Observation of High-Order Quantum Resonances in the Kicked Rotor

Quantum resonances in the kicked rotor are characterized by a dramatically increased energy absorption rate, in stark contrast to the momentum localization generally observed. These resonances occur when the scaled Planck’s constant $\tilde{\hslash }=\frac{r}{s}4\pi$, for any integers $r$ and $s$. However, only the $\tilde{\hslash }=r2\pi$ resonances are easily observable. We have observed high-order quantum resonances ($s>2$) utilizing a sample of low energy, noncondensed atoms and a pulsed optical standing wave. Resonances are observed for $\tilde{\hslash }=\frac{r}{16}4\pi$ for integers $r=2–6$. Quantum numerical simulations suggest that our observation of high-order resonances indicate a larger coherence length (i.e., coherence between different wells) than expected from an initially thermal atomic sample.

Figure 1

Experimental graphs showing energy in recoil energies, $E_r$, versus $\tilde{\hslash }/\pi$, and period, $T$, both with a kick number of 16 (a) Kick strength of $k=1.23$. High-order resonances are seen for $\tilde{\hslash }=\frac{r}{16}4\pi$ for $r=2–6$. (b) Same as (a) but with a kick strength of $k=1.64$. Background trend is from a general kick-strength dependent quantum diffusion rate which is shown in (b) in the inset for $k=1.6$.

Figure 2

Magnitude of the $\tilde{\hslash }=\frac{3}{4}\pi$ resonance as a function of the kick number. Solid and open circles indicate different sets of experimental data.

Figure 3

Numerical simulations showing energy in recoil energies, $E_r$, versus $\tilde{\hslash }/\pi$, and period, $T$, both with a kick number of 16 and $k=1.6$. (a): The rms spatial distribution of the initial wave function is $x_0=0.109\mu \mathrm{m}$ ($v_0=3.43\mathrm{mm}/\mathrm{s}$). (b): Initial distributions with coherence throughout the sample. The gray line has an initial distribution of a sum of in-phase Gaussians, each separated by the lattice spatial period, $l=0.94\mu \mathrm{m}$, while that of the black line has a wide initial rms spatial width of $x_0=1.09\mu \mathrm{m}$.