Deflation-based Identification of Nonlinear Excitations of the 3D Gross–Pitaevskii equation

04/22/2020 ∙ by N. Boullé, et al. ∙ University of Massachusetts Amherst University of Oxford Cal Poly 0

We present previously unknown solutions to the 3D Gross-Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures exhibit striking topological features, involving combinations of vortex rings and multiple, possibly bent vortex lines. Although unstable, many of them persist for long times in dynamical simulations. These solutions were identified by a state-of-the-art numerical technique called deflation, which is expected to be applicable to many problems from other areas of physics.



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Appendix A Supplemental Material

Solutions emanating from the 1st excited state. Deflation identified three solutions emanating from the first excited state at . Fig. 6(a) shows a Cartesian state with one cut along the -axis. Fig. 6(b) shows corresponding to a single vortex line with topological charge . The rotations of these solutions along the , , and axes such as the and Cartesian states are also obtained by deflation but are not reported.

[width=2.6cm]Figure/argument/arg_89.png (a)[width=2.6cm]Figure/special/sp_91.png (b)[width=2.6cm]Figure/vortices/vort_90.png (c)

Figure 6: Solutions emanating from the first excited state . Panels (a) and (b) show a dipole and single vortex line solution. The colours represent the argument of the solutions, ranging from to (blue and red represent a phase of and , respectively). Panel (c) corresponds to the density isosurfaces of the Chladni soliton at densities and .

Bifurcation diagram of the 2nd excited states. As discussed in the paper, the steady-state solutions to the nonlinear Schrödinger equation are identified by the deflation method at . The branches are then continued backward in down to the linear limit by a standard zero-order continuation method (Seydel, 2010, §4.4.2). A typical example of a relevant solution is shown in Fig. 6(c) and represents a so-called Chladni soliton, previously identified in cylindrical geometry in Mateo and Brand (2014, 2015). We present the bifurcation diagram of the solutions emanating from the second excited state in Fig. 7. Our diagnostic functional is the total number of atoms (or squared norm):


The inset panel in the top-left corner of Fig. 7 uses the atom number difference between the branches 2(a) and 2(a)* to illustrate a bifurcation in the diagram. The latter branch bifurcates from the former around and is shown in Fig. 8.


Figure 7: Bifurcation diagram of the solutions emanating from the 2nd excited state at . The labels indicate the solutions represented in the different panels of Figs. 1 and 2 of the paper. The main panel corresponds to the total number of atoms as a function of , while the top-left inset shows the atom number difference between the branch 2(a) and 2(a)*, coloured in red.

[width=2.6cm]Figure/argument/arg_82.png       [width=2.6cm]Figure/vortices/vort_82.png

Figure 8: Left: argument of the solution 2(a)* at . This branch bifurcates from branch 2(a) at (see Fig. 7). Right: density isosurfaces of the state at densities and . The parent branch 2(a) is plotted in Fig. 2(a) of the paper.

Bogoliubov-de Gennes stability analysis. The stability of a stationary solution to the NLS equation is determined by perturbing it with the following ansatz:


where is a small perturbation parameter, is the eigenfrequency, and the corresponding eigenvector. After substituting (8) into the time-dependent NLS equation:


we obtain the following complex eigenvalue problem


where is the eigenvalue and the matrix elements are given by equationparentequation


We decompose the eigenvector into real and imaginary components as , , and rewrite (10) as


where and . The eigenvalues of the matrix on the right-hand side of (12) are (with multiplicity two). Therefore, solving a real eigenvalue problem with the left-hand matrix of (12) yields the same eigenvalues and eigenvectors as the complex eigenvalue problem (10). We use a Krylov–Schur algorithm with a shift-and-invert spectral transformation Stewart (2002), implemented in the SLEPc library Hernandez et al. (2005), to solve the following eigenvalue problem:


where the matrices are real and is complex. This problem is discretized with the same piecewise cubic finite element method used to find multiple solutions with deflation. The spectra of the solutions emanating from the 2nd excited states, presented in Figs. 1 and 2 of the paper, are respectively displayed in the different panels of Figs. 10 and 9. It is interesting to observe that some of the states such as the doubly charged vortex line of Fig. 1(e) can be dynamically stable for large values of the chemical potential , while others such as the anisotropic ring of Fig. 1(f) can be stable for sufficiently low .

[width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/2_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/6_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/4_r.png

[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/2_c.png (a)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/6_c.png (b)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/4_c.png (c)

[width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/0_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/8_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/1_r.png

[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/0_c.png (d)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/8_c.png (e)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/1_c.png (f)

Figure 9: Spectra of the solutions presented in Fig. 1 of the paper (e.g., the panel (a) corresponds to the branch 1(a), illustrated in the panel (a) of Fig. 1). The real and imaginary parts of the corresponding eigenfrequencies are respectively depicted in the top and bottom panels.

Dynamics. The NLS equation (9) is integrated in time until using the following perturbed stationary solution as initial state:


where is a stationary solution discovered by deflation and is its most unstable eigendirection. The eigenvector is normalized so that


and the perturbation parameter is chosen to be . We use a modified Crank-Nicolson method in time Delfour et al. (1981), which preserves to machine precision the square of the norm (i.e., atom number) and the energy of the solutions


and piecewise cubic finite elements in space. Given the solution at time , is obtained by solving the following nonlinear PDE:


where is the time step used in the numerical simulations. The nonlinear problem is solved with Newton’s method. We present different snapshots of the two vortex rings solution (see Fig. 2(f) of the paper) in Fig. 11. We observe that the original two rings collapse around to form a single vortex line (see also the relevant movie in the Supplemental Material).

[width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/5_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/3_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/9_r.png

[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/5_c.png (a)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/3_c.png (b)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/9_c.png (c)

[width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/11_r.png [width=2.6cm,trim=50 100 50 40, clip]Figure/Stability/13_r.png

[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/11_c.png (d)[width=2.6cm,trim=50 75 50 40, clip]Figure/Stability/13_c.png (e)

Figure 10: Spectra of the solutions presented in Fig. 2 (a)-(e) of the paper.

[width=2.6cm]Figure/Dynamics/85/sol_85_0.png [width=2.6cm]Figure/Dynamics/85/sol_85_1.png [width=2.6cm]Figure/Dynamics/85/sol_85_2.png

[width=2.6cm]Figure/Dynamics/85/sol_85_3.png [width=2.6cm]Figure/Dynamics/85/sol_85_4.png [width=2.6cm]Figure/Dynamics/85/sol_85_5.png

Figure 11: Snapshots of the two vortex rings state of Fig. 2(f) of the paper, obtained by solving the time-dependent NLS equation with the modified Crank-Nicolson time-stepping scheme.