  We demonstrate phase locking of two ultraviolet laser sources by modulating a
fundamental infrared laser with 4th-order sidebands using an electro-optic
modulator and phase locking of one sideband to a second fundamental infrared
laser. Subsequent sum frequency generation and second harmonic generation
successfully translates the frequency offset to the ultraviolet domain. The
phase lock at 139 GHz is confirmed through stimulated Raman transitions for
thermometry of $^9$Be$^+$ ions confined in a cryogenic Penning trap. This
technique might be used for sideband cooling of single $^9$Be$^+$ ions as well
as sympathetic cooling schemes and quantum logic based measurements in Penning
traps in the future.
