Highlights

## Quantum stirring of particles in closed devices (2005-2008)()

We study the quantum analog of stirring of water inside a cup using a spoon. This can be regarded as a prototype example for quantum pumping in closed devices. The current in the device is induced by translating a scatterer. Its calculation is done using the Kubo formula approach. The transported charge is expressed as a line integral that encircles chains of Dirac monopoles. For simple systems the results turn out to be counter intuitive: e.g. as we move a small scatterer "forward" the current is induced "backwards". One should realize that the route towards quantum-classical correspondence has to do with "quantum chaos" considerations, and hence assumes greater complexity of the device. We also point out the relation to the familiar S matrix formalism which is used to analyze quantum pumping in open geometries.

In [5,6] we propose a BEC stirring device which can be regarded as the incorporation of a quantum pump into a closed circuit: it produces a DC circulating current in response to a cyclic adiabatic change of two control parameters of an optical trap. We demonstrate the feasibility of this concept and point out that such device can be utilized in order to probe the interatomic interactions.

Figure: A scatterer (represented by a black circle) is translated through a systems that has a Fermi occupation of spineless non-interacting electrons. In (a) the system is a simple ring.  In (b) it is a chaotic ring (Sinai billiard). In (c) and in (d) we have network systems that are of the same type of (a) and (b) respectively. In (c)-(e) the scatterer is a delta function (represented by a big circle), while the current is measured at a section (represented by a dotted vertical line). In (e) we have an open geometry with left and right leads that are attached to reservoirs that have the same chemical potential.

[1] D. Cohen, T. Kottos and H. Schanz, Phys. Rev. E 71, 035202(R) (2005). [arXiv] [pdf]
[2] G. Rosenberg and D. Cohen, J. Phys. A 39, 2287 (2006). [arXiv] [pdf]
[3] I. Sela and D. Cohen, Phys. Rev. B 77, 245440 (2008). [arXiv] [pdf]
[4] I. Sela and D. Cohen, Phys. Rev. B 78, 155404 (2008). [arXiv] [pdf]
[5] M. Hiller, T. Kottos, D. Cohen, Europhysics Letters 82, 40006 (2008). [arXiv] [pdf]
[6] M. Hiller, T. Kottos, D. Cohen, Phys. Rev. A 78, 013602 (2008). [arXiv] [pdf]

## Operating a quantum pump in a closed circuit (2002-2006)

During an adiabatic pumping cycle a conventional two barrier quantum device takes an electron from the left lead and ejects it to the right lead. Hence the pumped charge per cycle is naively expected to be Q < e. This zero order adiabatic point of view is in fact misleading. For a closed device we can get Q > e and even Q >> e.  Using the Kubo formula for the geometric conductance, and applying the Dirac chains picture, we derive practical estimates for Q.

[1] I. Sela and D. Cohen, J. Phys. A 39, 3575 (2006). [arXiv] [pdf]
[2] D. Cohen, Phys. Rev. B 68, 201303(R) (2003). [arXiv] [pdf]
[3] D. Cohen, Phys. Rev. B 68, 155303 (2003). [arXiv] [pdf]
[4] D. Cohen (2002). [arXiv] [pdf]

## Counting statistics in closed and multiple path geometries (2007-2008)

The amount $\blue Q$ of particles that are transported via a path of motion is characterized by its expectation value $\blue \langle Q \rangle$ and by its variance $\blue \text{Var}(Q)$.  In [1] we consider the full counting statistics which is associated with a two level coherent transition in the context of a continuous quantum measurement process. The conventional probabilistic point of view implies that if a particle has a probability $\blue p$ to make a transition from one site to another site, then the average transport should be $\blue \langle Q \rangle=p$ with a variance $\blue \text{Var}(Q)=(1-p)p$. In the quantum mechanical context this observation becomes a non-trivial manifestation of restricted quantum-classical correspondence. In particular we test the possibility of getting a valid result for $\blue \text{Var}(Q)$ within the framework of the adiabatic picture, analyzing the simplest non-trivial example of a Landau-Zener crossing.

In [2] we analyze what happens if a particle has two optional paths available to get from one site to another site, and in particular what is $\blue \text{Var}(Q)$ for the current which is induced in a quantum stirring device. Both the coherent splitting of a wavepacket and the quantum stirring effect are intimately related and cannot be understood within a classical or stochastic framework. Due to the multiple path geometry there is no longer a simple relation between the "counting statistics" and the "occupation statistics". In particular we demonstrate that interference shows up differently in the second moment calculation.

[1] M. Chuchem and D. Cohen, Phys. Rev. A 77, 012109 (2008). [arXiv] [pdf]
[2] M. Chuchem and D. Cohen, J. Phys. A 41, 075302 (2008). [arXiv] [pdf]
[3] M. Chuchem and D. Cohen, Physica E, Proc. of FQMT (Prague, 2008). [pdf]