Force-dependent Fluxes of Adiabatic Classical and Quantum Rocking Ratchets

We consider the adiabatic mode of Brownian particle motion in a periodic potential under the action of symmetric dichotomous fluctuations of an external force F with zero mean value (rocking ratchet), in which the fluctuation frequency is much less than the inverse relaxation time of the particle in each of the states of the dichotomous process. Expressions are given for force-dependent fluxes of an adiabatic classical rocking ratchet. In the absence of thermal fluctuations, within the semiclassical approximation, analytical expressions are obtained for the rocking-ratchet tunneling flux in a sawtooth periodic potential of arbitrary asymmetry and in the potential of two sinusoids. It is shown that the tunneling flux has a linear asymptotics in modulus of small F due to (i) the absence of reverse tunneling fluxes with respect to the direction F and (ii) the root dependence of the integrand of the Gamow formula on the potential energy. The main parameters of the model are the energy barrier V0 and the period L of the potentials, as well as the rocking force F and the asymmetry parameter ξ = l / L of the sawtooth potential with the width of one of its teeth equal to l. It is shown that the direction of quantum ratchet motion is opposite to the motion direction of the corresponding classical ratchet in a limited range of values of the rocking force |F| L / V0 < αc , where the parameter αc changes from the value α1 = (√5 − 1) / 2 ≈ 0.618 for the extremely asymmetric sawtooth potential (ξ = 1) to the value α2 = 2 / 3 ≈ 0.667 for the symmetric potential (ξ= 1 / 2). In the range of values α2 < αc < α2, the sign of the tunneling flux changes with the change in the asymmetry parameter ξ. Numerical calculations for the potential of two sinusoids corresponding to the effective value ξ ≈ 0.655 of the asymmetry lead to similar results with αc ≈ 0.81.