Optical Lattice

General Idea

By creating a standing wave of light you use the same effect as that outlined in Optical Dipole Trap to hold the atoms at the high intensity part of the wave. You can do this in multiple dimensions and these will be referred to a 1D, 2D or 3D lattice. Here is a photo

Formulas

Trap Frequencies

To derive the axial frequency one considers the potential. The potential is given by

$U_0{\sin }^2(kx)$

Where $U_0$ is $4U_{0,singlebeam}$ because we have twice the electric field, and can be found in Optical Dipole Trap. To find the trap frequency we match this with a harmonic oscillator by Taylor expanding around a minima. So consider

$U_0\sin (kx)\approx U_0(1-\frac{(kx{)}^2}{2}{)}^2\approx U_0(1-(kx{)}^2)$

Where we neglect higher order terms because we only care about quadratic terms. So now we know

$\frac{1}{2}m{\omega }^2{x}^2=U_0{k}^2{x}^2$

Thus

${\omega }_{axial}=k\sqrt{\frac{2U_0}{m}}$

The radial trap frequencies are the same