Controlling vortical motion of particles in two-dimensional driven superlattices

We demonstrate the control of vortical motion of neutral classical particles in driven superlattices. Our superlattice consists of a superposition of individual lattices whose potential depths are modulated periodically in time but with different phases. This driving scheme breaks the spatial reflection symmetries and allows an ensemble of particles to rotate with an average angular velocity. An analysis of the underlying dynamical attractors provides an efficient method to control the angular velocities of the particles by changing the driving amplitude. As a result, spatially periodic patterns of particles showing different vortical motions can be created. Possible experimental realizations include holographic optical lattice based setups for colloids or cold atoms.

Figure 1

Schematic representation of superlattice setups A and B formed by the superposition of four square sublattices driven with an amplitude $V$ but at different phases ${\varphi }_i=\frac{(i-1)\pi }{2}, i=1,2,3,4$. Each red circle denotes the position of an individual Gaussian well. The thick black dashed lines denote the boundary of the lattice unit cells. The spatial period of setup A is (2,2,0), whereas that of setup B is (3,3,0) due to the presence of empty sites without any wells. The blue and green regions in (a) denote plaquettes with clockwise and counterclockwise chiralities with respect to the spatial orientation of the wells with driving phases ${\varphi }_i$. The remaining parameters are $V=0.41, \alpha =3, \gamma =0.1$.

Figure 2

Typical trajectories exhibiting rotational motion in (a) setup A and (c) setup B over one time period of rotation (color bars). The colored circles denote the positions of individual Gaussian wells with different driving phases ${\varphi }_i$. (b) and (d) The fraction of particles $\rho \left(\overline{\mathrm{\Omega }}\right)$ possessing mean angular velocity $\overline{\mathrm{\Omega }}$ for different noise strengths $D$ in setups A and B, respectively. The insets show the variation of the net rotational current ${\mathbf{J}}_{\mathrm{\Omega }}$ with $D$. The remaining parameters are the same as in Fig. 1.

Figure 3

(a) Bifurcation diagram of ${\mathrm{\Omega }}^{\prime }\left(t\right)$ as a function of the driving amplitude $V$ depicting the chaotic (broad blue bands) and regular (thin blue lines) attractors of setup B [see Fig. 1]. (b) The mean angular velocity $\overline{\mathrm{\Omega }}$ of the attractors in Fig. 3 as a function of $V$. The values of $\overline{\mathrm{\Omega }}$ for the regular attractors denoting rotational motion and the turning number $\tau$ of the corresponding closed curves are labeled with arrows. The remaining parameters are the same as in Fig. 1.

Figure 4

(a) Schematic representation of one unit cell of our setup consisting of four plaquettes, ${\mathcal{D}}_1, {\mathcal{D}}_2, {\mathcal{D}}_3$, and ${\mathcal{D}}_4$, with the thick dashed lines denoting the plaquette boundaries. The shaded circles denote the positions of individual Gaussian wells driven with amplitudes $V_1=0.51$ or $V_2=0.078$ and phases ${\varphi }_i$. ${\mathcal{D}}_1$ and ${\mathcal{D}}_4$ (${\mathcal{D}}_2$ and ${\mathcal{D}}_3$) have counterclockwise (clockwise) chirality with respect to the spatial orientation of the wells with driving phases ${\varphi }_i$. Trajectories of particles exhibiting vortical motion for $D=0$ with positive (red) and negative (blue) $\overline{\mathrm{\Omega }}$ have been superimposed on the unit cell. The trajectories in ${\mathcal{D}}_1, {\mathcal{D}}_2, {\mathcal{D}}_3$, and ${\mathcal{D}}_4$ have $\overline{\mathrm{\Omega }}=-1$, 1, $-\frac{1}{3}$, and $\frac{1}{3}$, respectively. (b) and (c) An extract of the spatial arrangements of the trajectories exhibiting vortical motion within different plaquettes for $D={10}^{-4}$ and $D={10}^{-3}$, respectively. The remaining parameters are the same as in Fig. 1.