Some analytical and intuitive results in the quantum theory of mixing

Abstract

This paper is an elaboration upon Tucker’s quantum theory of mixing, which has proven very valuable in interpreting recent experiments in superconducting quasiparticle mixing. Tucker’s formula for the conversion gain of a three-frequency quantum mixer in the low intermediate frequency (IF) limit is generalized to include arbitrary source reactance; one choice of source reactance is found to resonate out the quantum reactance in the conversion expression. The signal reflection gain is evaluated; it becomes infinite simultaneously with the IF conversion gain. Infinite gain occurs only within the region of negative dc differential resistance. A number of relationships among the complex elements of the small-signal admittance matrix is presented; the nonlinear reactive elements are shown to be small under certain conditions when the conversion gain is maximized. The local oscillator (LO) power required for quantum mixing is calculated and found to be related to the gain denominator. This implies that certain quantum mixers at high gain will exhibit a constant gain-powerwidth product and a gain-proportional noise temperature component. A particularly simple expression for the conversion gain of an optimized quantum mixer is developed, which agrees with previous experiments. The origin of nonclassical behavior, including conversion gain and the nonlinear quantum reactance, is discussed. We conclude that the nonlinear quantum reactance, though it certainly must exist, is not itself responsible for conversion gain and has a minor, perhaps insignificant, effect upon the properties of a quantum mixer.