Relativistic cyclotron resonance shape in magnetic bottle geonium

The thermally excited axial oscillation of the electron through the weak magnetic bottle needed for the continuous Stern-Gerlach effect modulates the cyclotron frequency and produces a characteristic approximately 12-kHz-wide vertical rise-exponential decline line shape of the cyclotron resonance. At the same time the relativistic mass shift decreases the frequency by approximately 200 Hz per cyclotron motion quantum level n. Nevertheless, our analysis of the complex line shape shows that it should be possible to produce an abrupt rise in the cyclotron quantum number n from 0 to approximately 20 over a small fraction of 200 Hz, when the 160-GHz microwave drive approaches the n = 0 --> 1 transition, and a jump of 14 levels over a frequency increment of 200 Hz has already been observed in preliminary work. This realizes an earlier proposal to generate a very sharp cyclotron resonance feature by quasithermal excitation with a square noise band and should provide a way to detect spin flips when a weak bottle is used to reduce the broadening of the g - 2 resonance by a factor of 20.