Quasi-two-dimensional soliton in a self-repulsive spin-orbit-coupled dipolar binary condensate

We study the formation of solitons in a uniform quasi-two-dimensional (quasi-2D) spin-orbit (SO) coupled self-repulsive binary dipolar and nondipolar Bose-Einstein condensate (BEC) using the mean-field Gross-Pitaevskii equation. For a weak SO coupling, in a nondipolar BEC, one can have three types of degenerate solitons: a multi-ring soliton with intrinsic vorticity of angular momentum projection $+1$ or $-1$ in one component and 0 in the other, a circularly-asymmetric soliton and a stripe soliton with stripes in the density. For an intermediate SO couplings, the multi-ring soliton ceases to exist and there appears a square-lattice soliton with a spatially-periodic pattern in density on a square lattice, in addition to the degenerate circularly-asymmetric and stripe solitons. In the presence of a dipolar interaction, with the polarization direction aligned in the quasi-2D plane, only the degenerate circularly-asymmetric and stripe solitons appear.

💡 Research Summary

This paper investigates the formation of self‑localized wave packets (solitons) in a uniform quasi‑two‑dimensional (quasi‑2D) Bose‑Einstein condensate (BEC) that is both spin‑orbit (SO) coupled and either non‑dipolar or dipolar, with all atomic interactions being repulsive (positive scattering lengths). The authors employ the mean‑field Gross‑Pitaevskii (GP) framework, reducing the full three‑dimensional equations to an effective 2D system by assuming a tight harmonic confinement along the axial (z) direction. The SO coupling is taken in the Rashba–Dresselhaus form, characterized by a strength parameter γ and a sign η = ±1 distinguishing Rashba (η = +1) from Dresselhaus (η = –1). Contact interactions are described by intra‑ and inter‑species scattering lengths a and a12, while the dipolar part is expressed through the dipolar length a_dd and a polarization vector fixed in the x‑y plane (chosen along the y‑axis).

The analysis proceeds in two stages. First, the authors set all nonlinear terms to zero and solve the linear Hamiltonian H0^2D. In polar coordinates they identify two degenerate eigenstates with angular‑momentum projections (∓1, 0) and (0, ±1). The study focuses on the (∓1, 0) half‑quantum‑vortex state, which carries a spin‑½ contribution and thus a total angular‑momentum projection of ±½ℏ.

Next, the full nonlinear GP equations are solved numerically using a split‑step Fourier method. The dipolar convolution is evaluated efficiently in momentum space. By scanning the SO‑coupling strength γ and the ratio a_dd/a, the authors map out the possible soliton families.

Non‑dipolar (contact‑only) BEC:

• Weak SO coupling (γ ≲ 1): Three degenerate soliton types appear.
  1. Multi‑ring soliton: A half‑quantum‑vortex whose density exhibits concentric rings in one component while the other component remains vortex‑free.
  2. Stripe soliton: Both components display a periodic stripe pattern; the total density stays uniform. This pattern has been observed experimentally in Rashba‑Dresselhaus coupled 23Na condensates.
  3. Circularly asymmetric soliton: The densities of each component and the total density are elongated or otherwise break circular symmetry, forming an elliptic‑like profile.
• Intermediate SO coupling (γ ≳ 1): The multi‑ring solution disappears and a fourth family emerges: a square‑lattice soliton with a 2D periodic density modulation forming a square crystal‑like pattern. This lattice soliton coexists with the stripe and circularly asymmetric families, all sharing the same energy and magnetization (zero magnetization on average).

Dipolar BEC (polarization in‑plane, along y):
The long‑range anisotropic dipolar interaction stretches the condensate along the polarization direction, breaking the rotational symmetry of the system. Consequently, the square‑lattice soliton is no longer supported. Only two families survive for any γ:

1. Circularly asymmetric dipolar soliton: The density elongates along y, producing a pronounced anisotropy.
2. Stripe dipolar soliton: The stripe orientation aligns preferentially with the dipole polarization, and the stripe spacing becomes asymmetric.

In all cases, the Rashba and Dresselhaus versions have identical energies and magnetizations; only the relative phase (sign of η) between the two spin components differs. Magnetization remains essentially zero because the SO coupling does not conserve spin, yet the spin‑orbit term creates nontrivial phase winding that stabilizes the structures.

The paper emphasizes that the SO coupling provides an effective attractive mechanism that can counterbalance the repulsive contact interaction, thereby allowing stable 2D solitons even in a purely repulsive medium—something impossible without SO coupling due to the well‑known collapse instability in 2D. When dipolar forces are added, they introduce a new anisotropic channel for stabilization, leading to qualitatively different soliton shapes.

From an experimental perspective, the authors suggest realistic implementations using dysprosium isotopes (e.g., 164Dy–162Dy) for the dipolar case, or 87Rb hyperfine states for the non‑dipolar case. The required SO coupling can be generated with Raman laser schemes, while the scattering lengths can be tuned via magnetic Feshbach resonances. The predicted soliton families—especially the square‑lattice and multi‑ring configurations—should be observable through in‑situ imaging of the component densities after releasing the axial confinement.

In summary, the work demonstrates that a quasi‑2D binary BEC with repulsive contact interactions can host a rich variety of stable solitons when an artificial SO coupling is present. The addition of an in‑plane dipolar interaction further enriches the phenomenology by suppressing the square‑lattice state and enforcing anisotropic shapes. These findings open new avenues for exploring exotic nonlinear excitations, supersolid‑like density patterns, and topological textures in ultracold atomic gases where both spin‑orbit and dipolar physics can be engineered simultaneously.