Vortex and Translational Currents due to Broken Time-Space Symmetries

We consider the classical dynamics of a particle in a ($d=2,3$)-dimensional space-periodic potential under the influence of time-periodic external fields with zero mean. We perform a general time-space symmetry analysis and identify conditions, when the particle will generate a nonzero averaged translational and vortex currents. We perform computational studies of the equations of motion and of corresponding Fokker-Planck equations, which confirm the symmetry predictions. We address the experimentally important issue of current control. Cold atoms in optical potentials and magnetic traps are among possible candidates to observe these findings experimentally.

Figure 1

(a) Dependence of the current components, $J_x$(solid line) and $J_y$(dashed line), on $\theta$ for (1, 10, 11) with $D=1$, $E_x^{(1)}=-E_x^{(2)}=2$, $E_y^{(1)}=-E_y^{(2)}=2.5$. Data are for the overdamped ($\mu =0$, $\gamma =1$ thick lines) and underdamped ($\mu =1$, $\gamma =0.1$ thin lines) cases, respectively; (b) the time evolution of the mean particle position, $\overline{\mathbf{r}}(t)=\int \mathbf{r}P(\mathbf{r},\mathbf{v},t)d\mathbf{r}d\mathbf{v}$, for $\theta =0$. The trajectories are superimposed on the contour plot of the potential (10). Curve (i) corresponds to $E_x^{(1)}=3$, $E_x^{(2)}=E_y^{(1)}=0$, $E_y^{(2)}=3.5$ and curve (ii)—to the parameters of panel (a). The other parameters are $D=\mu =1$, $\gamma =0.1$; (c) the phase lag ${\theta }_x^{(0)}$ as a function of the dissipation strength $\gamma$.

Figure 2

(a) Dependence $J_{\Omega }(\theta )$, Eq. (12), for (1, 16, 17), with $\mu =1$, $D=0.5$, $E_x^{(1)}=0.4$, $E_y^{(1)}=0.8$ and $\gamma =0.2$ (solid line), $\gamma =0.05$ (dashed line), and $\gamma =2$ (dash-dotted line). Insets: the trajectory (left inset) and the corresponding attractor solution, $\overline{\mathbf{r}}(t)$, (right inset) for the case $\gamma =0.2$ and $\theta =\pi /2$. We have used $N={10}^5$ independent stochastic realizations to perform the noise averaging; (b) the phase lag ${\theta }_0$ as a function of the dissipation strength $\gamma$.