Butterfly Floquet Spectrum in Driven SU(2) Systems

The Floquet spectrum of a class of driven SU(2) systems is shown to display a butterfly pattern with multifractal properties. The level crossing between Floquet states of the same parity or different parities is studied. The results are relevant to studies of fractal statistics, quantum chaos, coherent destruction of tunneling, and the validity of mean-field descriptions of Bose-Einstein condensates.

Figure 1

Eigenphase spectrum (denoted $ϵ$) of the Floquet operator $F$ in Eq. (3). $J=20$ in (a), 20.5 in (b), and 100 in (c). $\alpha /{\hslash }_{\mathrm{eff}}=\alpha J=1.0$ in all panels.

Figure 2

Poincaré surfaces of a section (with $J_x>0$) of the classical or mean-field limit of $F$ in Eq. (3), with $\alpha =0.05$ [the same as in Fig. 1a] and $\eta =5$ in (a), 30 in (b), and 75 in (c).

Figure 3

(a) The cumulative Floquet state density and (b)–(d) the Floquet state density distribution at different scales, for ${\hslash }_{\eta }=(\sqrt{5}-1)\pi /2$, $\alpha /{\hslash }_{\mathrm{eff}}=1$, and $J=2999$. Panel (e) shows the generalized fractal dimension $D_q$. Crosses and circles are for odd-parity and even-parity states, respectively. Triangles represent the result for a standard kicked-top model.

Figure 4

(a) Level crossings between three even-parity states (dashed lines) and two odd-parity states (solid lines), for $J=10$ and $\alpha /{\hslash }_{\mathrm{eff}}=1.0$. (b) Number of level crossings versus $J$, for ${\hslash }_{\eta }\in [0,4\pi )$ and $\alpha /{\hslash }_{\mathrm{eff}}=1.0$. The cross (square) symbols are for crossings between different-parity (same-parity) states, and the fitting suggests a power law scaling $J^{3.0}$ ($J^{2.7}$).