Theoretical description of two ultracold atoms in finite three-dimensional optical lattices using realistic interatomic interaction potentials

A theoretical approach is described for an exact numerical treatment of a pair of ultracold atoms interacting via a central potential and that are trapped in a finite three-dimensional optical lattice. The coupling of center-of-mass and relative-motion coordinates is treated using an exact diagonalization (configuration-interaction) approach. The orthorhombic symmetry of an optical lattice with three different but orthogonal lattice vectors is explicitly considered as is the fermionic or bosonic symmetry in the case of indistinguishable particles.

Figure 1

Two particles 1 and 2 in the absolute and c.m.–relative-motion Cartesian coordinate systems.

Figure 2

(a) The ${\sin }^2\left(x\right)$ function (black solid line) together with the 2nd- (blue dashed line), 3rd- (red dashed line), 5th- (blue solid line), 7th- (red solid line) and 11th-order (green solid line) expansion of the Taylor series. (b) The ${\cos }^2\left(x\right)$ function (black line) together with the 6th- (green line), 7th- (red line) and 8th-order (blue line) expansion of the Taylor series.

Figure 3

Symmetry elements of the two particles interacting by a central potential in an orthorhombic ${\sin }^2$-like trap. The list is complete with the identity element $E$ added.

Figure 4

Energy spectrum of two bosonic atoms confined in a harmonic trap of anisotropy ${\omega }_x/{\omega }_z=10$. The numerical calculation (blue dots) is compared to the analytical prediction of the pseudopotential approximation [65] (red lines).

Figure 5

Energy of the first five trap states for different values of $l_{\max }$ in the strongly repulsive regime and for $d_x/a_{\mathrm{sc}}=1.46$. The deeper-lying bound state is not shown here, because it is already converged for $l_{\max }=5$. (To guide the eye, the discrete points of the numerical calculation are connected by continuous lines.)