. At one parameter set ( $g_1=1$ , $g_2=0.8g_1$ , $g_{12}=1.2g_1$ , equal trap frequencies, $N_1=N_2$ ) they found: reflection with oscillation confined to one component when the soliton reaches the wall slowly; transmission with oscillation between components when it arrives fast (transmission failing for release at $-6.1\xi$ and succeeding at $-6.2\xi$ ); sound radiation in both components in either case; and a striking atom-carrying mechanism — the soliton drags atoms of the source component through the wall into the destination component (i.e., it becomes a dark-bright soliton, a dark notch in one component filled by a trapped bright blob of the other), causing the soliton depth and energy to decay over repeated crossings until the soliton disappears. They noted qualitatively that larger $g_{12}$ or strong atom-number asymmetry makes transfer harder, but provided no quantitative mapping. Their treatment is strictly one-dimensional, zero-temperature GP dynamics, excluding finite temperature, explicit dissipation, and the transverse snake instability (the decay of a planar dark soliton into vortices in higher dimensions). Arazo et al. (Phys. Rev. A 104, 043312 (2021)) subsequently swept $g_{12}/g$ in 1D, showed dark-bright soliton generation at the wall with a semi-analytical oscillation frequency, and found that the outcome of a second collision cannot be explained by the critical-velocity argument alone — the wall's deformation matters. Ye et al. (Chin. Phys. B, 2026) extended the study to quasi-2D with a phase diagram over interspecies coupling and initial displacement. The quantitative, parameter-resolved theory of this interfacial transport is nevertheless still missing.

Problem Statement

For two immiscible Bose–Einstein condensates forming a symmetry-breaking state in a trap, with a dark soliton launched in one component toward the domain wall, establish a quantitative theory of the soliton–domain-wall interaction, comprising jointly: (i) the critical transmission threshold — the critical velocity at the wall, or equivalently in a trap the critical release position — as a function of the system parameters (trap strength, intraspecies interactions $g_1,g_2$ , interspecies interaction $g_{12}$ , and relative atom numbers $N_1,N_2$ ); (ii) quantitative laws for the long-time evolution of the transmitted soliton — how its depth and energy decay with the number of wall crossings and with the mass of the other component entrained per crossing, and the resulting long-time fate of the soliton; (iii) the modification of this picture, and in particular of the atom-carrying mechanism and the sound radiation, by transverse (snake) instability and vortex formation beyond strictly one-dimensional Gross–Pitaevskii dynamics; and (iv) the characterization of the domain-wall back-action — how the wall deformation and centroid oscillations induced by each crossing feed back into the critical conditions for subsequent crossings (e.g., hysteresis or cumulative deformation of the wall). (i) and (ii) are to be established in the strictly one-dimensional, zero-temperature Gross–Pitaevskii setting of the benchmark described above, with (iii) giving the modification of that picture beyond one dimension; the threshold in (i) is the pristine, first-crossing threshold of an undeformed wall, while (iv) asks for the deformation-corrected condition governing later crossings. The parameter list in (i) names the dependence to be characterized rather than requiring every parameter to be varied independently, and the threshold is to be characterized as a function of the interspecies interaction $g_{12}$ together with one of the remaining parameters (trap strength or relative atom number). These deliverables share one objective: a predictive, parameter-resolved account of nonlinear matter-wave transport across the magnetic interface of a phase-separated binary superfluid, of the kind initiated by Öhberg and Santos (2001) and Zheng, Zhang, and Liu (2019) at single parameter sets.

References

Lang Zheng, Yi-Cai Zhang, and Chao-Fei Liu, 'Propagation of dark soliton interacting with domain wall in two immiscible Bose–Einstein condensates', Chinese Physics B 28, 116701 (2019). DOI: 10.1088/1674-1056/ab457e, https://doi.org/10.1088/1674-1056/ab457e
P. Öhberg and L. Santos, 'Dark Solitons in a Two-Component Bose-Einstein Condensate', Physical Review Letters 86, 2918–2921 (2001). DOI: 10.1103/PhysRevLett.86.2918, https://doi.org/10.1103/PhysRevLett.86.2918
P. Öhberg and L. Santos, 'Solitons in two-component Bose-Einstein condensates', Journal of Physics B: Atomic, Molecular and Optical Physics 34, 4721–4735 (2001). DOI: 10.1088/0953-4075/34/23/316, https://doi.org/10.1088/0953-4075/34/23/316
Maria Arazo, Montserrat Guilleumas, Ricardo Mayol, and Michele Modugno, 'Dynamical generation of dark-bright solitons through the domain wall of two immiscible Bose-Einstein condensates', Physical Review A 104, 043312 (2021). DOI: 10.1103/PhysRevA.104.043312, arXiv:2107.03209, https://arxiv.org/abs/2107.03209
Chan Ye, Chen Jiao, Qi Wang, and Tao Yang, 'Interfacial Dynamics of a Dark Soliton in a Two-Dimensional Phase-Separated Bose–Einstein Condensate', Chinese Physics B (published online 2026). DOI: 10.1088/1674-1056/ae37fd, https://doi.org/10.1088/1674-1056/ae37fd
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Xiaoquan Yu and P. B. Blakie, 'Dark-soliton-like magnetic domain walls in a two-dimensional ferromagnetic superfluid', Physical Review Research 3, 023043 (2021). DOI: 10.1103/PhysRevResearch.3.023043, arXiv:2008.08175, https://arxiv.org/abs/2008.08175