Environmental rocking ratchet: Environmental rectification by a harmonically driven avoided crossing

We propose a rocking ratchet designed as a symmetric quantum two-state system driven by a single periodic harmonic force and influenced symmetrically by thermal fluctuations. We show that the necessary broken symmetry can dynamically be achieved by a thermal environment that couples to the energy difference between the two states and the tunnel coupling between them. The quantum two-state system is driven by the harmonic periodic drive through its avoided crossing. The correspondingly driven dissipative quantum dynamics results on average in a finite population difference between both states. This then causes directed particle transport.

Figure 1

Landau-Zener probability versus sweep speed $v$ plotted for various mixings, i.e., ${\beta }_x$ and ${\beta }_z=\sqrt{1-{\beta }_x^2}$ of the environmental fluctuations. Stars show results for forward sweeps, whereas triangles show results for the corresponding backward sweep.

Figure 2

Plot of $\mathrm{\Xi }$ (color coded) versus ${\beta }_x$ and driving frequency $\mathrm{\Omega }$ for five different temperatures and three different driving amplitudes.

Figure 3

Plot of $\mathrm{\Xi }$ (color coded) shown versus ${\beta }_x$ and driving amplitude $A$ for $T=4\mathrm{\Delta }$ and $\mathrm{\Omega }=5\times {10}^{-4}\mathrm{\Delta }$.