Achieving optimal rectification using underdamped rocked ratchets

An underdamped rocked ratchet operated at very low temperatures and damping is shown: (i) to be capable of rectifying the ac input signal more efficiently than in the overdamped regime; (ii) to be insensitive to the initial conditions, at variance with noiseless, or deterministic, ratchets; and (iii) to be characterized by a wide damping “window,” where its efficiency is appreciable also for weak input amplitudes. All these properties are rather robust, irrespective of the wave form of the drive and the ratchet potential. Our results relate to recent experiments on current-biased annular Josephson junctions and also on rectifiers of magnetic flux quanta in superconductors.

Figure 1

(Color online) Response function, $\gamma v∕A\equiv \gamma \mu$ vs damping $\gamma$, of the inertial ratchet (1) with drive (4) and harmonic potential (5) for (a) $A=2$, (b) $A=1$, (c) $A=0.5$, and different values of $kT$. Other simulation parameters are $L=1$ and $\Omega ={10}^{-3}$.

Figure 2

(Color online) Response function, $\gamma v∕A\equiv \gamma \mu$ vs damping $\gamma$, of the inertial ratchet (1) with drive (4) and piecewise linear potential (6) for (a) $A=2$, (b) $A=1$, (c) $A=0.3$, and different values of $kT$. Other simulation parameters: $Q=0.35$, $L_1=0.8$, $L_2=0.2$, and $\Omega ={10}^{-3}$. The potential profile is drawn in the inset of panel (a).

Figure 3

(Color online) Underdamped particle $(\gamma =0.03)$ in the symmetric washboard potential $V\left(x\right)={\omega }_0^2\cos x$ with ${\omega }_0=1$ and $L=2\pi$. (a) The locked-to-running states transition. The thresholds $F_1=(4∕\pi ){\omega }_0\gamma$ (pinning) and $F_3={\omega }_0^2$ (depinning) of the hysteretic loop (dashed curves) and the zero-temperature step at $F_2\simeq 3.36{\omega }_0\gamma$ (solid curve) are marked explicitly; (b) the locked-to-running transition threshold $F_{\mathrm{th}}$ at $kT=0.3$ is plotted vs $\gamma$ (solid circles); simulation data bridge the zero-temperature asymptotes $F_2$ ($\gamma \to 0$, dashed tangent line) and $F_3$ ($\gamma \to \infty$, solid line) [19]. Simulation data (empty circles) for $F_1$ vs $\gamma$ are plotted for a comparison; the dashed tangent line at the origin represents the analytical approximation $F_1=(4∕\pi ){\omega }_0\gamma$ for $\gamma \to 0$.

Figure 4

(Color online) Underdamped particle $(\gamma =0.03)$ in the asymmetric potential (5) with $L=2\pi$: Mobility curves $\mu \left(A\right)$ for (a) $T=0+$, Eq. (12), and (b) $T=0$, Eq. (13). Note that the rectification curve in (b) matches recent experimental data in Ref. 24.

Figure 5

(Color online) Temperature dependence of $\gamma \mu$ for different drive amplitudes $A$ and two damping regimes: underdamped $\gamma =0.1{\omega }_0$ (open dots) and overdamped $\gamma =10{\omega }_0$ (solid dots). The ratchet dynamics (1) has been simulated for the harmonic potential (5) with $L=1$ and up-down ramp (4) with $\Omega ={10}^{-3}$.

Figure 6

(Color online) Curves $\gamma \mu$ and ${\rho }_E$ vs $\gamma$ for the inertial ratchet (1) with drive (3) and harmonic potential (5). Other simulation parameters are: $kT=0.03$, $A=0.5$, $L=1$, and $\Omega ={10}^{-3}$.