The Quantum Three Wave Instability

The three wave interaction, the lowest order nonlinear interaction in plasma physics, has both a well-known classical description [1] and a recently investigated quantum description [2]; however, the relationship between these descriptions has not been fully investigated. The classical three wave interaction is subject to a parametric linear instability which would initially seem at odds with a quantum description. The present work describes how a time-independent finite-dimensional quantum system, which is Hermitian with all real eigenvalues, can give rise to a linear instability in the corresponding classical system. It is found that the distribution of eigenvalues in the quantum description of the classically unstable system has a richer spectrum than for the quantum description of the classically stable system, and the instability is realized in the quantum regime as a cascade of probability from higher to lower probability states. Finally, the nonlinear regime of classical instability is compared with the corresponding quantum regime, and the conditions for quantum instability are described.

[1] M. Rosenbluth, R. White, and C. Liu, Phys. Rev. Lett. 31, 1190 (1973).

[2] Y. Shi, H. Qin, and N. Fisch, Phys. Plamsas 28, 042104 (2021).