A quantum chaologist´s view of quartz resonators

Physicists have gained many non-trivial insights into the behaviour of quantum systems whose corresponding classical dynamics are chaotic; the study of such systems is called “quantum chaology”. The main function of this paper is (i) to tell the reader about the existence of this knowledge and (ii) to explain how it relates to the discipline of frequency control. Resonators are analogous to closed quantum systems, in the sense that the trajectory of a high-frequency wavepacket bouncing between the bounding surfaces of a resonator is similar to the trajectory of a quantum wave-particle bouncing inside the walls of a confining potential. Through this analogy, the same universal laws that control the behaviour of closed quantum systems should be applicable to the sorts of resonators that are used for frequency-control applications. Any finite resonator has a discrete set of normal modes. Quantum chaology places certain restrictions both on the way in which the frequencies of these normal modes are distributed in frequency space and on the way in which these frequencies change as the body is perturbed (by, say, a change in its operating temperature). These restrictions motivate the use of “chaotic” resonators, which are inherently immune to activity dips and sudden shifts in frequency. An experiment is described, which verifies the predictions of quantum chaology for a matchbox-sized block of crystalline quartz