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Double kicked rotors (DKRs) appear to be the simplest nonintegrableHamiltonian systems featuring classical translational symmetry in phase space(i.e., in angular momentum) for an \emph{infinite} set of values (the rationalones) of a parameter $\eta$. The experimental realization of quantum DKRs byatom-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 afinite strip of coupled lattice chains. Exact connections are then obtainedbetween quasienergy spectra of $\beta $-DKRs for all $\beta $ in a genericallyinfinite set. The general conditions of quantum resonance for $\beta $-DKRs areshown to be the simultaneous rationality of $\eta $, $\beta$, and a scaledPlanck constant $\hbar _{\mathrm{S}}$. For rational $\hbar _{\mathrm{S}}$ andgeneric values of $\beta $, the quasienergy spectrum is found to have astaggered-ladder structure. Other spectral structures, resembling Hofstadterbutterflies, are also found. Finally, we show the existence of particular DKPwave-packets whose quantum dynamics is \emph{free}, i.e., the evolutionfrequencies of expectation values in these wave-packets are independent of thenonintegrability. All the results for rational $\hbar _{\mathrm{S}}$ exhibitunique number-theoretical features involving $\eta $, $\hbar _{\mathrm{S}}$,and $\beta $.

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