# Forces in the Interaction of Light with Matter
## 3 Optical Forces in the Laser Cooling and Trapping of Atoms
### The Dipole Force  
The second force is the so-called _dipole force_ (known also as _gradient force_ or _optical dipole force_). To correctly represent the lineshape of laser light, it can be considered as a mixture of a large number of electromagnetic modes with slightly different frequencies. Under such irradiation, an atom can thereby absorb a photon from one component and release a photon by stimulated emission into another. The net effect of these absorption-stimulated emission cycles on the atomic gross motion is the dipole force, which, for a two-level atom, is given by  
\[\mathbf{F}_{dip}=-\frac{\hbar\Delta}{4}\frac{\
abla\Omega^{2}(\mathbf{R})}{ \Delta^{2}(\mathbf{R},\mathbf{V})+\Omega^{2}(\mathbf{R})/2+\Gamma^{2}/4}. \tag{22}\]  
For blue detuning (\(\omega_{L}>\omega_{0}\), \(\Delta>0\)), this force repels atoms from high-intensity regions, while for red detuning (\(\omega_{L}<\omega_{0}\), \(\Delta<0\)), atoms are attracted towards high intensities. By comparing Equation (20) with Equation (22), we observe that the dipole force increases in intensity, while the scattering force saturates at a maximum value. The dipole force is a conservative force which may be regarded as resulting from an optical dipole potential given by  
\[U_{dip}=\frac{\hbar\Delta}{2}\ln\biggl{(}1+\frac{\Omega^{2}(\mathbf{R})/2}{ \Delta^{2}(\mathbf{R},\mathbf{V})+\Gamma^{2}/4}\biggr{)}. \tag{23}\]  
In the case of very large detuning, the optical dipole potential assumes the approximate form \(U_{dip}=\hbar\Omega^{2}/\Delta\). The dipole force and its associated potential are responsible for the trapping of atomic motion. Using vortex-structured light, red detuning can be deployed for trapping atoms in an annular high-intensity region, whereas blue detuning enables trapping in the counterpart intensity minima. Such traps can capture single atoms or up to several million in trapping areas with dimensions ranging from a few micrometers to several millimeters. Figure 1: For a two-level atom moving with velocity \(V\) in the standing wave of two counter-propagating waves, the frequencies of the counter-propagating waves are Doppler shifted with different frequencies \(\omega_{\pm}=\omega\pm k_{L}v\). The radiation pressure force is the sum of the two forces separately, namely the atomic force \(F_{sc,+}\), parallel to the direction of beam propagation, and the force \(F_{sc,-}\), which acts in the direction anti-parallel to the beam. These two force components are represented by dotted and dashed lines respectively, and their sum is represented by the unbroken line. Arbitrary units. One of the most significant applications of dipole trapping is found in the generation of _optical lattices_. These are light fields whose intensity structure exhibits periodicity in one, two, or three dimensions, resulting in a landscape of optical dipole traps resembling the structure of crystalline lattices. Here, optical vortices provide for the generation of optical lattices with cylindrical symmetries such as the _optical Ferris wheel_[139] and _helical optical tubes_[70; 72]. Although the scattering force is not conservative, due to its dissipative nature, it can be combined with spatially varying magnetic fields created by current-carrying coils to form a robust atom-trapping mechanism. Devices based on this principle, known as the _magneto-optical trap_ (MOT), represent the most widely used form of trap in atomic physics laboratories. Similarly, the conservative dipole force can be exploited in so-called _high-intensity Sisyphus cooling_[140] (in contrast to a polarization mechanism discussed in the following Section 3.3). This effect is experienced by blue-detuned atoms, moving in a high-intensity standing laser beam, with a velocity large enough to travel over several laser wavelengths during its lifetime in the excited state \(\Gamma^{-1}\) but also low enough to neglect non-adiabatic transitions between atom field-dressed states.