Deterministic Ratchet from Stationary Light Fields

Ratchets are dynamic systems where particle transport is induced by zero-average forces due to the interplay between nonlinearity and asymmetry. Generally, they rely on the effect of a strong external driving. We show that stationary optical lattices can be designed to generate particle flow in one direction while requiring neither noise nor driving. Such optical fields must be arranged to yield a combination of conservative (dipole) and nonconservative (radiation pressure) forces. Under strong friction all paths converge to a discrete set of limit periodic trajectories flowing in the same direction.

Figure 1

Equipotential lines are plotted with interstitial colors growing from blue (potential minima) to white (maxima). Red arrows show the individual trajectories of particles subject to the total force field, which includes the conservative force deriving from the shown potential and a nonconservative force obtained from the curl of a not shown vector potential. The square $(-1,3)\times (-1,3)$ is a unit cell of the resulting periodic structure. Points $P$ and $Q$ are described in the caption of Fig. 2. (a) Single-laser case [$\delta =0$, $\beta =1.4$ in Eqs. (2, 4)]; particles are confined within a quarter unit cell but even a minor amount of thermal noise allows them to diffuse isotropically throughout the periodic structure. (b) Double-laser case ($\delta =1$, $\beta =1.4$); all trajectories evolve to the left, showing a ratchet effect which stems from purely deterministic and stationary forces.

Figure 2

Grey arrows show particle trajectories under the effect of the total (nonconservative plus conservative) force. The force is horizontal/vertical at the blue/red lines. (a) $\delta =0$, $\beta =1.4$; (b) $\delta =1$, $\beta =0.35$; (c) $\delta =1$, $\beta =0.45$; (d) $\delta =1$, $\beta =1.4$. Points $P$ and $Q$ become critical (zero force) at the transition between flow regimes, defined by the presence or absence of the various classes of attractive fixed points.

Figure 3

A set of 100 trajectories is shown for the setup of Figs. 1b, 2d. Initial positions are random within the chosen unit cell. All paths converge to two limit periodic trajectories which flow to the left.

Figure 4

Average velocity $v$ of the limit periodic trajectories, in units of ${\alpha }^{\prime }{E}_0^{2}k/4\eta$ [see Eqs. (2, 3, 4)]. (a) $v/(1+2\beta )$ as a function of $\beta$ for second-laser intensity $\delta =1$. (b) $v$ as a function of $\delta$ for $\beta =1.4$.