Directed anomalous diffusion without a biased field: A ratchet accelerator

Directed classical current that increases linearly with time without using a biased external field is obtained in a simple model Hamiltonian system derived from a modified kicked rotor model, by breaking the spatial symmetry of the transporting regular islands in the classical phase space. A parallel study of the corresponding quantum dynamics suggests that although quantum coherence effects suppress the directed current and can even induce a current reversal, directed quantum current that increases linearly with time can nonetheless be realized in a quantum system that is far from the classical limit.

Figure 1

Classical phase space structures of (a) the standard map [${\kappa }_1=3.8$, $f\left(n\right)=1$, ${\kappa }_2=0$], (b) a modified kicked rotor model [${\kappa }_1=3.8$, $f\left(n\right)=g\left(n\right)$, ${\kappa }_2=0$], and (c) a ratchet accelerator [${\kappa }_1=3.8$, $f\left(n\right)=g\left(n\right)$, ${\kappa }_2=0.2$]. All variables are in dimensionless units. Note that the regular islands seen in (b) and (c) are transporting while those in (a) are not.

Figure 2

(a) The linear time dependence of the classical directed current defined as the ensemble averaged value of the momentum, with the initial condition $\tilde{p}=0$ and $\tilde{x}$ uniformly distributed between $-\infty$ and $+\infty$. (b) The associated time dependence of the momentum variance in a $\log \text{−}\log$ plot. The system parameters are the same as those in the case of Fig. 1c. All variables are in dimensionless units.

Figure 3

The time dependence of the quantum dimensionless directed current defined as the expectation value of $\widehat{\tilde{p}}$ for two values of the effective Planck constant $\tau$, with the initial condition given by $\mid 0⟩$ and other parameters are the same as in Fig. 1c. Results here should be compared with those in Fig. 1c.

Figure 4

The probability $P\left(m\right)$ of finding the quantum system in state $\mid m⟩$ after 4000 kicks, with the initial state given by $\mid 0⟩$. The solid line is for $\tau =0.1$ and the dashed line is for $\tau =0.2$. Other parameters are the same as those in the case of Fig. 1c.