# Forces in the Interaction of Light with Matter
## 3 Optical Forces in the Laser Cooling and Trapping of Atoms
### Scattering Force, Doppler Cooling and Optical Molasses  
The scattering force (or _dissipative force_) is generated by the atom scattering input photons, repeated cycles giving rise to a huge number absorption-spontaneous emission processes--typically about \(10^{8}\) such cycles per second. As the resonant scattering removes photons from the input beam, conservation of the linear momentum in the beam--atom system results in a loss of momentum for the beam and a corresponding gain for the atom in the direction of beam propagation. For example, at typical levels of laser intensity and vapour density, one might expect a sodium atom, excited on its \(3^{2}S_{(1/2)}-3^{2}P_{(3/2)}\) transition, to experience an acceleration \(a=N\hbar k/M\approx 10^{5}g\), where \(g\) is the acceleration due to gravity. The general formula for the scattering force is:  
\[\mathbf{F}_{sc}=\hbar\Gamma\Omega^{2}(\mathbf{R})\frac{\
abla\Theta(\mathbf{R} )/4}{\Delta^{2}(\mathbf{R},\mathbf{V})+\Omega^{2}(\mathbf{R})/2+\Gamma^{2}/4}, \tag{20}\]  
where \(\Omega(\mathbf{R})\) is the Rabi frequency, \(\Delta(\mathbf{R},\mathbf{V})\) is the detuning (which, in general, may depend both on the atom's position and velocity, due to a Doppler shift), and \(\Theta(\mathbf{R})\) is the phase of the electric field of the input beam [93]. When the laser input has the form of a Gaussian beam, then \(\
abla\Theta(\mathbf{R})\approx\mathbf{k}\), and the scattering force is mainly along the beam propagation direction. However, if the laser beam is in the form of a vortex of topological charge (winding number) \(\ell\), then \(\
abla\Theta(\mathbf{R})\approx\mathbf{k}+\ell\hat{\phi}/\rho\); in this case, the scattering force is also responsible for the generation of a torque acting on the atom. Here, when the laser intensity is relatively strong, the scattering force saturates to the expression \(\mathbf{F}_{sc,max}=\hbar\Gamma k\hat{\mathbf{k}}/2\), while the torque reduces to the expression \(\mathbf{N}_{sc,max}=\hbar\Gamma\ell\hat{\mathbf{k}}/2\). If the atom moves with a velocity \(\mathbf{V}\) and is irradiated by a laser beam of angular frequency \(\omega_{L}\), it experiences a Doppler-shifted frequency \(\omega_{L}-\
abla\Theta\cdot\mathbf{V}\), irradiation by a Gaussian beam thus results in a Doppler-shifted frequency \(\omega_{L}-\mathbf{k}\cdot\mathbf{V}\). However, in the case of irradiation by an optical vortex, the shifted frequency is approximately equal to \(\omega_{L}-\mathbf{k}\cdot\mathbf{V}+\ell V_{\phi}/\rho\), where the last term represents an _azimuthal Doppler shift_[138]. When a moving atom is subject to two weak counter-propagating Gaussian beams, it experiences Doppler-shifted frequencies for the two beams approximately given by \(\omega_{L}\mp\mathbf{k}\cdot\mathbf{V}\). This frequency imbalance is responsible for the appearing of a damping force that is always proportional and opposite to the atomic velocity. This Doppler cooling can result in the formation of an ostensibly viscous atomic system known as _optical molasses_ in one, two, or three dimensions [121]--see Figure 1. In this case, the total scattering force on the atom is given by \(\mathbf{F}_{sc}\approx-a_{z}V_{z}\), with the damping coefficient given by \(a_{z}=2\hbar k^{2}s\frac{(-\Delta\Gamma)}{(\Delta^{2}+\Gamma^{2}/4)}\), where \(\Delta<0\), and \(s\) is the so-called saturation parameter given by \(s=\frac{\Omega^{2}/2}{(\Delta^{2}+\Gamma^{2}/4)}\). By such means, optical molasses can cool atomic motion up to a kinetic temperature known as a _Doppler temperature_[122], which is given by  
\[T_{D}=\frac{\hbar\Gamma}{2k_{B}}, \tag{21}\]  
where \(k_{B}\) is the Boltzmann constant. The mechanism for optical molasses has a very interesting counterpart in the case of an atom interacting with optical vortices. If we assume that the moving atom is irradiated by two co-propagating optical vortices with opposite winding numbers, then we may observe so-called _azimuthal molasses_.