Underdamped quantum ratchets

We investigate the quantum ratchet effect under the influence of weak dissipation which we treat within a Floquet-Markov master equation approach. A ratchet current emerges when all relevant symmetries are violated. Using time-reversal symmetric driving we predict a purely dissipation-induced quantum ratchet current. This directed quantum transport results from bath-induced superpositions of non-transporting Floquet states. Copyright c © EPLA, 2009 An intriguing phenomenon in non-equilibrium transport is the ratchet effect [1–3], i.e., the emergence of directed motion in the absence of any net bias. Net transport results from an interplay between ac driving, spatiotemporal asymmetries, and non-linearities in a periodic potential. This mechanism provides the basis for an increasing number of experiments ranging from particle transport in biological systems [4] and nano-engines [5] to charge transport in semiconductor heterostructures [6,7], superconductors [8] and spin transport [9]. Symmetry investigations revealed the necessary conditions on the ac force and the static potential, such that a ratchet current can emerge [10–13]. A widely employed model for studying the ratchet effect is a one-dimensional periodic potential in which classical Brownian particles move [1–5]. It describes also the motion of a thermal cloud of cold atoms in an ac-driven optical potential [13]. As the atom cloud is cooled down further, one expects quantum effects to become relevant [14]. The Hamiltonian limit of such quantum ratchets has been studied recently [15–17]. A more realistic description of quantum ratchets necessitates the inclusion of the ubiquitous decoherence and quantum dissipation [18–21]. For moderate-to-strong dissipation, incoherent tunneling transitions prevail and the quantum ratchet current can be studied within quantum rate theory [18–21], while in the high-temperature limit, one can employ a Fokker-Planck equation with quantum corrections [22]. For very strong friction, a description in terms of an effective Smoluchowski equation comprising leading-order quantum corrections is appropriate [23]. By contrast, the crossover towards the coherent quantum regime, i.e., the underdamped regime [24], in which already weak decoherence significantly alters the Hamiltonian dynamics, still represents an ambitious challenge. In this letter we study ac-driven quantum ratchet transport in the technically demanding regime of weak quantum dissipation where quantum coherence and relaxation affect each other. We analyze within a Floquet-Markov description [25] the dynamics on quantum attractors by expanding them into the Floquet states of the corresponding coherent time-dependent system. Then a most intriguing question is whether violation of time-reversal symmetry due to weak quantum dissipation is perceivable in the quantum attractor and in the quantum ratchet current. Model and quantum master equation. – A quantum particle in a time-dependent periodic potential obeys the Schrödinger equation