Experimental verification of a one-parameter scaling law for the quantum and “classical” resonances of the atom-optics kicked rotor

We present experimental measurements of the mean energy in the vicinity of the first and second quantum resonances of the atom-optics kicked rotor for a number of different experimental parameters. Our data are rescaled and compared with the one-parameter $\left(ϵ\right)$ classical scaling function developed to describe the quantum resonance peaks. Additionally, experimental data are presented for the “classical” resonance which occurs in the limit as the kicking period goes to zero. This resonance is found to be analogous to the quantum resonances, and a similar one-parameter classical scaling function is derived, and found to match our experimental results. The widths of the quantum and classical resonance peaks are compared, and their sub-Fourier nature examined.

Figure 1

Schematic diagram of our experimental setup. A standard six-beam magneto-optical trap (MOT) of about ${10}^5$ Cs atoms is formed inside a vacuum cell at the intersection of three retroreflected “trapping” beams [vertical beams and (anti-)Helmholtz coils are not shown]. A standing wave is formed across the cloud of atoms by retroreflecting light from a “kicking laser,” which is transported to the MOT by means of a single-mode fiber (SMF). This light is pulsed on and off by an acousto-optic modulator (AOM) which is gated by a programmable pulse generator (PPG). The PPG’s pulse train is uploaded from a computer, which also controls the timing of the experiment (e.g., when the trapping AOM and anti-Helmholtz coils are turned on and off).

Figure 2

Experimentally measured mean energies around the first quantum resonance at $\tau =2\pi$ after (a) 5, (b) 10, and (c) 15 kicks. Error bars show an average over three independent experiments. The kicking strength and initial momentum standard deviation were measured to be $k=4.1\pm 0.6$ and ${\sigma }_p=5.9\pm 0.2$, respectively. Note that the estimated errors in these parameters do not take into account systematic drifts which take place over the course of experimental runs. The solid line joins the experimental points to aid the eye.

Figure 3

Experimentally measured mean energies around the second quantum resonance at $\tau =4\pi$ for (a) 5; (b) 10, and (c) 15 kicks. The kicking strength and initial momentum standard deviation were measured to be $k=5.0\pm 0.5$ and ${\sigma }_p=6.3\pm 0.1$, respectively. Error bars as in Fig. 2. We note both in this figure and in Fig. 2 that the resonances exhibit some asymmetry, which is thought to be of purely experimental origin (see the discussion in Sec. 4).

Figure 4

Experimental mean energies around $\tau =2\pi$ taken from Fig. 2 and rescaled as $(⟨E_{t,ϵ}⟩-{\sigma }_p^2∕2)∕(t{k}^2∕4)$. Triangles are for $t=5$, squares for $t=10$, and circles for $t=15$. Error bars represent statistical fluctuations over three experiments, and do not take into account fluctuations in $k$ or ${\sigma }_p$. The solid line shows the numerically evaluated scaling function $R\left(x\right)$ of Eq. (8). We note that, for 10 and 15 kicks, data for $\mid ϵ\mid <0.03$ have been omitted due to our inability to accurately resolve atomic energies for fast atoms this close to resonance. Experimental data for both positive and negative values of $ϵ$ are plotted. We would like to note the good correspondence between the $ϵ$ classical prediction and the experimental data for over one order of magnitude in the scaling variable $x$.

Figure 5

Scaled experimental mean energies around $\tau =4\pi$ taken from Fig. 3; triangles are for $t=5$, squares for $t=10$, and circles for $t=15$ kicks. The solid line shows the scaling function $R\left(x\right)$ from Eq. (8). Again, for 10 and 15 kicks, data too close to resonance, i.e., for $\mid ϵ\mid <0.03$, have been omitted.

Figure 6

(a) Circles show experimentally measured mean energies as $\tau \to 0$ after 5 kicks. The measured value of $k$ is $4.9\pm 0.2$. The solid line is classical data for $k=4.9$, as generated by the map (9), using practically the same initial momentum distribution as in the experiment. The thermal energy ${\sigma }_p^2∕2$ has been subtracted to facilitate comparison with the quantum resonance curve in (b). In (b), circles show experimental data after 5 kicks near the second quantum resonance for positive $ϵ=\tau -4\pi$ and the experimental parameters are as given for Fig. 3. The thermal energy ${\sigma }_p^2∕2$ has been subtracted. The solid line shows $ϵ$ classical data as generated by the map (6).

Figure 7

(a) Circles show experimental data as $\tau \to 0$ for 10 kicks. The other experimental parameters are the same as those given for Fig. 6a. The circles in (b) show experimental data once again for the second quantum resonance after 10 kicks this time. Other experimental parameters are the same as those given for Fig. 6b. We note that for the quantum resonance in (b), the simulation and experimental results differ most markedly near the resonance peak. In this region $(ϵ\lesssim 0.03)$, some fast, resonant atoms are being lost from the experimental viewing area leading to a lower energy growth rate than predicted theoretically (see discussion in Secs. 1, 2). Note that in (a) it is not possible to probe low values of $\tau =ϵ$ due to the finite width of the pulses.

Figure 8

Rescaled experimental mean energies for $\tau =0.033–0.284$ (corresponding to $0.32–2.75\mu \mathrm{s}$). The data are for $k=4.9$ with $t=3$ (circles), 7 (diamonds), and 16 (stars). Error bars indicate statistical fluctuations over three experiments, and do not include variations in $k$ or ${\sigma }_p$. The solid line shows the classical scaling function of Eq. (12). The dash-dotted line shows the scaling function from Eq. (8) (valid for the quantum resonances) for comparison.