Observation of robust quantum resonance peaks in an atom optics kicked rotor with amplitude noise

The effect of pulse train noise on the quantum resonance peaks of the atom optics kicked rotor is investigated experimentally. Quantum resonance peaks in the late time mean energy of the atoms are found to be surprisingly robust against all levels of noise applied to the kicking amplitude, while even small levels of noise on the kicking period lead to their destruction. The robustness to amplitude noise of the resonance peak and of the fall-off in mean energy to either side of this peak are explained in terms of the occurrence of stable, $ϵ$ classical dynamics [S. Wimberger, I. Guarneri, and S. Fishman, Nonlinearity 16, 1381 (2003)] around each quantum resonance.

Figure 1

Measured energies (a) and (b) near the first and second primary quantum resonances and associated simulation results (c) and (d) for various levels of amplitude noise. We have taken $\kappa ∕\bar{k}=3.77$, as calculated from Eq. (7) and the spontaneous emission rate used for simulations is 2.5%. The dotted line represents the no-noise case; circles, ${\mathcal{L}}_A=0.50$; diamonds, ${\mathcal{L}}_{\mathrm{A}}=1.0$; triangles, ${\mathcal{L}}_A=1.50$; and squares, ${\mathcal{L}}_A=2.0$. Sample error bars are shown on the second point for each curve.

Figure 2

Measured energies (a) and (b) near the first and second primary quantum resonances and associated simulation results (c) and (d) for various levels of period noise. We have taken $\kappa ∕\bar{k}=3.61$, as calculated from Eq. (8) and the spontaneous emission rate used for simulations is 2.5%. The dotted line represents the no-noise case; circles, ${\mathcal{L}}_P=0.01$; diamonds, ${\mathcal{L}}_P=0.02$; triangles, ${\mathcal{L}}_P=0.05$; and squares, ${\mathcal{L}}_P=0.1$. Sample error bars are shown on the second point for each curve. In (b), a straight line was fitted to the energies in the highest noise case. It gives the quasilinear energy as $66\pm 0.7$.

Figure 3

Analytical predictions of the effect of amplitude noise on the quantum resonance peak at $2\pi$ for $\kappa ∕\bar{k}=3.7$. (a) shows the behavior of the early energy growth rate $D$ [as given by Eq. (10)] of the classical and quantum kicked rotor as a function of $\bar{k}$ for amplitude noise levels of ${\mathcal{L}}_A=1$ and 2. Dashed and dotted lines are the classical rates for amplitude noise levels of 1 and 2, respectively. Solid and dash-dotted lines are the quantum rates for ${\mathcal{L}}_A=1$ and 2, respectively. (b) shows the resonant peak produced by 20 iterations of Eq. (12) about $\bar{k}=2\pi$ with ${\mathcal{L}}_A=0$ (circles), 0.5 (solid line), 1.0 (dash-dotted line), 1.5 (dashed line), and 2.0 (dotted line). Each point is an average of 10 different noise realizations. The letters $A-C$ indicate values of $ϵ$ as referenced from Fig. 4.

Figure 4

Phase space portraits for the $ϵ$ classical standard map for $\bar{k}=2\pi$ and $k=3.7$. The figures in the first row (a), (b), and (c) are for an amplitude noise level of 0 and for values of $ϵ$ of 0.001, 0.02, and 0.04, respectively. The second row (d), (e), and (f) shows the $ϵ$ classical phase space for an amplitude noise level of 2 and the same values of $ϵ$. In Fig. 3b, the values of $\bar{k}$ corresponding to $ϵ=0.001$, 0.02, and 0.04 are labeled $A,B$, and $C$, respectively.