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Thermalization of Neutrons in Gels of Ultracold Nanoparticles and Application of Such Ultracold Neutrons in Fundamental Physics Valery V. Nesvizhevsky Institut Laue-Langevin, Grenoble, France The new method for producing ultracold neutrons (UCN) consists in the equilibrium cooling of faster neutrons owing to their many collisions with ultracold nanoparticles made from low-absorbing materials (D2 O, D2 , O2 etc) down to the temperature of these nanoparticles of about 1 mK during the diffusion motion of these neutrons in a macroscopically large ensemble of nanoparticles. Gels of ultracold nanoparticles can provide unique oppotunity for such extremely efficient cooling of neutrons, as they possess all needed properties: weak bonds between the nanoparticles, the nanoparticles sizes are close to the optimum, high porosity that provides high contrast of the corresponding effective potential for scattering of neutrons, low absorption of neutrons when the nanoparticles material is properly chosen. UCN are used in a broad range of investigations in fundamental physics. Thus, this is just thanks to their unique properties, that the gravitationally bound quantum states of matter were observed for the first time (with the effective temperature of thier vertical motion equal to about 20 nK). We discuss the last results, the current status and the future of this experiment, in particular the resonance transitions between the gravitationally bound neutron quantum states. Sorting category: Ee Materials, techniques, and applications Keywords: ultracold, neutron, gels, thermalization INVITED PAPER LT1111