Optical Lattice Calculations

Trap Depth

We know for a single beam ODT the trap depth is given by

$U_0=\frac{\alpha }{c{ϵ}_0}\frac{P}{\pi r^2}$

As seen in Optical Dipole Trap. Now because we retroreflect the beam to make the lattice the electric field becomes twice as large compared to a single beam. This looks like 4 times the intensity or 4 times the power. Thus the trap depth for a lattice is

$U_0=4\frac{\alpha }{c{ϵ}_0}\frac{P}{\pi r^2}$

Lets do a calculation for the trap depth for our beam. If we assume a $30\mu m$ $1/e^2$ waist and 30W of power at the chamber we find

mass = 87*1.66e-27 # mass of Rb-87 in kg
polarizability = 711*4*epsilon_0*pi*(5.29177210544e-11)**3 # polarizability of Rb in C*m^2/V
waist = 100e-6 # beam waist in meters
power = 18 #Watts
 
 
trap_depth = 4*(polarizability/(c*epsilon_0))*power/(pi*waist**2)
 
print("Trap Depth (in uKelvin):", trap_depth/k/1e-6)

Outputs

Trap Depth (in uKelvin): 1191.718953560015

Trap Frequencies

From here we can calculate the trap frequency which in the radial direction is the same as a single beam ODT

${\omega }_r=\sqrt{\frac{4U_0}{\pi {\omega }_0^2}}$

So using the trap depth lets calculate the radial trap frequency

trap_frequency_r=sqrt(4*trap_depth/(mass*waist**2))/(2*pi)
 
print("Radial Trap Frequency (in Hz):", trap_frequency_r)

Outputs

Radial Trap Frequency (in Hz): 12087.617649467995