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

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 $.