Current reversals in a rocking ratchet: Dynamical versus symmetry-breaking mechanisms

Directed transport in ratchets is determined by symmetry breaking in a system out of equilibrium. A hallmark of rocking ratchets is current reversals: an increase in the rocking force changes the direction of the current. In this work for a biharmonically driven spatially symmetric rocking ratchet we show that a class of current reversal is precisely determined by symmetry breaking, thus creating a link between dynamical and symmetry-breaking mechanisms.

Figure 1

(Color online) Average atomic velocity as a function of the relative phase $\varphi$ between harmonics of the ac drive for several values of the driving amplitude $F_0$ and $\omega ={\omega }_v=k{\left(2U_0/m\right)}^{1/2}$. Top panel: simulation data in the weakly damped regime. The friction and the noise strength values are fixed to $\alpha =0.15{\alpha }_0$ and $D=1.944D_0$, respectively, where ${\alpha }_0=mk{v}_0$, $D_0={\alpha }_0^2{v}_0/k$, and $v_0={\left(U_0/m\right)}^{1/2}/10$. The values near $\varphi =\pi /2$ (or $\varphi =3\pi /2$) show a current reversal as the driving amplitude is increased. Bottom panel: simulation data in the Hamiltonian regime $\left(\alpha =D=0\right)$. The initial conditions were chosen within the chaotic sea shown in Fig. 3. Lines are a guide to the eye.

Figure 2

(Color online) Simulation data in the weakly damped regime: current as a function of the driving amplitude $F_0$ for several values of the friction $\alpha$ and a fixed driving phase $\varphi =\pi /2$; rest of the parameters as in Fig. 1. As the dissipation is decreased, the current reversal position is shifted to lower values of the driving amplitude $F_0$. The diamonds show the results for the Hamiltonian system. Lines are a guide to the eye.

Figure 3

(Color online) Simulation data in the Hamiltonian regime: the current as a function of the initial conditions $v\left(0\right)$ and $x\left(0\right)$ is represented using a color density plot. The driving parameters are $\omega ={\omega }_v$, $F_0=0.1U_{0}k$, and $\varphi =\pi /2$. A black circumference indicates the boundary of a regular circular region in which the current is zero throughout.

Figure 4

(Color online) Simulation data in the overdamped regime: current as a function of the driving phase $\varphi$ for several values of the driving amplitude $F_0$, with $\omega =0.01{\omega }_v$ and a friction $\alpha =100{\alpha }_0$. Top panel: $D=1.944\times {10}^3{D}_0$, with lines being a guide to the eye. Bottom panel: $D=0.1944\times {10}^{-3}{D}_0$. The inset in the bottom panel shows the current as a function of $F_0$ for a fixed driving phase $\varphi =1.01$, displaying multiple current reversals.

Figure 5

Experimental results for 1D rocking ratchet for cold atoms. (a)–(f) Average atomic velocity, rescaled by the recoil velocity $v_r$ ($v_r=5.88\text{mm}/\text{s}$ for ${}^{87}\text{R}\text{b}$), as a function of the relative phase $\varphi$ between harmonics of the ac drive, for different values of the driving force amplitude. (g) Dissipation-induced phase lag ${\varphi }_0$, as obtained by fitting data as those in (a)–(f) with the function $v/v_r=A\text{sin}\left(\varphi -{\varphi }_0\right)$, as a function of the driving amplitude. The amplitude $A$ is reported in (h). The parameters of the optical lattice are detuning from resonance $\Delta =-15\Gamma$ and intensity per lattice beam $I_L=105\text{mW}/{\text{cm}}^2$. The rocking force is of the form of Eq. (2), with $\omega /\left(2\pi \right)=100\text{kHz}$, $A=1$, $B=2$, and $F_0=-m{f}_0\omega /k$, where $m$ is the atomic mass, $k$ is the laser field wave vector, and the values of $f_0$ (in MHz) for the different sets of data are reported in the figures.