  The recent demonstration of a terahertz (THz) molecular gas laser pumped by a
mid-infrared quantum cascade laser (QCL) has opened up new perspectives for
this family of sources, traditionally relying on CO2-laser pumping. A so far
open question concerning QCL-pumped THz molecular lasers (MLs) is related to
their spectral purity. Indeed, assessing their frequency/phase noise is crucial
for a number of applications potentially exploiting these sources as local
oscillators. Here this question is addressed by reporting the measurement of
the frequency noise power spectral density (PSD) of a THz ML pumped by a
10.3{\mu}m-wavelength QCL, and emitting 1mW at 1.1THz in continuous wave. This
is achieved by beating the ML frequency with the 1080th harmonic of the
repetition rate of a 1560nm frequency comb. We find a frequency noise PSD <
10Hz2/Hz (-95dBc/Hz) at 100kHz from the carrier. To demonstrate the effect of
the stability of the pump laser on the spectral purity of the THz emission we
also measure the frequency noise PSD of a CO2-laser-pumped 2.5THz ML, reaching
0.1Hz2/Hz (-105dBc/Hz) at 40kHz from the carrier, limited by the frequency
noise of the frequency comb harmonic. Finally, we show that it is possible to
actively phase-lock the QCL-pumped molecular laser to the frequency comb
repetition rate harmonic by controlling the QCL current, demonstrating a sub-Hz
linewidth.
