Fourier methods for the perturbed harmonic oscillator in linear and nonlinear Schrödinger equations

We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since the system can be split into the kinetic and remaining part, and each part can be solved efficiently using fast Fourier transforms. Splitting the system into the quantum harmonic-oscillator problem and the remaining part allows us to get higher accuracies in many cases, but it requires us to change between Hermite basis functions and the coordinate space, and this is not efficient for time-dependent frequencies or strong nonlinearities. We show how to build methods that combine the advantages of using Fourier methods while solving the time-dependent harmonic oscillator exactly (or with a high accuracy by using a Magnus integrator and an appropriate decomposition).

Figure 1

Error in logarithmic scale for the integration of the ground state of the harmonic potential using the split (2.28) for $T\in [-\pi ,\pi ]$ (integration forward and backward in time). The left and right panels show the 2-norm error and a zoom about $T=\pi$, respectively.

Figure 2

Exact evolution at $t=T$ (solid red line) from the initial conditions given by $\psi (x,0)=\rho e^{-(x-1){}^2/2}$ (dashed blue line). The number of grid points is given by $N$.

Figure 3

Error vs the number of basis changes (BC), i.e., Fourier or Hermite transforms, in logarithmic scale for different splittings for the leapfrog method (2.4).

Figure 4

Comparison of second-order (upper panel) and fourth-order (lower panel) methods for the different splittings and decompositions discussed in the text and $\sigma =0.01$. The curves for ${\mathrm{LF}}_H$ and ${\mathrm{LF}}_{H}30$ overlap in the lower panel; the dashed curves are identical with the solid ones that correspond to the same composition method $X$, i.e., $X_H$ overlaps with $X_{H}50$.

Figure 5

Same as Fig. 4 for $\sigma =1$. In the upper panel, ${\mathrm{LF}}_H\hspace{0.5em}$ and $(8,2){}_H$ coincide with ${\mathrm{LF}}_{H}30$ and $(8,2){}_{H}30$, respectively.

Figure 6

Same as Fig. 4 for $\sigma =100$. F splittings (blue dashed) overlap the corresponding (solid red) Fourier-Hermite curves.

Figure 7

Comparison of Fourier and Fourier-Hermite splittings for two fourth order methods. The (red) dashed line indicates the two-exponential approximation (2.14), the (green) solid line corresponds to the sixth-order Magnus approximation presented in the text (4.4). The inset shows the evolution of the harmonic trap frequency $\omega (t){}^2$ and the parameters used for the Hamiltonian are $w=1/2,A=4,\epsilon =0.1,{\varepsilon }_Q=0.01$.

Figure 8

Compare Fig. 7. The parameters used are $w_0=4$, $A=0.25$, and $B=2$ with a small anharmonicity ${\varepsilon }_Q=0.01$.