(Solved): 3. (Dipole trapping) For dipole trapping sodium atoms, a Neodymium:YAG laser with a wavelength = ...

3. (Dipole trapping) For dipole trapping sodium atoms, a Neodymium:YAG laser with a wavelength $$?=$$ $$1064\mathrm{nm}$$ and a Gaussian intensity profile $$I(r)=\frac{2P}{?w_0^2}{e}^{?2r^2/w_0^2}$$ is used, where $$P$$ is the laser power and $$w_0=5?\mathrm{m}$$ the waist of the laser beam. The wavelength of the $$D_2$$ line of $${}^{23}\mathrm{Na}$$ is $${?}_0=598.2\mathrm{nm}$$ and the corresponding spontaneous emission rate $$?/2?=9.8\mathrm{MHz}$$ (a) Use the expression for the dipole force at large detuning (slides section 7) $${\mathbf{F}}_{\mathrm{dip}}=??U_{\mathrm{dip}}??\frac{??\left({?}_0^2\right)}{4?}$$ to determine the dipole trapping potential $$U_{\text{dip }}(r)$$ in the transverse direction as a function of the distance $$r$$ from the centre of the laser beam. (b) The trap depth is given by the potential $$U_{\text{dip }}(0)$$ at the centre. Derive an expression for the minimum laser power $$P$$ required to trap sodium atoms cooled to the Doppler temperature $$k_B{T}_D=??/2$$. (Hint: an atomic sample at a temperature $$T$$ can be trapped when the thermal energy $$k_{B}T$$ is lower than the trap depth.) (c) What laser power is required to trap Doppler-cooled sodium atoms which have a saturation intensity $$I_s=6.26\mathrm{mW}/{\mathrm{cm}}^2$$ ?