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Current progress in laser cooling of antihydrogen

We discuss laser cooling methods of (anti)hydrogen and its importance for current and future experiments. The exploration of antimatter presents a great interest for $CERN$ and $GSI$ experiments aimed at check of quantum mechanics laws, fundamental symmetries of nature and gravity and investigations in atomic and nuclear physics. The spectral transition $1S\rightarrow 2P$ in $\bar{H} (H)$ atom is the most suitable for laser cooling due to a small lifetime of $2P$ state and insignificant ionization losses. However the pulsed and continuous laser sources at Lyman-$α$ wavelength do not possess enough power for fast and efficient cooling. The small power of laser sources at $λ=121.6\ \nm$ is poor technical problem associated with a complexity of generation scheme of such radiation, which arises due to absence of nonlinear $BBO$ crystals at this wavelength. The advances in this area will completely destine the future progress of the experiments aimed at study of antimatter.