To extract directed transport from random fluctuations is a problem at the heart of statistical mechanics with a long history, including links to the Maxwell demon. In far from equilibrium systems, in presence, for instance, of unbiased ac-driving, noise and dissipation, a directed transport, also known as ratchet effect, can be generated. The appearance of ratchet transport has recently gained renewed attention due to its possible relevance for biological transport, molecular motors and the prospects of nanotechnology. Using the method of quantum trajectories, we study the quantum dynamics of dissipative systems which are chaotic in the classical limit. We demonstrate [1] a quantum chaotic dissipative ratchet appearing for particles in a pulsed asymmetric potential in the presence of a dissipative environment. The system is characterized by directed transport emerging from a quantum strange attractor. This model exhibits, in the limit of small effective Planck constant, a transition from quantum to classical behavior, in agreement with the correspondence principle. We also discuss parameter values suitable for implementation of the quantum ratchet effect with cold atoms in optical lattices. Finally, we show the transition from wave packet collapse to explosion, induced by varying the dissipation rate [2]. For strong dissipation the quantum wave function in the phase space collapses onto a compact packet which follows classical chaotic dynamics and whose area is proportional to the Planck constant. At weak dissipation the exponential instability of quantum dynamics on the Ehrenfest time scale dominates and leads to wave packet explosion. The transition from collapse to explosion takes place when the dissipation time scale exceeds the Ehrenfest time. For integrable nonlinear dynamics the explosion practically disappears leaving place to collapse.

[1] G.G. Carlo, G. Benenti, G. Casati and D.L. Shepelyansky, ``Quantum ratchets in dissipative chaotic systems'', Phys. Rev. Lett. 94, 164101 (2005).

[2] G.G. Carlo, G. Benenti and D.L. Shepelyansky, ``Dissipative quantum chaos: transition from wave packet collapse to explosion'', Phys. Rev. Lett. 95, 164101 (2005).