Numerical Scheme Used

quTARANG uses a pseudo specral scheme, TSSP (Time Splitting Spectral method) [BJM03] to solve the dynamics of the GPE.

The main advantage of using the TSSP scheme is that it is unconditionally stable scheme. The dimensionless for of GPE is given by

(1)\[\iota \frac{\partial\psi(\vec{r},t)}{\partial t}= -\frac{1}{2}\nabla^2\psi(\vec{r},t) + V(\vec{r},t)\psi(\vec{r},t) + g|\psi(\vec{r},t)|^2\psi(\vec{r},t)\]

For time interval \(\Delta t\) between \(t=t_n\) and \(t=t_{n+1}\), one can solve above equation numerically by splitting it into two steps. The first step is

\[\iota \partial_t\psi = -\frac{1}{2}\nabla^2\psi\]

The second step is

\[\iota \partial_t\psi = V\psi + g|\psi|^2\psi\]

By taking a fourier trasnform of (1) , one can convert the PDE into a list of PDEs which can be solved exactly in Fourier space and the wavefunction in real space can be retrieved by taking an inverse fourier transform. For \(t \ \epsilon \ [t_n,t_{n+1}]\), \(|\psi|^2\) remains almost constant therefore, eq(ref{eq:sstep2}), now just an ODE, can be solved exactly in \(t_n\) and \(t_{n+1}\)

Between \(t_n\) and \(t_{n+1}\), the two steps are connected through strang splitting:

\[\begin{split}\psi_n^{(1)} = \psi_n e^{-\iota(V + g|\psi_n|^2)\frac{\Delta t}{2}} \\ \hat{\psi}_n^{(2)} = \hat{\psi}_n^{(1)}e^{-\iota\frac{\vec{k}^2}{2}\Delta t} \\ \psi_{n+1} = \psi_n^{(2)} e^{-\iota(V + g|\psi_n^{(2)}|^2)\frac{\Delta t}{2}}\end{split}\]

where, \(\hat{\psi}^{(1)}\) is Fourier transform of \(\psi^{(1)}\) and \(\psi_n^{(2)}\) is inverse Fourier transform of \(\hat{\psi}_n^{(2)}\).

One can calculate the ground state for a given system by using imaginary time proppogation method wherein all the eigenstates except the groundstate of the system decay with time. In imaginary time propagation method, \(t\) is replaced by -\(\iota t\) and then evolved.

BJM03

Weizhu Bao, Dieter Jaksch, and Peter A Markowich. Numerical solution of the gross–pitaevskii equation for bose–einstein condensation. Journal of Computational Physics, 187(1):318–342, 2003.