Signatures of Quantum Stability in a Classically Chaotic System

We experimentally and numerically investigate the quantum accelerator mode dynamics of an atom optical realization of the quantum $\delta$-kicked accelerator, whose classical dynamics are chaotic. Using a Ramsey-type experiment, we observe interference, demonstrating that quantum accelerator modes are formed coherently. We construct a link between the behavior of the evolution’s fidelity and the phase space structure of a recently proposed pseudoclassical map, and thus account for the observed interference visibilities.

Figure 1

Experimental momentum distributions as the microwave phase difference $\theta$ is varied in a $\pi /2$—$20$ kick—$\pi /2$ sequence where ${\delta }_L^b\sim 35\mathrm{G}\mathrm{H}\mathrm{z}$, with (a) $T=60.5\mu \mathrm{s}$ (QAM at $-17\hslash G$), and (b) $T=74.5\mu \mathrm{s}$ (QAM at $20\hslash G$). Corresponding numerical momentum distributions, where ${\delta }_L^b=35\mathrm{G}\mathrm{H}\mathrm{z}$, are in (c) and (d). Population arbitrarily normalized to $\mathrm{\text{maximum value}}=1$.

Figure 2

Numerical plots of $A/2$ against ${\varphi }_d$ with $N=20$, for (a) $T=60.5\mu \mathrm{s}$ and (c) $T=74.5\mu \mathrm{s}$. Plots of the noninterfering population $\int dqC(q)(|{\psi }_a^q{|}^2+|{\psi }_b^q{|}^2)/4$ for (b) $T=60.5\mu \mathrm{s}$ and (d) $T=74.5\mu \mathrm{s}$. Dashes demarcate the experimental range of ${\varphi }_d$.

Figure 3

Stroboscopic Poincaré sections determined by Eqs. (4, 5) for $T=60.5\mu \mathrm{s}$ ($\Rightarrow ϵ=-0.58$), and for (a) ${\varphi }_d=0.35\pi$, (b) the average experimental value $0.8\pi$, and (c) $1.5\pi$. (d), (e), and (f) show corresponding plots for $T=74.5\mu \mathrm{s}$ ($\Rightarrow ϵ=0.73$). Units are dimensionless.

Figure 4

Numerical plots of $\cos ({\varphi }_I+N\delta {\varphi }_d)$ against ${\varphi }_d$ ($N=20$) for (a) $T=60.5\mu \mathrm{s}$ and (b) $T=74.5\mu \mathrm{s}$. Crosses demarcate the experimental range of ${\varphi }_d$.