Magnetic soliton molecules in binary condensates

TL;DR

This work analyzes magnetic solitons in two-component Bose-Einstein condensates to reveal how oppositely polarized solitons form bound molecular states. By deriving a spin-based bound-state criterion and a semi-analytic intersoliton potential, the authors show that opposite polarization yields an attractive minimum at zero separation, while like polarization yields repulsion; the dissociation energy and the critical width b_c = 1 quantify the bound/unbound transition and align with full GPE simulations. Numerical experiments demonstrate rich dynamics, including dipole-like collisions between bound states and substitution reactions when a bound state interacts with a single soliton, with the bound-state oscillation period diverging as dissociation is approached. The results provide a predictive framework for observing magnetic soliton molecules and offer avenues for testing inter-soliton potentials in experiments, with extensions to dipolar interactions, Rabi-coupled systems, and higher-dimensional regimes.

Abstract

Two-component Bose-Einstein condensates in the miscible phase can support polarization solitary waves, known as magnetic solitons. By calculating the interaction potential between two magnetic solitons, we elucidate the mechanisms and conditions for the formation of bound states -- or molecules -- and support these predictions with dynamical simulations. We analytically determine the dissociation energy of bound states consisting of two oppositely polarized solitons and find good agreement with full numerical simulations. Collisions between bound states -- either with other bound states or with individual solitons -- produce intriguing dynamics. Notably, collisions between a pair of bound states exhibit a dipole-like behavior. We anticipate that such bound states, along with their rich collision dynamics, are within reach of current experimental capabilities.

Figures (6)

• Figure 1: (a) Density and (b) wavefunction phase as a function of position for two zero-velocity magnetic solitons with opposite polarizations at $t=0$, calculated using Gross-Pitaevskii theory. Component 1 (2) is shown with a solid (dot-dashed) line. (c) Spin density dynamics, $s=n_1-n_2$, starting from the above soliton pair, which forms an excited bound state. (d) Dynamics of a like-polarization pair, illustrating the contrasting case in which the solitons repel and are unbound.
• Figure 2: Relative-phase domain walls showing the phase profiles of component 1 (2) as solid (dot-dashed) curves for three different widths, $b=\{0.5, 1, 2\}$, from dark to light.
• Figure 3: (a) Interaction potential between two magnetic solitons with opposite polarization as a function of their separation, calculated using the GPEs (blue points) and the uniform-$n$ approximation $V^\text{uni}$ (black curve). Inset shows the oscillation period of the bound states as a function of their maximum separation. (b) Interaction potential for pairs with like polarization. When $\Delta x \to 0$, the potential approaches the energy of a single magnetic soliton (horizontal dashed line). Magenta circles highlight the two cases shown in Fig. \ref{['Fig1:DensPhase']}(c,d).
• Figure 4: Energy of relative-phase domain walls---interpreted as cospatial pairs of oppositely polarized magnetic solitons with opposite velocities---as a function of inverse width. Symbols show GPE results (circles: bound, squares: unbound), and curves show the uniform-$n$ prediction [Eq. (\ref{['Eq:SAE']})] (solid: bound, dashed: unbound). Magnetic soliton pairs are bounded (unbounded) when the width of the relative-phase domain wall is smaller (larger) than a critical value, $b_\text{c}=1$ (i.e., $E=0$). Insets show examples of unbound (left) and bound (right) dynamics starting from the initial states indicated by the open symbols.
• Figure 5: Collisions involving excited bound states under various conditions: (a) two tightly bound states with oscillating polarity that remain opposite (black arrows indicate polarity); (b) two tightly bound states with the same polarity; (c) two loosely bound states with the same polarity; (d) an isolated magnetic soliton colliding with a bound state.