Experimental investigation of early-time diffusion in the quantum kicked rotor using a Bose-Einstein condensate

We report measurements of the early-time momentum diffusion for the atom-optical delta-kicked rotor. In this experiment a Bose-Einstein condensate provides a source of ultracold atoms with an ultranarrow initial momentum distribution, which is then subjected to periodic pulses (or “kicks”) using an intense far-detuned optical standing wave. We characterize the effect of varying the effective Planck’s constant for the system, while keeping all other parameters fixed. The observed behavior includes both quantum resonances (ballistic energy growth) and antiresonances (re-establishment of the initial state). Our experimental results are compared with theoretical predictions.

Figure 1

(a) Typical time-of-flight images of atomic clouds separated in momentum by $2\hslash k_L$ for 1 (top) to 4 (bottom) kicks. (b) Corresponding momentum distributions. This particular case illustrates enhanced energy growth close to the quantum resonance at $\bar{k}=4\pi$.

Figure 2

Experimental energies (in dimensionless units) after 1 and 2 kicks, $E\left(1\right)$ and $E\left(2\right)$, at the lower kicking strength. After one kick the energies are approximately constant as a function of the effective Planck’s constant $\bar{k}$ [theory predicts $E\left(1\right)={\kappa }^2∕4$, a constant]. For a broad initial momentum distribution, theory predicts simply that $E\left(2\right)=2E\left(1\right)={\kappa }^2∕2$, but for a very narrow initial momentum distribution the energy $E\left(2\right)$ is heavily dependent on $\bar{k}$ and exhibits pronounced peaks and dips. The error bars reflect shot-to-shot variation in the atom number and fluctuation in the laser intensity.

Figure 3

Energy versus kick number for the lower kicking strength $\kappa =7.7$ (a) $\bar{k}=6.211$, (b) $\bar{k}=10.523$, and (c) $\bar{k}=12.573$. Crosses are experimental data, while the solid lines are the results of numerical simulations with $\kappa =7$. Plot (a) illustrates antiresonance behavior at $\bar{k}\simeq 2\pi$, while plot (b) demonstrates the presence of a period-4 antiresonance close to $\bar{k}=10.5$. Plot (c) illustrates enhanced energy growth close to the quantum resonance at $\bar{k}=4\pi$.

Figure 4

Energy versus kick number for the higher kicking strength, $\kappa =11.7$ (a) $\bar{k}=6.211$, (b) $\bar{k}=10.992$, and (c) $\bar{k}=12.573$. Similar resonance type behavior is observed in Fig. 3 as expected.

Figure 5

Numerical simulations of energy as a function of kick number for (a) $\bar{k}=6.211$ and (b) $\bar{k}=12.573$, with $\kappa =7$ and ${\overline{p}}_0=0$ (solid), ${\overline{p}}_0=0.1\hslash k_{\mathrm{L}}$ (dashed), and ${\overline{p}}_0=0.15\hslash k_{\mathrm{L}}$ (dot-dashed). The quantum antiresonance and resonance phenomena are clearly very sensitive to the initial mean momentum of the atomic cloud.

Figure 6

Theoretical predictions using Eq. (6) for the energy after 2 kicks, $E\left(2\right)$, versus effective Planck’s constant $\bar{k}$, for $\kappa =7$ and ${\overline{p}}_0=0$ (solid), ${\overline{p}}_0=0.25\hslash k_{\mathrm{L}}$ (dashed), and ${\overline{p}}_0=0.5\hslash k_{\mathrm{L}}$ (dot-dashed). These results further illustrate the sensitivity of the energy growth rate to the initial mean momentum of the atomic cloud, particularly at the quantum resonance $(\bar{k}=4\pi )$.