Horizontal-vertical coupling for three dimensional laser cooling

In order to achieve three dimensional crystal beam, laser cooling forces are required not only in the longitudinal direction, but also in the transverse directions. With the resonance coupling method, transverse temperature is transmitted into longitudinal direction, and we have already demonstrated the horizontal laser cooling experimentally. In the present paper, we will describe an approach to extend this result to three dimensional cooling. The vertical cooling requires that the horizontal oscillation couples with the vertical oscillation. For achieving horizontal-vertical coupling, the solenoid in electron beam cooling apparatus is utilized with an experiment (νx =2.07, νy=1.07). For various solenoidal magnetic fields from 0 to 100 Gauss, horizontal and vertical betatron tunes are measured by beam transfer function. For a certain region of the solenoidal magnetic field, these tunes are mixed up each other. (author)