Quantum properties of double kicked systems with classical translational invariance in momentum

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 $\eta$. 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 $\beta \hslash$, to a generalized DKR, the “$\beta \text{−}\mathrm{DKR}$.” We determine general quantum properties of $\beta \text{−}\mathrm{DKRs}$ 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 ${\hslash }_{\mathrm{S}}$. For rational ${\hslash }_{\mathrm{S}}$ 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 free, i.e., the evolution frequencies of expectation values in these wave-packets are independent of the nonintegrability. All the results for rational ${\hslash }_{\mathrm{S}}$ exhibit unique number-theoretical features involving $\eta ,{\hslash }_{\mathrm{S}}$, and $\beta$.

Figure 1

Some orbits of the DKR map (2) for $K=\tilde{K}=0.5,V\left(\theta \right)=\tilde{V}\left(\theta \right)=cos\left(\theta \right)$, and (a) $\eta =1/4$; (b) $\eta =1/3$. The periodicity in $L/\left(2\pi \right)$ with periods $c=4$ and $c=3$ in (a) and (b), respectively, is evident. The three basic orbits in the first unit cell, with $0\le L/\left(2\pi \right)<c$, are a vibrational orbit, a chaotic orbit in the main stochastic layer, and a rotational orbit. Each of these orbits starts from the same initial conditions in (a) and (b).

Figure 2

QE spectra $\omega$ of $\beta$-DKRs as functions of rational $\eta$ for ${\hslash }_{\mathrm{S}}=2/1 \left(\hslash =4\pi \right),K/\hslash =\tilde{K}/\hslash =1,V\left(\theta \right)=\tilde{V}\left(\theta \right)=cos\left(\theta \right)$, and (a) $\beta =0$, (b) $\beta =1/20$, (c) $\beta =1/7$. Cases (b) and (c) clearly feature a staggered-ladder structure with $g=10$ and $g=7$ stages, respectively, for most $\eta$'s; these values of $g$ coincide with the corresponding values of $g_{\max }$ in Eq. (33). In case (a), there is no staggered-ladder structure since $g=1$ for all $\eta$. The similarity of the spectral structure in this case to that of a Hofstadter butterfly was first pointed out in Ref. [45]; see also Sec. 5. In (a), (b), and (c), $\eta$ takes all rational values in $(0,0.5)$ with denominators $c\le 50,c\le 25$, and $c\le 25$, respectively.

Figure 3

Similar to Fig. 2 but with the following changes: (a) ${\hslash }_{\mathrm{S}}=1/2$ and $\beta =0$; (b) ${\hslash }_{\mathrm{S}}=3/2$ and $\beta =1/3$; (c) ${\hslash }_{\mathrm{S}}=3/2$ and $\beta =1/5$. In cases (a) and (b), there appear structures resembling Hofstadter butterflies. Case (c) features a staggered-ladder structure with $g=g_{\max }=5$ stages for most values of $\eta$, where $g_{\max }$ is given by Eq. (33). This structure does not emerge in cases (a) and (b), since $g=1$ for all $\eta$. In (a), (b), and (c), $\eta$ takes all rational values in $(0,1)$ with denominators $c\le 50,c\le 30$, and $c\le 20$, respectively.

Figure 4

Similar to Fig. 2 but with the following changes: (a) ${\hslash }_{\mathrm{S}}=2/3$ and $\beta =0$; (b) ${\hslash }_{\mathrm{S}}=2/3$ and $\beta =1/5$. Only the latter case features a staggered-ladder structure with $g=g_{\max }=5$ stages for most values of $\eta$. The spectral structure in case (a), with $g=1$ for all $\eta$, bears some resemblance to a Hofstadter butterfly. In cases (a) and (b), $\eta$ takes all rational values in $(0,1)$ with denominators $c\le 30$ and $c\le 20$, respectively.