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International Journal of Theoretical Physics

SU(1,1) Coherent States for the Generalized Two-Mode Time-Dependent Quadratic Hamiltonian System

Jeong Ryeol Choi, Kyu Hwang Yeon

Published Dec 17, 2007• 5 citations
Publisher
DOI
10.1007/s10773-007-9634-5

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We realized SU (1,1) Lie algebra in terms of the appropriate SU (1,1) generators for the time-dependent quadratic Hamiltonian system perturbed by a singularity. Exact quantum states of the system are investigated using SU (1,1) Lie algebra. Various expectation values in two kinds of the generalized SU (1,1) coherent states, that is, BG coherent states and Perelomov coherent states are derived. We applied our study to the CKOPS (Caldirola–Kanai oscillator perturbed by a singularity). Due to the damping constant γ, the probability density of the SU (1,1) coherent states for the CKOPS converged to the center with time. The time evolution of the probability density in SU (1,1) coherent states for the CKOPS are very similar to the classical trajectory.
We have used a generic two-mode Hamiltonian with two associated time scales to study the evolution of generalized SU(1,1) coherent states, in particular the pair and Perelomov coherent states, which have been realized in many systems such as radiation fields, trapped ions, and phonons. We have found that their dynamics does not depend on the ratio of these time scales but is instead determined crucially by the difference in photon numbers of the two modes. This is in stark contrast to the previously studied harmonic-oscillator coherent states and can be attributed to the different nature of the underlying algebra. We provide analytical results for their revival and fractional revival along with numerical plots of the autocorrelation function and the quadrature distribution and demonstrate the formation of Schr\"odinger cats. The results are extremely sensitive to the Casimir invariant and the complex parameters characterizing SU(1,1) coherent states.