Unified way for computing dynamics of Bose–Einstein condensates and degenerate Fermi gases

ABSTRACT

In this work we present a very simple and efficient numerical scheme which can be applied to study the dynamics of bosonic systems like, for instance, spinor Bose–Einstein condensates (BEC) with non-local interactions but equally well works for Fermi gases. The method we use is a modification of well known Split Operator Method (SOM). We carefully examine this algorithm in the case of F=1 spinor BEC without and with dipolar interactions and for strongly interacting two-component Fermi gas. Our extension of the SOM method has many advantages: it is fast, stable, and keeps constant all the physical constraints (constants of motion) at high level.

KEYWORDS:

• Split operator method
• SOM
• gross-Pitaevskii equation
• GP
• spinor condensate
• dipolar condensate
• Bose–Einstein condensate
• BEC
• degenerate Fermi gas
• nonlinear partial integro-differential set of equations
• PDE
• NoPDE
• PIDE

2010 AMS SUBJECT CLASSIFICATIONS:

• 35Q41
• 35Q55
• 65Z05
• 81-08
• 81V45

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The Baker–Hausdorff theorem reads ${\mathrm{e}}^{A+B}={\mathrm{e}}^A{\mathrm{e}}^B{\mathrm{e}}^{-\left(1/2\right)[A,B]}\text{ if }[A,[A,B\left]\right]=0\text{ and }[B,[A,B\left]\right]=0.$ In our case the definitions of a A and B are $A=i\left(\mathrm{\Delta }t/2\right){\mathrm{\nabla }}^2,B=-i\mathrm{\Delta }t({\displaystyle \frac{1}{2}{r}^2+g|\psi \left(t\right){|}^2+V_{\mathrm{dip}}\left(t\right))}$ and are $\mathrm{\Delta }t$-dependent, and ${\mathrm{e}}^{-\left(1/2\right)[A,B]}\propto I$ (because it's $\propto \mathrm{\Delta }{t}^2$), so one can write ${\mathrm{e}}^{A+B}\approx {\mathrm{e}}^A{\mathrm{e}}^B.$

2 In fact we copied the original idea of the SOM: to calculate the evolution due to the kinetic part of the Hamiltonian one has to switch to the basis, in which the Laplace operator is diagonal, that is, which is the momentum space. Because in the momentum space the differentiation becomes a simple multiplication by$-i\mathbf{k}$, the evolution due to the kinetic term (which includes the second derivative in position space) becomes very easy to compute - instead of the second derivative one has to multiply Fourier components by $\exp [-k^2\mathrm{\Delta }t/2]$. In the spinor version we have to do a similar trick in the position space: to facilitate calculations of the evolution according to the potential energy, first we go to the basis in which the position dependent part of the Hamiltonian $V(\mathbf{r},t)$ is diagonal, and only then we use the plane wave basis for evolution due to a kinetic part.

3 By spin dynamics we understand the situation with non-negligible transfer of atoms from one component to the other. From Equation (9(9) $\begin{array}{rl}& i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\begin{array}{c}{\psi }_1\\ {\psi }_0\\ {\psi }_{-1}\end{array}\right)\\ & =\left(\begin{array}{ccc}{\mathcal{H}}_0+{\mathcal{H}}_{c11}+{\mathcal{H}}_{d11}& {\mathcal{H}}_{c10}+{\mathcal{H}}_{d10}& 0\\ {\mathcal{H}}_{c10}^{\ast }+{\mathcal{H}}_{d10}^{\ast }& {\mathcal{H}}_0+{\mathcal{H}}_{c00}& {\mathcal{H}}_{c0-1}+{\mathcal{H}}_{d10}\\ 0& {\mathcal{H}}_{c0-1}^{\ast }+{\mathcal{H}}_{d10}^{\ast }& {\mathcal{H}}_0+{\mathcal{H}}_{c-1-1}-{\mathcal{H}}_{d11}\end{array}\right)\left(\begin{array}{c}{\psi }_1\\ {\psi }_0\\ {\psi }_{-1}\end{array}\right).\end{array}$(9) ) it follows that ${\psi }_1$ satisfies the equation $i\left(\mathrm{\partial }{\psi }_1/\mathrm{\partial }t\right)=({\mathcal{H}}_0+{\mathcal{H}}_{c11}){\psi }_1+{\mathcal{H}}_{c10}{\psi }_0=({\mathcal{H}}_0+(c_0+c_2\left)\right(|{\psi }_1{|}^2|{\psi }_0{|}^2)+(c_0-c_2\left)|{\psi }_{-1}{|}^2\right){\psi }_1+c_2{\psi }_{-1}^{\star }{\psi }_0{\psi }_0$ and if ${\psi }_1\left(0\right)={\psi }_{-1}\left(0\right)=0$ the right-hand side vanishes, that is, the ${\psi }_1$ field is constant what corresponds to a ‘spin frozen’ situation. To account for quantum fluctuations we need some small initial seeds in the fields ${\psi }_1$ and ${\psi }_{-1}$.

Additional information

Funding

The work was supported by the (Polish) National Science Center [Grant No. DEC-2012/04/A/ST2/00090]. Part of the results were obtained using computers at the Computer Center of University of Bialystok.