Exact solution of the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators

Abstract

We study the isotropic and anisotropic Hamiltonian of two coupled harmonic oscillators from an algebraic approach of the $SU(1,1)$ and $SU(2)$ groups. In order to obtain the energy spectrum and eigenfunctions of this problem, we write its Hamiltonian in terms of the boson generators of the $SU(1,1)$ and $SU(2)$ groups. We use the one boson and two boson realizations of the $su(1,1)$ Lie algebra, and the one boson realization of the $su(2)$ Lie algebra to apply three tilting transformations to diagonalize the original Hamiltonian. These transformations let us to obtain the exact solutions of the isotropic and the anisotropic cases, from which the particular expected results are obtained for the cases where the coupling is neglected.

arXiv comment13 pages. arXiv admin note: substantial text overlap with arXiv:2409.08179

References in corpus (11)

• Cavity Optomechanics
• Circuit Quantum Electrodynamics
• Bringing entanglement to the high temperature limit
• Coupled harmonic oscillators and their quantum entanglement
• Quantum entanglement of nanocantilevers
• General solution of the time evolution of two interacting harmonic oscillators
• The SU(1,1) Perelomov number coherent states and the non-degenerate parametric amplifier
• Two-mode generalization of the Jaynes-Cummings and Anti-Jaynes-Cummings models
• A generalized Jaynes-Cummings model: The relativistic parametric amplifier and a single trapped ion
• Matrix diagonalization and exact solution of the k-photon Jaynes-Cummings model
• $Sp(4,R)$ algebraic approach of the most general Hamiltonian of a two-level system in two-dimensional geometry