Motion of ferrodark solitons in trapped superfluids: spin corrections and emergent oscillators

TL;DR

This work develops a spin-dependent framework for soliton motion in trapped spinor superfluids by decomposing the total force on a soliton into buoyancy and spin-correction components, enabling a macroscopic equation of motion that remains valid for large-amplitude dynamics. The soliton energy is reframed as a function of local density, chemical potential, and velocity, leading to an inertial mass and an emergent force f = f_b + f_s; for ferrodark solitons in a spin-1 BEC, the spin correction can dominate and even reverse the total force, producing rich behaviors such as a mapping to a quartic oscillator in a harmonic trap and to a simple harmonic oscillator in a hard-wall trap. The analysis reveals type-I and type-II ferrodark solitons with divergent inertial mass at the transition and predicts equilibrium positions, local speed limits, and three distinct oscillation regimes in a linear potential, all captured by the emergent-potential mapping. The results provide analytic predictions for oscillation frequencies and amplitudes across regimes, extendable to other multicomponent superfluids, and offer a concrete route for experimentally probing spin-driven soliton dynamics in ultracold gases.

Abstract

We propose a framework for topological soliton dynamics in trapped spinor superfluids, decomposing the force acting on the soliton by the surrounding fluid into the buoyancy force and spin corrections arising from the density depletion at soliton core and the coupling between the orbital motion and the spin mixing, respectively. Our formulation applies to large-amplitude soliton motion in general superfluids with spin degrees of freedom under arbitrary external potentials. For ferrodark solitons (FDSs) in spin-1 Bose-Einstein condensates , the spin correction could diverge, change the direction of the total force and enable mapping the FDS motion in a harmonic trap to the atomic-mass particle dynamics in an emergent quartic potential. Initially placing a type-I FDS near the trap center, a single-sided oscillation happens, which maps to the particle moving around a local minimum of the emergent double-well potential. As the initial distance of a type-II FDS from the trap center increases, the motion exhibits three regimes: trap-centered harmonic and anharmonic oscallations followed by single-sided oscillations. Correspondingly the emergent quartic potential undergoes a transition from a single minimum to a double-well shape, where the particle motion shifts from oscillating around the single minimum to crossing between two minima via the local maximum, then the symmetry-breaking motion around one of the two minima. In a hard-wall trap with linear potential, the FDS motion maps to a harmonic oscillator.

Figures (8)

• Figure 1: Evolution of the inertial mass $M_{\rm in}$ (green), the constant buoyancy force $f_b$ (blue), the spin correction $f_s$ (red), the total force $f$ (dark red), and the FDS core trajectory (black solid) during a half period of the oscillation in a linear potential with initially placing a type-I FDS at $X_0$, evaluated using Eqs. \ref{['spincorrection']}\ref{['inertialmassFDS']} and \ref{['Buoyancy']}. When crossing the type transition point $X_{*}=\mu_N/k-\sqrt{(\mu_{N}/k-X_0-q/k)^2+q^2/4k^2}-q/2k$ (guided by the dash-dotted line), $M_{\rm in}$, $f_s$ and $f$ diverge and change sign. The acceleration $a$ (orange) keeps sign before reaching the nearby equilibrium point $X_{\rm eq}=X_0+q/2k>X_{*}$(guided by the dash line), where the total force $f$ vanishes and changes sign again. Here $X_0=-10\bar{\xi}_n$, $\tilde{q}=-q/2g_s\bar{n}_b=0.2$, $\mu_N=0.6 g_n \bar{n}_b$, $k=0.01 g_n \bar{n}_b/\bar{\xi}_n$ (for the parameters chosen, $X_{\rm eq}=0$), $\bar{\xi}_n=\sqrt{\hbar^2/ M g_n\bar{n}_b}$, $t_0=\hbar/g_n \bar{n}_b$, and $\bar{n}_b$ is the average density.
• Figure 2: FDS dynamics in a harmonically trapped spin-1 BEC: a FDS core trajectory evolution[(a1)-(a4)] [spin-1 GPE simulations (lines); macroscopic soliton EOM Eq. \ref{['solitonEOM']} (markers)] and the corresponding emergent particle quartic potential $\tilde{U}(X)$ [(b1)-(b4)](schematic). Here, $g_s/g_n =-1/2$, $\tilde{q} =-q/2g_s n_{\text{peak}}=0.1$, $\mu_N=201.2 \hbar \omega_h$, $a_{\rm ho} =\sqrt{\hbar/(M \omega_h)}$, $t_h = 1/\omega_h$ and $\omega_h$ is the trap frequency. (a1) A type-I FDS is initially placed near the trap center, it drifts away from the trap center and then returns, exhibiting single-sided oscillations, mapping to a particle rolling down from the local maximum and then oscillating around a local minimum of an emergent double-well potential [(b1)]; (a2)-(a4) When imprinting a type-II FDS, the motion depends on the initial distance to the trap center. (a2) Slightly away: harmonic oscillations of a type-II FDS near the trap center, mapping to particle oscillations around the single minimum of the emergent quartic potential [(b2)]; (a3) Further away: anharmonic oscillations centered at the trap origin, mapping to a particle going around the local maximum of the double-well potential [(b3)]; (a4) Far enough: single-sided oscillations [inverse process of (a1)], mapping to a particle moving around a local minimum of the double-well potential [(b4)]. Insets in (b1)(b3) and (b4) show the initial position dependence of oscillation frequencies: analytical predictions for small amplitude oscillations [Eqs. \ref{['oscillationfrequencytypeI']}, \ref{['oscillationfrequencyharmonic']} and \ref{['oscillationfrequencytypeII']}](red dash-dotted lines) and large amplitude oscillations evaluated via $\omega = 2 \pi /\left|2\int^{X_{\rm end}}_{X_0} 1/V(X) \, dX\right|$ (black solid lines) with $X_{\text{end}}$ being the turning position, and numerical simulations of spin-1 GPEs (markers).
• Figure S1: Comparison of three component densities for a spin-1 BEC confined by a harmonic trap $U(x)=M \omega^2 x^2/2$ between the Thomas-Fermi approximation (solid lines) given by Eqs. \ref{['TFdensity1']} and \ref{['TFdensity2']} and numerical results obtained using the gradient flow method Bao2008a (markers). Here $a_{\rm ho}= \sqrt{\hbar/M\omega}$, $g_s/g_n = -1/2$, the total particle number $N = 10^{6}$, $\tilde{q} =-q/2 g_s n_{\text{peak}}= 0.1, 0.2, 0.3$, and $n_{\text{peak}}$ is the total density at the trap center $x=0$.
• Figure S2: Time evolution of the total number density $n$. The parameters are the same as those adopted in Fig.2.
• Figure S3: Time evolution of the component density $n_0$. The parameters are the same as those adopted in Fig.2.