Scaling law and stability for a noisy quantum system

We show that a scaling law exists for the near-resonant dynamics of cold kicked atoms in the presence of a randomly fluctuating pulse amplitude. Analysis of a quasiclassical phase-space representation of the quantum system with noise allows a new scaling law to be deduced. The scaling law and associated stability are confirmed by comparison with quantum simulations and experimental data.

Figure 1

(a) Experimental measurements of quantum resonance peaks as a function of $ϵ$ for noise levels $L=0$ (⋆), $L=0.5$ (◻), $L=1.0$ (◇), $L=1.5$ (△), and $L=2.0$ (○). (b) Points show experimentally measured peak energies (circles), whilst the dashed line shows the theoretical formula Eq. (4). Crosses show simulation energies, which exactly agree with the theory. Sample error bars are plotted from shot-to-shot measurements, not taking into account systematic uncertainties in the absolute value of $k$.

Figure 2

(Color online) Phase-space diagrams showing the effect of amplitude noise on the pseudoclassical map (2). Left panel: without noise. Middle panel: noise level $L=1.5$. Right panel: pendulum trajectories for various initial conditions. The separatrix is shown by a thick, dashed line. The gray shaded area shows a region of $\pm ⟨\Delta J_{\mathrm{res}}^2⟩$ for $L=1.5$ about the separatrix, demonstrating the trajectories which lead to the correction in Eq. (7).

Figure 3

(Color online) Theoretical scaling function for $L=0$ (thick solid line), $L=0.5$ (dotted line), $L=1.0$ (dashed line), $L=1.5$ (dot-dashed line) and $L=2.0$ (thin line). Simulation data is also shown, rescaled by the factor $\frac{1}{4}{k}^{2}t$, for $L=0.5$ (◻), $L=1.0$ (◇), $L=1.5$ (△), and $L=2.0$ (○). Open symbols are produced for fixed $k=2.8$, varying $ϵ∊[{10}^{-3},0.1]$, $t∊[20,150]$, while filled symbols represent data for randomly chosen values of $k∊[1,10]$, $ϵ∊[{10}^{-3},0.1]$, $t∊[2,150]$.

Figure 4

Theoretical scaling function of Eq. (7) (as shown in Fig. 3) is compared with rescaled experimental data (as, e.g., from Fig. 1). Shown are data across more than one order of magnitude in the scaling variable $x$ for (a) $L=0$ (⋆), $L=1.0$ (◇), and $L=2.0$ (○) and (b) $L=0.5$ (◻) and $L=1.5$ (△). Theoretical curves are shown with the same line styles as in Fig. 3. Note that for $x<5$ the data comes from separate five kick experiments, and the error bars are the same size as the plotted points. Sample error bars are calculated as described in the caption of Fig. 1. Data for $\mid ϵ\mid >0.15$ is excluded since the pseudoclassical theory breaks down in this region.