Quantum Properties of Double Kicked Systems with Classical Translational Invariance in Momentum (1410.7362v2)

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 $\eta$. 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 $\beta\hbar$, to a generalized DKR, the \textquotedblleft $\beta $-DKR\textquotedblright . We determine general quantum properties of $\beta $-DKRs and DKPs for arbitrary rational $\eta $. The quasienergy problem of $\beta $-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 $\beta $-DKRs for all $\beta $ in a generically infinite set. The general conditions of quantum resonance for $\beta $-DKRs are shown to be the simultaneous rationality of $\eta $, $\beta$, and a scaled Planck constant $\hbar _{\mathrm{S}}$. For rational $\hbar _{\mathrm{S}}$ and generic values of $\beta $, 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 $\hbar _{\mathrm{S}}$ exhibit unique number-theoretical features involving $\eta $, $\hbar _{\mathrm{S}}$, and $\beta $.