Deviations from early-time quasilinear behavior for the atom-optics kicked rotor near the classical limit

We present experimental measurements of the mean energy for the atom-optics kicked rotor after just two kicks. The energy is found to deviate from the quasilinear value for small kicking periods. The observed deviation is explained by recent theoretical results which include the effect of a nonuniform initial momentum distribution, previously applied only to systems using much colder atoms than ours.

Figure 1

Energy after two kicks as a function of $\bar{k}$ as given by Eq. (5). The solid line is for $\kappa ∕\bar{k}=7$ and ${\sigma }_p=1.8$ two-photon recoils. The dashed line is for $\kappa ∕\bar{k}=7$ and ${\sigma }_p=3.2$ two-photon recoils. Simulation results for the same parameters (crosses and circles, respectively), but for kicking with rectangular pulses of constant width ${\tau }_p$, are overlayed. The ratio ${\tau }_p∕T$ varied from about 0.015 for $\bar{k}\approx 3$ to 0.5 for $\bar{k}\approx 0.1$. The vertical lines show the value of ${\bar{k}}_{\mathrm{crit}}$ as determined from Eq. (6). The dot-dashed line is for ${\sigma }_p=1.8$ and the dotted line is for ${\sigma }_p=3.2$.

Figure 2

Experimentally measured energies after one and two kicks (circles and crosses, respectively). The value of $\kappa ∕\bar{k}$ was found to be $5.2\pm 0.4$ and ${\sigma }_p$ was measured to be 4.2 two-photon recoils. The solid lines give the theoretical values for the energy after one and two kicks, respectively. The initial thermal energy has been subtracted to allow a more instructive comparison between the energies.

Figure 3

Experimentally measured energies after two kicks for various values of $\kappa ∕\bar{k}$ and ${\sigma }_p=3.35\pm 0.06$. The measured values of $\kappa ∕\bar{k}$ were (a) 7.5, (b) 6.4, (c) 5.6, and (d) 4.5 each with an experimental error of $\pm 0.1$. The solid line through the data shows the energy given by Eq. (5) The dotted and dashed lines show the energy given by the quasilinear theory after one and two kicks, respectively.

Figure 4

Experimentally measured energies after two kicks for various values of ${\sigma }_p$. The measured values of ${\sigma }_p$ were (a) 3.3, (b) 4.2, (c) 5.3, and (d) 6.0 in two-photon recoils. We estimate the error in these measurements to be on the order of 0.9 two-photon recoil. For these curves, the measured value of $\kappa ∕\bar{k}$ is $5.8\pm 0.2$. The solid lines show the analytical results and the dashed lines show the energy predicted by the quasilinear theory.

Figure 5

Phase-space diagrams for $T$ corresponding to $\bar{k}=\left(\text{top}\right)0.001$, (middle) 0.3, and (bottom) 3.0 and $\kappa$ such that $\kappa ∕\bar{k}=5.5$. The plots were generated from 100 iterations of the map in Eq. (2) using a uniform initial distribution in the $\varphi$ coordinate and a Gaussian initial distribution in $p$ (momentum in experimental units) with ${\sigma }_p=3.6$.