Variational determination of approximate bright matter-wave soliton solutions in anisotropic traps

We consider the ground state of an attractively interacting atomic Bose-Einstein condensate in a prolate, cylindrically symmetric harmonic trap. If a true quasi-one-dimensional limit is realized, then for sufficiently weak axial trapping this ground state takes the form of a bright soliton solution of the nonlinear Schrödinger equation. Using analytic variational and highly accurate numerical solutions of the Gross-Pitaevskii equation, we systematically and quantitatively assess how solitonlike this ground state is, over a wide range of trap and interaction strengths. Our analysis reveals that the regime in which the ground state is highly solitonlike is significantly restricted and occurs only for experimentally challenging trap anisotropies. This result and our broader identification of regimes in which the ground state is well approximated by our simple analytic variational solution are relevant to a range of potential experiments involving attractively interacting Bose-Einstein condensates.

Figure 1

Comparison of quasi-1D variational and numerical solutions: (a) Energy-minimizing axial lengths ${\ell }_{\mathrm{G}}$ (Gaussian Ansatz, squares) and ${\ell }_{\mathrm{S}}$ (soliton Ansatz, circles) for the quasi-1D GPE. (b) Minimum variational energy compared with the numerically calculated ground-state energy $E_{1\mathrm{D}}$ (black line) for each Ansatz: for low $\gamma$ we show $H_{1\mathrm{D}}$ (solid symbols), which tends to $-1/24$ as $\gamma \to 0$; for high $\gamma$ we show $H_{1\mathrm{D}}^{\prime }=H_{1\mathrm{D}}/\gamma$ (hollow symbols), which tends to $1/2$ as $\gamma \to \infty$ ($H_{1\mathrm{D}}^{\prime }$ is equal to the energy expressed in the “harmonic units” $\hslash =m={\omega }_x=1$). (c) Relative error in the variational energy, $\Delta =(H_{1\mathrm{D}}-E_{1\mathrm{D}})/E_{1\mathrm{D}}$. (d) Normalized maximum deformation of the best-fitting Ansatz wave function ${\psi }_{Ansatz}$ with respect to the numerical ground state ${\psi }_0$, $\Delta \psi =\max \left(\right|{\psi }_{Ansatz}-{\psi }_0\left|\right)/\max \left({\psi }_0\right)$, expressed as a percentage. For clarity in (a),(b) [(c)], every 16th [20th] datum is marked by a symbol.

Figure 2

Energy-minimizing variational parameters for the 3D GPE using a Gaussian Ansatz: (a) axial length ${\ell }_{\mathrm{G}}$ as a function of the radial length $k_{\mathrm{G}}^{-1}$ and the parameter $\gamma$ [Eq. (23)]. Lines show the simultaneous solutions of Eqs. (22) and (24) for the axial length ${\ell }_{\mathrm{G}}$ and radial length $k_{\mathrm{G}}^{-1}$, for different anisotropies $\kappa$ and values of $\gamma$. Projections of these solutions on the $\gamma$-${\ell }_{\mathrm{G}}$ plane are also shown; here the black line indicates the quasi-1D result [from Fig. 1a]. (b)–(d) Illustration of the intersections of Eqs. (22) [lines with vertical asymptote ${\ell }_{\mathrm{G}}=1/{\left(2\pi \gamma \right)}^{1/2}\kappa$ shown with fine dashes] and (24) for various $\kappa$: the higher-${\ell }_{\mathrm{G}}$ intersection, which corresponds to a physical solution for the axial length ${\ell }_{\mathrm{G}}$ and radial length $k_{\mathrm{G}}^{-1}$, can be found using a “staircase” method starting from $k_{\mathrm{G}}=1$. The numerical solutions obtained in this way, and shown by points in (a), are shown by crosses in (b)–(d). The lowest values of $\gamma$ plotted in (b)–(d) are the lowest for which a self-consistent Gaussian Ansatz solution is found.

Figure 3

Energy-minimizing variational parameters for the 3D GPE using a soliton Ansatz: (a) axial length ${\ell }_{\mathrm{S}}$ as a function of the radial length $k_{\mathrm{S}}^{-1}$ and the parameter $\gamma$ [Eq. (30)]. Lines show the simultaneous solutions of Eqs. (33) and (34) for the axial length ${\ell }_{\mathrm{S}}$ and radial length $k_{\mathrm{S}}^{-1}$, for different anisotropies $\kappa$ and values of $\gamma$. Projections of these solutions on the $\gamma$-${\ell }_{\mathrm{S}}$ plane are also shown; here the black line indicates the quasi-1D result [from Fig. 1a]. (b)–(d) Illustration of the intersections of Eqs. (33) [lines with vertical asymptote ${\ell }_{\mathrm{S}}=(\pi /3)/{\left(2\pi \gamma \right)}^{1/2}\kappa$ shown with fine dashes] and (34) for various $\kappa$: the higher-${\ell }_{\mathrm{S}}$ intersection, which corresponds to a physical solution for the axial length ${\ell }_{\mathrm{S}}$ and radial length $k_{\mathrm{S}}^{-1}$, can be found using a staircase method starting from $k_{\mathrm{S}}=1$. The numerical solutions obtained in this way, and shown by points in (a), are shown by crosses in (b)–(d). The lowest values of $\gamma$ plotted in (b)–(d) are the lowest for which a self-consistent soliton Ansatz solution is found.

Figure 4

Comparison of 3D variational and numerical solutions in a waveguide configuration (${\omega }_x=0$): (a) Energy-minimizing axial length ${\ell }_{\mathrm{S}}$ and radial length $k_{\mathrm{S}}^{-1}$ for the soliton Ansatz. Solutions, given by Eq. (41), exist for all $\Gamma =\kappa \gamma >3^{1/2}/4$. (b) Relative error in the minimum variational energy of the soliton Ansatz, $\Delta =(H_{3\mathrm{D}}-E_{3\mathrm{D}})/E_{3\mathrm{D}}$, where $E_{3\mathrm{D}}$ is the numerically determined ground-state energy.

Figure 5

Comparison of 3D variational and numerical solutions: (a) Energy-minimizing axial lengths ${\ell }_{\mathrm{G}}$ (Gaussian Ansatz, solid symbols) and ${\ell }_{\mathrm{S}}$ (soliton Ansatz, hollow symbols). (b) Scaled variational energies $H_{3\mathrm{D}}^{\prime }=\kappa H_{3\mathrm{D}}/\gamma (\kappa +1/2)$ [a similarly scaled ground-state energy $E_{3\mathrm{D}}^{\prime }=\kappa E_{3\mathrm{D}}/\gamma (\kappa +1/2)$ tends to 1 in the limit $\gamma \to \infty$ for all anisotropies $\kappa$] compared with the numerically calculated ground-state energies $E_{3\mathrm{D}}$ (black dots). (c),(d) Normalized relative error in the variational energy $\Delta =(H_{3\mathrm{D}}-E_{3\mathrm{D}})/E_{3\mathrm{D}}$ for the Gaussian (c) and soliton (d) Ansätze. For clarity every fourth datum is marked by a symbol in (a)–(d).