Numerical method for the projected Gross-Pitaevskii equation in an infinite rotating two-dimensional Bose gas

We present a method for evolving the projected Gross-Pitaevskii equation in an infinite rotating Bose-Einstein condensate, the ground state of which is a vortex lattice. We use quasiperiodic boundary conditions to investigate the behavior of the bulk superfluid in this system, in the absence of boundaries and edge effects. We also give the Landau gauge expression for the phase of a BEC subjected to these boundary conditions. Our spectral representation uses the eigenfunctions of the one-body Hamiltonian as basis functions. Since there is no known exact quadrature rule for these basis functions we approximately implement the projection associated with the energy cutoff, but we show that by choosing a suitably fine spatial grid the resulting error can be made negligible. We show how the convergence of this model is affected by simulation parameters such as the size of the spatial grid and the number of Landau levels. Adding dissipation, we use our method to find the lattice ground state for $N$ vortices. We can then perturb the ground-state, to investigate the melting of the lattice.

Figure 1

A sketch of the system: (a) A large, oblate, harmonically trapped (${\omega }_x={\omega }_y={\omega }_{\perp }\ll {\omega }_z={\omega }_{\parallel }$) condensate rotating with angular frequency $\mathrm{\Omega }$. (b) In the centrifugal limit ($\mathrm{\Omega }\to {\omega }_{\perp }$) a small cell in the bulk of the now-infinite condensate can be approximated using the Landau gauge with quasiperiodic (twisted) boundary conditions. The height of the surface represents the density of the wave function, while the color represents the phase of the superfluid.

Figure 2

The projection error, $\delta$ as a function of $Q$ for varying values of $M$. We have set $a=b=2^6$ to be the cell size, fixed $p_{\text{max}}=10$, and set $N=4$. The dotted lines are added as a guide to the eye.

Figure 3

The truncation error $\delta$, for varying $p_{\text{max}}$ with fixed $M$ and $Q$. For $M\in \{2^6,2^8,2^{10}\}$ we used $Q=2^7$ grid-points, while for $M>2^{10}$, it is necessary to use $Q=2^8$ grid-points to achieve a meaningful result. Note that $a=b=2^6$ and $N=4$ in this analysis. The dotted lines are added as a guide to the eye.

Figure 4

Example configuration of vortices using the method described in Sec. 4. The color bar indicates the phase of the superfluid. (a) A single, positively charged, vortex is placed at the center of the cell. (b) Two positively charged vortices are placed at $(a/2,3b/4)$ and $(a/2,b/4)$. (c) Three positively charged and one negatively charged vortices create a dipole pair in the cell.

Figure 5

Evolution error for the quantities ${\mathcal{N}}_R$, row (a); ${\mathcal{N}}_C$, row (b); and $\mathcal{E}$, row (c). Column (i): Varying $M$ for $Q=2^6$, blue crosses; $Q=2^7$, red circles; $Q=2^8$, yellow squares; $Q=2^9$, purple asterisks. Column (ii): Varying $Q$ for $M=2^6$, blue crosses; $M=2^7$, red circles; $M=2^8$, yellow squares. For columns (i) and (ii), $\epsilon ={10}^{-10}$. Column (iii): Varying $\epsilon$ for $M=2^6$ and $Q=2^8$. In all cases, $a=b=2^6$.

Figure 6

The hexagonal lattice ground states. (a) a system with $N=6$ vortices, (b) a system with $N=8$ vortices, and (c) a system with $N=18$ vortices. The primitive vectors of a hexagonal lattice, ${\mathit{L}}_1$ and ${\mathit{L}}_2$, are added as a guide to the eye. In each case, $a=32\sqrt{3}$ and $b=32$.

Figure 7

Rows (a)(i)–(a)(v): Instantaneous density profile at $t=5000$. Rows (b)(i)–(b)(v): Instantaneous phase profile at $t=5000$. Rows (c)(i)–(c)(v): Time and ensemble averaged density profile, $\overline{\rho }$. Rows (d)(i)–(d)(v): Time and ensemble averaged phase profile, $\overline{\theta }$. The initial configurations are given by: Columns (a)(i)–(d)(i): $\eta =0.982$, columns (a)(ii)–(d)(ii): $\eta =0.986$, columns (a)(iii)–(d)(iii): $\eta =0.990$, columns (a)(iv)–(d)(iv): $\eta =0.994$, and columns (a)(v)–(d)(v): $\eta =0.998$. See the Supplemental Material [80], which contains movies of the time evolution.

Figure 8

The time and ensemble averaged occupation of $c_{n,k}$ as a function of Landau levels for $\eta =0.982$, solid red line, $\eta =0.990$, dashed blue line, and $\eta =0.998$, dot-dashed black line. The equipartition of energy, $1/E$, green dashed line, is added as a guide to the eye.