Isotropy of three-dimensional quantum lattice Boltzmann schemes

Numerical simulations with previous formulations of the quantum lattice Boltzmann (QLB) scheme in three spatial dimensions showed significant lack of isotropy. In two or more spatial dimensions the QLB approach relies upon operator splitting to decompose the time evolution into a sequence of applications of the one-dimensional QLB scheme along coordinate axes. Each application must be accompanied by a rotation of the wave function into a basis of chiral eigenstates aligned along the relevant axis. The previously observed lack of isotropy was due to an inconsistency in the application of these rotations. Once this inconsistency is removed, the QLB scheme is shown to exhibit isotropic behavior to within a numerical error that scales approximately linearly with the lattice spacing. This establishes the viability of the QLB approach in two and three spatial dimensions.

Figure 1

Dispersion of a wave packet with initial spread ${\Delta }_0=14$, particle mass $m=0.35,$ and no potential in a cube with side length $\ell =100$ and a computational grid with ${128}^3$ points. The three computed spreads ${\Delta }_x$, ${\Delta }_y$, ${\Delta }_z$ collapse onto a single curve, which differs by a superimposed high frequency oscillation from the Schrödinger solution (28) (smooth curve).

Figure 2

Evolution of the average difference $\mathrm{\Delta ̃}$ between the three spreads for a wave packet with initial spread ${\Delta }_0=14$, mass $m=0.35,$ and no potential in a cube with side length $\ell =100$. The cube was discretized using ${128}^3$, ${256}^3$, ${512}^3$, and ${1024}^3$ points.

Figure 3

Evolution of $n^2\mathrm{\Delta ̃}$ for a wave packet with initial spread ${\Delta }_0=14$, mass $m=0.35,$ and no potential in a cube with side length $\ell =100$. The scaled mean anisotropies $n^2\mathrm{\Delta ̃}$ collapse onto a single curve for four different computations using $n^3$ points with $n=128,256,512,1024$.

Figure 4

Evolution of the computed spreads ${\Delta }_x$, ${\Delta }_y$, ${\Delta }_z$ for a wave packet with initial spread ${\Delta }_0=14$, particle mass $m=0.1$, potential ${\omega }_0=1/(2m{\Delta }_0^2)$, in a cube with side length $\ell =200$ discretized with ${128}^3$ points.

Figure 5

Evolution of $\mathrm{\Delta ̃}$ for a wave packet with initial spread ${\Delta }_0=14$, mass $m=0.1$, potential ${\omega }_0=1/(2m{\Delta }_0^2)$, in a cube with side length $\ell =200$. The cube was discretized using ${128}^3$, ${256}^3$, ${512}^3$, and ${1024}^3$ points.

Figure 6

Evolution of $n\mathrm{\Delta ̃}$ for a wave packet with initial spread ${\Delta }_0=14$, mass $m=0.1$, potential ${\omega }_0=1/(2m{\Delta }_0^2)$, in a cube with side length $\ell =200$. The cube was discretized using ${128}^3$, ${256}^3$, ${512}^3$, and ${1024}^3$ points. The scaled anistropies $n\mathrm{\Delta ̃}$ collapse into a single curve for $t\lesssim 60$.

Figure 7

Plot of $\sigma (r)$ at $t=100$ for $n=128,256,512,1024$ in a cube of side $\ell =200$. The feature at $r\approx 100$ is due to coupling with negative energy states that have passed through the periodic boundaries. The ${128}^3$ grid contains too few points to compute $\sigma$ for $r<3$.

Figure 8

Plot of $n\sigma (r)$ at $t=100$ for $n=128,256,512,1024$ confirming first-order convergence to spherical symmetry under grid refinement in the region $r\lesssim 50$. The nonconverging feature centered at $r\approx 100$ is due to coupling with negative energy states that have passed through the periodic boundaries.

Figure 9

Dispersion of a wave packet with initial spread ${\Delta }_0=14$, particle mass $m=0.1$, potential ${\omega }_0=1/(2m{\Delta }_0^2)$, in a cube with side length $\ell =100$ and a computational grid with ${128}^3$ points. Computed spreads ${\Delta }_x$, ${\Delta }_y$, ${\Delta }_z$.

Figure 10

Evolution of $n\mathrm{\Delta ̃}$ for a wave packet with initial spread ${\Delta }_0=14$, mass $m=0.1$, potential ${\omega }_0=1/(2m{\Delta }_0^2)$, in a cube with side length $\ell =100$. The computational grids contained $n^3$ points for $n=128,256,512,1024$.

Figure 11

Contours of $|{\varphi }_1^{+}(x,y,0)|$ on the plane $z=0$ from a ${512}^3$ simulation in a cube of side length $\ell =100$ (values increase from outer circles to inner circles). The contours are noticeably distorted away from circular symmetry by the boundaries of the cube.