Dark spherical shell solitons in three-dimensional Bose-Einstein condensates: Existence, stability, and dynamics

In this work we study spherical shell dark soliton states in three-dimensional atomic Bose-Einstein condensates. Their symmetry is exploited in order to analyze their existence, as well as that of topologically charged variants of the structures, and, importantly, to identify their linear stability Bogoliubov–de Gennes spectrum. We compare our effective one-dimensional spherical and two-dimensional cylindrical computations with the full three-dimensional numerics. An important conclusion is that such spherical shell solitons can be stable sufficiently close to the linear limit of the isotropic condensates considered herein. We have also identified their instabilities leading to the emergence of vortex line and vortex ring cages. In addition, we generalize effective particle pictures of lower-dimensional dark solitons and ring dark solitons to the spherical shell solitons concerning their equilibrium radius and effective dynamics around it. In this case too, we favorably compare, qualitatively and quantitatively, the resulting predictions such as the shell equilibrium radius with full numerical solutions in three dimensions.

Figure 1

The two pairs of the top panels show the modulus $\left|\mathrm{\Psi }\right|$ (left panels) and argument $\kappa$ (right panels) of a dark soliton shell state at $\mu =12$; the first row illustrates the $z=0$ plane, while the second row shows the $x=0$ plane. The bottom panel shows two density isocontours for this state. The inner sphere corresponds to an isocontour at high density, while another isocontour at low density corresponds to the outer sphere (edge of the condensate) and the middle spheres (low densities on each side next to the dark soliton shell).

Figure 2

Top: A radial profile of the dark soliton shell state in the TF limit for $\mu =80$. Bottom: Number of particles as a function of chemical potential $\mu$, showing the continuation of states from the linear limit to the nonlinear regime.

Figure 3

Stability spectra for the dark soliton shell as a function of the chemical potential $\mu$. (a) and (b) Spectra computed using our effectively 1D method (in the radial variable, using a spherical harmonic decomposition in the transverse directions). Identical results are obtained when using the 2D variant of the method in cylindrical coordinates (and decomposing only the azimuthal direction in suitable Fourier modes). Both 1D and 2D numerics use a spacing of $h=0.01$. (c) Spectra computed using the full 3D numerics with spacing $h=0.2$. Orange lines depict the real part of $\lambda$ (i.e., unstable parts of the eigenvalues), while blue lines show the imaginary part of $\lambda$.

Figure 4

Comparison near the linear limit of the spectrum of the degenerate perturbation method and the numerical result of the effectively 1D method involving the spherical harmonic decomposition. Thick blue and orange lines depict the imaginary and real parts of the numerical eigenvalues, while the thinner cyan and green lines depict the theoretical results of the degenerate perturbation method. The eigenvalue depicted in green (decreasing thin line starting at $\lambda =1$ and crossing zero around $\mu =5.25$) is the one responsible for the destabilization of the DSS. The large red dots correspond to the two frequencies extracted from the oscillations of the perturbed DSS depicted in Fig. 5. The smaller frequency corresponds to the oscillatory mode of the DSS, while the larger one corresponds to the breathing mode of the background density.

Figure 5

The top set of panels shows 2D density cuts ($z=0$) from the full 3D numerical solution for $\mu =4.5$. The initial condition corresponds to the stationary DSS at its equilibrium radius. The second set of panels shows the evolution of a DSS shifted from its equilibrium position undergoing oscillatory motion over one oscillation period $T$. The white dashed circle denotes the starting location of the DSS at the beginning of the period. The third panel shows the radial profiles for the oscillating DSS over half a period. The bottom panel shows oscillations of the DSS radius vs time. Blue dots represent the radius extracted from full 3D numerics, and the solid red line is the best (least-squares) fit to a linear combination of oscillations with frequencies $w_1\approx 1.7581$ and $w_2=2.0671$ (see red dots in Fig. 4) corresponding, respectively, to the frequencies of the DSS oscillations and the breathing mode of the background cloud. For a movie depicting the stable oscillations of the DSS please see the Supplemental Material.

Figure 6

(a) The VL6 cage and (b) the VR6 cage that transiently emerge as a result of the instability of the DSS state. The states are captured from the numerical integration of the GPE (1) dynamics at $t=18$ and $t=7$ for $\mu =5.8$ and $8.2$, respectively. In particular, the VR6 cage seems to be robust and appears in the dynamics for a wide range of chemical potentials from the onset of the instability around $\mu =6.2$.

Figure 7

The equilibrium radius of the spherical shell dark soliton, scaled by $\sqrt{\mu }/\omega$, as a function of $\mu$ (solid red line). Note that the numerical values reach the asymptotic value of $\sqrt{4/5}$ (dashed black line), as per the second particle picture, when $\mu$ is large. The first particle picture slightly overestimates $r_c$ (dash-dotted blue line).