Quantum Properties of Double Kicked Systems with Classical Translational Invariance in Momentum

Itzhack Dana

doi:10.1103/PhysRevE.91.012914

paper

Quantum Properties of Double Kicked Systems with Classical Translational Invariance in Momentum

arXiv:1410.7362 · doi:10.1103/PhysRevE.91.012914

Abstract

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 \emph{infinite} set of values (the rational ones) of a parameter $η$ . 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 $β ℏ$ , to a generalized DKR, the \textquotedblleft $β$ -DKR\textquotedblright . We determine general quantum properties of $β$ -DKRs and DKPs for arbitrary rational $η$ . The quasienergy problem of $β$ -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 $β$ -DKRs for all $β$ in a generically infinite set. The general conditions of quantum resonance for $β$ -DKRs are shown to be the simultaneous rationality of $η$ , $β$ , and a scaled Planck constant $ℏ_{S}$ . For rational $ℏ_{S}$ and generic values of $β$ , 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 \emph{free}, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational $ℏ_{S}$ exhibit unique number-theoretical features involving $η$ , $ℏ_{S}$ , and $β$ .

10 pages, 4 figures

Quantum Properties of Double Kicked Systems with Classical Translational Invariance in Momentum

Abstract

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