  The fast detection of terahertz radiation is of great importance for various
applications such as fast imaging, high speed communications, and spectroscopy.
Most commercial products capable of sensitively responding the terahertz
radiation are thermal detectors, i.e., pyroelectric sensors and bolometers.
This class of terahertz detectors is normally characterized by low modulation
frequency (dozens or hundreds of Hz). Here we demonstrate the first fast
semiconductor-based terahertz quantum well photodetectors by carefully
designing the device structure and microwave transmission line for high
frequency signal extraction. Modulation response bandwidth of gigahertz level
is obtained. As an example, the 6.2-GHz modulated terahertz light emitted from
a Fabry-P\'{e}rot terahertz quantum cascade laser is successfully detected
using the fast terahertz quantum well photodetector. In addition to the fast
terahertz detection, the technique presented in this work can also facilitate
the frequency stability or phase noise characterizations for terahertz quantum
cascade lasers.
