Non-monotonic dependence of the radiation line width on pumping in a phonon laser

A phonon laser is an important device that can generate coherent sound waves at gigahertz frequency. In this paper, we demonstrate that in a phonon laser, the radiation line width can depend on the pumping non-monotonically. This is because there are two different types of solutions whose spectra depend differently on the noise amplitude. The first solution is a zero solution, which is stable before the generation threshold and is nondegenerate. The second solution is a nonzero solution, which is stable above the generation threshold and is degenerate with respect to the phase. The line width of peaks in the spectrum of the zero solution does not depend on the noise amplitude, while the line width of the peak in the spectrum of the nonzero solution increases with increasing noise amplitude. The noise amplitude increases with temperature, and as a result, there is a temperature above which the transition from the zero solution to the nonzero solution leads to an increase in the radiation line width. In this case, the radiation line width depends non-monotonically on the pumping.