Abstract
We study the topological configurations of the two-component condensates of bosons with the $3$D $\vec{\sigma}\cdot \vec{p}$ Weyl-type spin-orbit coupling subject to a harmonic trapping potential. The topology of the condensate wavefunctions manifests in the quaternionic representation. In comparison to the $U(1)$ complex phase, the quaternionic phase manifold is $S^3$ and the spin orientations form the $S^2$ Bloch sphere through the 1st Hopf mapping. The spatial distributions of the quaternionic phases exhibit the 3D skyrmion configurations, and the spin distributions possess non-trivial Hopf invariants. Spin textures evolve from the concentric distributions at the weak spin-orbit coupling regime to the rotation symmetry breaking patterns at the intermediate spin-orbit coupling regime. In the strong spin-orbit coupling regime, the single-particle spectra exhibit the Landau-level type quantization. In this regime, the three-dimensional skyrmion lattice structures are formed when interactions are below the energy scale of Landau level mixings. Sufficiently strong interactions can change condensates into spin-polarized plane-wave states, or, superpositions of two plane-waves exhibiting helical spin spirals.
Highlights
Pattern formation and the presence of coexisting phases in spatially separated domains are an emergent feature of diverse dynamical systems throughout physics [1], chemistry [2], and biology [3]
We study the sensitivity of coupled condensate formation dynamics on the history of initial stochastic domain formation in the context of instantaneously quenched elongated harmonically trapped immiscible two-component atomic Bose gases
We numerically map out diverse key aspects of these competing growth dynamics, focusing on the role of shot-to-shot fluctuations and global parameter changes, with our findings qualitatively confirmed by realistic finite-duration quenches
Summary
Pattern formation and the presence of coexisting phases in spatially separated domains are an emergent feature of diverse dynamical systems throughout physics [1], chemistry [2], and biology [3]. Providing strong evidence that the density profiles emerging over experimentally relevant time scales are determined by the history of spontaneous defect formation and subsequent dynamics (Fig. 1). This long-term memory is facilitated by the added stability provided by formation, during coarse-graining dynamics, of a composite dark-bright solitary wave [54,55,56,57,58,59] defect, a process found to be robust to perturbations, but sensitive to shot-to-shot fluctuations and global parameter changes. Our work offers insight into the complexities of two-component BEC formation, and suggests caution when concluding that an experimental immiscible two-component BEC has reached a true equilibrium state
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