2014/10/31 by Itzhack Dana
Mathematics · Physics and Astronomy · #Angular momentum #Classical mechanics #Cold Atom Physics and Bose-Einstein Condensates #Eigenvalues and eigenvectors #Hamiltonian (control theory) #Mathematical physics #Mathematics #Phase space #Physics #Planck constant #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #Quantum, superfluid, helium dynamics #Theoretical physics #Total angular momentum quantum number #Wave packet #math-ph #math.MP #nlin.CD #quant-ph
paper · pdf · doi:10.1103/physreve.91.012914
published as Physical Review E 91, 012914 (2015) · 10 pages, 4 figures
openalex publication_date 2015/01/20 · arxiv created 2015/01/21 · arxiv updated 2015/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Double kicked rotors (DKRs) appear to be the simplest nonintegrable Hamiltonian systems featuring classical translational symmetry in phase space (i.e., in angular momentum) for an infinite set of values (the rational ones) of a parameter \ensuremathη. The experimental realization of quantum DKRs by atom-optics methods motivates the study of the double kicked particle (DKP). The latter reduces, at any fixed value of the conserved quasimomentum \ensuremathβ\ensuremathℏ, to a generalized DKR, the ``\ensuremathβ\text\ensuremath-DKR.'' We determine general quantum properties of \ensuremathβ\text\ensuremath-DKRs and DKPs for arbitrary rational \ensuremathη. The quasienergy problem of \ensuremathβ-DKRs is shown to be equivalent to the energy eigenvalue problem of a finite strip of coupled lattice chains. Exact connections are then obtained between quasienergy spectra of \ensuremathβ-DKRs for all \ensuremathβ in a generically infinite set. The general conditions of quantum resonance for \ensuremathβ-DKRs are shown to be the simultaneous rationality of \ensuremathη,\phantom\rule0.16em0ex\ensuremathβ, and a scaled Planck constant \ensuremathℏS. For rational \ensuremathℏS and generic values of \ensuremathβ, the quasienergy spectrum is found to have a staggered-ladder structure. Other spectral structures, resembling Hofstadter butterflies, are also found. Finally, we show the existence of particular DKP wave-packets whose quantum dynamics is free, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational \ensuremathℏS exhibit unique number-theoretical features involving \ensuremathη,\phantom\rule0.16em0ex\ensuremathℏS, and \ensuremathβ.