Stability of matter-wave solitons in a density-dependent gauge theory

We consider the linear stability of chiral matter-wave solitons described by a density-dependent gauge theory. By studying the associated Bogoliubov–de Gennes equations both numerically and analytically, we find that the stability problem effectively reduces to that of the standard Gross-Pitaevskii equation, proving that the solitons are stable to linear perturbations. In addition, we formulate the stability problem in the framework of the Vakhitov-Kolokolov criterion and provide supplementary numerical simulations which illustrate the absence of instabilities when the soliton is initially perturbed. These results justify the production of chiral solitons in ultracold experiments and their potential application for practical transport dynamics in interferometry and atomtronics.

Figure 1

(a) Numerically obtained eigenvalue spectrum of the Bogoliubov–de Gennes equations with discrete states (green-dash) and continuous states (gray-shaded). The band edge of the continuous spectrum is highlighted in red, for both numerical (solid) and analytical (dots) results. The soliton parameters are taken as $g_{1\mathrm{D}}m\ell /{\hslash }^2=-1$, and $vm\ell /\hslash =1$. (b) Subset of (a) taken at $a_1/\hslash =1$. All energies are scaled to units of ${\epsilon }_0={\hslash }^2/m{\ell }^2$.

 

Figure 2

Bogoliubov–de Gennes eigenvectors ${\left(uv\right)}^{\mathrm{T}}$ for $g_{1\mathrm{D}}m\ell /{\hslash }^2=-1, vm\ell /\hslash =1$, and $a_1/\hslash =1$. Pictured are the degenerate bound states, (a–b) and (c–d), corresponding to the discrete spectrum and the first continuous state (e–f). All eigenvectors are scaled to units of ${\ell }^{-1/2}$ for both numerical (solid line) and analytical (dots) results.

Figure 3

Propagation of a chiral soliton, in the moving frame, whose initial envelope is perturbed due to a change in number of atoms. Shown are the predicted trajectories from the variational equations (white-solid) in comparison to the full numerics (shaded) and the unperturbed case (red-dash). The soliton parameters are $g_{1\mathrm{D}}m\ell /{\hslash }^2=-1, vm\ell /\hslash =1$, and $a_1/\hslash =1$, with the mismatch parameters $\mathrm{\Delta }\phi =+0.01$ (a), and $\mathrm{\Delta }\phi =-0.01$ (b). The condensate density is scaled to units of ${\ell }^{-1}$, with the units of time given by $t_0=m{\ell }^2/\hslash$.

Figure 4

Numerical convergence of the Bogoliuvbov de-Gennes eigenvalues in the continuum limit. (a) The continuous state band-edge $(q=0)$ eigenvalue compared to the chemical potential for increasing domain length, and (b) discrete eigenvalues for decreasing domain spacing. The soliton parameters are fixed at $g_{1\mathrm{D}}m\ell /{\hslash }^2=-1, vm\ell /\hslash =1$, and $a_1/\hslash =1$, with the gray vertical line of each plot corresponding to the spectrum shown in the respective inset with the eigenvalues color- and shape-coded. All energies are scaled to units of ${\epsilon }_0={\hslash }^2/m{\ell }^2$.