# Laser Cooling and Trapping of Rare-Earth-Doped Particles
## 1 Introduction  
Cooling or refrigeration is a physical process permitting the maintaining of a sample at a temperature below that of its surroundings. This technique dates back thousands of years to when people first tried to preserve food. The reliable source of coherent radiation (laser), demonstrated by [PERSON] in 1960, has revolutionized the cooling process. Laser cooling (optical refrigeration) and trapping have since been developed.  
In late 1950 and the early 1960s, [PERSON] and [PERSON] introduced the first ion traps. [PERSON] with colleagues developed the three-dimensional radiofrequency (RF) trap, known also as the Paul trap, from a linear quadrupole mass filter. It can be considered as a two-dimensional ion trap [1]. [PERSON] and a number of groups working independently proposed a magnetic field for trapping [2; 3; 4; 5; 6; 7]. [PERSON] named this trap the Penning trap after [PERSON] who, in 1936, proposed the use of a magnetic field to an electrical discharge in order to increase the lifetime of electrons using the confining effect of the field [7].  
In 1968, [PERSON] proposed the use of the dipole force to trap atoms [8]. The electric dipole interaction between the atoms and a standing laser wave was used to form a periodic lattice of potential wells. In this lattice, minima coincided with the nodes or antinodes of the standing laser wave.  
In 1975, Doppler cooling was proposed by two teams: [PERSON] and [PERSON] for free atoms [9], and [PERSON] and [PERSON] for trapped particles [10]. Doppler cooling occurs when atoms are irradiated with counter-propagating laser beams detuned below resonance. Following the Doppler mechanism, a moving atom absorbs photons into the laser wave counterpropagating its velocity rather than into the copropagating wave; thus, it encounters a force opposed to its velocity and undergoes cooling.  
In 1988, it was realized that, using multilevel atoms and the spatial variation of light polarization, a temperature lower than the Doppler cooling limit can be achieved [11]. This technique is known as sub-Doppler, polarization gradient, or Sisyphus cooling.  
In 1995, a fifth, low-energy form of matter (Bose-Einstein condensate) was demonstrated by [PERSON] and [PERSON] in a gas of rubidium atoms, which was cooled to 170 nK, [12] and by [PERSON] in a gas of sodium atoms [13].  
All these developed techniques cause laser cooling of the center-of-mass motion of atoms, ions, or particles. Indeed, the energy of molecules in gases almost completely belong to translational degrees of freedom. Using one of the mentioned methods, the translational temperature of a nanocrystal can be decreased. This process is known as translational cooling. In 1929, [PERSON] proposed the use of anti-Stokes fluorescence for cooling [14]. Following this approach, the internal energy included in the crystal lattice vibrations can be removed from a number of solids. Laser trapping and cooling rare-earth (RE)-doped particles are considered in this review. The theory of laser cooling with anti-Stokes fluorescence is briefly discussed in Section 2. Recent achievements in the development of optical trapping and cooling of micro- and nanoparticles are presented in Section 3.