Ferrodark soliton collisions: Breather formation, pair reproduction, and spin-mass separation

Technical Summary: Ferrodark Soliton Collisions in Spin-1 Bose-Einstein Condensates

Problem Statement
The paper investigates the collision dynamics of topological solitons, specifically ferrodark solitons (FDSs) and their anti-solitons (antiFDSs), within the easy-plane phase of ferromagnetic spin-1 Bose-Einstein condensates (BECs). While soliton collisions in integrable systems (e.g., KdV, sine-Gordon) are well-understood as shape-preserving, non-dissipative events, collisions in non-integrable systems like superfluids exhibit complex behaviors such as annihilation, reflection, and bound state formation. The study focuses on FDSs, which are propagating $Z_2$ kinks in the spin order, to explore how these topological defects interact, annihilate, or reproduce, and to characterize the resulting non-equilibrium states.

Methodology
The authors employ numerical simulations based on the mean-field Gross-Pitaevskii equations (GPEs) for a spin-1 BEC. The system is modeled in a uniform ferromagnetic phase with quadratic Zeeman energy $q$ and spin-dependent interaction strength $g_s$ (specifically at the exactly solvable point $g_s/g_n = -1/2$, with qualitative checks at $g_s/g_n = -0.01$). The study initializes pairs of FDSs and antiFDSs with opposite velocities and varying initial speeds ($V_{in}$). The pairs are categorized into three collision scenarios:

1. Type-I Pair: Collision between a Type-I FDS and a Type-I antiFDS (both possessing positive inertial mass).
2. Type-II Pair: Collision between a Type-II FDS and a Type-II antiFDS (both possessing negative inertial mass).
3. Mixed Pair: Collision between a Type-I FDS and a Type-II antiFDS.

The evolution of the transverse magnetization ($F_\perp$) and total mass superfluid density ($n$) is tracked to determine the post-collision states, energy dissipation rates, and topological structure preservation.

Key Contributions and Results

• Type-I Pair Dynamics (Annihilation and Breather Formation):

  • Low Velocity: At low incoming velocities, the Type-I pair annihilates, destroying the $Z_2$ kink structure in the spin order. Instead of dispersing into linear waves, the system forms an extremely long-lived, spatially localized dissipative breather. This breather is characterized by out-of-phase oscillations of the magnetization density and the mass superfluid density.
  • Energy Dissipation: The breather loses energy by periodically emitting spin and density waves (identified as gapless Bogoliubov modes). The authors find that the breather energy decays logarithmically over time ($E \propto 1/\log(t)$), distinguishing it from breathers in integrable systems which are typically non-dissipative.
  • Critical Velocity and Pair Reproduction: As the incoming velocity increases, a critical velocity ($V_c$) is identified. Near $V_c$, the lifetime of a stationary Type-I pair formed after collision exhibits a power-law divergence ($\tau \propto |V_c - V_{in}|^{-\alpha}$ with $\alpha \approx 0.5$), resembling critical behavior.
  • High Velocity: Above $V_c$, the pair does not form a stationary state but is reproduced as a propagating pair with a finite separating velocity (effectively a reflection), provided the remaining energy exceeds the threshold for a stationary pair.

• Type-II Pair Dynamics (Reflection):

  • In contrast to Type-I pairs, Type-II pairs never undergo annihilation. The kink structure is preserved throughout the collision.
  • The interaction results purely in reflection. Due to the negative inertial mass of Type-II FDSs (where energy decreases with velocity), the reflected velocity is higher than the incoming velocity, despite weak energy dissipation.

• Mixed Pair Dynamics (Spin-Mass Separation):

  • When a Type-I FDS collides with a Type-II antiFDS, a phenomenon of "spin-mass separation" is observed.
  • Spin Order: The topological structures in the spin order ($Z_2$ kinks) reflect off each other, consistent with general $Z_2$ kink behavior.
  • Mass Density: Conversely, the mass superfluid density profiles pass through each other.
  • Collision Shift: The collision center is shifted toward the incoming side of the Type-I FDS. This asymmetry arises from the inelastic nature of the collision and the distinct dispersion relations of the two soliton types, causing the Type-I FDS to slow down and the Type-II antiFDS to speed up.

Significance and Claims
The paper claims to provide a unique platform for studying the rich dynamics of kink collisions in superfluids, a regime where propagating kinks are rare. Key theoretical advancements include:

1. Novel Dissipative Breather: The identification of a long-lived, weakly dissipative breather formed from soliton annihilation in a non-integrable system, distinct from known breathers on unstable backgrounds (e.g., Akhmediev breathers).
2. Critical Behavior: The observation of power-law divergence in the lifetime of a stationary soliton pair near a critical velocity, a feature not previously reported in $Z_2$ kink collisions in the $\phi^4$ model.
3. Spin-Mass Separation: The demonstration that in spinor superfluids, spin and mass degrees of freedom can decouple during collisions, with spin orders reflecting while mass densities transmit.
4. Experimental Relevance: The authors suggest these results may aid in interpreting scaling laws observed in the quench dynamics of spin-1 BECs, where ferrodark solitons are proposed as relevant topological defects. The findings are presented as a motivation for future experimental investigations in this regime.