Directed transport in microscopic systems

A great challenge for the future technology on microscopic scale is the design and construction of microscopic motors that can use input energy to drive directed motion in the face of inescapable thermal and other noise. Driving such directed motion is what protein motors -perfected over the course of millions of years by evolution- do in every cell in our bodies [1-3].

Modern technologies go down to smaller and smaller scales, where quantum effects are expected to play a very important role. Indeed, along with thermal activation, quantum tunneling provides an alternative mechanism to overcome energy barriers and lead to directed motion. Therefore, for future technological progress it is very important to be able to create directed transport and to be able to control its direction in the presence of noise, decoherence, thermal fluctuations, and quantum effects. Such systems with directed transport, called ratchets, find applications in a very broad range of physical systems: mesoscopic systems with antidot lattices, experimental realization of quantum ratchets in semiconductor heterostructures, directed transport of cold atoms, thermal rectifiers in molecular chains, biomotors and directed transport in molecular wires and proteins.
Quantum ratchets are potentially useful in a number of technological applications such as novel rectifiers, pumps, molecular switches, and transistors.
The understanding of the physics of such systems might also bring important insights for the treatment and operation of quantum information at the nanoscale.
Quantum ratchets in dissipative chaotic systems
Using the method of quantum trajectories we have studied [4] 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.
The role of transporting islands in the chaotic sea for the corresponding classical model has been  investigated in [5]
We have also studied a model, consisting of two series of spatially periodic kicks, that offers a clear-cut way to implement directed transport with cold atoms in optical lattices [6].



Phase space pictures for the model discussed in Ref. [4] after 100 kicks: classical Poincare sections (left) and quantum Husimi functions at effective Planck constant equal to 0.012 (right).
The strange attractor is asymmetric and leads, independently of the initial conditions, to stationary directed transport (<p> different from zero).
Magnifications of the plots are shown in the second and third rows (the phase-space area is reduced by a factor 1/9 and 1/81, respectively):  the fractal structure of the classical strange attractor is smoothed on the scale of Planck's cell.
The color is proportional to the density: blue for zero and red for maximal density.
Many-body quantum ratchet in a Bose-Einstein condensate

Quantum Hamiltonian ratchets are relevant in systems such as cold atoms in which the high degree of quantum control may allow experimental implementations near to the dissipationless limit. Moreover, the realization of Bose-Einstein condensates (BECs) of dilute gases has opened new opportunities for the study of dynamical systems in the presence of many-body interactions. Indeed, it is possible to prepare initial states with high precision and to tune over a wide range the many-body atom-atom interaction. From the viewpoint of directed transport, the study of many-body quantum system is, to our knowledge, at the very beginning.

We have studied [7] the dynamics of a dilute Bose-Einstein condensate confined in a toroidal trap and exposed to a pair of periodically flashed optical lattices. We first proved that in the noninteracting case this system can present a quantum symmetry which forbids the ratchet effect classically expected. We then showed how the interaction between atoms in the condensate, studied in the mean-field approximation, can break the quantum symmetry present in our model in the noninteracting limit, thus giving rise to the ratchet effect. The role of noise, the validity of the mean-field description and the possibility to observe experimentally our ratchet model have been discussed as well.

Momentum versus time for different values of the dimensionless atom-atom interaction strength g:  g=0 (dashed line), g=0.5 (continuous curve), g=1(dotted curve). Note that directed transport is induced by interactions (see Ref. [7]).


Steering Bose-Einstein condensates despite time symmetry
We have considered [8] a non-dissipative, i.e. Hamiltonian system in which the time-reversal symmetry is not broken and the system is driven smoothly. We have show that, starting out from initial conditions symmetric both in momentum and space, interactions, treated in the limits of a mean-field approximation, induce directed current in a BEC. In a full many-body analysis of the system, symmetry considerations imply that this directed current is asymptotically decaying. Despite this fact we show that a finite current persists over a time scale which increases  with increasing atom number  in the condensate.


References:

[1] R.D. Astumian and P. Hanggi, Brownian motors, Physics Today 55 (11), 33 (2002).
[2] P. Reimann, Brownian Motors: Noisy transport far from equilibrium, Phys. Rep. 361, 57 (2002).
[3] F. Julicher, A. Ajdari, and J. Prost, Modeling molecular motors, Rev. Mod. Phys. 69, 1269 (1997).
[4] G.G.  Carlo,  G. Benenti,  G. Casati, and D.L. Shepelyansky, Quantum ratchets in dissipative chaotic systems, Phys. Rev. Lett. 94, 164101 (2005).
[5] L. Wang, G. Benenti, G. Casati and B. Li, Ratchet effect and the transporting islands in the chaotic sea, Phys. Rev. Lett. 99, 244101 (2007).
[6] G.G. Carlo, G. Benenti, G. Casati, S. Wimberger, O. Morsch, R. Mannella, and E. Arimondo,  Quantum ratchet with cold atoms in a pair of pulsed optical lattices, Phys. Rev. A 74, 033617 (2006).
[7] D. Poletti, G. Benenti, G. Casati and B. Li, Interaction-induced quantum ratchet in a Bose-Einstein condensate, Phys. Rev. A 76, 023421 (2007).
[8] D. Poletti, G. Benenti, G. Casati, P. Hanggi and B. Li, Steering Bose-Einstein condensates despite time symmetry, Phys. Rev. Lett. 102, 130604 (2009).