Ratchet Effect for Cold Atoms in an Optical Lattice

The possibility of realizing a directed current for a quantum particle in a flashing asymmetric potential is investigated. It is found that quantum resonances, where the value of the effective Planck constant is equal to an integer or half-integer multiple of $\pi$, give rise to a directed current. The effect should be readily observable in experiments.

Figure 1

Time development of the wave number $⟨k⟩$ and the square wave number $⟨k^2⟩$ for a number of parameter choices. In all cases, the potential strength $P=0.5$ and the asymmetry $\alpha =0.3$. In the topmost two panels, the lines with circles represent the choice $\mathrm{k̵}=0.711\pi$ and the lines with crosses are $\mathrm{k̵}=0.232\pi$. In the middle panels, the lines with circles represent $\mathrm{k̵}=1.0\pi$ and the lines with crosses $\mathrm{k̵}=1.001\pi$. In the bottom two panels, the lines with circles represent $\mathrm{k̵}=0.5\pi$ and the lines with crosses $\mathrm{k̵}=0.501\pi$.

Figure 2

Mean wave number of the wave packet over time, as a function of the effective Planck constant $\mathrm{k̵}$. The asymmetry of the potential has been chosen to $\alpha =0.3$, and the potential strength $P=0.5$. In the topmost panel, the initial state is a homogeneous zero-momentum state and the averaging is done over 20 temporal periods for the dashed line and 100 periods for the solid line. In the lower panel, the initial state was chosen to be the ground state of the potential.

Figure 3

Momentum distribution of the wave function. The top left panel displays the momentum distribution for the resonant case $\mathrm{k̵}=0.5\pi$ after one (dotted line) and 100 kicks (solid line). The initial state was a homogeneous zero-momentum state. The top right panel shows the corresponding result for the case $\mathrm{k̵}=2\pi$. The bottom left panel shows the corresponding result for $\mathrm{k̵}=2\pi$ for inhomogeneous initial conditions, i.e., when the initial state is taken to be the ground state of the potential. The bottom right panel shows the difference between the momentum distributions for $\mathrm{k̵}=2\pi$ with inhomogeneous and homogeneous initial conditions, $\Delta n=n_{\mathrm{inhom}}-n_{\mathrm{hom}}$.