Unique Floquet decomposition theory near resonance in quantum systems

A response theory in quantum systems, which can discuss the stability of the total system and the appearance of subharmonics, is proposed. The propagator of the system in the Liouville space is decomposed into the periodic factor with the same frequency as the external field and the remaining factor by using the unique Floquet decomposition theory. The relation between the usual Floquet operator and the remaining factor is studied. It is shown that the remaining factor is responsible for the stability of the total system and the appearance of subharmonics. A quantum version of Mathieuʹs equation is studied as an application of the general theory. The stability condition of subharmonics is shown to coincide with the classical one. It is also shown that the drastic change of energy spectrum occurs. If the subharmonic oscillation is unstable, the energy spectrum is continuous, while in the stable case the energy spectrum is discrete.