Chaotic ratchet dynamics with cold atoms in a pair of pulsed optical lattices

We present a very simple model for realizing directed transport with cold atoms in a pair of periodically flashed optical lattices. The origin of this ratchet effect is explained and its robustness demonstrated under imperfections typical of cold atom experiments. We conclude that our model offers a clear-cut way to implement directed transport in an atom optical experiment.

Figure 1

Phase space pictures for $\varphi =\pi ∕2$, at $n=20$: classical Poincaré sections (a) and quantum Husimi functions at ${\hslash }_{\mathrm{eff}}\simeq 0.16$ (b) and ${\hslash }_{\mathrm{eff}}\simeq 1$ (c). The displayed region is given by $I=pT∊[-20,20]$ (vertical axis) and $x∊[0,2\pi )$ (horizontal axis). Note that, to draw the attractor, $x$ is taken modulus $2\pi$. The brightness is inversely proportional to the density: black (white) regions correspond to maximal (zero) density.

Figure 2

Average rescaled momentum $⟨I⟩=⟨p⟩T$ as a function of the discrete time $n$, for the same parameter values as in Fig. 1. The solid curve corresponds to the classical case, while the dashed curve corresponds to quantum results for ${\hslash }_{\mathrm{eff}}\simeq 0.16$ and the dot-dashed one to ${\hslash }_{\mathrm{eff}}\simeq 1$.

Figure 3

Average momentum $⟨p⟩$ as a function of $n$, for $\varphi =\pi ∕2$ (positive values), $\varphi =0$ (zero values), and $\varphi =-\pi ∕2$ (negative values). Both the classical (solid curves) and the quantum case (dot-dashed curves, ${\hslash }_{\mathrm{eff}}\simeq 1$) are shown. Note that at $\varphi =0$ quantum and classical curves are almost superimposed.

Figure 4

Average momentum $⟨p⟩$ as a function of the discrete time $n$, for the same parameters as in Fig. 2, at ${\hslash }_{\mathrm{eff}}\simeq 1$. At each kick, the quasimomentum can jump to any other possible value with probabilities 0 (dot-dashed curve), 0.2 (dashed curve), and 0.5 (long dashed curve). The solid curve corresponds to the classical case.

Figure 5

Average current ${⟨p⟩}_{\mathrm{av}}$ as a function of the noise $\delta K$ in the kick strength $K$, for parameter values as in Fig. 2, at ${\hslash }_{\mathrm{eff}}\simeq 1$.

Figure 6

Average current ${⟨p⟩}_{\mathrm{av}}$ as a function of the relative error $\delta T∕T$ in the kicking period $T$, for parameter values as in Fig. 5.

Figure 7

Average current ${⟨p⟩}_{\mathrm{av}}$ as a function of the relative error $\delta p_c∕p_c$ in the cutoff momentum $p_c$, for parameter values as in Fig. 5. A magnification of the figure for small values of $\delta p_c∕p_c$ is shown in the inset.