The noise figure or limiting sensitivity for both traveling-wave and resonant-cavity quantum-mechanical amplifiers, sensitive to one direction of propagation, is derived with spontaneous emission as the limiting noise. The concept of effective temperature is introduced as an analytical parameter; thus negative temperatures appear in a natural fashion. It is pointed out that the results of this calculation can be considered the solution to the problem of linear counting of coherent particles. In this case the least count is one and the signal-to-noise ratio (for constant photon flux) increases as the reciprocal of the band width. The limiting temperature sensitivity of properly designed quantum-mechanical amplifiers is given as $\frac{h\nu }{k}$ degrees because of the drastic difference between negative and positive temperatures.

• Received 6 February 1957

DOI:https://doi.org/10.1103/PhysRev.106.617