Dynamics of SU(1, 1) coherent states for the time-dependent quadratic Hamiltonian system

Abstract

The dynamics of SU(1, 1) coherent states introduced by Perelomov are investigated for the time-dependent quadratic Hamiltonian system. SU(1, 1) generators we employed are closely related to the invariant operator theory while those of the previous work of Gerry et al. [C.C. Gerry, P.K. Ma, E.R. Vrscay, Phys. Rev. A 39 (1989) 668] are associated to the simple harmonic oscillator. This is the main difference between the two approaches. The merit of the method used in this paper is that it admits wide sphere of analytical description for quantum features of time-dependent quadratic Hamiltonian system. Our development is applied to the Caldirola–Kanai oscillator and compared the corresponding results with those of the Gerry et al. after correcting some miscalculations of theirs. We showed that the results of our theory are in good agreement with the results of the corrected work of Gerry et al. even if the form of the SU(1, 1) generators we employed are somewhat different from those of their work. The nontrivial zero-point energy plays a dominant role in the very low energy limit ( ξ ˜ 0 → 0 ) for the Caldirola–Kanai oscillator, leading the system to exhibit pure quantum effects as expected. On the other hand, it turn out for sufficiently high energy limit ( ξ ˜ 0 → 1 ) that the characteristic feature of dissipating quantum energy become very much the same as that of the classical energy.