Quantum stirring, ratchets, and pumping

A pump is an alternating current-driven device that generates a direct current (DC).

In such open geometry, the pump takes particles from one reservoir and emits them into the other.

Stirring is the operation of inducing a circulating current with a non-vanishing DC component in a closed system.

The simplest geometry is obtained by integrating a pump in a closed circuit.

More generally one can consider any type of stirring mechanism such as moving a spoon in a cup of coffee.

Pumping and stirring effects in quantum physics have counterparts in purely classical stochastic and dissipative processes.

is affected by inter-particle interactions, disorder, chaos, noise and dissipation.

This property can be used to induce spin polarization in conventional semiconductors by purely electric means.

Pumping and Stirring devices are close relatives of ratchet systems.

[7] The latter are defined in this context as AC driven spatially periodic arrays, where DC current is induced.

In contrast to that a quantum pumping mechanism produces a DC current in response to a cyclic deformation of the confining potential.

In order to have a DC current from an AC driving, time reversal symmetry (TRS) should be broken.

In the absence of magnetic field and dissipation it is the driving itself that can break TRS.

The best known example is the peristaltic mechanism that combines a cyclic squeezing operation with on/off switching of entrance/exit valves.

are indexes over the mechanical degrees of freedom and the leads respectively, and the subindex "

Integrating the above equation for a system with two leads yields the well known relation between the pumped charge per cycle

is an Aharonov Bohm magnetic flux through the ring, then by Faraday law

Though this formula is written using quantum mechanical notations it holds also classically if the commutator is replaced by Poisson brackets.

we can rewrite the formula for the amount of pumped particles as where we define the normal vector

The ``Berry phase" which is acquired by a wavefunction at the end of a closed cycle is Accordingly, one can argue that the "magnetic charge" that generates (so to say) the

It follows from gauge invariance that the degeneracies of the system are arranged as vertical Dirac chains.

Optionally it is possible to evaluate the transported charge per pumping cycle from the Berry phase by differentiating it with respect to the Aharonov–Bohm flux through the device.

[14] The Ohmic conductance of a mesoscopic device that is connected by leads to reservoirs is given by the Landauer formula: in dimensionless units the Ohmic conductance of an open channel equals its transmission.

The extension of this scattering point of view in the context of quantum pumping leads to the Brouwer-Buttiker-Pretre-Thomas (BPT) formula[2] which relates the geometric conductance to the

is a projector that restrict the trace operations to the open channels of the lead where the current is measured.

[16] A very recent work considers the role of interactions in the stirring of Bose condensed particles.

In the Kondo regime, as the temperature is lowered, the pumping effect is modified.

There are also works that consider interactions over the whole system (including the leads) using the Luttinger liquid model.

In such a configuration, the pump works similarly to an Adiabatic quantum motor.

[19] The mentioned paradigm has been generalized to include non-linear effects and stochastic fluctuations.