$2\delta$-Kicked Quantum Rotors: Localization and `Critical' Statistics

Abstract

The quantum dynamics of atoms subjected to pairs of closely-spaced $\delta$-kicks from optical potentials are shown to be quite different from the well-known paradigm of quantum chaos, the singly-$\delta$-kicked system. We find the unitary matrix has a new oscillating band structure corresponding to a cellular structure of phase-space and observe a spectral signature of a localization-delocalization transition from one cell to several. We find that the eigenstates have localization lengths which scale with a fractional power $L\sim {\hslash }^{-.75}$ and obtain a regime of near-linear spectral variances which approximate the `critical statistics' relation ${\Sigma }_2(L)\simeq \chi L\approx 1/2(1-\nu )L$, where $\nu \approx 0.75$ is related to the fractal classical phase-space structure. The origin of the $\nu \approx 0.75$ exponent is analyzed.

4 pages, 3 figs