The SU(1, 1) Perelomov number coherent states and the non-degenerate parametric amplifier

Abstract

We construct the Perelomov number coherent states for any three su(1, 1) Lie algebra generators and study some of their properties. We introduce three operators which act on Perelomov number coherent states and close the su(1, 1) Lie algebra. We show that the most general SU (1, 1) coherence-preserving Hamiltonian has the Perelomov number coherent states as eigenfunctions, and we obtain their time evolution. We apply our results to obtain the non-degenerate parametric amplifier eigenfunctions, which are shown to be the Perelomov number coherent states of the two-dimensional harmonic oscillator.

Lewis-Riesenfeld quantization and SU(1, 1) coherent states for 2D damped harmonic oscillator

“…In the literature, various coherent states [58][59][60] are contructed for different Lie algebra. To construct the appropriate coherent states for this system whose eigenfunction is expressed in terms of the generalized Laguerre functions as in [61][62][63][64][65][66], we factorise this eigenfunction to find the hidden symmetry of the system through the establishment of an appropriate Lie algebra.…”

Algebraic approach to the Tavis-Cummings model with three modes of oscillation

“…In this SU(1, 1) representation, the group numbers n, k are related with the physical numbers n l , m n as n = n l and k = 1 2 (m n + 1) [34]. Thus, from these results we obtain that the energy spectrum of the most general case of the Tavis-Cummings model is…”

“…Thus, Ψ = D(ξ)Ψ ′ = ψ a (x) ⊗ D(ξ)ψ ′ n l ,mn (ρ, φ). By using equation (76) of Appendix, we find that the action of the displacement operator D(ξ) on ψ ′ n l ,mn (ρ, φ) are the Perelomov number coherent states for the two-dimensional harmonic oscillator ψ ζ,n l ,k [34] ψ ζ,n l ,k = ρ, φ|ζ, k, n l =…”

“…The SU(1, 1) Perelomov number coherent state |ζ, k, n is defined as the action of the displacement operator D(ξ) onto an arbitrary excited state |k, n [34] |ζ, k, n =…”

Unified algebraic method to non-Hermitian systems with Lie algebraic linear structure

“…The Hermitian version of Hamiltonian (24) is obtained by setting special case: ω is real parameter, and λ = µ * , which has been widely used in the fields of atomic and optical physics, quantum optics, and the weakly interacting Bose system. [36][37][38][39] Imitating general diagonalization strategy of Hermitian case, we take a non-unitary operator ansatz for the similarity transformation aŝ…”