  The appearance of spontaneous coherence is a fundamental feature of a
Bose-Einstein condensate and an essential requirement for possible applications
of the condensates for data processing and quantum computing. In the case of a
magnon condensate in a magnetic crystal, such computing can be performed even
at room temperature. So far, the process of coherence formation in a magnon
condensate was inaccessible. We study the evolution of magnon radiation spectra
by direct detection of microwave radiation emitted by magnons in a
parametrically driven yttrium iron garnet crystal. By using specially shaped
bulk samples, we show that the parametrically overpopulated magnon gas evolves
to a state, whose coherence is only limited by the natural magnon relaxation
into the crystal lattice.
