Geometry of classical periodic orbits and quantum coherent states in coupled oscillators with SU(2) transformations

Abstract

The geometry of classical dynamics in coupled oscillators with SU(2) transformations is explored and found to be relevant to a family of continuous-transformation orbits between Lissajous and trochoidal curves. The quantum wave-packet coherent states are derived analytically to correspond exactly to the transformation geometry of classical dynamics. By using the quantum wave-packet coherent states derived herein, stationary coherent states are constructed and are shown to possess spatial patterns identical to the transformation geometry between Lissajous and trochoidal orbits.

Model of commensurate harmonic oscillators with SU(2) coupling interactions: Analogous observation in laser transverse modes

“…We would like to remark that, in view of the case (p,q) = (1 , 1) for the coupled isotropic HO, the wave functions have been demonstrated previously on a group theory level via the SU(2) transformation [28,31]. Likewise, it enables us to derive the wave functions by employing the transformation of the SU(2) symmetry group.…”

“…(1). The coupling termĤ c is introduced as an SU(2) coupling interaction [27,28], which can be modeled aŝ…”

“…More recently, the isotropic HOs with SU(2) coupling interactions have been used to investigate the generation and evolution of quantum vortex states [27] and the transformation geometry between Lissajous and trochoidal orbits [28]. It has been shown [29,30] that the commensurate HOs can be mapped into the isotropic HOs via the canonical transformation.…”

Generation of polygonal vortex beams in quasi-frequency- degenerate states of Yb:CALGO laser

“…In common non-degenerate states, it has been confirmed that the high-order HG mode can be experimentally generated under off-axis pumping with a sufficiently large off-axis displacement [35,38]. Whereas in degenerate states, it has been theoretically and experimentally confirmed that the emission mode can be characterized by the quantum coherent states with SU(2) Lie algebra [33]: where M + 1 stands for the number of HG modes in family of ψ (HG) n 0 +Q·K,0,s 0 −P ·K (x, y, z) with K = 1, 2, 3, · · · , which constitute a family of frequency-degenerate modes, where n 0 and s 0 represent the minimum transverse and maximum longitudinal orders in the degenerate family respectively. Note that all the discussion corresponding to the off-axis pumping with (∆x, 0) can be applied to the case of (0, ∆y) for the degenerate family in terms of ψ (HG) 0,m 0 +Q·K,s 0 −P ·K (x, y, z).…”

Reduced dynamics of two oscillators collectively coupled to a thermal bath

“…Complex poles corresponding to the unstable oscillator then arise in the complex energy plane. From another point of view, the a i 's can be regarded as the normal modes of two degenerate oscillators coupled through the SU(2) coupling interactions [19]. The system Hamiltonian H is then unitarily related to a family of Hamiltonians by SU (2) transformation.…”